Recent zbMATH articles in MSC 35Bhttps://www.zbmath.org/atom/cc/35B2022-05-16T20:40:13.078697ZWerkzeugSpectral clustering revisited: information hidden in the Fiedler vectorhttps://www.zbmath.org/1483.310382022-05-16T20:40:13.078697Z"DePavia, Adela"https://www.zbmath.org/authors/?q=ai:depavia.adela"Steinerberger, Stefan"https://www.zbmath.org/authors/?q=ai:steinerberger.stefanSummary: We study the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters. We argue that the vertices for which the eigenvector has the largest and the smallest entries, respectively, are unusually strongly connected to their own cluster and more reliably classified than the rest. This can be regarded as a discrete version of the Hot Spots conjecture and should be a useful heuristic for evaluating 'strongly clustered' versus 'liminal' nodes in applications. We give a rigorous proof for the stochastic block model and discuss several explicit examples.Unique continuation for first order systems of PDEshttps://www.zbmath.org/1483.350012022-05-16T20:40:13.078697Z"Berhanu, Shiferaw"https://www.zbmath.org/authors/?q=ai:berhanu.shiferawFrom the introduction: This article is a survey of some of the results on weak and strong unique continuation for systems of linear and nonlinear first order partial differential equations. The linear PDEs arise as sections of a vector subbundle \(\mathcal V\) of the complexified tangent bundle \(\mathbb C TM\) of a connected manifold \(M\).Radial symmetry of positive solutions to a class of fractional Laplacian with a singular nonlinearityhttps://www.zbmath.org/1483.350112022-05-16T20:40:13.078697Z"Cao, Linfen"https://www.zbmath.org/authors/?q=ai:cao.linfen"Wang, Xiaoshan"https://www.zbmath.org/authors/?q=ai:wang.xiaoshanSummary: In this paper, we consider the following nonlocal fractional Laplacian equation with a singular nonlinearity
\[
(-\Delta)^su(x)=\lambda u^{\beta}(x)+a_0u^{-\gamma}(x),\ x\in \mathbb{R}^n,
\]
where \(0<s<1\), \(\gamma>0\), \(1<\beta\leq\frac{n+2s}{ n-2s}\), \(\lambda>0\) are constants and \(a_0\geq0\).
We use a direct method of moving planes which introduced by Chen-Li-Li to prove that positive solutions \(u(x)\) must be radially symmetric and monotone increasing about some point in \(\mathbb{R}^n \).Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operatorshttps://www.zbmath.org/1483.350122022-05-16T20:40:13.078697Z"Anh, Nguyen Thi Van"https://www.zbmath.org/authors/?q=ai:anh.nguyen-thi-van"Ke, Tran Dinh"https://www.zbmath.org/authors/?q=ai:ke.tran-dinh"Lan, Do"https://www.zbmath.org/authors/?q=ai:lan.doSummary: In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.A singular perturbed problem with critical Sobolev exponenthttps://www.zbmath.org/1483.350132022-05-16T20:40:13.078697Z"Chen, Mengyao"https://www.zbmath.org/authors/?q=ai:chen.mengyao"Li, Qi"https://www.zbmath.org/authors/?q=ai:li.qi.1|li.qiSummary: This paper deals with the following nonlinear elliptic problem
\[
\tag{1} -\varepsilon^2\Delta u+\omega V(x)u=u^p +u^{2^{\ast}-1},\quad u>0\quad\text{in }\mathbb{R}^N,
\]
where \(\omega\in\mathbb{R}^+\), \(N\geq 3\), \(p\in (1,2^{\ast}-1)\) with \(2^{\ast}=2N/(N-2), \varepsilon> 0\) is a small parameter and \(V(x)\) is a given function. Under suitable assumptions, we prove that problem (1) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small \(\varepsilon\), which concentrate at local minimum points of potential function \(V(x)\). Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.Design of a mode converter using thin resonant ligamentshttps://www.zbmath.org/1483.350142022-05-16T20:40:13.078697Z"Chesnel, Lucas"https://www.zbmath.org/authors/?q=ai:chesnel.lucas"Heleine, Jérémy"https://www.zbmath.org/authors/?q=ai:heleine.jeremy"Nazarov, Sergei A."https://www.zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: The goal of this work is to design an acoustic mode converter. The wave number is fixed so that two modes can propagate. We explain how to construct geometries such that the energy of the modes is completely transmitted and additionally the mode 1 is converted into the mode 2 and conversely. To proceed, we work in a symmetric waveguide made of two branches connected by two thin ligaments whose lengths and positions are carefully tuned. The approach is based on asymptotic analysis for thin ligaments around resonance lengths. We also provide numerical results to illustrate the theory.An optimal control problem in a tubular thin domain with rough boundaryhttps://www.zbmath.org/1483.350152022-05-16T20:40:13.078697Z"Nakasato, Jean Carlos"https://www.zbmath.org/authors/?q=ai:nakasato.jean-carlos"Pereira, Marcone Corrêa"https://www.zbmath.org/authors/?q=ai:pereira.marcone-correaSummary: In this paper we analyze the asymptotic behavior of a control problem set by a convection-reaction-diffusion equation with mixed boundary conditions and defined in a tubular thin domain with rough boundary. The control term acts on a subset of the rough boundary where a Robin-type boundary condition and a catalyzed reaction mechanism are set. The reaction mechanism depends on a parameter \(\alpha \in \mathbb{R} \). Such parameter establishes different regimes which also depend on the profile and geometry of the tube defined by a periodic function \(g : \mathbb{R}^2 \mapsto \mathbb{R} \). We see that, if \(\partial_2 g\) is not null (that is, when \(g\) really depends on the second variable), then three regimes with respect to \(\alpha\) are established: \( \alpha < 2\), \(\alpha = 2\) (the critical value) and \(\alpha > 2\). On the other hand, if \(\partial_2 g \equiv 0\), similar regimes are obtained but now with a different critical value. Indeed, if \(\partial_2 g \equiv 0\), then the critical value is achieved at \(\alpha = 1\). For each one of these six regimes, we obtain the asymptotic behavior of the control problem when the cylindrical thin domain degenerates to the unit interval. We show that the problem is asymptotically controllable just when \(\alpha\) assumes the critical values. Our analysis is mainly performed using the periodic unfolding method adapted to cylindrical coordinates in \(\mathbb{R}^3\).Enhancement of elasto-dielectrics by homogenization of active chargeshttps://www.zbmath.org/1483.350162022-05-16T20:40:13.078697Z"Francfort, Gilles A."https://www.zbmath.org/authors/?q=ai:francfort.gilles-a"Gloria, Antoine"https://www.zbmath.org/authors/?q=ai:gloria.antoine"Lopez-Pamies, Oscar"https://www.zbmath.org/authors/?q=ai:lopez-pamies.oscarThis paper investigates a PDE system resulting from even electromechanical coupling in elastomers with periodic microstructure, consisting of a dielectric part and a elastic-dielectric part. Under proper smoothness assumptions, the homogenized system is derived, based on the uniform \(W^{1, q}\) estimate. The results depend crucially on periodicity (or adequate randomness) and on the type of two-phase microstructure under consideration. It also shows electric enhancement if the charges are carefully tailored to the homogenized electric field and explicits that enhancement, as well as the corresponding electrostrictive enhancement in a dilute regime.
Reviewer: Yao Xu (Beijing)A classical approach for the \(p\)-Laplacian in oscillating thin domainshttps://www.zbmath.org/1483.350172022-05-16T20:40:13.078697Z"Nakasato, Jean Carlos"https://www.zbmath.org/authors/?q=ai:nakasato.jean-carlos"Pereira, Marcone Corrêa"https://www.zbmath.org/authors/?q=ai:pereira.marcone-correaSummary: In this work we study the asymptotic behavior of solutions to the \(p\)-Laplacian equation posed in a 2-dimensional open set which degenerates into a line segment when a positive parameter \(\varepsilon\) goes to zero (a thin domain perturbation). Also, we notice that oscillatory behavior on the upper boundary of the region is allowed. Combining methods from classic homogenization theory and monotone operators we obtain the homogenized equation proving convergence of the solutions and establishing a corrector function which guarantees strong convergence in \(W^{1,p}\) for \(1< p< +\infty\).Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditionshttps://www.zbmath.org/1483.350182022-05-16T20:40:13.078697Z"Mallick, Mohan"https://www.zbmath.org/authors/?q=ai:mallick.mohan"Sasi, Sarath"https://www.zbmath.org/authors/?q=ai:sasi.sarath"Shivaji, R."https://www.zbmath.org/authors/?q=ai:shivaji.ratnasingham"Sundar, S."https://www.zbmath.org/authors/?q=ai:sundar.subbiahSummary: We study the structure of positive solutions to steady state ecological models of the form:
\[
\begin{cases}
-\Delta u = \lambda uf(u) & \text{in }\Omega, \\
\alpha (u) \dfrac{\partial u}{\partial \eta}+[1-\alpha (u)]u = 0 & \text{on }\partial\Omega,
\end{cases}
\]
where \(\Omega\) is a bounded domain in \(\mathbb{R}^n;\, n>1\) with smooth boundary \(\partial\Omega\) or \(\Omega = (0,1), \frac{\partial}{\partial\eta}\) represents the outward normal derivative on the boundary, \(\lambda\) is a positive parameter, \(f\colon [0,\infty)\to \mathbb{R}\) is a \(C^2\) function such that \(\frac{f(s)}{k-s}>0\) for some \(k>0\), and \(\alpha \colon [0,k]\to [0,1]\) is also a \(C^2\) function. Here \(f(u)\) represents the per capita growth rate, \(\alpha (u)\) represents the fraction of the population that stays on the patch upon reaching the boundary, and \(\lambda\) relates to the patch size and the diffusion rate. In particular, we will discuss models in which the per capita growth rate is increasing for small \(u\), and models where grazing is involved. We will focus on the cases when \(\alpha^{\prime} (s)\geq 0; [0,k]\), which represents negative density dependent dispersal on the boundary. We employ the method of sub-super solutions, bifurcation theory, and stability analysis to obtain our results. We provide detailed bifurcation diagrams via a quadrature method for the case \(\Omega = (0,1)\).Bifurcation analysis of dissipative rogue wave in electron-positron-ion plasma with relativistic ions and superthermal electronshttps://www.zbmath.org/1483.350192022-05-16T20:40:13.078697Z"Shahein, R. A."https://www.zbmath.org/authors/?q=ai:shahein.r-a"El-Shehri, Jawaher H."https://www.zbmath.org/authors/?q=ai:el-shehri.jawaher-hSummary: In this manuscript, a modified nonlinear Schrodinger equation (MNLSE) is derived for an unmagnetized collisionless three components plasma containing superthermal electrons, Boltzmann distribution of positrons and relativistic ions. By bifurcation of dynamical system, we determined the stable and unstable regions and predicted the kinds of solutions of MNLSE. This solutions reveal dark soliton in heteroclinic areas and rogue wave in unstable regions. A novel form of rogue wave is obtained also the effects of viscosity, superthermal electron \(\kappa\), ratio of electron to positron temperature, ratio of ion to electron temperature and the density of positron are illustrated with some graphics in two-dimensional and three-dimensional. The derived results have numerous applications in plasma physics and it could be much importance in predicting and enriching rogue wave happening in dissipation plasma physics.Rigorous validation of a Hopf bifurcation in the Kuramoto-Sivashinsky PDEhttps://www.zbmath.org/1483.350202022-05-16T20:40:13.078697Z"van den Berg, Jan Bouwe"https://www.zbmath.org/authors/?q=ai:van-den-berg.jan-bouwe"Queirolo, Elena"https://www.zbmath.org/authors/?q=ai:queirolo.elenaSummary: We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto-Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto-Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton-Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.Spectral stability of hydraulic shock profileshttps://www.zbmath.org/1483.350212022-05-16T20:40:13.078697Z"Sukhtayev, Alim"https://www.zbmath.org/authors/?q=ai:sukhtayev.alim"Yang, Zhao"https://www.zbmath.org/authors/?q=ai:yang.zhao"Zumbrun, Kevin"https://www.zbmath.org/authors/?q=ai:zumbrun.kevin-rSummary: By reduction to a generalized Sturm-Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint-Venant equations for inclined shallow-water flow, over the full parameter range of their existence, for both smooth-type profiles and discontinuous-type profiles containing subshocks. Together with work of Mascia-Zumbrun and Yang-Zumbrun, this yields linear and nonlinear \(H^2 \cap L^1 \to H^2\) stability with sharp rates of decay in \(L^p\), \(p \geq 2\), the first complete stability results for large-amplitude shock profiles of a hyperbolic relaxation system.Rotating periodic patterns in reaction diffusion systemshttps://www.zbmath.org/1483.350222022-05-16T20:40:13.078697Z"Zhang, He"https://www.zbmath.org/authors/?q=ai:zhang.he"Xu, Fei"https://www.zbmath.org/authors/?q=ai:xu.fei.1|xu.fei|xu.fei.2|xu.fei.3"Li, Yong"https://www.zbmath.org/authors/?q=ai:li.yong.1Summary: We investigate a special class of spatio-temporal patterns in reaction diffusion systems, namely, \((Q,T)\)-periodic patterns, or rotating ones. We rigorously prove the existence of such patterns by using the Leray-Schauder degree theory and the method of lower and upper solutions. A couple of examples of reaction diffusion systems that exhibit \((Q,T)\)-periodic patterns are identified and used to illustrate the properties of such spatio-temporal structures.Existence and general decay estimates for a Petrovsky-Petrovsky coupled system with nonlinear strong dampinghttps://www.zbmath.org/1483.350232022-05-16T20:40:13.078697Z"Beniani, Abderrahmane"https://www.zbmath.org/authors/?q=ai:beniani.abderrahmane"Bahri, Noureddine"https://www.zbmath.org/authors/?q=ai:bahri.noureddine"Zennir, Khaled"https://www.zbmath.org/authors/?q=ai:zennir.khaledSummary: In this paper, we consider a coupled system of Petrovsky-Petrovsky equations with nonlinear dissipative terms. We proved the existence and stability of solution to the coupled system (1) under some assumptions (7)--(9) based on the work \textit{A. Benaissa} et al. [J. Math. Phys. 53, No. 12, 123514, 19 p. (2012; Zbl 1282.35243)]. The approach adopted is the Faedo-Galerkin method. Furthermore, by applying the multiplier method and some weighted integral inequalities, we strictly proved the decay properties (23).Fractional oscillon equations; solvability and connection with classical oscillon equationshttps://www.zbmath.org/1483.350242022-05-16T20:40:13.078697Z"Bezerra, Flank D. M."https://www.zbmath.org/authors/?q=ai:bezerra.flank-david-morais"Figueroa-López, Rodiak N."https://www.zbmath.org/authors/?q=ai:figueroa-lopez.rodiak-n"Nascimento, Marcelo J. D."https://www.zbmath.org/authors/?q=ai:nascimento.marcelo-jose-diasSummary: In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation
\[
u_{tt} - \mu (t) \Delta u+ \omega (t)u_t = f(u),\, x \in \Omega,\, t \in \mathbb{R},
\]
subject to Dirichlet boundary condition on \(\partial \Omega\), where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(N \geq 3\), the function \(\omega\) is a time-dependent damping, \(\mu\) is a time-dependent squared speed of propagation, and \(f\) is a nonlinear functional. Under structural assumptions on \(\omega\) and \(\mu\) we establish the existence of time-dependent attractor for the fractional models in the sense of \textit{A. N. Carvalho} et al. [Attractors for infinite-dimensional non-autonomous dynamical systems. Berlin: Springer (2013; Zbl 1263.37002)], and \textit{F. Di Plinio} et al. [Discrete Contin. Dyn. Syst. 29, No. 1, 141--167 (2011; Zbl 1223.37100)].Optimal decay rate for the Cauchy problem of the standard linear solid model with Gurtin-Pipkin thermal lawhttps://www.zbmath.org/1483.350252022-05-16T20:40:13.078697Z"Bounadja, Hizia"https://www.zbmath.org/authors/?q=ai:bounadja.hizia"Khader, Maisa"https://www.zbmath.org/authors/?q=ai:khader.maisaSummary: In this paper, we consider the Cauchy problem for the standard linear solid model in the whole space \(\mathbb{R}^n\), where the heat conduction is given by the Gurtin-Pipkin thermal law. The energy method in the Fourier space is used to obtain the optimal decay rate of the \(L^2\)-norm of the solution. More precisely, we prove that the decay rate of the solution is of the form \((1 + t)^{- n / 4}\) and of regularity-loss type. Hence, we improve and extend the result obtained in [\textit{D. Wang} and \textit{W. Liu}, Asymptotic Anal. 123, No. 1--2, 181--201 (2021; Zbl 1473.35557)].Decay rates for Kelvin-Voigt damped wave equations. II: The geometric control conditionhttps://www.zbmath.org/1483.350262022-05-16T20:40:13.078697Z"Burq, Nicolas"https://www.zbmath.org/authors/?q=ai:burq.nicolas"Sun, Chenmin"https://www.zbmath.org/authors/?q=ai:sun.chenminAuthors' abstract: ``We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. When the damping coefficient is sufficiently smooth (C1 vanishing nicely, see the following equation: \(|\nabla a| \leq C a^{1/2}\)) we show that exponential decay follows from geometric control conditions (see [\textit{N. Burq} and \textit{H. Christianson}, Commun. Math. Phys. 336, No. 1, 101--130 (2015; Zbl 1320.35062); \textit{L. Tebou}, C. R., Math., Acad. Sci. Paris 350, No. 11--12, 603--608 (2012; Zbl 1255.35039)] for similar results under stronger assumptions on the damping function).''
For Part I, see [\textit{N. Burq}, SIAM J. Control Optim. 58, No. 4, 1893--1905 (2020; Zbl 1452.35030)].
Reviewer: Kaïs Ammari (Monastir)General decay of solutions in one-dimensional porous-elastic with memory and distributed delay termhttps://www.zbmath.org/1483.350272022-05-16T20:40:13.078697Z"Choucha, Abdelbaki"https://www.zbmath.org/authors/?q=ai:choucha.abdelbaki"Ouchenane, Djamel"https://www.zbmath.org/authors/?q=ai:ouchenane.djamel"Zennir, Khaled"https://www.zbmath.org/authors/?q=ai:zennir.khaledSummary: As a continuity to the study by \textit{T. A. Apalara} [J. Math. Anal. Appl. 469, No. 2, 457--471 (2019; Zbl 1402.35042)], we consider a one-dimensional porous-elastic system with the presence of both memory and distributed delay terms in the second equation. Using the well known energy method combined with Lyapunov functionals approach, we prove a general decay result given in Theorem 2.1.The Bramson correction for integro-differential Fisher-KPP equationshttps://www.zbmath.org/1483.350282022-05-16T20:40:13.078697Z"Graham, Cole"https://www.zbmath.org/authors/?q=ai:graham.coleSummary: We consider integro-differential Fisher-KPP equations with nonlocal diffusion. For typical equations, we establish the logarithmic Bramson delay for solutions with step-like initial data. That is, these solutions resemble a front at position \(c_\ast t-\frac{3}{2\lambda_\ast} \log t+\mathcal{O}(1)\) for explicit constants \(c_\ast\) and \(\lambda_\ast\). Certain strongly asymmetric diffusions exhibit more exotic behaviour.Existence and general decay for a viscoelastic equation with logarithmic nonlinearityhttps://www.zbmath.org/1483.350292022-05-16T20:40:13.078697Z"Ha, Tae Gab"https://www.zbmath.org/authors/?q=ai:ha.tae-gab"Park, Sun-Hye"https://www.zbmath.org/authors/?q=ai:park.sunhyeSummary: In the present work, we investigate a viscoelastic equation involving a logarithmic nonlinear source term. After proving the existence of solutions, we establish a general decay estimate of the solution using energy estimates and theory of convex functions. This result extends and complements some previous results of us [Adv. Difference Equ. 2020, Paper No. 235, 17 p. (2020; Zbl 1482.35047)] and \textit{M. I. Mustafa} [Math. Methods Appl. Sci. 41, No. 1, 192--204 (2018; Zbl 1391.35058)].Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domainhttps://www.zbmath.org/1483.350302022-05-16T20:40:13.078697Z"Ivanovici, Oana"https://www.zbmath.org/authors/?q=ai:ivanovici.oanaSummary: We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay than the wave, unlike in the flat space.Existence and global asymptotic stability in a fractional double parabolic chemotaxis system with logistic sourcehttps://www.zbmath.org/1483.350312022-05-16T20:40:13.078697Z"Lei, Yuzhu"https://www.zbmath.org/authors/?q=ai:lei.yuzhu"Liu, Zuhan"https://www.zbmath.org/authors/?q=ai:liu.zuhan"Zhou, Ling"https://www.zbmath.org/authors/?q=ai:zhou.lingSummary: This paper studies a double parabolic chemotaxis system with logistic source and a fractional diffusion of order \(\alpha\in(0,2)\)
\[
\begin{cases}
u_t=-\Lambda^\alpha u-\chi\nabla\cdot (u\nabla v)+au-bu^2, \\
v_t=\Delta v-v+u
\end{cases}
\]
on two dimensional periodic torus \(\mathbb{T}^2\). In contrast to the well-known Neumann heat semigroup \(\{e^{t\Delta}\}_{t\geq 0}\) estimates in a smoothly bounded domain \(\Omega\subset \mathbb{R}^n\) introduced by Winkler (2010), we obtain the spatio-temporal estimates of the analytic semigroup \(\{T_t^\alpha(x)\}_{t\geq 0}\) and \(\{T_t(x)\}_{t\geq 0}\) which are generated by \(-(-\Delta)^{\frac{\alpha}{2}}-I\) and \(\Delta-I\) respectively over periodic torus \(\mathbb{T}^2\). With the help of these conclusions, we can use the semigroup method to study the global existence and the asymptotic behavior of the above fractional chemotaxis model, which has not been studied yet. It is proved that for any nonnegative initial data \((u_0,v_0)\in H^4 (\mathbb{T}^2)\times H^5(\mathbb{T}^2)\), if \(1<\alpha<2\), there admits a unique globally classical solution. Furthermore, if \(a=0\), we can obtain that the solution components \(u\) and \(v\) converge to zero with respect to the norm in \(L^\infty(\mathbb{T}^2)\) as \(t\to\infty\).Spreading speeds of nonlocal KPP equations in heterogeneous mediahttps://www.zbmath.org/1483.350322022-05-16T20:40:13.078697Z"Liang, Xing"https://www.zbmath.org/authors/?q=ai:liang.xing"Zhou, Tao"https://www.zbmath.org/authors/?q=ai:zhou.tao.2|zhou.tao|zhou.tao.3|zhou.tao.1Summary: This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media. The kernel \(K\) is assumed to depend on the media. First, we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues. Second, we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.Long-time behavior of solutions to von Karman equations with variable sourceshttps://www.zbmath.org/1483.350332022-05-16T20:40:13.078697Z"Li, Fang"https://www.zbmath.org/authors/?q=ai:li.fang.1|li.fang.5|li.fang.3|li.fang.2|li.fang|li.fang.6|li.fang.4"Li, Xiaolei"https://www.zbmath.org/authors/?q=ai:li.xiaoleiSummary: The interest of this paper is to deal with long-time behavior of the solutions to the following Von Karman equation involving variable sources and clamped boundary conditions:
\[
u_{tt}+\Delta^2 u+a|u_t|^{m(x)-2}u_t=[u,F(u)]+b|u|^{p(x)-2}u,\quad \Delta^2F(u)=-[u,u].
\]
First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships among initial energy value, the term \(\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}\mathrm{d}x\) and the Airy stress functions, which ensure that the energy functional are nonnegative with respect to time variable. And then, some energy estimates and Komornik inequality is used to prove a uniform estimate of decay rates of the solution which provides an estimation of long-time behavior of solutions. As we know, such results are seldom seen for the variable exponent case.Two-species competition model with chemotaxis: well-posedness, stability and dynamicshttps://www.zbmath.org/1483.350342022-05-16T20:40:13.078697Z"Li, Guanlin"https://www.zbmath.org/authors/?q=ai:li.guanlin"Yao, Yao"https://www.zbmath.org/authors/?q=ai:yao.yaoConvergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesishttps://www.zbmath.org/1483.350352022-05-16T20:40:13.078697Z"Liu, Qingqing"https://www.zbmath.org/authors/?q=ai:liu.qingqing"Peng, Hongyun"https://www.zbmath.org/authors/?q=ai:peng.hongyun"Wang, Zhi-An"https://www.zbmath.org/authors/?q=ai:wang.zhianSummary: In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis. Under some suitable structural assumption on the pressure function, we first predict and derive the system admits a nonlinear diffusion wave in \(\mathbb{R}\) driven by the damping effect. Then we show that the solution of the concerned system will locally and asymptotically converge to this nonlinear diffusion wave if the wave strength is small. By using the time-weighted energy estimates, we further prove that the convergence rate of the nonlinear diffusion wave is algebraic.Asymptotic stabilization for a class of chemotaxis-consumption systems with generalized logistic sourcehttps://www.zbmath.org/1483.350362022-05-16T20:40:13.078697Z"Lyu, Wenbin"https://www.zbmath.org/authors/?q=ai:lyu.wenbinSummary: This paper is concerned with a chemotaxis-consumption system
\[
\left\{u_t=\nabla\cdot(\nabla u-uS(x,u,v)\cdot\nabla v)+\rho u-\mu u^l, v_t=\Delta v-uv,\right.
\]
under no-flux boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb{R}^n\) \((n\geqslant 1)\), where the chemotactic sensitivity tensor \(S\in C^2(\overline{\Omega}\times [0,+\infty)^2;\mathbb{R}^{n\times n})\) fulfills that there exists some nondecreasing function \(S_0\) on \([0,+\infty)\) such that
\[
|S(x,u,v)|\leqslant S_0(v)\text{ for all }(x,u,v)\in\overline{\Omega}\times [0,+\infty)\times [0,+\infty).
\]
We show that for any \(\rho,\mu>0\) and \(l>1\), any generalized solution of the above system asymptotically approaches to the nontrivial spatially homogeneous steady state
\[\bigg(\left(\frac{\rho}{\mu}\right)^{\frac{1}{l-1}},0\bigg)
\]
as \(t\to+\infty\).Large time behavior for a Hamilton-Jacobi equation in a critical coagulation-fragmentation modelhttps://www.zbmath.org/1483.350372022-05-16T20:40:13.078697Z"Mitake, Hiroyoshi"https://www.zbmath.org/authors/?q=ai:mitake.hiroyoshi"Tran, Hung V."https://www.zbmath.org/authors/?q=ai:tran.hung-vinh"Van, Truong-Son"https://www.zbmath.org/authors/?q=ai:van.truong-sonThe large time behavior of viscosity solutions to
\begin{align*}
& \partial_t F(x,t) + \frac{1}{2} (\partial_x F(x,t)-1) (\partial_x F(x,t)-2) + \frac{F(x,t)}{x} - 1 = 0 \;\;\text{ in }\;\; (0,\infty)^2\,, \\
& 0 \le F(x,t) \le x \;\;\text{ on }\;\; [0,\infty)^2\,, \\
& F(x,0) = F_0(x) \;\;\text{ in }\;\; [0,\infty)\,,
\end{align*}
is investigated, when the initial condition \(F_0\) is sublinear with \(0\le F_0(x)\le x\) and \(0 \le \partial_x F_0\le 1\), and shown to depend upon the behavior of \(F_0(x)x^{-2/3}\) as \(x\to\infty\). More precisely, assuming that \(F_0(x)x^{-2/3}\to L\) as \(x\to\infty\) for some \(L\in [0,\infty]\), the family \(\{F(t) : t\ge 0\}\) converges as \(t\to\infty\) to a limit \(F_\infty\) uniformly on compact subsets of \((0,\infty)\) with \(F_\infty\equiv 0\) when \(L=0\), \(F_\infty(x)=x\) when \(L=\infty\), and \(F_\infty\) is a uniquely determined sublinear stationary solution (depending on \(L\)) when \(L\in (0,\infty)\). It is also proved that the family \(\{F(t) : t\ge 0\}\) need not have a limit as \(t\to\infty\) when \(F_0(x)x^{-2/3}\) does not have a limit as \(x\to\infty\). The proofs rely on the representation of viscosity solutions provided by optimal control theory and characteristics.
This problem is connected with the analysis of the large time behavior of weak solutions to the coagulation-fragmentation equation with multiplicative coagulation kernel, constant fragmentation kernel, and total mass equal to one, see [\textit{H. V. Tran} and \textit{T.-S. Van}, ``Coagulation-fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel'', Comm. Pure Appl. Math. (to appear)].
Reviewer: Philippe Laurençot (Toulouse)Feedback exponential stabilization of the semilinear heat equation with nonlocal initial conditionshttps://www.zbmath.org/1483.350382022-05-16T20:40:13.078697Z"Munteanu, Ionuţ"https://www.zbmath.org/authors/?q=ai:munteanu.ionutSummary: The present paper is devoted to the problem of stabilization of the one-dimensional semilinear heat equation with nonlocal initial conditions. The control is with boundary actuation. It is linear, of finite-dimensional structure, given in an explicit form. It allows to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel, then to apply a fixed point argument in a convenient space.Polynomial stability of a suspension bridge model by indirect dampingshttps://www.zbmath.org/1483.350392022-05-16T20:40:13.078697Z"Nicaise, Serge"https://www.zbmath.org/authors/?q=ai:nicaise.serge"Hadjsaleh, Monia Bel"https://www.zbmath.org/authors/?q=ai:hadjsaleh.monia-belSummary: In this paper, we study the indirect stabilization of a coupled string-beam system related to the well-known Lazer-McKenna suspension bridge model. We prove some decay results of the energy of the system with either interior dampings or boundary ones. Our method is based on observability estimates of the undamped system and on the spectral analysis of the spatial operator.Stochastic Euler-Bernoulli beam driven by additive white noise: global random attractors and global dynamicshttps://www.zbmath.org/1483.350402022-05-16T20:40:13.078697Z"Chen, Huatao"https://www.zbmath.org/authors/?q=ai:chen.huatao"Guirao, Juan Luis García"https://www.zbmath.org/authors/?q=ai:garcia-guirao.juan-luis"Cao, Dengqing"https://www.zbmath.org/authors/?q=ai:cao.dengqing"Jiang, Jingfei"https://www.zbmath.org/authors/?q=ai:jiang.jingfei"Fan, Xiaoming"https://www.zbmath.org/authors/?q=ai:fan.xiaomingSummary: This paper concerns with the long time dynamical behavior of a stochastic Euler-Bernoulli beam driven by additive white noise. By verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam, the existence of global random attractors that attracts all bounded sets in phase space is proved. Furthermore, the finite Hausdorff dimension for the global random attractors is attained. In light of the relationship between global random attractor and random invariant probability measure, the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors.Existence and asymptotic behavior of solutions to a class of semilinear degenerate parabolic equations with nonlinearities of arbitrary orderhttps://www.zbmath.org/1483.350412022-05-16T20:40:13.078697Z"Chi, Tran Thi Quynh"https://www.zbmath.org/authors/?q=ai:chi.tran-thi-quynh"Thuy, Le Thi"https://www.zbmath.org/authors/?q=ai:le-thi-thuy.|thuy.le-thi-hong"Tu, Nguyen Xuan"https://www.zbmath.org/authors/?q=ai:tu.nguyen-xuanSummary: We study the existence and asymptotic behavior of weak solutions to a class of semilinear degenerate parabolic equations with nonlinearities of arbitrary order. The main novelty of our result is that no restriction on the upper growth of the nonlinearities is imposed.Global dynamics of solutions for a sixth-order parabolic equation describing continuum evolution of film-free surfacehttps://www.zbmath.org/1483.350422022-05-16T20:40:13.078697Z"Duan, Ning"https://www.zbmath.org/authors/?q=ai:duan.ning"Zhao, Xiaopeng"https://www.zbmath.org/authors/?q=ai:zhao.xiaopengSummary: This paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces \(H^3(\Omega)\) and fractional-order spaces \(H^k(\Omega)\), where \(0 \leqslant k<\infty\).Rate of convergence of global attractors for some perturbed reaction-diffusion equations under smooth perturbations of the domainhttps://www.zbmath.org/1483.350432022-05-16T20:40:13.078697Z"Pires, Leonardo"https://www.zbmath.org/authors/?q=ai:pires.leonardo"Samprogna, Rodrigo A."https://www.zbmath.org/authors/?q=ai:samprogna.rodrigo-aSummary: In this paper we obtain a rate of convergence for the asymptotic behavior of some semilinar parabolic problems with Dirichlet boundary conditions relatively to smooth perturbations of the domain. We will obtain a rate of convergence dependent on convergence of domains for eigenvalues, eigenfunctions, invariant manifolds and continuity of attractors.Asymptotic autonomy of bi-spatial attractors for stochastic retarded Navier-Stokes equationshttps://www.zbmath.org/1483.350442022-05-16T20:40:13.078697Z"Zhang, Qiangheng"https://www.zbmath.org/authors/?q=ai:zhang.qiangheng"Li, Yangrong"https://www.zbmath.org/authors/?q=ai:li.yangrongSummary: We establish semi-convergence of a non-autonomous bi-spatial random attractor towards to an autonomous attractor under the topology of the regular space when time-parameter goes to infinity, where the criteria are given by forward compactness of the attractor in the terminal space as well as forward convergence of the random dynamical system in the initial space. We then apply to both non-autonomous and autonomous stochastic 2D Navier-Stokes equations with general delays (including variable and distribution delays). The forward-pullback asymptotic compactness in the space of continuous Sobolev-valued functions is proved by the method of spectrum decomposition.Nonlocal nonlinear reaction preventing blow-up in supercritical case of chemotaxis systemhttps://www.zbmath.org/1483.350452022-05-16T20:40:13.078697Z"Bian, Shen"https://www.zbmath.org/authors/?q=ai:bian.shen"Chen, Li"https://www.zbmath.org/authors/?q=ai:chen.li.1"Latos, Evangelos A."https://www.zbmath.org/authors/?q=ai:latos.evangelos-aSummary: This paper is devoted to the analysis of non-negative solutions for the chemotaxis model with nonlocal nonlinear source in bounded domain. The qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction. When the growth factor is stronger than the dampening effect, with the help of the nonlocal nonlinear term in the reaction, for appropriately chosen exponents and arbitrary initial data, the model admits a classical solution which is uniformly bounded. Moreover, when the growth factor has the same order with the dampening effect, the nonlocal nonlinear exponents can prevent the chemotactic collapse.Local well-posedness and finite time blow-up of solutions to an attraction-repulsion chemotaxis system in higher dimensionshttps://www.zbmath.org/1483.350462022-05-16T20:40:13.078697Z"Hosono, Tatsuya"https://www.zbmath.org/authors/?q=ai:hosono.tatsuya"Ogawa, Takayoshi"https://www.zbmath.org/authors/?q=ai:ogawa.takayoshiSummary: We consider the Cauchy problem for an attraction-repulsion chemotaxis system in \(\mathbb{R}^n\) with the chemotactic coefficients of the attractant \(\beta_1\) and the repellent \(\beta_2\). In particular, these coefficients are important role in the global existence and blow up of the solutions. In this paper, we show the local well-posedness of solutions in the critical spaces \(L^{n / 2}( \mathbb{R}^n)\) and the finite time blow-up of the solution under the condition \(\beta_1 > \beta_2\) in higher dimensional spaces.Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reactionhttps://www.zbmath.org/1483.350472022-05-16T20:40:13.078697Z"Iagar, Razvan Gabriel"https://www.zbmath.org/authors/?q=ai:iagar.razvan-gabriel"Sánchez, Ariel"https://www.zbmath.org/authors/?q=ai:sanchez.arielSummary: We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation:
\[
\partial_tu=\Delta u^m+|x|^\sigma u^m,
\]
posed for \(x\in\mathbb{R}^N\), \(t\geq 0\), where \(m>1\), dimension \(N\geq 2\) and \(\sigma>0\). A new and explicit critical exponent
\[
\sigma_c=\frac{2(m-1)(N-1)}{3m+1}
\]
is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent \(\sigma_c\) and also depends strongly on whether the dimension \(N\geq 4\) or \(N\in\{2,3\}\). These results extend previous works of the authors in dimension \(N=1\).Finite time blow-up for the heat flow of \(H\)-surface with constant mean curvaturehttps://www.zbmath.org/1483.350482022-05-16T20:40:13.078697Z"Li, Haixia"https://www.zbmath.org/authors/?q=ai:li.haixiaSummary: We consider an initial boundary value problem for the heat flow of the equation of surfaces with constant mean curvature which was investigated in [\textit{T. Huang} et al., Manuscr. Math. 134, No. 1--2, 259--271 (2011; Zbl 1210.53012)], where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.Lifespan of solutions to a hyperbolic type Kirchhoff equation with arbitrarily high initial energyhttps://www.zbmath.org/1483.350492022-05-16T20:40:13.078697Z"Yang, Hui"https://www.zbmath.org/authors/?q=ai:yang.hui.1"Han, Yuzhu"https://www.zbmath.org/authors/?q=ai:han.yuzhuSummary: In this paper, an initial boundary value problem for a hyperbolic type Kirchhoff equation with a strong dissipation and a general nonlinearity is considered. First, local existence and uniqueness of weak solutions are obtained with the help of Banach fixed point theorem. Then, by constructing an auxiliary functional and adopting the concavity argument, we give a new finite time blow-up criterion for this problem, which in particular implies that the problem admits blow-up solutions with arbitrarily high initial energy. Meanwhile, a bound for the blow-up time is derived from above. Further, we obtain a lower bound for the blow-up time when blow-up occurs. From methods to results, we partially extend the ones obtained in earlier literature.Global gradient estimates for a general type of nonlinear parabolic equationshttps://www.zbmath.org/1483.350502022-05-16T20:40:13.078697Z"Cavaterra, Cecilia"https://www.zbmath.org/authors/?q=ai:cavaterra.cecilia"Dipierro, Serena"https://www.zbmath.org/authors/?q=ai:dipierro.serena"Gao, Zu"https://www.zbmath.org/authors/?q=ai:gao.zu"Valdinoci, Enrico"https://www.zbmath.org/authors/?q=ai:valdinoci.enricoSummary: We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations, possibly in a Riemannian geometry setting. Our result is new in comparison with the existing ones in the literature, in light of the validity of the estimates in the global domain, and it detects several additional regularity effects due to special parabolic data. Moreover, our result comprises a large number of nonlinear sources treated by a unified approach, and it recovers many classical results as special cases.Chiti-type reverse Hölder inequality and torsional rigidity under integral Ricci curvature conditionhttps://www.zbmath.org/1483.350512022-05-16T20:40:13.078697Z"Chen, Hang"https://www.zbmath.org/authors/?q=ai:chen.hang|chen.hang.1Summary: In this paper, we prove a reverse Hölder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of \textit{N. Gamara} et al. [Open Math. 13, 557--570 (2015; Zbl 06632233)] and \textit{D. Colladay} et al. [J. Geom. Anal. 28, No. 4, 3906--3927 (2018; Zbl 1410.58016)] from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to \(p\)-Laplacian.Boundedness and asymptotic behavior in a Keller-Segel(-Navier)-Stokes system with indirect signal productionhttps://www.zbmath.org/1483.350522022-05-16T20:40:13.078697Z"Dai, Feng"https://www.zbmath.org/authors/?q=ai:dai.feng"Liu, Bin"https://www.zbmath.org/authors/?q=ai:liu.bin.6|liu.bin.2|liu.bin.3|liu.bin.9|liu.bin.7|liu.bin.4|liu.bin.5|liu.bin.1|liu.bin.8|liu.binSummary: This paper deals with the Keller-Segel(-Navier)-Stokes system with indirect signal production
\[
\begin{cases}
n_t + u \cdot \nabla n = \Delta n - \nabla \cdot ( n \nabla v ) + r n - \mu n^2, \\
v_t + u \cdot \nabla v = \Delta v - v + w, \\
w_t + u \cdot \nabla w = \Delta w - w + n, \\
u_t + \kappa ( u \cdot \nabla ) u = \Delta u - \nabla P + n \nabla \Phi, \quad \nabla \cdot u = 0
\end{cases} \eqno{(\star)}
\]
in a bounded and smooth domain \(\Omega \subset \mathbb{R}^N\) (\(N = 2, 3\)) with no-flux boundary for \(n, v, w\) and no-slip boundary for \(u\), where \(r \in \mathbb{R}\), \(\mu \geq 0\), \(\kappa \in \{0, 1 \}\) and \(\Phi \in W^{2 , \infty}(\Omega)\). In the case without logistic source (\(r = \mu = 0\)), it is proved that for all suitably regular initial data, the associated initial-boundary value problem for the spatially two-dimensional Navier-Stokes system (\( \star\)) admits a globally bounded classical solution. This result improves and extends the previously known ones. We point out that the same result to the corresponding two-dimensional Navier-Stokes system with direct signal production holds necessarily imposing some saturated chemotactic sensitivity, logistic damping or small total initial population mass. In the case coupled with logistic source (\(r \in \mathbb{R}\), \(\mu > 0\)), it is shown that for any reasonably regular initial data, the corresponding initial-boundary value problem for the spatially three-dimensional Stokes system (\( \star\)) possesses a globally bounded classical solution, and that this solution stabilizes toward the corresponding spatially homogeneous equilibrium with the explicit convergence rates for the cases \(r < 0\), \(r = 0\) and \(r > 0\). We underline that the global boundedness of classical solution to the corresponding three-dimensional Stokes system with direct signal production was obtained only for \(\mu \geq 23\) (or sublinear signal production), and that the convergence result to the corresponding system with direct signal production was established only for \(r = 0\) and \(\mu \geq 23\). Our results rigorously confirm that the indirect signal production mechanism genuinely contributes to the global boundedness of classical solution to the Keller-Segel(-Navier)-Stokes system.Global \(C^2 \)-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary conditionhttps://www.zbmath.org/1483.350532022-05-16T20:40:13.078697Z"Gao, Zhenghuan"https://www.zbmath.org/authors/?q=ai:gao.zhenghuan"Wang, Peihe"https://www.zbmath.org/authors/?q=ai:wang.peiheSummary: In this paper, we establish global \(C^2 \) a priori estimates for solutions to the uniformly parabolic equations with Neumann boundary condition on the smooth bounded domain in \(\mathbb R^n\) by a blow-up argument. As a corollary, we obtain that the solutions converge to ones which move by translation. This generalizes the viscosity results derived before by Da Lio.The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growthhttps://www.zbmath.org/1483.350542022-05-16T20:40:13.078697Z"Ho, Ky"https://www.zbmath.org/authors/?q=ai:ho.ky"Kim, Yun-Ho"https://www.zbmath.org/authors/?q=ai:kim.yunho"Winkert, Patrick"https://www.zbmath.org/authors/?q=ai:winkert.patrick"Zhang, Chao"https://www.zbmath.org/authors/?q=ai:zhang.chao.1Summary: In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of \textit{X. Fan} and \textit{D. Zhao} [Nonlinear Anal., Theory Methods Appl. 36, No. 3, 295--318 (1999; Zbl 0927.46022)] and \textit{P. Winkert} and \textit{R. Zacher} [Discrete Contin. Dyn. Syst., Ser. S 5, No. 4, 865--878 (2012; Zbl 1261.35061)] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the \(C^{1, \alpha}\)-regularity follows immediately.Uniform Sobolev estimates in \(\mathbb{R}^n\) involving singular potentialshttps://www.zbmath.org/1483.350552022-05-16T20:40:13.078697Z"Huang, Xiaoqi"https://www.zbmath.org/authors/?q=ai:huang.xiaoqi"Sogge, Christopher D."https://www.zbmath.org/authors/?q=ai:sogge.christopher-dSummary: We generalize the Stein-Tomas in [\textit{P. A. Tomas}, Bull. Am. Math. Soc. 81, 477--478 (1975; Zbl 0298.42011)] \(L^2\)-restriction theorem and the uniform Sobolev estimates of \textit{C. E. Kenig} et al. [Duke Math. J. 55, 329--347 (1987; Zbl 0644.35012)] by allowing critically singular potential. We also obtain Strichartz estimates for Schrödinger and wave operators with such potentials. Due to the fact that there may be nontrivial eigenfunctions we are required to make certain spectral assumptions, such as assuming that the solutions only involve sufficiently large frequencies.A priori estimates for a weakly coercive system of minimal differential polynomials in the anisotropic Sobolev spacehttps://www.zbmath.org/1483.350562022-05-16T20:40:13.078697Z"Limanskii, D. V."https://www.zbmath.org/authors/?q=ai:limanskii.dmitrii-vSummary: In the paper there are found necessary and sufficient conditions for a system of minimal differential polynomials in the anisotropic Sobolev space \(\stackrel{_\circ}{W_\infty^l}(\mathbb{R}^n)\), where \(l=(l_1,\ldots,l_n)\) is the vector of positive integers such that the least common multiple of \(l_1,\ldots,l_n\) coincides with the maximal number among them.Global boundedness of a chemotaxis model with logistic growth and general indirect signal productionhttps://www.zbmath.org/1483.350572022-05-16T20:40:13.078697Z"Liu, Suying"https://www.zbmath.org/authors/?q=ai:liu.suying"Wang, Li"https://www.zbmath.org/authors/?q=ai:wang.li.4|wang.li.3|wang.li.1|wang.li.5|wang.li|wang.li.6|wang.li.2In this paper, the authors consider the solutions to the following chemotaxis-growth systems with signal production
\[
\begin{cases}
\begin{aligned}
u_t &= \Delta u - \nabla \cdot (u \nabla v_1) + \mu (u - u^\alpha), \\
v_{1,t} &= \Delta v_1 - v_1 + v_2, \\
v_{2,t} &= \Delta v_2 - v_2 + v_3, \\
&\cdots \\
v_{k,t} &= \Delta v_k - v_k + u, \qquad x \in \Omega\subset\mathbb{R}^n,\ t > 0,
\end{aligned} \\
\partial_\nu u=\partial_\nu v_1=\dots=\partial_\nu v_k=0,\\
u(x,0)=u_0(x),\ v_i(x,0)=v_{i,0}(x),\ i=1,2,\dots,k.
\end{cases}
\]
The bounded domain \(\Omega\) is in \(n\)-dimension space. The main result is that, if \(\alpha > \min \{2, \max \{1, \frac{n+2}{2k}\} \}\), then the solutions are globally bounded. The heat semigroup estimates on the bounded domain obtained in [\textit{M. Winkler}, J. Differ. Equations 248, No. 12, 2889--2905 (2010; Zbl 1190.92004)] are applied. The authors can control the chemical gradient \(\nabla v_1\) and obtain global boundedness through detailed analysis.
Reviewer: Siming He (Durham)Global existence and boundedness of solutions to a parabolic attraction-repulsion chemotaxis system in \(\mathbb{R}^2\): the repulsive dominant casehttps://www.zbmath.org/1483.350582022-05-16T20:40:13.078697Z"Yamada, Tetsuya"https://www.zbmath.org/authors/?q=ai:yamada.tetsuyaSummary: This paper deals with the initial value problem for a parabolic attraction-repulsion chemotaxis system in \(\mathbb{R}^2\):
\[
\begin{cases}
\partial_t u = \Delta u - \nabla \cdot ( u \nabla ( \beta_1 v_1 - \beta_2 v_2 ) ), & t > 0, \;\;x \in \mathbb{R}^2, \\
\partial_t v_j = \Delta v_j - \lambda_j v_j + u, & t > 0,\;\;x \in \mathbb{R}^2\;\; ( j = 1 , 2 ), \\
u ( 0 , x ) = u_0 ( x ), \;\; v_j ( 0 , x ) = v_{j 0} ( x ) , & x \in \mathbb{R}^2\;\; ( j = 1 , 2 )
\end{cases}
\]
with positive parameters \(\beta_1, \beta_2, \lambda_1, \lambda_2\) and nonnegative initial data \(u_0, v_{10}, v_{20}\). It is well known that the sign of \(\beta_1 - \beta_2\) and the mass on \(u_0\) play a crucial role in the global boundedness of the system. Specifically, in the attractive dominant case \(\beta_1 > \beta_2\), the uniform boundedness of nonnegative solutions has been guaranteed under the condition \(\int_{\mathbb{R}^2} u_0 \,d x < 4 \pi \). Also, in the balance case \(\beta_1 = \beta_2\), it has been proven that every nonnegative solution is bounded uniformly in time without any restriction on the mass \(u_0\). In this paper, we discuss the global existence and boundedness of nonnegative solutions to the system in the repulsive dominant case \(\beta_1 < \beta_2\).Li-Yau multipLier set and optimal Li-Yau gradient estimate on hyperboLic spaceshttps://www.zbmath.org/1483.350592022-05-16T20:40:13.078697Z"Yu, Chengjie"https://www.zbmath.org/authors/?q=ai:yu.chengjie"Zhao, Feifei"https://www.zbmath.org/authors/?q=ai:zhao.feifeiThis paper is motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound. To reach this aim, the authors first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold, then, an optimal Li-Yau-type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Lastly, as an application of the previous results, a sharp and interesting Harnack inequalities on hyperbolic spaces is shown.
Reviewer: Vincenzo Vespri (Firenze)On some nonlocal problems in the calculus of variationshttps://www.zbmath.org/1483.350602022-05-16T20:40:13.078697Z"Chipot, Michel"https://www.zbmath.org/authors/?q=ai:chipot.michel"Mikayelyan, Hayk"https://www.zbmath.org/authors/?q=ai:mikayelyan.haykSummary: The goal of this paper is to investigate some nonlocal problems of the calculus of variations where pointwise comparisons principles fail but where some kind of nonlocal maximum principle or monotonicity property remains true for some quantities attached to the problem. A special attention will be given to a nonlocal energy recently introduced in [\textit{H. Mikayelyan}, ESAIM, Control Optim. Calc. Var. 24, No. 2, 859--872 (2018; Zbl 1402.49007)].Maximum principle for solutions of a Cauchy problem and its connection to Schoenberg's problemhttps://www.zbmath.org/1483.350612022-05-16T20:40:13.078697Z"Manov, A. D."https://www.zbmath.org/authors/?q=ai:manov.a-dSummary: In this paper we prove maximum principle for solutions of a Cauchy problem and find its connection to positive definite functions and in particular Schoenberg's problem.Strong unique continuation for two-dimensional anisotropic elliptic systemshttps://www.zbmath.org/1483.350622022-05-16T20:40:13.078697Z"Kuan, Rulin"https://www.zbmath.org/authors/?q=ai:kuan.rulin"Nakamura, Gen"https://www.zbmath.org/authors/?q=ai:nakamura.gen"Sasayama, Satoshi"https://www.zbmath.org/authors/?q=ai:sasayama.satoshiSummary: In this paper, we give the strong unique continuation property for a general two-dimensional anisotropic elliptic system with real coefficients in a Gevrey class under the assumption that the principal symbol of the system has simple characteristics. The strong unique continuation property is derived by obtaining some Carleman estimate. The derivation of the Carleman estimate is based on transforming the system to a larger second order elliptic system with diagonal principal part which has complex coefficients.Geometric regularity theory for a time-dependent Isaacs equationhttps://www.zbmath.org/1483.350632022-05-16T20:40:13.078697Z"Andrade, Pêdra D. S."https://www.zbmath.org/authors/?q=ai:andrade.pedra-d-s"Rampasso, Giane C."https://www.zbmath.org/authors/?q=ai:rampasso.giane-c"Santos, Makson S."https://www.zbmath.org/authors/?q=ai:santos.makson-sSummary: The purpose of this work is to produce a regularity theory for a class of parabolic Isaacs equations. Our techniques are based on approximation methods which allow us to connect our problem with a Bellman parabolic model. An approximation regime for the coefficients, combined with a smallness condition on the source term unlocks new regularity results in Sobolev and Hölder spaces.Higher regularity for the Signorini problem for the homogeneous, isotropic Lamé systemhttps://www.zbmath.org/1483.350642022-05-16T20:40:13.078697Z"Rüland, Angkana"https://www.zbmath.org/authors/?q=ai:ruland.angkana"Shi, Wenhui"https://www.zbmath.org/authors/?q=ai:shi.wenhuiSummary: In this note we discuss the (higher) regularity properties of the Signorini problem for the homogeneous, isotropic Lamé system. Relying on an observation by \textit{R. Schumann} [Manuscr. Math. 63, No. 3, 255--291 (1989; Zbl 0692.73076)], we reduce the question of the solution's and the free boundary regularity for the homogeneous, isotropic Lamé system to the corresponding regularity properties of the obstacle problem for the half-Laplacian.Eternal solutions for a reaction-diffusion equation with weighted reactionhttps://www.zbmath.org/1483.350652022-05-16T20:40:13.078697Z"Iagar, Razvan Gabriel"https://www.zbmath.org/authors/?q=ai:iagar.razvan-gabriel"Sánchez, Ariel"https://www.zbmath.org/authors/?q=ai:sanchez.arielSummary: We prove existence and uniqueness of \textit{eternal solutions} in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation
\[
\partial_tu = \Delta u^m+|x|^{\sigma}u^p,
\]
posed in \(\mathbb{R}^N \), with \( m>1,0<p<1 \) and the critical value for the weight
\[
\sigma = \frac{2(1-p)}{m-1}.
\]
Existence and uniqueness of some specific solution holds true when \(m+p\geq2 \). On the contrary, no eternal solution exists if \(m+p<2 \). We also classify exponential self-similar solutions with a different interface behavior when \(m+p>2 \). Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivityhttps://www.zbmath.org/1483.350662022-05-16T20:40:13.078697Z"Gallay, Thierry"https://www.zbmath.org/authors/?q=ai:gallay.thierry"Mascia, Corrado"https://www.zbmath.org/authors/?q=ai:mascia.corradoThis paper performed a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby-Gawlinski model. The traveling wave profile and its asymptotic states at \(\pm \infty\) are investigated in the regimes are called, respectively, homogeneous invasion and heterogeneous invasion. In both cases, we prove that a propagating front exists whenever the speed parameter \(c\) is strictly positive. We also derive an accurate approximation of the front profile in the singular limit \(c \rightarrow 0.\) This paper also shows that there exists no minimal speed for the propagation fronts of a given system. Finally, all the results are summarized with some conclusions.
Reviewer: Lingeshwaran Shangerganesh (Ponda)Stability of traveling wave in a PDE approximation of coupled arrays of Chua's circuithttps://www.zbmath.org/1483.350672022-05-16T20:40:13.078697Z"Li, Ji"https://www.zbmath.org/authors/?q=ai:li.ji.1|li.ji.2"Yu, Qing"https://www.zbmath.org/authors/?q=ai:yu.qingSummary: Coupled arrays of Chua's circuit have been studied extensively. The system admits traveling front, back, pulse as well as periodic and chaotic waves. However, the stability problem of these waves has rarely been investigated. In this paper, we consider the PDE approximation of the system and prove for the PDE that the traveling back is nonlinearly stable. The question of stability can be reduced to an eigenvalue problem in \(\mathbb{R}^4\). The Evans function method is applied, along with the geometric singular perturbation theory and the smooth linearization.On a family of torsional creep problems in Finsler metricshttps://www.zbmath.org/1483.350692022-05-16T20:40:13.078697Z"Fărcăşeanu, Maria"https://www.zbmath.org/authors/?q=ai:farcaseanu.maria"Mihăilescu, Mihai"https://www.zbmath.org/authors/?q=ai:mihailescu.mihai"Stancu-Dumitru, Denisa"https://www.zbmath.org/authors/?q=ai:stancu-dumitru.denisaSummary: The asymptotic behavior of solutions to a family of Dirichlet boundary value problems, involving differential operators in divergence form, on a domain equipped with a Finsler metric is investigated. Solutions are shown to converge uniformly to the distance function to the boundary of the domain, which takes into account the Finsler norm involved in the equation. This implies that a well-known result in the analysis of problems modeling torsional creep continues to hold in this more general setting.Balanced viscosity solutions to a rate-independent coupled elasto-plastic damage systemhttps://www.zbmath.org/1483.350722022-05-16T20:40:13.078697Z"Crismale, Vito"https://www.zbmath.org/authors/?q=ai:crismale.vito"Rossi, Riccarda"https://www.zbmath.org/authors/?q=ai:rossi.riccardaIn nonlinear elasticity, rate-independent systems are idealized models where internal oscillations and viscous dissipations are neglected, since the (slower) scale of external loadings is dominant. On the other hand, in the latter time scale the system presents time discontinuities, corresponding to fast transitions between equilibria. In such transitions, a major role is played by the viscous dissipations.
A well-known method to study e.g.\ damage models is to consider a system where the flow rule for the damage variable is viscously regularized; next, one passes to the limit as the viscosity parameter tends to zero. The time discontinuities of the resulting evolution can be interpolated by means of transitions governed again by viscosity.
In this paper the authors study a model for damage coupled with plasticity, affected by viscosity both in the damage evolution and in the elastoplastic evolution. Moreover, a further dissipation source may come from a hardening process. Viscosity and hardening provide regularizing terms in the PDE system.
In the rate-independent idealization, one would neglect both viscosity and hardening. To rigorously see this, the authors consider a singularly perturbed PDE system where the terms related to viscosity and hardening are modulated by small parameters tending to zero. By tuning the speed of the convergence of such coefficients, one may model a system where the elastic and the plastic strain converge to rate-independent evolution with the same rate, or with a faster rate, than the damage variable.
Specifically, the system analyzed by the authors features three coefficients: a hardening parameter \(\mu\), a viscosity parameter \(\varepsilon\) related to damage, and a viscosity coefficient \(\varepsilon\nu\) related to plasticity. In fact, \(\nu\) is a rate parameter that modulates the rate of convergence of the damage variable with respect to the plastic strain. The authors study the convergence of the system as \(\varepsilon\to0\) while \(\nu,\mu\) are fixed, or as \(\varepsilon,\nu\to0\), or as all parameters \(\varepsilon,\nu,\mu\) converge to zero. In the limit, they obtain different notions of solutions, showing in the time discontinuities a single-rate or a multi-rate character. Studying various notions of rate-independent solutions is important in order to understand which of them captures the behavior of the system for small viscosities.
Reviewer: Giuliano Lazzaroni (Firenze)On the regularity and partial regularity of extremal solutions of a Lane-Emden systemhttps://www.zbmath.org/1483.350752022-05-16T20:40:13.078697Z"Hajlaoui, Hatem"https://www.zbmath.org/authors/?q=ai:hajlaoui.hatemSummary: In this paper, we consider the system \(-\Delta u =\lambda (v+1)^p\), \(-\Delta v = \gamma (u+1)^\theta \) on a smooth bounded domain \( \Omega \) in \( \mathbb{R}^N\) with Dirichlet boundary condition \( u=v=0\) on \( \partial \Omega \). Here \( \lambda ,\gamma \) are positive parameters and \( 1 < p \le \theta \). Let \(x_0\) be the largest root of the polynomial
\[
\begin{aligned} H(x) = x^4 - \frac {16p\theta (p+1)(\theta +1)}{(p\theta -1)^2}x^2 + \frac {16p\theta (p+1)(\theta +1)(p+\theta +2)}{(p\theta -1)^3}x\\ -\frac {16p\theta (p+1)^2(\theta +1)^2}{(p\theta -1)^4}.\end{aligned}
\]
We show that the extremal solutions associated to the above system are bounded provided \(N<2+2x_0\). This improves the previous work by \textit {C. Cowan} [Methods Appl. Anal. 22, No. 3, 301--312 (2015; Zbl 1332.35115)]. We also prove that if \( N\geq 2+2x_0\), then the singular set of any extremal solution has Hausdorff dimension less than or equal to \(N-(2+2x_0)\).Nonexistence of nonnegative entire solutions of semilinear elliptic systemshttps://www.zbmath.org/1483.350902022-05-16T20:40:13.078697Z"Gladkov, Alexander"https://www.zbmath.org/authors/?q=ai:gladkov.aleksandr-lvovich|gladkov.alexander-l"Sergeenko, Sergey"https://www.zbmath.org/authors/?q=ai:sergeenko.sergeySummary: We consider the second-order semilinear elliptic system \(\Delta u = p(x)v^\alpha\), \(\Delta v = q(x)u^\beta\), where \(x \in \mathbf{R}^N\), \(N \geq 3\), \(\alpha\) and \(\beta\) are positive constants, \(p\) and \(q\) are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions \(p\) and \(q\) are of slow decay.Calderón-Zygmund estimates for non-uniformly elliptic equations with discontinuous nonlinearities on nonsmooth domainshttps://www.zbmath.org/1483.350982022-05-16T20:40:13.078697Z"Byun, Sun-Sig"https://www.zbmath.org/authors/?q=ai:byun.sun-sig"Lim, Minkyu"https://www.zbmath.org/authors/?q=ai:lim.minkyuIn this paper, the authors consider a general class of the so-called double phase problems and study a global Calderón-Zygmund estimate for nonlinear elliptic equations with possibly discontinuous coefficients in non-smooth bounded domains. They also study regularity results in non-smooth domains going beyond the Lipschitz category.
Reviewer: Said El Manouni (Berlin)Maximal regularity of parabolic transmission problemshttps://www.zbmath.org/1483.351062022-05-16T20:40:13.078697Z"Amann, Herbert"https://www.zbmath.org/authors/?q=ai:amann.herbertSummary: Linear reaction-diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal \(L_{\mathrm{p}}\) regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally.Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear dampinghttps://www.zbmath.org/1483.351092022-05-16T20:40:13.078697Z"Lyu, Wenbin"https://www.zbmath.org/authors/?q=ai:lyu.wenbinThe existence of global non-negative weak solutions to the Keller-Segel chemotaxis system with logistic term
\begin{align*}
\partial_t u & = \mathrm{div}(D(v) \nabla u - u S(v)\nabla v) + f(u) \text{ in } (0,\infty)\times\Omega, \\
\partial_t v & = \Delta v - v + u\text{ in } (0,\infty)\times\Omega,
\end{align*}
supplemented with homogeneous Neumann boundary conditions and initial conditions \(u_0\in C(\bar{\Omega})\), \(u_0\ge 0\), \(u_0\not\equiv 0\), and \(v_0\in W^{2,\infty}(\Omega)\), \(v_0>0\), is investigated. Here, \(\Omega\) is a bounded domain of \(\mathbb{R}^n\), \(n\ge 1\), with smooth boundary, the functions \(D\) and \(S\) belong to \(C^3([0,\infty))\) and satisfy \[ 0 < k \le D(s) \le K\,, \quad |S(s)|\le K\,, \qquad s\ge x_0\,, \] for some \(x_0>0\), and the logistic term \(f\in C^1([0,\infty))\) is such that \(f(0)=0\) and \(f(s) s^{-l} \to -\infty\) as \(s\to\infty\) for some \(l>\max\left\{ \frac{2(n+2)}{n+4} , \frac{2(n-1)}{n} \right\}\). The proof is performed by a compactness approach, the key tool being the use of the superlinear dissipation induced by the logistic term to derive suitable estimates on \(v\) and then on \(u\).
Reviewer: Philippe Laurençot (Toulouse)Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal productionhttps://www.zbmath.org/1483.351112022-05-16T20:40:13.078697Z"Tao, Xueyan"https://www.zbmath.org/authors/?q=ai:tao.xueyan"Zhou, Shulin"https://www.zbmath.org/authors/?q=ai:zhou.shulin"Ding, Mengyao"https://www.zbmath.org/authors/?q=ai:ding.mengyaoSummary: This work is concerned with a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production: \[
u_t 0= \nabla \cdot ((1+u)^\alpha \nabla u) - \nabla u)- \nabla \cdot (u(1+u)^{\beta-1}\nabla v)+ f(u),\quad v_t = \Delta v- v +u^\gamma,
\]
with nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain, where \(\alpha, \beta, \in \mathbb{R}\) and \(\gamma > 0\). The logistic type source term \(f(u)\) satisfies that either \(f(u)\equiv 0\) or \(f(u)=ru-\mu u^k\) with \(r \in \mathbb{R}, \mu > 0\) and \(k>1\). The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameters conditions, which improves the known results.Blow up rate and blow up sets for degenerate parabolic equations coupled via nonlinear boundary fluxhttps://www.zbmath.org/1483.351122022-05-16T20:40:13.078697Z"Xu, Si"https://www.zbmath.org/authors/?q=ai:xu.siSummary: This paper deals with the blow up rate estimates of solutions for a degenerate parabolic system coupled via nonlinear boundary flux. The upper and lower bounds of blow up rate are established. With the aid of blow-up rates, we also obtain the blow up sets.Asymptotic behaviour of nonlocal \(p\)-Laplacian reaction-diffusion problemshttps://www.zbmath.org/1483.351152022-05-16T20:40:13.078697Z"Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomas"Herrera-Cobos, Marta"https://www.zbmath.org/authors/?q=ai:herrera-cobos.marta"Marín-Rubio, Pedro"https://www.zbmath.org/authors/?q=ai:marin-rubio.pedroSummary: In this paper, we focus on studying the existence of attractors in the phase spaces \(L^2(\Omega)\) and \(L^p(\Omega)\) (among others) for time-dependent \(p\)-Laplacian equations with nonlocal diffusion and nonlinearities of reaction-diffusion type. Firstly, we prove the existence of weak solutions making use of a change of variable which allows us to get rid of the nonlocal operator in the diffusion term. Thereupon, the regularising effect of the equation is shown applying an argument of a posteriori regularity, since under the assumptions made we cannot guarantee the uniqueness of weak solutions. In addition, this argument allows to ensure the existence of an absorbing family in \(W_0^{1,p}(\Omega)\). This leads to the existence of the minimal pullback attractors in \(L^2(\Omega)\), \(L^p(\Omega)\) and some other spaces as \(L^{p^\ast-\epsilon}(\Omega)\). Relationships between these families are also established.Curved fronts of bistable reaction-diffusion equations in spatially periodic mediahttps://www.zbmath.org/1483.351162022-05-16T20:40:13.078697Z"Guo, Hongjun"https://www.zbmath.org/authors/?q=ai:guo.hongjun"Li, Wan-Tong"https://www.zbmath.org/authors/?q=ai:li.wan-tong"Liu, Rongsong"https://www.zbmath.org/authors/?q=ai:liu.rongsong"Wang, Zhi-Cheng"https://www.zbmath.org/authors/?q=ai:wang.zhi-cheng.2|wang.zhi-cheng.1The authors studies the curved fronts of spatially periodic R-D equations of bistable type: \[u_t=\Delta u+f(x,y,u), \quad (t,x,y)\in\mathbb{R}\times\mathbb{R}^2, \] where \(\Delta u:=u_{xx}+u_{yy}\) denotes the diffusion term, and the reaction term \(f(x,y,u)\) is assumed to be periodic in \((x,y)\) and bistable in \(u\). Under the a priori assumption that there exist pulsating fronts in every direction, the authors give some sufficient and some necessary conditions (along with some concrete examples) of the existence of curved fronts. Moreover, they prove the uniqueness and stability of such curved front. Finally, the authors construct a curved front with varying interfaces and give an example.
Reviewer: Jia-Bing Wang (Wuhan)High-dimensional spatial patterns in fractional reaction-diffusion system arising in biologyhttps://www.zbmath.org/1483.351172022-05-16T20:40:13.078697Z"Owolabi, Kolade M."https://www.zbmath.org/authors/?q=ai:owolabi.kolade-matthewSummary: The concept of fractional derivative has been demonstrated to be successful when applied to model a range of physical and real life phenomena, be it in engineering and science related fields. It is a known fact that reaction-diffusion equation permits the use of different numerical methods in space and time. As a result, we introduce the Fourier spectral method for the discretization of space fractional derivative and adapt the modified version of the exponential time-integrator to advance in time in attempt to explore the dynamic richness of fractional reaction-diffusion equations in two and three dimensions. This approach gives a full diagonal representation of the fractional derivative operator and yields a better spectral convergence irrespective of the value of fractional order chosen in the experiment. Recommendations are made based on some amazing results which arise from the computational experiments. We intend to answer the question `why is wildlife animals going into extinction in Africa?' to a reasonable extent. We believe that the spatial patterns obtained in the simulation framework to mimic the ones found in wildlife would provide a measure and serves as a good alternative to an act of killing of wildlife animals for ornamental and decorative purposes, also would serve as a guild to textile industries on pattern formations.Blow-up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearityhttps://www.zbmath.org/1483.351182022-05-16T20:40:13.078697Z"Han, Yuzhu"https://www.zbmath.org/authors/?q=ai:han.yuzhuSummary: In this short note, the author establishes a blow-up result for a semilinear heat equation with logarithmic nonlinearity, by using the logarithmic Sobolev inequality. This improves a recent blow-up result obtained in [\textit{H. Chen} et al. [J. Math. Anal. Appl. 422, No. 1, 84--98 (2015; Zbl 1302.35071)].Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearityhttps://www.zbmath.org/1483.351192022-05-16T20:40:13.078697Z"Jaquette, Jonathan"https://www.zbmath.org/authors/?q=ai:jaquette.jonathan"Lessard, Jean-Philippe"https://www.zbmath.org/authors/?q=ai:lessard.jean-philippe"Takayasu, Akitoshi"https://www.zbmath.org/authors/?q=ai:takayasu.akitoshiSummary: In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapping region around the zero equilibrium.On stability with respect to boundary conditions for anisotropic parabolic equations with variable exponentshttps://www.zbmath.org/1483.351222022-05-16T20:40:13.078697Z"Zhan, Huashui"https://www.zbmath.org/authors/?q=ai:zhan.huashuiSummary: The anisotropic parabolic equations with variable exponents are considered. If some of diffusion coefficients \(\{b_{i}(x)\}\) are degenerate on the boundary, the others are always positive, then how to impose a suitable boundary value condition is researched. The existence of weak solutions is proved by the parabolically regularized method. The stability of weak solutions, based on the partial boundary value condition, is established by choosing a suitable test function.The Dirichlet-to-Neumann operator associated with the 1-Laplacian and evolution problemshttps://www.zbmath.org/1483.351262022-05-16T20:40:13.078697Z"Hauer, Daniel"https://www.zbmath.org/authors/?q=ai:hauer.daniel"Mazón, José M."https://www.zbmath.org/authors/?q=ai:mazon-ruiz.jose-mSummary: In this paper, we present first insights about the Dirichlet-to-Neumann operator in \(L^1\) associated with the 1-Laplace operator or total variational flow operator. This operator is the main object, for example, in studying inverse problems related to image processing, but also admits an important relation to geometry. We show that this operator can be represented by the sub-differential in \(L^1\times L^\infty\) of a convex, homogeneous, and continuous functional on \(L^1\). This is quite surprising since it implies a type of stability or compactness result even though the singular Dirichlet problem governed by the 1-Laplace operator might have infinitely many weak solutions (if the given boundary data is not continuous). As an application, we obtain well-posedness and long-time stability of solutions of a singular coupled elliptic-parabolic initial boundary-value problem.Nonnegative solutions to a doubly degenerate nutrient taxis systemhttps://www.zbmath.org/1483.351272022-05-16T20:40:13.078697Z"Li, Genglin"https://www.zbmath.org/authors/?q=ai:li.genglin"Winkler, Michael"https://www.zbmath.org/authors/?q=ai:winkler.michaelSummary: This paper deals with the doubly degenerate nutrient taxis system
\[
\begin{cases}
u_t = (uvu_x)_x - (u^2 vv_x)_x + \ell uv, & x\in \Omega, t>0, \\
v_t = v_{xx} -uv, & x\in \Omega, t>0,
\end{cases}
\]
in an open bounded interval \(\Omega\subset \mathbb{R}\), with \(\ell \geq 0\), which has been proposed to model the formation of diverse morphological aggregation patterns observed in colonies of \textit{Bacillus subtilis} growing on the surface of thin agar plates. It is shown that under the mere assumption that
\[
\begin{cases}
u_0\in W^{1,\infty}(\Omega) \text{ is nonnegative with } u_0\not\equiv 0 \qquad \text{and} \\
v_0\in W^{1,\infty}(\Omega) \text{ is positive in } \overline{\Omega}, \end{cases}
\tag{\(\star\)}
\]
an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution \((u,v)\), where \(u\geq 0\) and \(v>0\) in \(\overline{\Omega}\times [0,\infty)\), and that moreover there exists \(u_{\infty}\in C^0 (\overline{\Omega})\) such that the solution \((u,v)\) approaches the pair \((u_{\infty},0)\) in the large time limit with respect to the topology \((L^{\infty}(\Omega))^2\). This extends comparable results recently obtained in [17], the latter crucially relying on the additional requirement that \(\int_{\Omega} \ln u_0 >-\infty\), to situations involving nontrivially supported initial data \(u_0\), which seems to be of particular relevance in the addressed application context of sparsely distributed populations.The boundary value condition of a generate parabolic equationhttps://www.zbmath.org/1483.351282022-05-16T20:40:13.078697Z"Zhan, Huashui"https://www.zbmath.org/authors/?q=ai:zhan.huashuiSummary: Consider the following strong degenerate parabolic equation
\[
\frac{\partial u}{\partial t} = \frac{\partial}{\partial x_i}\left(a_{ij}(u)\frac{\partial}{\partial x_j}\right) + \sum_{i=1}^N \frac{\partial b_i(u)}{\partial x_i},\quad (x,t)\in \Omega\times (0,T),
\]
with the homogeneous boundary value, where \(\Omega\subset \mathbb{R}^N\) is an open bounded domain with \(C^2\) boundary \(\partial \Omega\), \((a_{ij})\) is a symmetric matrix with nonnegative characteristic values. If the equation is not only degenerate in the interior of \(\Omega\), but also degenerate on the boundary \(\partial \Omega\) in some sense, the paper discusses how to quote the suitable initial boundary values to assure the well-posedness of the problem. It can be shown that the part of the boundary, on which the homogeneous boundary value is posed, is varying with the constants quoted in the corresponding entropy solution.On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditionshttps://www.zbmath.org/1483.351302022-05-16T20:40:13.078697Z"Ngoc, Le Thi Phuong"https://www.zbmath.org/authors/?q=ai:le-thi-phuong-ngoc."Uyen, Khong Thi Thao"https://www.zbmath.org/authors/?q=ai:uyen.khong-thi-thao"Nhan, Nguyen Huu"https://www.zbmath.org/authors/?q=ai:nhan.nguyen-huu"Long, Nguyen Thanh"https://www.zbmath.org/authors/?q=ai:nguyen-thanh-long.Summary: In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operatorshttps://www.zbmath.org/1483.351322022-05-16T20:40:13.078697Z"Belluzi, Maykel"https://www.zbmath.org/authors/?q=ai:belluzi.maykel"Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomas"Nascimento, Marcelo J. D."https://www.zbmath.org/authors/?q=ai:nascimento.marcelo-jose-dias"Schiabel, Karina"https://www.zbmath.org/authors/?q=ai:schiabel.karinaSummary: In this paper we consider the nonautonomous semilinear parabolic problems with time-dependent linear operators
\[
u_t + A ( t ) u = f ( t , u ), \;\;t > \tau; \;\;u ( \tau ) = u_0,
\]
in a Banach space \(X\). Under suitable conditions, we obtain regularity results for \(u_t(t, x)\) with respect to its spatial variable \(x\) and estimates for \(u_t\) in stronger spaces \(( X^\alpha)\). We then apply those results to a nonautonomous reaction-diffusion equation
\[
u_t - d i v(a(t, x) \nabla u) + u = f(t, u)
\]
with Neumann boundary condition and time-dependent diffusion. From the regularity of \(u_t\) we derive the existence of classical solutions and from the estimates for \(u_t\) we prove that the variation of the solution \(u\) is bounded in the long-time dynamics. We also prove the existence of pullback attractor, as well as the existence of a compact set that contains the long-time dynamics of the derivatives \(u_t\), without requiring any assumption concerning monotonicity or decay in time of \(a(t, x)\).Parabolic quaternionic Monge-Ampère equation on compact manifolds with a flat hyperKähler metrichttps://www.zbmath.org/1483.351332022-05-16T20:40:13.078697Z"Zhang, Jiaogen"https://www.zbmath.org/authors/?q=ai:zhang.jiaogenSummary: The quaternionic Calabi conjecture was introduced by Alesker-Verbitsky, analogous to the Kähler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperKähler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic Monge-Ampère equation. Our goal is to prove the long time existence and \(C^{\infty}\) convergence for normalized solutions as \(t\rightarrow\infty \). As a consequence, we show that the limit function is exactly the solution of quaternionic Monge-Ampère equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.Well-posedness and stability for Kirchhoff equation with non-porous acoustic boundary conditionshttps://www.zbmath.org/1483.351372022-05-16T20:40:13.078697Z"Vicente, A."https://www.zbmath.org/authors/?q=ai:vicente.andre|vicente.antonio-cascales|vicente.avelinoSummary: In this paper we prove the well-posedness to the wave equation of Kirchhoff type. Under a portion of the boundary, we consider the acoustic boundary conditions. We also prove the exponential stability of the energy associated to the problem. Our result generalize the previous literature where only the Carrier model was considered (when the nonlinearity involves the \(L^2(\Omega)\) norm) or the Kirchhoff model with porous acoustic boundary conditions. The main tool is the Faedo-Galerkin method. Due to the presence of acoustic boundary conditions, we can not use special basis and new estimates are necessary. To prove the stability we use integral estimates.Stability properties of dissipative evolution equations with nonautonomous and nonlinear dampinghttps://www.zbmath.org/1483.351382022-05-16T20:40:13.078697Z"Nicaise, Serge"https://www.zbmath.org/authors/?q=ai:nicaise.sergeSummary: In this paper, we obtain some stability results of (abstract) dissipative evolution equations with a nonautonomous and nonlinear damping using the exponential stability of the retrograde problem with a linear and autonomous feedback and a comparison principle. We then illustrate our abstract statements for different concrete examples, where new results are achieved. In a preliminary step, we prove some well-posedness results for some nonlinear and nonautonomous evolution equations.Singular matrix conjugacy problem with rapidly oscillating off-diagonal entries. Asymptotics of the solution in the case when a diagonal entry vanishes at a stationary pointhttps://www.zbmath.org/1483.351442022-05-16T20:40:13.078697Z"Budylin, A. M."https://www.zbmath.org/authors/?q=ai:budylin.a-mIn this paper, the author considers a \(2 \times 2\) matrix conjugation problem (the Riemann-Hilbert factorization problem) with rapidly oscillating off-diagonal inputs and a quadratic phase function, in particular when one of the diagonal inputs vanishes at a stationary point. The main result here is the determination of the leading term with respect to time asymptotics of the solution of this problem.
Reviewer: Philippe Briet (Toulon)Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditionshttps://www.zbmath.org/1483.351462022-05-16T20:40:13.078697Z"Wang, Deng-Shan"https://www.zbmath.org/authors/?q=ai:wang.dengshan"Guo, Boling"https://www.zbmath.org/authors/?q=ai:guo.boling"Wang, Xiaoli"https://www.zbmath.org/authors/?q=ai:wang.xiaoli.2|wang.xiaoli.1The Kundu-Eckhaus equation
\[
iq_t+\frac{1}{2}q_{xx}+|q|^2q+2\beta |q|^4q-2i\beta(|q|^2)_x+q=0
\]
is considered, with the initial condition
\[
q(x,0)\sim Ae^{i(\mu x+\theta_\pm)} \text{ when } x\to\pm\infty.
\]
Here \(\beta, A(>0),\mu\), and \(\theta_\pm\) are real constants.
The long-time asymptotics of the solution is established in three different sectors depending on the magnitude \(\xi=x/t\): plane wave sector \(\xi<\xi_1=\mu-2\beta A^2-2\sqrt{2}A\), plane wave sector \(\xi>\xi_2=\mu-2\beta A^2+2\sqrt{2}A\), and modulated genus 1 elliptic wave sector \(\xi_1<\xi<\xi_2\).
Reviewer: Ilya Spitkovsky (Williamsburg)Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domainshttps://www.zbmath.org/1483.351512022-05-16T20:40:13.078697Z"Falocchi, Alessio"https://www.zbmath.org/authors/?q=ai:falocchi.alessio"Gazzola, Filippo"https://www.zbmath.org/authors/?q=ai:gazzola.filippoThe Navier-Stokes equations equipped with the Navier boundary conditions are considered in three-dimensional Lipschitz domains enjoying some symmetries, called ``sectors''. These domains are of interest in real-life applications. The regularity of solutions is studied using delicate symmetrization techniques.
Reviewer: Piotr Biler (Wrocław)Modeling, approximation, and time optimal temperature control for binder removal from ceramicshttps://www.zbmath.org/1483.351612022-05-16T20:40:13.078697Z"Chicone, Carmen"https://www.zbmath.org/authors/?q=ai:chicone.carmen-c"Lombardo, Stephen J."https://www.zbmath.org/authors/?q=ai:lombardo.stephen-j"Retzloff, David G."https://www.zbmath.org/authors/?q=ai:retzloff.david-gSummary: The process of binder removal from green ceramic components -- a reaction-gas transport problem in porous media -- has been analyzed with a number of mathematical techniques: 1) non-dimensionalization of the governing decomposition-reaction ordinary differential equation (ODE) and of the reaction gas-permeability partial differential equation (PDE); 2) development of a pseudo steady state approximation (PSSA) for the PDE, including error analysis via \(L^2\) norm and singular perturbation methods; 3) derivation and analysis of a discrete model approximation; and 4) development of a time optimal control strategy to minimize processing time with temperature and pressure constraints. Theoretical analyses indicate the conditions under which the PSSA and discrete models are viable approximations. Numerical results indicate that under a range of conditions corresponding to practical binder burnout conditions, utilization of the optimal temperature protocol leads to shorter cycle times as compared to typical industrial practice.A rigorous derivation and energetics of a wave equation with fractional dampinghttps://www.zbmath.org/1483.351652022-05-16T20:40:13.078697Z"Mielke, Alexander"https://www.zbmath.org/authors/?q=ai:mielke.alexander"Netz, Roland R."https://www.zbmath.org/authors/?q=ai:netz.roland-r"Zendehroud, Sina"https://www.zbmath.org/authors/?q=ai:zendehroud.sinaThe authors consider the coupled system accounting for the motion of longitudinal elastic waves of a membrane coupled to viscous flows in the enclosing half-space. This system is written as \(\rho _{memb}\overset{..}{U} =\kappa \Delta _{x}U-\mu \partial _{z}v\), \(\overset{.}{U}(t,x)=v(t,x,0)\), for \(t>0\), \(x\in \Sigma \subset \mathbb{R}^{d-1}\), where \(U\) is the horizontal displacement of the membrane, \(\Sigma \) represents the membrane, and \(v\) is the horizontal velocity of the viscous fluid which satisfies \( \rho _{bulk}\overset{.}{v}=\mu \Delta _{x,z}v\), for \(t>0\), \((x,z)\in \Omega =\Sigma \times (-\infty ,0)\).\ The authors introduce the associated energy \( \mathbb{E}(U,\overset{.}{U},v)=\int_{\Sigma }(\frac{\rho _{memb}}{2}\overset{ ..}{U}^{2}+\frac{\kappa }{2}\left\vert \nabla U\right\vert ^{2})dx+\int_{\Omega }\frac{\rho _{bulk}}{2}v^{2}dzdx\) and the functional space \(\mathbf{H}=H^{1}(\Sigma )\times L^{2}(\Sigma )\times L^{2}(\Omega )\), and they prove that \(\mathbb{E}\) acts as a Lyapunov function and that it is a bounded quadratic form on \(\mathbf{H}\). They introduce a non-dimensional form of this system \(\overset{..}{U}=\Delta _{x}U-\partial _{z}v\mid _{z=0}\) , \(\overset{.}{U}=v\mid _{z=0}\), for \(t>0\), \(x\in \Sigma \), \(\overset{.}{v} =\varepsilon ^{2}\Delta _{x}v+\partial _{z}^{2}v\) for \(t>0\), \((x,z)\in \Omega \), with \(\varepsilon =\mu /\sqrt{\rho _{memb}k}\), and the operator \( A_{\varepsilon }:D(A_{\varepsilon })\subset \mathbf{H}\rightarrow \mathbf{H}\) through \(A_{\varepsilon }\left( \begin{array}{c} U \\
V \\
v \end{array} \right) =\left( \begin{array}{c} V \\
\Delta _{x}U-\partial _{z}v\mid _{z=0} \\
\varepsilon ^{2}\Delta _{x}v+\partial _{z}^{2}v \end{array} \right) \). The first main result proves that the operator \(A_{\varepsilon }\) is the generator of a strongly continuous semigroup. The second main result of the paper describes the asymptotic behavior of the solution to the original problem when \(\varepsilon \) goes to 0: for every initial condition \( w_{0}\in \mathbf{H}\), the semi-group solution \(w^{\varepsilon }(t)=e^{tA_{\varepsilon }}w_{0}\) strongly converges to some limit \(w^{0}\). In the last part of their paper, the authors prove that the damped wave equation \(\overset{..}{U}(t,x)+\int_{0}^{t}\frac{1}{\sqrt{\pi (t-\tau )}} \overset{..}{U}(\tau ,x)d\tau =\Delta U(t,x)\) on \(\Sigma \) carries a natural energy-dissipation structure.
Reviewer: Alain Brillard (Riedisheim)Bound states in the continuum and Fano resonances in subwavelength resonator arrayshttps://www.zbmath.org/1483.351672022-05-16T20:40:13.078697Z"Ammari, Habib"https://www.zbmath.org/authors/?q=ai:ammari.habib-m"Davies, Bryn"https://www.zbmath.org/authors/?q=ai:davies.bryn"Hiltunen, Erik Orvehed"https://www.zbmath.org/authors/?q=ai:hiltunen.erik-orvehed"Lee, Hyundae"https://www.zbmath.org/authors/?q=ai:lee.hyundae"Yu, Sanghyeon"https://www.zbmath.org/authors/?q=ai:yu.sanghyeonAuthors' abstract: When wave scattering systems are subject to certain symmetries, resonant states may decouple from the far-field continuum; they remain localized to the structure and cannot be excited by incident waves from the far field. In this work, we use layer-potential techniques to prove the existence of such states, known as bound states in the continuum, in systems of subwavelength resonators. When the symmetry is slightly broken, this resonant state can be excited from the far field. Remarkably, this may create asymmetric (Fano-type) scattering behavior where the transmission is fundamentally different for frequencies on either side of the resonant frequency. Using asymptotic analysis, we compute the scattering matrix of the system explicitly, thereby characterizing this Fano-type transmission anomaly.
Reviewer: Konstantin Merz (Braunschweig)Long-time behavior of solutions to cubic Dirac equation with Hartree type nonlinearity in \(\mathbb{R}^{1+2}\)https://www.zbmath.org/1483.351722022-05-16T20:40:13.078697Z"Tesfahun, Achenef"https://www.zbmath.org/authors/?q=ai:tesfahun.achenefSummary: In this paper we study the long-time behavior of solutions to the Dirac equation
\[
\big(-i\gamma^\mu\partial_\mu+m\big)\psi=\left(V\ast(\overline{\psi}\psi)\right)\psi\text{ in }\mathbb{R}^{1+2},
\]
where \(V\) is the Yukawa potential in \(\mathbb{R}^2\). It is proved that if \(m>0\) and the initial data is small in \(H^s(\mathbb{R}^2)\) for \(s>0\), the corresponding initial value problem is globally well posed and the solution scatters to free waves asymptotically as \(t\rightarrow\pm\infty\). The main ingredients in the proof are Strichartz estimates and space-time \(L^2\)-bilinear null-form estimates for free waves.First-order reduction and emergent behavior of the one-dimensional kinetic Cucker-Smale equationhttps://www.zbmath.org/1483.351732022-05-16T20:40:13.078697Z"Kim, Jeongho"https://www.zbmath.org/authors/?q=ai:kim.jeonghoSummary: In this paper, we introduce the kinetic description of the first-order Cucker-Smale (CS) flocking model on the real line. We reveal the equivalent relation between the measure-valued solution to the first- and second-order kinetic CS equations. The emergent behavior of the first-order kinetic CS equation and the characterization of the asymptotic solution are studied. We also provide the equivalent relation between classical/measure-valued solutions to the first-order kinetic CS equation and a classical solution to the second-order hydrodynamic CS equations, and present the corresponding analysis on the large-time behavior. The numerical experiments support our analysis and provide an efficient algorithm to obtain the asymptotic solution without simulating the model for a long time.Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasmahttps://www.zbmath.org/1483.351772022-05-16T20:40:13.078697Z"Du, Xia-Xia"https://www.zbmath.org/authors/?q=ai:du.xia-xia"Tian, Bo"https://www.zbmath.org/authors/?q=ai:tian.bo"Qu, Qi-Xing"https://www.zbmath.org/authors/?q=ai:qu.qixing"Yuan, Yu-Qiang"https://www.zbmath.org/authors/?q=ai:yuan.yu-qiang"Zhao, Xue-Hui"https://www.zbmath.org/authors/?q=ai:zhao.xue-huiSummary: Electron-positron-ion plasmas are found in the primordial Universe, active galactic nuclei, surroundings of black holes and peripheries of neutron stars. We focus our attention on a modified Zakharov-Kuznetsov (mZK) equation which describes the ion acoustic drift solitary waves in an electron-positron-ion magnetoplasma. Lie symmetry generators and groups are presented by virtue of the Lie symmetry method. Optimal system of the one-dimensional subalgebras is presented, which is influenced via the ratio of the unperturbed ion density to electron density \(n_{io}/n_{eo}\), the ratio of the unperturbed positron density to electron density \(n_{po}/n_{eo}\), the ratio of the electron temperature to positron temperature \(T_e/T_p\) and the normalized ion drift velocity \(v_o^*\). Based on the optimal system, we construct the power-series, multi-soliton, breather-like and periodic-wave solutions. Two types of the elastic interactions, including the overtaking and head-on interactions between (among) two (three) solitons are discussed. We find that the amplitudes of the solitons and periodic waves are positively related to the electron Debye length \(\lambda_{De}\) and negatively related to \(|\rho_i|\) with \(\rho_i\) as the ion Larmor radius. Besides, we find that the mZK equation is not only strictly self-adjoint but also nonlinearly self-adjoint. Condition for the nonlinear self-adjointness is related to \(n_{io}/n_{eo}\), \(n_{po}/n_{eo}\), \(T_e/T_p\) and \(v_o^*\). Based on the nonlinear self-adjointness of the mZK equation, conservation laws, which are related to \(n_{io}/n_{eo}\), \(n_{po}/n_{eo}\), \(T_e/T_p\), \(v_o^*\), \(\lambda_{De}\), \(\rho_i\) and may be associated with the conservation of momentum and energy, are obtained.The translating soliton equationhttps://www.zbmath.org/1483.351782022-05-16T20:40:13.078697Z"López, Rafael"https://www.zbmath.org/authors/?q=ai:lopez.rafael-beltran|lopez-camino.rafaelSummary: We give an analytic approach to the translating soliton equation with a special emphasis in the study of the Dirichlet problem in convex domains of the plane.
For the entire collection see [Zbl 1473.53006].Recurrent solutions of the Korteweg-de Vries equation with boundary forcehttps://www.zbmath.org/1483.351832022-05-16T20:40:13.078697Z"Chen, Mo"https://www.zbmath.org/authors/?q=ai:chen.moSummary: In this paper, we will establish the existence of the bounded solution, periodic solution, quasi-periodic solution and almost periodic solution for the Korteweg-de Vries equation with boundary force.New traveling wave solutions and interesting bifurcation phenomena of generalized KdV-mKdV-like equationhttps://www.zbmath.org/1483.351842022-05-16T20:40:13.078697Z"Chen, Yiren"https://www.zbmath.org/authors/?q=ai:chen.yiren"Li, Shaoyong"https://www.zbmath.org/authors/?q=ai:li.shaoyongSummary: Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.Generalized solutions of an inhomogeneous inviscid Burgers equationhttps://www.zbmath.org/1483.351852022-05-16T20:40:13.078697Z"Engu, Satyanarayana"https://www.zbmath.org/authors/?q=ai:engu.satyanarayana"Manasa, M."https://www.zbmath.org/authors/?q=ai:manasa.m"Venkatramana, P. B."https://www.zbmath.org/authors/?q=ai:venkatramana.p-bSummary: We derive generalized solutions of an inhomogeneous inviscid Burgers equation using vanishing viscosity method. This is achieved with the classical solution of a concerned viscous inhomogeneous Burgers equation. We then study Riemann problem for a de-coupled system. The weak solutions of the system are explicitly obtained by Volpert product concept. There are infinitely many real valued solutions for the system in the case of rarefaction wave and the weak solutions consist of \(\delta\)-measures in the case of shock wave. Motivated by the structure of weak solutions, we construct the explicit generalized solutions for a more general de-coupled system.Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaceshttps://www.zbmath.org/1483.351882022-05-16T20:40:13.078697Z"Yan, Wei"https://www.zbmath.org/authors/?q=ai:yan.wei"Zhang, Yimin"https://www.zbmath.org/authors/?q=ai:zhang.yimin"Li, Yongsheng"https://www.zbmath.org/authors/?q=ai:li.yongsheng"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiaoSummary: We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation
\[
\partial_x \left( u_t -\beta\partial_x^3 u +\partial_x (u^2)\right)+\partial_y^2 u-\gamma u = 0
\]
in the anisotropic Sobolev spaces \(H^{s_1,s_2} (\mathbb{R}^2)\). When \(\beta <0\) and \(\gamma >0\), we prove that the Cauchy problem is locally well-posed in \(H^{s_1, s_2}(\mathbb{R}^2)\) with \(s_1 >-\frac{1}{2}\) and \(s_2 \geq 0\). Our result considerably improves the Theorem 1.4 of [\textit{R. M. Chen} et al., Trans. Am. Math. Soc. 364, No. 7, 3395--3425 (2012; Zbl 1291.35276)]. The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in \(H^{s_1,0} (\mathbb{R}^2)\) with \(s_1<-\frac{1}{2}\) in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not \(C^3\). When \(\beta <0,\gamma >0\), by using the \(U^p\) and \(V^p\) spaces, we prove that the Cauchy problem is locally well-posed in \(H^{-\frac{1}{2},0}(\mathbb{R}^2)\).Exact solutions of cubic-quintic modified Korteweg-de-Vries equationhttps://www.zbmath.org/1483.351892022-05-16T20:40:13.078697Z"Zemlyanukhin, Alexander I."https://www.zbmath.org/authors/?q=ai:zemlyanukhin.alexander-i"Bochkarev, Andrey V."https://www.zbmath.org/authors/?q=ai:bochkarev.andrey-vThe aim of this paper is to study a non-integrable modified Korteweg-de-Vries (mKdV) equation containing a combination of third and fifth degree nonlinear terms that simulate waves in a three layer fluid, as well as in spatially one-dimensional nonlinear elastic deformable systems. Using the Painlevé analysis [\textit{R. M. Conte} and \textit{M. Musette}, The Painlevé handbook. Dordrecht: Springer (2008; Zbl 1153.34002)], the authors study the analytical structure of equation, obtained from the mKdV 3-5 equation by transition to a traveling wave variable, build its solution, expressed in terms of the Weierstrass elliptic function [\textit{A. V. Porubov}, J. Phys. A, Math. Gen. 26, No. 17, L\, 797-L\, 800 (1993; Zbl 0803.35132); \textit{R. Racke}, Appl. Anal. 58, No. 1--2, 85--100 (1995; Zbl 0832.35097)], classify exact and approximate partial solitary-wave and periodic solutions and plot the corresponding graphs. It is established that mKdV equation passes the Painlevé test in a weak form. After the traveling wave transformation, this equation reduces to a generalized Weierstrass elliptic function equation, the right side of which is determined by a sixth order polynomial in the dependent variable. Determined by the structure of the polynomial roots, the general solution of the equation is expressed in terms of the Weierstrass elliptic function or its successive degenerations rational functions depending on the exponential functions of the traveling wave variable or directly on traveling wave variable. The classification of exact solitary wave and periodic solutions is carried out, and the ranges of parameters necessary for their physical feasibility are revealed. An approach is proposed for constructing approximate solitary wave and periodic solutions to generalized Weierstrass elliptic equation with a polynomial right hand side of high orders. This paper is organized as follows: The first section is an introduction to the subject. The second section deals with Painlevé analysis. The third and forth sections are devoted to periodic and soliton solutions. The fifth section deals with kink-shaped solution. The sixth section is devoted to approximate solution.
For the entire collection see [Zbl 1471.74003].
Reviewer: Ahmed Lesfari (El Jadida)Energy conservation for the incompressible inhomogeneous Euler-Korteweg equations in a bounded domainhttps://www.zbmath.org/1483.351902022-05-16T20:40:13.078697Z"Zhang, Zhipeng"https://www.zbmath.org/authors/?q=ai:zhang.zhipengSummary: In this paper, we investigate the principle of the energy conservation for the weak solutions of the incompressible inhomogeneous Euler-Korteweg equations in a bounded domain. We provide two sufficient conditions on the regularity of the weak solutions to ensure the energy conservation. Due to the presence of the boundary, we need to impose the boundedness for the velocity and the Besov-type continuity for the normal component of the velocity and the gradient of the density near the boundary.
{\copyright 2021 American Institute of Physics}Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in \(\mathbb{R}^2\)https://www.zbmath.org/1483.351922022-05-16T20:40:13.078697Z"Alouini, Brahim"https://www.zbmath.org/authors/?q=ai:alouini.brahimSummary: We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in \(\mathbb{R}^2\) that reads
\[
u_t +i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right) +ig(|u|^2)u+\gamma u = f\, ,\quad (t,(x,y))\in \mathbb{R}\times \mathbb{R}^2.
\]
We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.Instability of ground states for the NLS equation with potential on the star graphhttps://www.zbmath.org/1483.351952022-05-16T20:40:13.078697Z"Ardila, Alex H."https://www.zbmath.org/authors/?q=ai:ardila.alex-hernandez"Cely, Liliana"https://www.zbmath.org/authors/?q=ai:cely.liliana"Goloshchapova, Nataliia"https://www.zbmath.org/authors/?q=ai:goloshchapova.nataliiaSummary: We study the nonlinear Schrödinger equation with an arbitrary real potential \(V(x)\in (L^1 +L^{\infty})(\Gamma)\) on a star graph \(\Gamma\). At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength \(-\gamma <0\). We show the existence of ground states \(\varphi_{\omega} (x)\) as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on \(V(x)\). Moreover, for \(V(x)=-\dfrac{\beta}{x^{\alpha}}\), in the supercritical case, we prove that the standing waves \(e^{i\omega t}\varphi_{\omega} (x)\) are orbitally unstable in \(H^1 (\Gamma)\) when \(\omega\) is large enough. Analogous result holds for an arbitrary \(\gamma \in \mathbb{R}\) when the standing waves have symmetric profile.Regularity properties of the cubic biharmonic Schrödinger equation on the half linehttps://www.zbmath.org/1483.351972022-05-16T20:40:13.078697Z"Başakoğlu, Engin"https://www.zbmath.org/authors/?q=ai:basakoglu.enginSummary: In this paper we study the regularity properties of the cubic biharmonic Schrödinger equation posed on the right half line. We prove local well-posedness and obtain a smoothing result in the low-regularity spaces on the half line. In particular we prove that the nonlinear part of the solution on the half line is smoother than the initial data obtaining a full derivative gain in certain cases. Moreover, in the defocusing case, we establish global well-posedness and global smoothing in the higher order regularity spaces by making use of the global-wellposedness result of \textit{T. Özsarı} and \textit{N. Yolcu} [Commun. Pure Appl. Anal. 18, No. 6, 3285--3316 (2019; Zbl 1479.35816)] in the energy space. Also this paper improves the well-posedness result of Özsarı and Yolcu [loc. cit.] in the case of cubic nonlinearity.Scattering in the weighted \( L^2 \)-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearityhttps://www.zbmath.org/1483.351982022-05-16T20:40:13.078697Z"Bensouilah, Abdelwahab"https://www.zbmath.org/authors/?q=ai:bensouilah.abdelwahab"Dinh, Van Duong"https://www.zbmath.org/authors/?q=ai:dinh.van-duong"Majdoub, Mohamed"https://www.zbmath.org/authors/?q=ai:majdoub.mohamedSummary: We investigate the defocusing inhomogeneous nonlinear Schrödinger equation
\[ i \partial_tu + \Delta u = |x|^{-b} \left(\mathrm{e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0) = u_0, \quad x \in \mathbb{R}^2, \]
with \( 0<b<1 \) and \( \alpha = 2\pi(2-b) \). First we show the decay of global solutions by assuming that the initial data \( u_0 \) belongs to the weighted space \( \Sigma(\mathbb{R}^2) = \{u\in H^1(\mathbb{R}^2) : |x|u\in L^2(\mathbb{R}^2)\} \). Then we combine the local theory with the decay estimate to obtain scattering in \( \Sigma \) when the Hamiltonian is below the value \( \frac{2}{(1+b)(2-b)} \).A priori bounds for the kinetic DNLShttps://www.zbmath.org/1483.351992022-05-16T20:40:13.078697Z"Cacciafesta, Federico"https://www.zbmath.org/authors/?q=ai:cacciafesta.federico"Tsutsumi, Yoshio"https://www.zbmath.org/authors/?q=ai:tsutsumi.yoshioSummary: In this note, we consider the kinetic derivative nonlinear Schrödinger equation (KDNLS), which arises as a model of propagation of a plasma taking the effect of the resonant interaction between the wave modulation and the ions into account. In contrast to the standard derivative NLS equation, KDNLS does not conserve the mass and the energy. Nevertheless, the dissipative structure of KDNLS enables us to show an a priori bound in the energy space and a lower bound of the \(L^2\) norm for its solution, as we see in this note. Combined with the local wellposedness result, which we plan to show in a forthcoming paper, these bounds will give a global existence result in the energy space for small initial data.
For the entire collection see [Zbl 1459.37002].The soliton solutions and long-time asymptotic analysis for an integrable variable coefficient nonlocal nonlinear Schrödinger equationhttps://www.zbmath.org/1483.352002022-05-16T20:40:13.078697Z"Chen, Guiying"https://www.zbmath.org/authors/?q=ai:chen.guiying"Xin, Xiangpeng"https://www.zbmath.org/authors/?q=ai:xin.xiangpeng"Zhang, Feng"https://www.zbmath.org/authors/?q=ai:zhang.fengSummary: An integrable variable coefficient nonlocal nonlinear Schrödinger equation (NNLS) is studied; by employing the Hirota's bilinear method, the bilinear form is obtained, and the \(N\)-soliton solutions are constructed. In addition, some singular solutions and period solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions, the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic.Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on \(\mathbb{T}^4\)https://www.zbmath.org/1483.352012022-05-16T20:40:13.078697Z"Chen, Xuwen"https://www.zbmath.org/authors/?q=ai:chen.xuwen"Holmer, Justin"https://www.zbmath.org/authors/?q=ai:holmer.justinSummary: We consider the \(\mathbb{T}^4\) cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross-Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [\textit{X. Chen} and \textit{J. Holmer}, Invent. Math. 217, No. 2, 433--547 (2019; Zbl 1422.35148); \textit{S. Herr} and \textit{V. Sohinger}, Commun. Contemp. Math. 21, No. 7, Article ID 1850058, 33 p. (2019; Zbl 1428.35516)] and does not require the existence of a solution in Strichartz-type spaces. We prove \(U\)-\(V\) multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel-Born expansion. The new combinatorics and the \(U\)-\(V\) estimates then seamlessly conclude the \(H^1\) unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove \(H^1\) uniqueness for the \(\mathbb{R}^3/\mathbb{R}^4/\mathbb{T}^3/\mathbb{T}^4\) energy-critical Gross-Pitaevskii hierarchies and thus the corresponding NLS.Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponentshttps://www.zbmath.org/1483.352022022-05-16T20:40:13.078697Z"Chen, Yongpeng"https://www.zbmath.org/authors/?q=ai:chen.yongpeng"Guo, Yuxia"https://www.zbmath.org/authors/?q=ai:guo.yuxia"Tang, Zhongwei"https://www.zbmath.org/authors/?q=ai:tang.zhongweiSummary: This paper is concerned with the critical quasilinear Schrödinger systems in \(\mathbb{R}^N\):
\[\begin{cases}
-\Delta w+(\lambda a(x)+1)w-(\Delta|w|^2)w = \frac{p}{p+q}|w|^{p-2}w|z|^q+\frac{\alpha}{\alpha+\beta}|w|^{\alpha-2}w|z|^\beta \\
-\Delta z+(\lambda b(x)+1)z-(\Delta|z|^2)z = \frac{q}{p+q}|w|^p|z|^{q-2}z+\frac{\beta}{\alpha+\beta}|w|^\alpha|z|^{\beta-2}z,
\end{cases}\]
where \( \lambda>0 \) is a parameter, \( p>2, q>2, \alpha>2, \beta>2, 2\cdot(2^*-1) < p+q<2\cdot2^* \) and \( \alpha+ \beta = 2\cdot2^* \). By using variational method, we prove the existence of positive ground state solutions which localize near the set \( \Omega = int \left\{a^{-1}(0)\right\}\cap int \left\{b^{-1}(0)\right\} \) for \( \lambda \) large enough.A note on global existence for the Zakharov system on \( \mathbb{T} \)https://www.zbmath.org/1483.352042022-05-16T20:40:13.078697Z"Compaan, E."https://www.zbmath.org/authors/?q=ai:compaan.erinSummary: We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in \( L^2 \)-based Sobolev spaces. We also obtain global well-posedness in \( H^{\frac12+} \times L^2 \), which is sharp (up to endpoints) in the class of \( L^2 \)-based Sobolev spaces.On the Cauchy problem associated with a nonequilibrium Bose-Einstein condensate of exciton polaritonshttps://www.zbmath.org/1483.352052022-05-16T20:40:13.078697Z"Corcho, Adán J."https://www.zbmath.org/authors/?q=ai:corcho.adan-j"Hajaiej, Hichem"https://www.zbmath.org/authors/?q=ai:hajaiej.hichemThis paper provides local and global existence results and numerical results for the rescaled Cauchy problem: \[ i\partial_t\psi=-\frac{1}{2}(\Delta+i\lambda)\psi+\alpha|\psi|^2\psi+(\beta+ir/2)\psi\,\eta \] \[ \partial_t\eta+\gamma_r \eta=-r|\psi|^2(\eta+p/\gamma_r) \] where \(\alpha>0\) (defocusing cubic term) and \(\beta>0\). The space variable \(x\) belongs to \(X\) with \(X=\mathbb{R}^d\) or \(X=\mathbb{T}^d\) (the d-dimensional \(2\pi\) periodic tore).
This system accounts for the interaction of an excited polariton in a boson condensate with density state \(\psi\) with cavity photons of density \(\eta\). The rate of optical pumping is p, the rate of amplification is \(r\) and \(\gamma_r\) is the relaxation rate.
First, the authors show that for \(r>0\) and \(p\gamma_r>0\) then \(\|(\psi,s\eta)\|_{L^2(X)}\) with \(s=\sqrt{\gamma_r/(2p)}\) is decreasing if \(\lambda\geq 0\) (exponentially if \(\lambda>0)\) otherwise it is a subexponential function.
Then a standard proof shows local and global existence (under small data asumptions and \(\lambda>0\) and \(\gamma_r>0\)) in \(H^s(X)\) with \(s>d/2\).
When \(d=1\) using Strichartz estimates a local existence result follows in C\(([0,T],L^2(X))\). Globality is achieved under the conditions providing the decrease of the \(L^2\) norm.
When \(p<0\) and \(\lambda<0\) while \(r>0\) and \(\gamma_r>0\) a third result states that for big initial data \((\psi,\eta)\in L^2\times L^1\cap (H^s(X))^2\) with \(s>d/2\) the function \(\eta\) either blows up in finite time in \(H^s(X)\) or its \(L^{\infty}_{t,x}\) norm exceeds \(p/(2\gamma_r)\).
Finally the authors provide numerical computations for the system using a classical second order Strang split-step method and fast Fourier transform. In the 1d and 2d case the numerical results are in agreement with the global result when \(p>0\) and \(\lambda>0\) (decay). When \(p>0\) and \(\lambda<0\) the results for \(d=1\) hint towards blow up for big enough initial data.
Reviewer: Vincent Lescarret (Gif-sur-Yvette)Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on torihttps://www.zbmath.org/1483.352082022-05-16T20:40:13.078697Z"Feola, Roberto"https://www.zbmath.org/authors/?q=ai:feola.roberto"Iandoli, Felice"https://www.zbmath.org/authors/?q=ai:iandoli.feliceSummary: We prove a local in time well-posedness result for quasi-linear Hamiltonian Schrödinger equations on \(\mathbb{T}^d\) for any \(d\geq 1\). For any initial condition in the Sobolev space \(H^s\), with \(s\) large, we prove the existence and uniqueness of classical solutions of the Cauchy problem associated to the equation. The lifespan of such a solution depends only on the size of the initial datum. Moreover we prove the continuity of the solution map.Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLShttps://www.zbmath.org/1483.352102022-05-16T20:40:13.078697Z"Genovese, Giuseppe"https://www.zbmath.org/authors/?q=ai:genovese.giuseppe"Lucà, Renato"https://www.zbmath.org/authors/?q=ai:luca.renato"Tzvetkov, Nikolay"https://www.zbmath.org/authors/?q=ai:tzvetkov.nikolaySummary: The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on \(L^2(\mathbb{T})\) with covariance \([1+(-\Delta)^s]^{- 1}\) under these transformations for any \(s > \frac{1}{2}\). This extends previous achievements by \textit{A. R. Nahmod} et al. [Math. Res. Lett. 18, No. 5, 875--887 (2011; Zbl 1250.60018)] and the first author et al. [Math. Ann. 374, No. 3--4, 1075--1138 (2019; Zbl 1420.35354)], who proved the result for integer values of the regularity parameter \(s\).Dynamical and variational properties of the NLS-\( \delta'_s\) equation on the star graphhttps://www.zbmath.org/1483.352112022-05-16T20:40:13.078697Z"Goloshchapova, Nataliia"https://www.zbmath.org/authors/?q=ai:goloshchapova.nataliiaSummary: We study the nonlinear Schrödinger equation with \(\delta'_s\) coupling of intensity \(\beta \in \mathbb{R} \smallsetminus \{0 \}\) on the star graph \(\Gamma\) consisting of \(N\) half-lines. The nonlinearity has the form \(g(u) = | u |^{p - 1} u\), \(p > 1\). In the first part of the paper, under certain restriction on \(\beta \), we prove the existence of the ground state solution as a minimizer of the action functional \(S_\omega\) on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of \(N\) profiles (one symmetric and \(N - 1\) asymmetric). In particular, for the attractive \(\delta'_s\) coupling \(( \beta < 0)\) and the frequency \(\omega\) above a certain threshold, we managed to specify the ground state. The second part is devoted to the study of orbital instability of the critical points. We prove spectral instability of the critical points using Grillakis/Jones Instability Theorem. Then orbital instability for \(p > 2\) follows from the fact that data-solution mapping associated with the equation is of class \(C^2\) in \(H^1(\Gamma)\). Moreover, for \(p > 5\) we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.Scattering for the quadratic nonlinear Schrödinger system in \(R^5\) without mass-resonance conditionhttps://www.zbmath.org/1483.352122022-05-16T20:40:13.078697Z"Hamano, Masaru"https://www.zbmath.org/authors/?q=ai:hamano.masaru"Inui, Takahisa"https://www.zbmath.org/authors/?q=ai:inui.takahisa"Nishimura, Kuranosuke"https://www.zbmath.org/authors/?q=ai:nishimura.kuranosukeSummary: We consider the quadratic nonlinear Schrödinger system (NLS system) on \(R^5\). The scattering below the ground state for NLS system was obtained by the first author under the mass-resonance condition. In this paper, we prove scattering below the ground state without the mass-resonance condition under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by \textit{C. E. Kenig} and \textit{F. Merle} [Invent. Math. 166, No. 3, 645--675 (2006; Zbl 1115.35125)]. Moreover, we discuss the concentration compactness and the rigidity for non-radial solutions.Global existence and scattering for quadratic NLS with potential in three dimensionshttps://www.zbmath.org/1483.352142022-05-16T20:40:13.078697Z"Léger, Tristan"https://www.zbmath.org/authors/?q=ai:leger.tristanSummary: We study the asymptotic behavior of a quadratic NLS equation with small, time-dependent potential and small spatially localized initial data. We prove global existence and scattering of solutions. The two main ingredients of the proof are the space-time resonance method and the boundedness of wave operators for the linear Schrödinger equation with potential.Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimensionhttps://www.zbmath.org/1483.352152022-05-16T20:40:13.078697Z"Li, Chunhua"https://www.zbmath.org/authors/?q=ai:li.chunhua"Nishii, Yoshinori"https://www.zbmath.org/authors/?q=ai:nishii.yoshinori"Sagawa, Yuji"https://www.zbmath.org/authors/?q=ai:sagawa.yuji"Sunagawa, Hideaki"https://www.zbmath.org/authors/?q=ai:sunagawa.hideakiSummary: We consider a two-component system of cubic nonlinear Schrödinger equations in one space dimension. We show that each component of the solutions to this system behaves like a free solution in the large time and that the scattering profiles satisfy a crucially restricted condition. This restriction results from non-trivial long-range nonlinear interactions.On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative termhttps://www.zbmath.org/1483.352162022-05-16T20:40:13.078697Z"Li, Kelin"https://www.zbmath.org/authors/?q=ai:li.kelin"Di, Huafei"https://www.zbmath.org/authors/?q=ai:di.huafeiSummary: Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term \(iu_t+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0\), where \(t\in\mathbb{R}\) and \(x\in \mathbb{R}^n\). First of all, for initial data \(\varphi(x)\in H^2(\mathbb{R}^n)\), we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value \(\varphi(x)\), we demonstrate the global well-posedness of the solutions by applying two different methods, and at the same time give the scattering behavior of the solutions. Finally, based on founded a priori estimates, we investigate the stability of solutions by the short-time and long-time perturbation theories, respectively.Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potentialhttps://www.zbmath.org/1483.352182022-05-16T20:40:13.078697Z"Mukherjee, Debangana"https://www.zbmath.org/authors/?q=ai:mukherjee.debangana"Nam, Phan Thành"https://www.zbmath.org/authors/?q=ai:phan-thanh-nam."Nguyen, Phuoc-Tai"https://www.zbmath.org/authors/?q=ai:nguyen-phuoc-tai.Summary: We consider the focusing nonlinear Schrödinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide a complete characterization for the minimal mass blow-up solutions to the time dependent problem.Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearityhttps://www.zbmath.org/1483.352212022-05-16T20:40:13.078697Z"Okamoto, Mamoru"https://www.zbmath.org/authors/?q=ai:okamoto.mamoru"Uriya, Kota"https://www.zbmath.org/authors/?q=ai:uriya.kotaSummary: We consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.On three-wave interaction Schrödinger systems with quadratic nonlinearities: global well-posedness and standing waveshttps://www.zbmath.org/1483.352222022-05-16T20:40:13.078697Z"Pastor, Ademir"https://www.zbmath.org/authors/?q=ai:pastor.ademirSummary: Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1-dimensional three-component systems of nonlinear Schrödinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.Inhomogeneous coupled non-linear Schrödinger systemshttps://www.zbmath.org/1483.352232022-05-16T20:40:13.078697Z"Saanouni, Tarek"https://www.zbmath.org/authors/?q=ai:saanouni.tarek"Ghanmi, Radhia"https://www.zbmath.org/authors/?q=ai:ghanmi.radhiaSummary: This work studies an inhomogeneous Schrödinger coupled system in the mass-super-critical and energy-sub-critical regimes. In the focusing sign, a sharp dichotomy of global existence and scattering vs finite time blow-up of solutions is obtained using some variational methods, a sharp Gagliardo-Nirenberg-type inequality, and a new approach of \textit{B. Dodson} and \textit{J. Murphy} [Proc. Am. Math. Soc. 145, No. 11, 4859--4867 (2017; Zbl 1373.35287)]. In the defocusing sign, using a classical Morawetz estimate, the scattering of global solutions in the energy space is proved.
{\copyright 2021 American Institute of Physics}Qualitative properties of stationary solutions of the NLS on the hyperbolic space without and with external potentialshttps://www.zbmath.org/1483.352262022-05-16T20:40:13.078697Z"Selvitella, Alessandro"https://www.zbmath.org/authors/?q=ai:selvitella.alessandro|selvitella.alessandro-mariaSummary: In this paper, we prove some qualitative properties of stationary solutions of the NLS on the Hyperbolic space. First, we prove a variational characterization of the ground state and give a complete characterization of the spectrum of the linearized operator around the ground state. Then we prove some rigidity theorems and necessary conditions for the existence of solutions in weighted spaces. Finally, we add a slowly varying potential to the homogeneous equation and prove the existence of non-trivial solutions concentrating on the critical points of a reduced functional. The results are the natural counterparts of the corresponding theorems on the Euclidean space. We produce also the natural virial identity on the Hyperbolic space for the complete evolution, which however requires the introduction of a weighted energy, which is not conserved and so does not lead directly to finite time blow-up as in the Euclidean case.Ginzburg-Landau patterns in circular and spherical geometries: vortices, spirals, and attractorshttps://www.zbmath.org/1483.352322022-05-16T20:40:13.078697Z"Dai, Jia-Yuan"https://www.zbmath.org/authors/?q=ai:dai.jia-yuan"Lappicy, Phillipo"https://www.zbmath.org/authors/?q=ai:lappicy.phillipoThe authors consider the (time-dependent) Ginzburg-Landau equation (containing a positive bifurcation parameter) on compact surfaces of revolution such as the unit disk or the unit 2-sphere and investigate how topological structure of the surface affects the dynamics of vortex solutions. They first show that all the bifurcation curves of time-independent vortex solutions (which are called vortex equilibria and satisfy the elliptic version of the Ginzburg-Landau equation) are global. Then the existence of the spiral wave solutions for the complex version of the considered Ginzburg-Landau equation is proved by perturbing vortex solutions. The authors finally construct global attractor of vortex equilibria. They prove the results by adapting the shooting method (usually used in this context) to the new situation. In this way, the needed hyperbolicity of vortex equilibria can be established.
Reviewer: Catalin Popa (Iaşi)On the stability of radial solutions to an anisotropic Ginzburg-Landau equationhttps://www.zbmath.org/1483.352332022-05-16T20:40:13.078697Z"Lamy, Xavier"https://www.zbmath.org/authors/?q=ai:lamy.xavier"Zuniga, Andres"https://www.zbmath.org/authors/?q=ai:zuniga.andresThe essential spectrum of periodically stationary solutions of the complex Ginzburg-Landau equationhttps://www.zbmath.org/1483.352342022-05-16T20:40:13.078697Z"Zweck, John"https://www.zbmath.org/authors/?q=ai:zweck.john-w"Latushkin, Yuri"https://www.zbmath.org/authors/?q=ai:latushkin.yuri"Marzuola, Jeremy L."https://www.zbmath.org/authors/?q=ai:marzuola.jeremy-l"Jones, Christopher K. R. T."https://www.zbmath.org/authors/?q=ai:jones.christopher-k-r-tSummary: We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically stationary pulses in ultrafast fiber lasers.Solutions to a nonlinear Maxwell equation with two competing nonlinearities in \(\mathbb{R}^3\)https://www.zbmath.org/1483.352352022-05-16T20:40:13.078697Z"Bieganowski, Bartosz"https://www.zbmath.org/authors/?q=ai:bieganowski.bartoszSummary: We are interested in the nonlinear, time-harmonic Maxwell equation
\[
\nabla\times (\nabla\times\mathbf{E}) + V(x) \mathbf{E} = h(x, \mathbf{E})\quad \text{in } \mathbb{R}^3
\]
with sign-changing nonlinear term \(h\), i.e. we assume that \(h\) is of the form
\[
h(x, \alpha w) = f(x, \alpha) w - g(x, \alpha) w
\]
for \(w \in \mathbb{R}^3, |w|=1\) and \(\alpha \in \mathbb{R}\). In particular, we can consider the nonlinearity consisting of two competing powers, \(h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}\) with \(2 < q < p < 6\). Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of a special form are in one-to-one correspondence with weak solutions to a Schrödinger equation with a singular potential. Using this equivalence result we show the existence of a least energy solution among cylindrically equivariant solutions to the Maxwell equation of a particular form, as well as to the Schrödinger equation.Stability of the Walker wall in two dimensionshttps://www.zbmath.org/1483.352362022-05-16T20:40:13.078697Z"Han, Fangyu"https://www.zbmath.org/authors/?q=ai:han.fangyu"Tan, Zhong"https://www.zbmath.org/authors/?q=ai:tan.zhong.1|tan.zhong"Yang, Ganshan"https://www.zbmath.org/authors/?q=ai:yang.ganshanDomain wall motion in axially symmetric spintronic nanowireshttps://www.zbmath.org/1483.352382022-05-16T20:40:13.078697Z"Rademacher, Jens D. M."https://www.zbmath.org/authors/?q=ai:rademacher.jens-d-m"Siemer, Lars"https://www.zbmath.org/authors/?q=ai:siemer.larsModal approximation for strictly convex plasmonic resonators in the time domain: the Maxwell's equationshttps://www.zbmath.org/1483.352422022-05-16T20:40:13.078697Z"Ammari, Habib"https://www.zbmath.org/authors/?q=ai:ammari.habib-m"Millien, Pierre"https://www.zbmath.org/authors/?q=ai:millien.pierre"Vanel, Alice L."https://www.zbmath.org/authors/?q=ai:vanel.alice-lSummary: We study the possible expansion of the electromagnetic field scattered by a strictly convex metallic nanoparticle with dispersive material parameters placed in a homogeneous medium in a low-frequency regime as a sum of \textit{modes} oscillating at complex frequencies (diverging at infinity), known in the physics literature as the \textit{quasi-normal modes} expansion. We show that such an expansion is valid in the static regime and that we can approximate the electric field with a finite number of modes. We then use perturbative spectral theory to show the existence, in a certain regime, of plasmonic resonances as poles of the resolvent for Maxwell's equations with non-zero frequency. We show that, in the time domain, the electric field can be written as a sum of modes oscillating at complex frequencies. We introduce renormalised quantities that do not diverge exponentially at infinity.Global well-posedness of incompressible elastodynamics in three-dimensional thin domainhttps://www.zbmath.org/1483.352492022-05-16T20:40:13.078697Z"Cai, Yuan"https://www.zbmath.org/authors/?q=ai:cai.yuan"Wang, Fan"https://www.zbmath.org/authors/?q=ai:wang.fanAuthors' abstract: In this article, we prove global existence of classical solutions to the incompressible isotropic Hookean elastodynamics in three-dimensional thin domain \(\Omega_\delta = \mathbb{R}^2 \times [0, \delta ]\) with periodic boundary condition. This system essentially consists of two-dimensional quasilinear wave-type equations. Following the classical vector field theory and the generalized energy method of Klainerman, we introduce the anisotropic weighted Sobolev-type inequalities, the anisotropic generalized energy, and the anisotropic weighted \(L^2\) norm adapted to the thin domain. The main issue in the anisotropic generalized energy estimate is that the pressure gradient brings an extra singular \(\delta^{- 1}\) factor arising from the \(\partial_3^2\) derivative. Based on the inherent cancellation structure, we introduce an auxiliary anisotropic generalized energy to overcome this difficulty. The ghost weight technique introduced by \textit{S. Alinhac} [Invent. Math. 145, No. 3, 597--618 (2001; Zbl 1112.35341)] and the strong null condition introduced by \textit{Z. Lei} [Commun. Pure Appl. Math. 69, No. 11, 2072--2106 (2016; Zbl 1351.35216)]play important roles for temporal decay estimates.
Reviewer: Kaïs Ammari (Monastir)Further results on generalized Holmgren's principle to the Lamé operator and applicationshttps://www.zbmath.org/1483.352502022-05-16T20:40:13.078697Z"Diao, Huaian"https://www.zbmath.org/authors/?q=ai:diao.huaian"Liu, Hongyu"https://www.zbmath.org/authors/?q=ai:liu.hongyu"Wang, Li"https://www.zbmath.org/authors/?q=ai:wang.li.6|wang.li.4|wang.li|wang.li.5|wang.li.3|wang.li.1|wang.li.2Summary: In our earlier paper [Calc. Var. Partial Differ. Equ. 59, No. 5, Paper No. 179, 49 p. (2020; Zbl 1450.35290)], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lamé system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticity into the GHP study with the soft-clamped, simply-supported as well as the associated impedance-type conditions. We derive a comprehensive and complete characterization of the GHP associated with all of the aforementioned physical conditions. As significant applications, we establish novel unique identifiability results by a few scattering measurements not only for the inverse elastic obstacle problem but also for the inverse elastic diffraction grating problem within polygonal geometry in the most general physical scenario. We follow the general strategy from [loc. cit.] in establishing the results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups. It is particularly worth noting that in [loc. cit.], the impedance parameters were assumed to be constant whereas in this work they can be variable functions.Transient thermoelastic stress analysis of a thin circular plate due to uniform internal heat generationhttps://www.zbmath.org/1483.352512022-05-16T20:40:13.078697Z"Gaikwad, Kishor R."https://www.zbmath.org/authors/?q=ai:gaikwad.kishor-r"Naner, Yogesh U."https://www.zbmath.org/authors/?q=ai:naner.yogesh-uSummary: The present work aims to analyzed the transient thermoelastic stress analysis of a thin circular plate with uniform internal heat generation. Initially, the plate is characterized by a parabolic temperature distribution along the z-direction given by \(T = T_0(r, z)\) and perfectly insulated at the ends \(z = 0\) and \(z = h\). For times \(t > 0\), the surface \(r = a\) is subjected to convection heat transfer with convection coefficient \(h_c\) and fluid temperature \(T_\infty\). The integral transform method used to obtain the analytical solution for temperature, displacement, and thermal stresses. The associated thermoelastic field is analyzed by making use of the temperature and thermoelastic displacement potential function. Numerical results are carried out with the help of computational software PTC Mathcad Prime-3.1 and shown in figures.Minnaert resonances for bubbles in soft elastic materialshttps://www.zbmath.org/1483.352542022-05-16T20:40:13.078697Z"Li, Hongjie"https://www.zbmath.org/authors/?q=ai:li.hongjie"Liu, Hongyu"https://www.zbmath.org/authors/?q=ai:liu.hongyu"Zou, Jun"https://www.zbmath.org/authors/?q=ai:zou.junAuthors' abstract: Minnaert resonance is a widely known acoustic phenomenon, and it has many important applications, in particular in the effective realization of acoustic metamaterials using bubbly media in recent years. In this paper, motivated by the Minnaert resonance in acoustics, we consider the low-frequency resonance for acoustic bubbles embedded in soft elastic materials. This is a hybrid physical process that couples the acoustic and elastic wave propagations. By delicately and subtly balancing the acoustic and elastic parameters as well as the geometry of the bubble, we show that Minnaert resonance can occur (at least approximately) for rather general constructions. Our study highlights the great potential for the effective realization of negative elastic materials by using bubbly elastic media.
Reviewer: Kaïs Ammari (Monastir)Effective model for elastic waves in a substrate supporting an array of plates/beams with flexural and longitudinal resonanceshttps://www.zbmath.org/1483.352552022-05-16T20:40:13.078697Z"Marigo, Jean-Jacques"https://www.zbmath.org/authors/?q=ai:marigo.jean-jacques"Pham, Kim"https://www.zbmath.org/authors/?q=ai:pham.kim-son"Maurel, Agnès"https://www.zbmath.org/authors/?q=ai:maurel.agnes"Guenneau, Sébastien"https://www.zbmath.org/authors/?q=ai:guenneau.sebastien-r-l|guenneau.sebastienSummary: In a previous study [the authors, J. Mech. Phys. Solids 143, Article ID 104029, 22 p. (2020; \url{doi:10.1016/j.jmps.2020.104029})] we have studied the effect of a periodic array of subwavelength plates or beams over a semi-infinite elastic ground on the propagation of waves hitting the interface. The study was restricted to the low frequency regime where only flexural resonances take place. Here, we present a generalization to higher frequencies which allows us to account for both flexural and longitudinal resonances and to evaluate their interplay. An effective model is obtained using asymptotic analysis and homogenization techniques, which can be expressed in terms of the ground alone with an effective dynamic (frequency-dependent) boundary conditions of the Robin's type. For an in-plane wave at oblique incidence, the scattered displacement fields and the reflection coefficients are obtained in closed forms and their effectiveness to reproduce the actual scattering is inspected by comparison with direct numerics in a two-dimensional setting.Stabilization of swelling porous elastic soils with fluid saturation, time varying-delay and time-varying weightshttps://www.zbmath.org/1483.352572022-05-16T20:40:13.078697Z"Nonato, C. A. S."https://www.zbmath.org/authors/?q=ai:nonato.carlos-a-s"Ramos, A. J. A."https://www.zbmath.org/authors/?q=ai:ramos.anderson-j-a"Raposo, C. A."https://www.zbmath.org/authors/?q=ai:raposo.carlos-alberto"Dos Santos, M. J."https://www.zbmath.org/authors/?q=ai:dos-santos.manoel-jeremias"Freitas, M. M."https://www.zbmath.org/authors/?q=ai:freitas.mirelson-mSummary: In this article we study the well-posedness and exponential stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject with time-varying weights and time-varying delay. We prove existence of global solution for the problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the energy method without the equal wave speeds assumption.Analysis and control of stationary inclusions in contact mechanicshttps://www.zbmath.org/1483.352592022-05-16T20:40:13.078697Z"Sofonea, Mircea"https://www.zbmath.org/authors/?q=ai:sofonea.mircea|sofonea.mircea-tSummary: We start with a mathematical model which describes the frictionless contact of an elastic body with an obstacle and prove that it leads to a stationary inclusion for the strain field. Then, inspired by this contact model, we consider a general stationary inclusion in a real Hilbert space, governed by three parameters. We prove the unique solvability of the inclusion as well as the continuous dependence of its solution with respect to the parameters. We use these results in the study of an associated optimal control problem for which we prove existence and convergence results. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact and provide the corresponding mechanical interpretations.Global existence analysis of energy-reaction-diffusion systemshttps://www.zbmath.org/1483.352622022-05-16T20:40:13.078697Z"Fischer, Julian"https://www.zbmath.org/authors/?q=ai:fischer.julian"Hopf, Katharina"https://www.zbmath.org/authors/?q=ai:hopf.katharina"Kniely, Michael"https://www.zbmath.org/authors/?q=ai:kniely.michael"Mielke, Alexander"https://www.zbmath.org/authors/?q=ai:mielke.alexanderGreen's function for the anisotropic hyperbolic heat equation in bounded spatial and temporal domainshttps://www.zbmath.org/1483.352632022-05-16T20:40:13.078697Z"López Molina, J. A."https://www.zbmath.org/authors/?q=ai:lopez-molina.juan-antonio|molina.j-a-lopezSummary: We study the regularity properties of Green's function \(G_T(\mathbf{x},t|\mathbf{x}_0,t_0)\) associated to Robin's problem for a class of second order operators on \(\Omega\times]-T,T[,\Omega\subset\mathbb{R}^n\) a bounded open set with regular boundary, including the hyperbolic heat equation for anisotropic non homogeneous bodies with constant thermal properties as a particular case. We show that for every source point \((\mathbf{x}_0,t_0)\) and every \(k\in\mathbb{N}\) there is an open set \(\mathcal{O}_k\subset\Omega\times]-T,T[\) such that \(G_T(\mathbf{x},t|\mathbf{x}_0,t_0)\) is \(k\) times differentiable with continuity on \(\mathcal{O}_k\) and \(\mu\bigl((\Omega\times]-T,T[)\backslash\mathcal{O}_k\bigr)=0\).Macroscopic approximation of a Fermi-Dirac statistics: Unbounded velocity space settinghttps://www.zbmath.org/1483.352642022-05-16T20:40:13.078697Z"Masmoudi, Nader"https://www.zbmath.org/authors/?q=ai:masmoudi.nader"Tayeb, Mohamed Lazhar"https://www.zbmath.org/authors/?q=ai:tayeb.mohamed-lazharSummary: An approximation by diffusion of a nonlinear Boltzmann equation modeling a Fermi-Dirac statistics is analyzed for an unbounded velocity space and Poisson coupling. A careful analysis of the entropy and entropy-dissipation allows to control the distribution function and to pass to the limit using duality method.Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forceshttps://www.zbmath.org/1483.352672022-05-16T20:40:13.078697Z"Carrillo, José A."https://www.zbmath.org/authors/?q=ai:carrillo.jose-antonio"Choi, Young-Pil"https://www.zbmath.org/authors/?q=ai:choi.young-pil"Jung, Jinwook"https://www.zbmath.org/authors/?q=ai:jung.jinwookSharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small datahttps://www.zbmath.org/1483.352682022-05-16T20:40:13.078697Z"Duan, Xianglong"https://www.zbmath.org/authors/?q=ai:duan.xianglongSummary: In this paper, we present sharp decay estimates for small data solutions to the following two systems: the Vlasov-Poisson (V-P) system in dimension 3 or higher and the Vlasov-Yukawa (V-Y) system in dimension 2 or higher. We rely on a modification of the vector field method for transport equation as developed by \textit{J. Smulevici} [Ann. PDE 2, No. 2, Paper No. 11, 55 p. (2016; Zbl 1397.35033)] for the Vlasov-Poisson system. Using the Green's function in particular to estimate the bilinear terms, we improve Smulevici's result by removing the requirement of some \(v\)-weighted \(L^p\) integrability for the initial data and extend the result to the Vlasov-Yukawa system.Strong Lagrangian solutions of the (relativistic) Vlasov-Poisson system for nonsmooth, spherically symmetric datahttps://www.zbmath.org/1483.352712022-05-16T20:40:13.078697Z"Körner, Jacob"https://www.zbmath.org/authors/?q=ai:korner.jacob"Rein, Gerhard"https://www.zbmath.org/authors/?q=ai:rein.gerhardDecay estimates of solutions to the \(N\)-species Vlasov-Poisson system with small initial datahttps://www.zbmath.org/1483.352722022-05-16T20:40:13.078697Z"Wang, Yichun"https://www.zbmath.org/authors/?q=ai:wang.yichunSummary: This paper is concerned with the time decay rates of the \(N -\) species Vlasov-Poisson system with small data in the whole space. The global existence and large time behaviors are obtained in \(\mathbb{R}^3\) and more higher dimensional space. For the proof, the classical (for \(\mathbb{R}^n, n \geq 4)\) and the modified (for \(\mathbb{R}^3)\) vector field method and the bootstrap argument are mainly employed. Compared to the unipolar case, there are some crucial new ideas introduced to handle the multi-species case, such as a new bootstrap assumption with some necessary parameters and the multipolar version of vector field method with new coefficients corresponding to different species charged particles, respectively.Sharp decay estimates for the Vlasov-Poisson system with an external magnetic fieldhttps://www.zbmath.org/1483.352732022-05-16T20:40:13.078697Z"Wu, Man"https://www.zbmath.org/authors/?q=ai:wu.manSummary: In this paper, we establish sharp decay estimates for the Vlasov-Poisson system with an external magnetic field on \(\mathbb{R}_x^3 \times \mathbb{R}_v^3\). Our arguments are based on the modified vector field method developed in Smulevici (2016) for the classical Vlasov-Poisson system in the 3-D case, and hence we extend some results in [\textit{J. Smulevici}, Ann. PDE 2, No. 2, Paper No. 11, 55 p. (2016; Zbl 1397.35033)] to the Vlasov-Poisson system with an external magnetic field.Local well-posedness and sensitivity analysis for the self-organized kinetic modelhttps://www.zbmath.org/1483.352762022-05-16T20:40:13.078697Z"Jiang, Ning"https://www.zbmath.org/authors/?q=ai:jiang.ning"Zhang, Zeng"https://www.zbmath.org/authors/?q=ai:zhang.zengSummary: We consider the self-organized kinetic model (SOK), which was derived as the mean field limit of the Couzin-Vicsek algorithm. This model yields a singularity when the particle flux vanishes. By showing that the singularity does not happen in finite time, we obtain local existence and uniqueness of smooth solutions to SOK. Furthermore, considering uncertainties in the initial data and in the interaction kernel, we analyze the random SOK model (RSOK). We provide local sensitivity analysis to justify the regularity with respect to the random parameter and the stability of solutions to RSOK.Stability of global Maxwellian for fully nonlinear Fokker-Planck equationshttps://www.zbmath.org/1483.352772022-05-16T20:40:13.078697Z"Liao, Jie"https://www.zbmath.org/authors/?q=ai:liao.jie"Yang, Xiongfeng"https://www.zbmath.org/authors/?q=ai:yang.xiongfengSummary: This paper considers the stability of solutions around a global Maxwellian to the fully non-linear Fokker-Planck equation in the whole space. This model preserves mass, momentum and energy at the same time, and its dissipation is much weaker than that in the simplified model considered in [the authors and \textit{Q. Wang}, ibid. 173, No. 1, 222--241 (2018; Zbl 1398.35017)]. To overcome the new difficulties, the macro-micro decomposition of the solution around the \textit{local Maxwellian} and energy estimates introduced in [\textit{T.-P. Liu} et al., Physica D 188, No. 3--4, 178--192 (2004; Zbl 1098.82618)] and [\textit{T. Yang} and \textit{H.-J. Zhao}, J. Math. Phys. 47, No. 5, 053301, 19 p. (2006; Zbl 1111.82048)] for Boltzmann equation is used. That is, we reformulate the model into a fluid-type system coupled with an equation of the microscopic part. The a priori estimates of the solution could be obtained by the standard energy method. Especially, by careful computation, the viscosity and heat diffusion terms in the fluid-type system are derived from the microscopic part, which give the dissipative mechanism to the system.Optimizing noisy complex systems liable to failurehttps://www.zbmath.org/1483.352782022-05-16T20:40:13.078697Z"Lunz, Davin"https://www.zbmath.org/authors/?q=ai:lunz.davinBlow-up of solutions of semilinear wave equations in accelerated expanding Friedmann-Lemaître-Robertson-Walker spacetimehttps://www.zbmath.org/1483.352792022-05-16T20:40:13.078697Z"Tsutaya, Kimitoshi"https://www.zbmath.org/authors/?q=ai:tsutaya.kimitoshi"Wakasugi, Yuta"https://www.zbmath.org/authors/?q=ai:wakasugi.yutaTracking the critical points of curves evolving under planar curvature flowshttps://www.zbmath.org/1483.352802022-05-16T20:40:13.078697Z"Fehér, Eszter"https://www.zbmath.org/authors/?q=ai:feher.eszter"Domokos, Gábor"https://www.zbmath.org/authors/?q=ai:domokos.gabor"Krauskopf, Bernd"https://www.zbmath.org/authors/?q=ai:krauskopf.berndSummary: We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function \(r(\varphi)\) measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function \(r(\varphi)\) and of the curvature \(\kappa (\varphi)\) (characterized by \(dr/d\varphi = 0\) and \(d\kappa /d\varphi = 0\), respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.
We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically -- in the spirit of experimental mathematics -- we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.On some singular mean-field gameshttps://www.zbmath.org/1483.352812022-05-16T20:40:13.078697Z"Cirant, Marco"https://www.zbmath.org/authors/?q=ai:cirant.marco"Gomes, Diogo A."https://www.zbmath.org/authors/?q=ai:gomes.diogo-luis-aguiar"Pimentel, Edgard A."https://www.zbmath.org/authors/?q=ai:pimentel.edgard-a"Sánchez-Morgado, Héctor"https://www.zbmath.org/authors/?q=ai:sanchez-morgado.hectorSummary: Here, we prove the existence of smooth solutions for mean-field games with a singular mean-field coupling; that is, a coupling in the Hamilton-Jacobi equation of the form \(g(m) = -m^{- \alpha}\) with \(\alpha>0\). We consider stationary and time-dependent settings. The function \(g\) is monotone, but it is not bounded from below. With the exception of the logarithmic coupling, this is the first time that MFGs whose coupling is not bounded from below is examined in the literature. This coupling arises in models where agents have a strong preference for low-density regions. Paradoxically, this causes the agents move towards low-density regions and, thus, prevents the creation of those regions. To prove the existence of solutions, we consider an approximate problem for which the existence of smooth solutions is known. Then, we prove new a priori bounds for the solutions that show that \(\frac{1}{m}\) is bounded. Finally, using a limiting argument, we obtain the existence of solutions. The proof in the stationary case relies on a blow-up argument and in the time-dependent case on new bounds for \(m^{-1}\).Analysis of a model for tumor growth and lactate exchanges in a gliomahttps://www.zbmath.org/1483.352832022-05-16T20:40:13.078697Z"Cherfils, Laurence"https://www.zbmath.org/authors/?q=ai:cherfils.laurence"Gatti, Stefania"https://www.zbmath.org/authors/?q=ai:gatti.stefania.1"Miranville, Alain"https://www.zbmath.org/authors/?q=ai:miranville.alain-m"Guillevin, Rémy"https://www.zbmath.org/authors/?q=ai:guillevin.remySummary: Our aim in this paper is to study a mathematical model for tumor growth and lactate exchanges in a glioma. We prove the existence of nonnegative (i.e. biologically relevant) solutions and, under proper assumptions, the uniqueness of the solution. We also state the permanence of the tumor when necrosis is not taken into account in the model and obtain linear stability results. We end the paper with numerical simulations.Existence of a \(T\)-periodic solution for the monodomain model corresponding to an isolated ventricle due to ionic-diffusive relationshttps://www.zbmath.org/1483.352862022-05-16T20:40:13.078697Z"Fraguela, Andrés"https://www.zbmath.org/authors/?q=ai:fraguela.andres"Felipe-Sosa, Raúl"https://www.zbmath.org/authors/?q=ai:felipe-sosa.raul"Henry, Jacques"https://www.zbmath.org/authors/?q=ai:henry.jacques"Márquez, Manlio F."https://www.zbmath.org/authors/?q=ai:marquez.manlio-fSummary: In this paper, we find relations between the ionic parameters and the diffusion parameters which are sufficient to ensure the existence of a periodic solution for a well-known monodomain model in a weak sense. We make use of the method of approximation of Faedo-Galerkin to prove the existence of weak periodic solutions of the monodomain model for the electrical activity of the heart assuming that it is periodically activated in its boundaries. Actually, this periodic solution has the same period of activation. Finally, we reflect on how these ionic-diffusive relations are useful to explain the pathophysiology of some rhythm disorders.On a periodic age-structured mosquito population model with spatial structurehttps://www.zbmath.org/1483.352882022-05-16T20:40:13.078697Z"Lv, Yunfei"https://www.zbmath.org/authors/?q=ai:lv.yunfei"Pei, Yongzhen"https://www.zbmath.org/authors/?q=ai:pei.yongzhen"Yuan, Rong"https://www.zbmath.org/authors/?q=ai:yuan.rongSummary: This paper deals with a general age-structured model with diffusion. The existence and uniqueness of solutions of the equivalent integral equation are obtained in light of the contraction mapping theorem. By taking the mosquito population growth as a motivating example, we derive a periodic stage-structured model with diffusion, intra-specific competition and periodic delay. Next, we show that the solution is globally bounded for the setup we chose. Then, the basic reproduction number \(R_0\) for this model is introduced to establish the threshold dynamics on mosquito extinction and persistence in terms of \(R_0\). In the case where intra-specific competition among immature individuals is ignored, the adult equation is decoupled from the full equations, and the global stability of the positive periodic solution is then obtained by introducing a suitable phase space on which the periodic semiflow is eventually strongly monotone and strictly subhomogeneous.Quorum-sensing induced transitions between bistable steady-states for a cell-bulk ODE-PDE model with lux intracellular kineticshttps://www.zbmath.org/1483.352892022-05-16T20:40:13.078697Z"Ridgway, Wesley"https://www.zbmath.org/authors/?q=ai:ridgway.wesley"Ward, Michael J."https://www.zbmath.org/authors/?q=ai:ward.michael-j"Wetton, Brian T."https://www.zbmath.org/authors/?q=ai:wetton.brian-t-rSummary: Intercellular signaling and communication are used by bacteria to regulate a variety of behaviors. In a type of cell-cell communication known as quorum sensing (QS), which is mediated by a diffusible signaling molecule called an autoinducer, bacteria can undergo sudden changes in their behavior at a colony-wide level when the density of cells exceeds a critical threshold. In mathematical models of QS behavior, these changes can include the switch-like emergence of intracellular oscillations through a Hopf bifurcation, or sudden transitions between bistable steady-states as a result of a saddle-node bifurcation of equilibria. As an example of this latter type of QS transition, we formulate and analyze a cell-bulk ODE-PDE model in a 2-D spatial domain that incorporates the prototypical LuxI/LuxR QS system for a collection of stationary bacterial cells, as modeled by small circular disks of a common radius with a cell membrane that is permeable only to the autoinducer. By using the method of matched asymptotic expansions, it is shown that the steady-state solutions for the cell-bulk model exhibit a saddle-node bifurcation structure. The linear stability of these branches of equilibria are determined from the analysis of a nonlinear matrix eigenvalue problem, called the \textit{globally coupled eigenvalue problem}. The key role on QS behavior of a bulk degradation of the autoinducer field, which arises from either a Robin boundary condition on the domain boundary or from a constant bulk decay, is highlighted. With bulk degradation, it is shown analytically that the effect of coupling identical bacterial cells to the bulk autoinducer diffusion field is to create an effective bifurcation parameter that depends on the population of the colony, the bulk diffusivity, the membrane permeabilities, and the cell radius. QS transitions occur when this effective parameter passes through a saddle-node bifurcation point of the Lux ODE kinetics for an isolated cell. In the limit of a large but finite bulk diffusivity, it is shown that the cell-bulk system is well-approximated by a simpler ODE-DAE system. This reduced system, which is used to study the effect of cell location on QS behavior, is easily implemented for a large number of cells. Predictions from the asymptotic theory for QS transitions between bistable states are favorably compared with full numerical solutions of the cell-bulk ODE-PDE system.Well-posedness and unconditional uniqueness of mild solutions to the Keller-Segel system in uniformly local spaceshttps://www.zbmath.org/1483.352902022-05-16T20:40:13.078697Z"Suguro, Takeshi"https://www.zbmath.org/authors/?q=ai:suguro.takeshiThe parabolic-elliptic Keller-Segel system is studied in the whole space \(\mathbb R^n\) in the scale of uniformly local Lebesgue spaces with exponents \(r\ge n/2\). Results on local well-posedness and global-in-time well-posedness for small initial data are shown. Estimates of Bessel potentials are the key point of the analysis. The case \(r=n/2\) extends recent work [\textit{S. Cygan} et al., J. Evol. Equ. 21, No. 4, 4873--4896 (2021; Zbl 07452690)].
Reviewer: Piotr Biler (Wrocław)Dynamical properties in an SVEIR epidemic model with age-dependent vaccination, latency, infection, and relapsehttps://www.zbmath.org/1483.352912022-05-16T20:40:13.078697Z"Sun, Dandan"https://www.zbmath.org/authors/?q=ai:sun.dandan"Li, Yingke"https://www.zbmath.org/authors/?q=ai:li.yingke"Teng, Zhidong"https://www.zbmath.org/authors/?q=ai:teng.zhi-dong"Zhang, Tailei"https://www.zbmath.org/authors/?q=ai:zhang.tailei"Lu, Jingjing"https://www.zbmath.org/authors/?q=ai:lu.jingjingSummary: An SVEIR epidemic model with continuous age-dependent vaccination, latency, infection, and disease relapse is proposed and analyzed in this paper. The dynamical behaviors including the derivation of basic reproduction number \(\mathcal{R}_0\), the existence and the stability of steady states, and the uniform persistence of the model are investigated. The results indicate that if \(\mathcal{R}_0 \leq 1\), the disease-free steady state is globally asymptotically stable, and the disease dies out, whereas if \(\mathcal{R}_0 > 1\), the disease is uniformly persistent, and the endemic steady state is also globally asymptotically stable, and the disease remains at the endemic level. The research shows the global dynamics of the model are sharply determined by its basic reproduction number \(\mathcal{R}_0\). Finally, numerical examples support our main theoretical results.Dynamics of a diffusive nutrient-phytoplankton-zooplankton model with spatio-temporal delayhttps://www.zbmath.org/1483.352922022-05-16T20:40:13.078697Z"Tao, Yiwen"https://www.zbmath.org/authors/?q=ai:tao.yiwen"Campbell, Sue Ann"https://www.zbmath.org/authors/?q=ai:campbell.sue-ann"Poulin, Francis J."https://www.zbmath.org/authors/?q=ai:poulin.francis-jAnalysis of a diffusive HBV model with logistic proliferation and non-cytopathic antiviral mechanismshttps://www.zbmath.org/1483.352932022-05-16T20:40:13.078697Z"Wang, Jinliang"https://www.zbmath.org/authors/?q=ai:wang.jinliang"Wu, Xiaoqing"https://www.zbmath.org/authors/?q=ai:wu.xiaoqing"Kuniya, Toshikazu"https://www.zbmath.org/authors/?q=ai:kuniya.toshikazuSummary: Non-cytopathic antiviral mechanisms play a vital role in blocking up the HBV infection through the non-cytolytic processes that precede the immune elimination of infected hepatocytes. Due to the fact that the non-cytolytic processes contribute in eliminating at least 90\% of the viral DNA from the liver, in this paper, we incorporate non-cytopathic antiviral mechanisms into the classical hepatitis B virus infection model. We consider the situations that: (i) only the movement of virus particles is allowed in the liver; (ii) the interactions between cells and virus locally occur in the tissue such as lymph nodes; (iii) logistic proliferation terms are considered for both the uninfected and infected liver cells. Through overcoming some difficulties of dynamical behaviors caused by lacks of diffusion terms and space-dependent parameters, the global threshold result governed by the basic reproduction number are obtained. Numerical simulations are performed to validate the main results.Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems?https://www.zbmath.org/1483.352942022-05-16T20:40:13.078697Z"Winkler, Michael"https://www.zbmath.org/authors/?q=ai:winkler.michaelA two-dimensional cross-diffusion model for nutrient taxis system in liquid environment, consisting of parabolic chemotaxis with general sensitivity tensor-valued function coupled with the Stokes flow, is considered in planar domains. Results on the existence of global-in-time solutions are provided. Under natural structure assumptions long time asymptotics of solutions is shown to be simple (the stable homogeneous steady states) irrespective of possible complicate, rotational fluxes described by those general sensitivity functions.
Reviewer: Piotr Biler (Wrocław)A novel derivation of rigorous macroscopic limits from a micro-meso description of signal-triggered cell migration in fibrous environmentshttps://www.zbmath.org/1483.352952022-05-16T20:40:13.078697Z"Zhigun, Anna"https://www.zbmath.org/authors/?q=ai:zhigun.anna"Surulescu, Christina"https://www.zbmath.org/authors/?q=ai:surulescu.christinaOptimal control problems governed by 1-D Kobayashi-Warren-Carter type systemshttps://www.zbmath.org/1483.352972022-05-16T20:40:13.078697Z"Antil, Harbir"https://www.zbmath.org/authors/?q=ai:antil.harbir"Kubota, Shodai"https://www.zbmath.org/authors/?q=ai:kubota.shodai"Shirakawa, Ken"https://www.zbmath.org/authors/?q=ai:shirakawa.ken"Yamazaki, Noriaki"https://www.zbmath.org/authors/?q=ai:yamazaki.noriakiSummary: This paper is devoted to the study of a class of optimal control problems governed by 1-D Kobayashi-Warren-Carter type systems, which are based on a phase-field model of grain boundary motion, proposed by [Kobayashi et al, Physica D, 140, 141-150, 2000]. The class consists of an optimal control problem for a physically realistic state-system of Kobayashi-Warren-Carter type, and its regularized approximating problems. The results of this paper are stated in three Main Theorems 1--3. The first Main Theorem 1 is concerned with the solvability and continuous dependence for the state-systems. Meanwhile, the second Main Theorem 2 is concerned with the solvability of optimal control problems, and some semi-continuous association in the class of our optimal control problems. Finally, in the third Main Theorem 3, we derive the first order necessary optimality conditions for optimal controls of the regularized approximating problems. By taking the approximating limit, we also derive the optimality conditions for the optimal controls for the physically realistic problem.Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domainshttps://www.zbmath.org/1483.353002022-05-16T20:40:13.078697Z"Egidi, Michela"https://www.zbmath.org/authors/?q=ai:egidi.michela"Nakić, Ivica"https://www.zbmath.org/authors/?q=ai:nakic.ivica"Seelmann, Albrecht"https://www.zbmath.org/authors/?q=ai:seelmann.albrecht"Täufer, Matthias"https://www.zbmath.org/authors/?q=ai:taufer.matthias"Tautenhahn, Martin"https://www.zbmath.org/authors/?q=ai:tautenhahn.martin"Veselić, Ivan"https://www.zbmath.org/authors/?q=ai:veselic.ivanSummary: We survey recent results on the control problem for the heat equation on unbounded and large bounded domains. First we formulate new uncertainty relations, respectively spectral inequalities. Then we present an abstract control cost estimate which improves upon earlier results. The latter is particularly interesting when combined with the earlier mentioned spectral inequalities since it yields sharp control cost bounds in several asymptotic regimes. We also show that control problems on unbounded domains can be approximated by corresponding problems on a sequence of bounded domains forming an exhaustion. Our results apply also for the generalized heat equation associated with a Schrödinger semigroup.
For the entire collection see [Zbl 1467.93008].Optimisation of the total population size with respect to the initial condition for semilinear parabolic equations: two-scale expansions and symmetrisationshttps://www.zbmath.org/1483.353042022-05-16T20:40:13.078697Z"Mazari, Idriss"https://www.zbmath.org/authors/?q=ai:mazari.idriss"Nadin, Grégoire"https://www.zbmath.org/authors/?q=ai:nadin.gregoire"Toledo Marrero, Ana Isis"https://www.zbmath.org/authors/?q=ai:toledo-marrero.ana-isisA convection-diffusion model on a star-shaped graphhttps://www.zbmath.org/1483.353072022-05-16T20:40:13.078697Z"Cazacu, Cristian M."https://www.zbmath.org/authors/?q=ai:cazacu.cristian-m"Ignat, Liviu I."https://www.zbmath.org/authors/?q=ai:ignat.liviu-i"Pazoto, Ademir F."https://www.zbmath.org/authors/?q=ai:pazoto.ademir-fernando"Rossi, Julio D."https://www.zbmath.org/authors/?q=ai:rossi.julio-danielSummary: In this paper we study a convection-diffusion equation on a star-shaped graph composed by \(n\) incoming edges and \(m\) outgoing edges with a nonlinearity \(f\in C^1(\mathbb{R})\) satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by \(\partial_x(f(u(t, x))\) with \(f(s)=-a|s|^{q-1}s\) with \(q\ge 2\) and \(a\in\mathbb{R}\) verifying \((n-m)a\ge 0\), we analyze the long time behavior of the solutions. For \(q> 2\) we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. In the case \(q=2\), the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers' equation. Explicit representations of the limit profiles are obtained.Suppression of spiral wave turbulence by means of periodic plane waves in two-layer excitable mediahttps://www.zbmath.org/1483.353092022-05-16T20:40:13.078697Z"Wang, Zhen"https://www.zbmath.org/authors/?q=ai:wang.zhen.1|wang.zhen.2"Rostami, Zahra"https://www.zbmath.org/authors/?q=ai:rostami.zahra"Jafari, Sajad"https://www.zbmath.org/authors/?q=ai:jafari.sajad"Alsaadi, Fawaz E."https://www.zbmath.org/authors/?q=ai:alsaadi.fawaz-e"Slavinec, Mitja"https://www.zbmath.org/authors/?q=ai:slavinec.mitja"Perc, Matjaž"https://www.zbmath.org/authors/?q=ai:perc.matjazSummary: Spiral waves are relatively common, yet fascinating, visually appealing, and important phenomena in many nonlinear dynamical systems. The emergence of spiral waves in the heart's atrium, for example, signals abnormality that can lead to arrhythmias such as atrial flutter and atrial fibrillation. Spiral waves have also been associated with the disruption of resting states in the human brain, which are crucial for unimpaired cognitive ability and information processing. Here we consider two-layer excitable media, where spiral wave turbulence is triggered as the initial state. We study the effects of periodic plane waves on the dynamics of spiral wave turbulence, in particular by varying their spatial frequency. Our research shows that planes waves with low spatial frequency are in general too weak to overcome spiral wave turbulence. But when the spatial frequency is sufficiently increased, the plane waves can overcome spiral wave turbulence and impose a stripped spatial pattern over the excitable media. By increasing the spatial frequency of the plane waves even further, we show that it is possible to minimize the time needed to destroy spiral wave turbulence, although we also observe an upper limit beyond which the recurrence of turbulence is likely. This is linked to residual spirals that remain following a too rash elimination attempt, which then gradually regain footing across the whole medium.The nonlinear fractional relativistic Schrödinger equation: existence, multiplicity, decay and concentration resultshttps://www.zbmath.org/1483.353142022-05-16T20:40:13.078697Z"Ambrosio, Vincenzo"https://www.zbmath.org/authors/?q=ai:ambrosio.vincenzoSummary: In this paper we study the following class of fractional relativistic Schrödinger equations:
\[
\begin{cases}
(-\Delta +m^2)^s u + V(\varepsilon x) u = f(u) & \text{in } \mathbb{R}^N, \\
u\in H^s (\mathbb{R}^N), \quad u>0 & \text{in } \mathbb{R}^N,
\end{cases}
\]
where \(\varepsilon >0\) is a small parameter, \(s\in (0, 1), \, m>0, N>2s, (-\Delta+m^2)^s\) is the fractional relativistic Schrödinger operator, \(V: \mathbb{R}^N \rightarrow \mathbb{R}\) is a continuous potential satisfying a local condition, and \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for \(\varepsilon >0\) small enough, the above problem admits a weak solution \(u_{\varepsilon}\) which concentrates around a local minimum point of \(V\) as \(\varepsilon \rightarrow 0\). We also show that \(u_{\varepsilon}\) has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential \(V\) attains its minimum value.Symmetry analysis of initial and boundary value problems for fractional differential equations in Caputo sensehttps://www.zbmath.org/1483.353192022-05-16T20:40:13.078697Z"Iskenderoglu, Gulistan"https://www.zbmath.org/authors/?q=ai:iskenderoglu.gulistan"Kaya, Dogan"https://www.zbmath.org/authors/?q=ai:kaya.doganSummary: In this work, we study Lie symmetry analysis of initial and boundary value problems (IBVPs) for partial differential equations (PDE) with Caputo fractional derivative. According to Bluman's definition and theorem for the symmetry analysis of the PDE system, we determine the symmetries of the PDE with Caputo fractional derivative in general form and prove theorem for the above equation. We investigate the symmetry analysis of IBVP for a fractional diffusion and third-order fractional partial differential equation (FPDE). And as a result of applying the method, we get several solutions.Modeling the heat flow equation with fractional-fractal differentiationhttps://www.zbmath.org/1483.353212022-05-16T20:40:13.078697Z"Koca, Ilknur"https://www.zbmath.org/authors/?q=ai:koca.ilknurSummary: In this paper, modeling the heat flow equation with fractional-fractal differentiation is considered. This problem has opened a new viewpoint for modeling the classical and the fractional differentiation. We presented the existence of positive solution of the new model using the fixed-point approach and we established the uniqueness of the positive solution. Finally, we provide an example to illustrate one of the main results.Energy solutions and concentration problem of fractional Schrödinger equationhttps://www.zbmath.org/1483.353222022-05-16T20:40:13.078697Z"Li, Peiluan"https://www.zbmath.org/authors/?q=ai:li.peiluan"Yuan, Yuan"https://www.zbmath.org/authors/?q=ai:yuan.yuan.2|yuan.yuan.1|yuan.yuan.3|yuan.yuanSummary: In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results.Liouville theorems for fractional and higher-order Hénon-Hardy systems on \(\mathbb{R}^n\)https://www.zbmath.org/1483.353252022-05-16T20:40:13.078697Z"Peng, Shaolong"https://www.zbmath.org/authors/?q=ai:peng.shaolongSummary: In this paper, we are concerned with the Hénon-Hardy type systems on \(\mathbb{R}^N\):
\[
\begin{aligned}
(-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^av^p(x), \quad u(x)\geq0, x\in\mathbb{R}^n, \\
(-\Delta)^{\frac{\alpha}{2}}v(x)=|x|^au^p(x), \quad v(x)\geq0, x\in\mathbb{R}^n,
\end{aligned}
\]
where \(n\geq2\), \(n>\alpha\), \(0<\alpha\leq2\) or \(\alpha=2m\). We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon-Hardy systems. The arguments used in our proof is the method of scaling spheres developed in
[\textit{W. Dai} and \textit{G. Qin}, ``Liouville type theorems for fractional and higher-order Hénon-Hardy type equations via the method of scaling spheres'', Preprint, \url{arXiv:1810.02752}].
Our results generalize the Liouville theorems for single Hénon-Hardy equation on \(\mathbb{R}^n\) in
[\textit{M.-F. Bidaut-Véron} and \textit{S. Pohozaev}, J. Anal. Math. 84, 1--49 (2001; Zbl 1018.35040)],
[\textit{W. Chen}, \textit{W. Dai} and \textit{G. Qin}, ``Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in \(\mathbb{R}^n\)'', Preprint, \url{arXiv:1808.06609}],
[\textit{W. Dai}, \textit{S. Peng} and \textit{G. Qin}, ``Liouville type theorems, a priori estimates and existence of solutions for non-critical higher order Lane-Emden-Hardy equations'', Preprint, \url{arXiv:1808.10771}],
[\textit{W. Dai} and \textit{G. Qin}, Math. Nachr. 293, No. 6, 1084--1093 (2020; Zbl 1475.35161);
``Liouville type theorems for fractional and higher-order Hénon-Hardy type equations via the method of scaling spheres'', Preprint, \url{arXiv:1810.02752}],
[\textit{Y. Guo} and \textit{J. Liu}, Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 2, 339--359 (2008; Zbl 1153.35028)], and
[\textit{Phan Quoc Hung} and \textit{P. Souplet}, J. Differ. Equations 252, No. 3, 2544--2562 (2012; Zbl 1233.35093)]
to systems.Critical Kirchhoff-Choquard system involving the fractional \(p\)-Laplacian operator and singular nonlinearitieshttps://www.zbmath.org/1483.353272022-05-16T20:40:13.078697Z"Sang, Yanbin"https://www.zbmath.org/authors/?q=ai:sang.yanbinSummary: In this paper we study a class of critical fractional \(p\)-Laplacian Kirchhoff-Choquard systems with singular nonlinearities and two parameters \(\lambda\) and \(\mu\). By discussing the Nehari manifold structure and fibering maps analysis, we establish the existence of two positive solutions for above systems when \(\lambda\) and \(\mu\) satisfy suitable conditions.Accuracy controlled data assimilation for parabolic problemshttps://www.zbmath.org/1483.353352022-05-16T20:40:13.078697Z"Dahmen, Wolfgang"https://www.zbmath.org/authors/?q=ai:dahmen.wolfgang-a"Stevenson, Rob"https://www.zbmath.org/authors/?q=ai:stevenson.rob-p"Westerdiep, Jan"https://www.zbmath.org/authors/?q=ai:westerdiep.janSummary: This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a \textit{regularized least squares} formulation in a continuous \textit{infinite-dimensional} setting that is based on stable variational \textit{time-space} formulations of the parabolic partial differential equation. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable \textit{Fortin operators} which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.Stability estimates for the relativistic Schrödinger equation from partial boundary datahttps://www.zbmath.org/1483.353442022-05-16T20:40:13.078697Z"Senapati, Soumen"https://www.zbmath.org/authors/?q=ai:senapati.soumenA free boundary problem for binary fluidshttps://www.zbmath.org/1483.353452022-05-16T20:40:13.078697Z"Benzi, Roberto"https://www.zbmath.org/authors/?q=ai:benzi.roberto"Bertsch, Michiel"https://www.zbmath.org/authors/?q=ai:bertsch.michiel"Deangelis, Francesco"https://www.zbmath.org/authors/?q=ai:deangelis.francescoSummary: A free boundary problem for the dynamics of a glasslike binary fluid naturally leads to a singular perturbation problem for a strongly degenerate parabolic partial differential equation in 1D. We present a conjecture for an asymptotic formula for the velocity of the free boundary and prove a weak version of the conjecture. The results are based on the analysis of a family of local travelling wave solutions.A free boundary problem with log-term singularityhttps://www.zbmath.org/1483.353462022-05-16T20:40:13.078697Z"de Queiroz, Olivaine S."https://www.zbmath.org/authors/?q=ai:de-queiroz.olivaine-santana"Shahgholian, Henrik"https://www.zbmath.org/authors/?q=ai:shahgholian.henrikSummary: We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional
\[
\mathcal J(v)=\int_\Omega\left(\frac{|\nabla v|^2}{2} -v^+(\text{log}\: v-1)\right)dx\rightarrow \text{min}
\]
which should be minimized in some natural admissible class of non-negative functions. Here, \(v^+=\max\{0,v\}.\) The Euler-Lagrange equation associated with \(\mathcal J\) is
\[
-\Delta u= \chi_{\{u>0\}}\text{log}\: u,
\]
which becomes singular along the free boundary \(\partial\{u>0\}.\) Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like
\[
r^2|\text{log}\: r|.
\]
This estimate is crucial in the study of analytic and geometric properties of the free boundary.Two species nonlocal diffusion systems with free boundarieshttps://www.zbmath.org/1483.353472022-05-16T20:40:13.078697Z"Du, Yihong"https://www.zbmath.org/authors/?q=ai:du.yihong"Wang, Mingxin"https://www.zbmath.org/authors/?q=ai:wang.mingxin"Zhao, Meng"https://www.zbmath.org/authors/?q=ai:zhao.mengSummary: We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by ``nonlocal diffusion'' instead of ``local diffusion''. We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [\textit{J.-F. Cao} et al., J. Funct. Anal. 277, No. 8, 2772--2814 (2019; Zbl 1418.35229)], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.On some nonlinear parabolic equations with nonmonotone multivalued termshttps://www.zbmath.org/1483.353522022-05-16T20:40:13.078697Z"Otani, Mitsuharu"https://www.zbmath.org/authors/?q=ai:otani.mitsuharu"Staicu, Vasile"https://www.zbmath.org/authors/?q=ai:staicu.vasileSummary: We study the local and the global existence of solutions to a class of nonlinear parabolic initialboundary value problems driven by the equation
\[
\frac{\partial u(x,t)}{\partial t}-\Delta u(x,t)\in -\partial\Phi(u(x,t))+G(x,t,u(x,t)),(x,t)\in Q_T,
\]
where \(\partial\Phi\) denotes the subdifferential (in the sense of convex analysis) of a proper, convex and lower semicontinuous function \(\Phi:\mathbb{R}\to [0,\infty],\Omega\subseteq\mathbb{R}^N\) is a bounded open set, \(T> 0,Q_T:=\Omega\times[0,T]\), and \(G:Q_T\times\mathbb{R}\to 2^\mathbb{R}\) is a multivalued mapping whose growth order with respect to \(u\) is Sobolev sub-critical.
We prove two local existence results: one for the case where the multivalued mapping \(u\mapsto G(\cdot,\cdot,u)\) is upper semicontinuous with closed convex values and the second one deals with the case when \(u\mapsto G(\cdot,\cdot,u)\) is lower semicontinuous with closed (not necessarily convex) values. We also give
two types of results concerning the global continuation of local solutions. One is for any large data and the other one for small data. Finally, we exemplify the applicability of our results.\(L^p\) regularity estimates for a class of integral operators with fold blowdown singularitieshttps://www.zbmath.org/1483.353542022-05-16T20:40:13.078697Z"Bentsen, Geoffrey"https://www.zbmath.org/authors/?q=ai:bentsen.geoffreySummary: We prove sharp \(L^p\) regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated with a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain-Demeter for plate decompositions of thin neighborhoods of cones and \(L^2\) estimates for related oscillatory integrals.Almost-periodic response solutions for a forced quasi-linear Airy equationhttps://www.zbmath.org/1483.370922022-05-16T20:40:13.078697Z"Corsi, Livia"https://www.zbmath.org/authors/?q=ai:corsi.livia"Montalto, Riccardo"https://www.zbmath.org/authors/?q=ai:montalto.riccardo"Procesi, Michela"https://www.zbmath.org/authors/?q=ai:procesi.michelaThis paper is mainly concerned with the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. By combining the approach developed by
\textit{W. Craig} and \textit{C. E. Wayne} [Commun. Pure Appl. Math. 46, No. 11, 1409--1498 (1993; Zbl 0794.35104)] with a KAM reducibility scheme and pseudo-differential calculus on \(\mathbb{T}^{\infty}\), the authors establish some new results on the existence of this type of solutions for a quasi-linear PDE.
Reviewer: Chao Wang (Kunming)Gradient estimates of \(\omega\)-minimizers to double phase variational problems with variable exponentshttps://www.zbmath.org/1483.490492022-05-16T20:40:13.078697Z"Byun, Sun-Sig"https://www.zbmath.org/authors/?q=ai:byun.sun-sig"Lee, Ho-Sik"https://www.zbmath.org/authors/?q=ai:lee.ho-sikThis paper concerns the integral functionals involving non-uniformly elliptic operators of the type
\[
\mathcal{F}(u,\Omega):=\int_{\Omega}\Bigl( f_1(x,Du) +a(x)f_2(x,Du)\Bigr)\ dx
\]
whose model case is \(f_1(x,z)=|z|^{p(x)}\), \(f_2(x,z)=|z|^{q(x)}\). Regarding the model case, the functions \(p(x),q(x),a(x)\) are assumed to be continuous and satisfying,
\[
0\leq a(x)\in C^{0,\alpha},\;\;\;1<\gamma_1\leq p(x)\leq q(x)\leq \gamma_2<\infty, \;\;\frac{q(x)}{p(x)}\leq 1+\frac{\alpha}{n}
\]
for some constant \(\alpha \in (0,1]\) and \(\gamma_1,\gamma_2 \in \mathbb R^n\).
\\
The authors establish a local Calderón-Zygmund theory for \(\omega\)-minimizers of functional \(\mathcal{F}\).
Reviewer: Luca Esposito (Fisciano)An ODE reduction method for the semi-Riemannian Yamabe problem on space formshttps://www.zbmath.org/1483.530592022-05-16T20:40:13.078697Z"Fernández, Juan Carlos"https://www.zbmath.org/authors/?q=ai:fernandez.juan-carlos"Palmas, Oscar"https://www.zbmath.org/authors/?q=ai:palmas.oscarThe authors prove the existence of blowing-up and globally defined solutions of Yamabe-type partial differential equations on semi-Euclidean space and on the pseudosphere of dimension at least 3. In the proof they use isoparametric functions which allow the reduction to a generalized Emden-Fowler ordinary differential equation.
Reviewer: Hans-Bert Rademacher (Leipzig)Mean curvature type flows of graphs with nonzero Neumann boundary data in product manifold \(M^n \times \mathbb{R} \)https://www.zbmath.org/1483.531082022-05-16T20:40:13.078697Z"Chen, Xiao-Li"https://www.zbmath.org/authors/?q=ai:chen.xiaoli"Gao, Ya"https://www.zbmath.org/authors/?q=ai:gao.ya"Lu, Wei"https://www.zbmath.org/authors/?q=ai:lu.wei"Mao, Jing"https://www.zbmath.org/authors/?q=ai:mao.jingLet \((M^n,\sigma)\), \(n\ge 2\), be an \(n\)-dimensional complete Riemannian manifold with metric \(\sigma\) and let \(\Omega\subset M^n\) be a bounded domain with \(C^3\) boundary. The authors study the evolution of a graphic hypersurface of the form \((x,u(x,t))\), \(x\in\Omega\), \(t>0\), in the product space \(M^n\times\mathbb{R}\) with a nonparametric mean curvature type flow. They prove a gradient estimate for the flow under suitable conditions. As a consequence long time existence for the flow is obtained.
Reviewer: Shu-Yu Hsu (Chiayi)The impact of white noise on a supercritical bifurcation in the Swift-Hohenberg equationhttps://www.zbmath.org/1483.600882022-05-16T20:40:13.078697Z"Bianchi, Luigi Amedeo"https://www.zbmath.org/authors/?q=ai:bianchi.luigi-amedeo"Blömker, Dirk"https://www.zbmath.org/authors/?q=ai:blomker.dirkSummary: We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.A modified weak Galerkin finite element method for nonmonotone quasilinear elliptic problemshttps://www.zbmath.org/1483.651832022-05-16T20:40:13.078697Z"Guo, Liming"https://www.zbmath.org/authors/?q=ai:guo.liming"Sheng, Qiwei"https://www.zbmath.org/authors/?q=ai:sheng.qiwei"Wang, Cheng"https://www.zbmath.org/authors/?q=ai:wang.cheng.1"Huang, Ziping"https://www.zbmath.org/authors/?q=ai:huang.zipingSummary: A modified weak Galerkin finite element method is studied for nonmonotone quasilinear elliptic problems. Using the contraction mapping theorem, the uniqueness of the solution to the discrete problem is proved. Moreover, optimal order a priori error estimates are established in both a discrete \(H^1\) norm and the \(L^2\) norm. Numerical experiments are conducted to confirm the theoretical results.Discrete strong extremum principles for finite element solutions of diffusion problems with nonlinear correctionshttps://www.zbmath.org/1483.651932022-05-16T20:40:13.078697Z"Wang, Shuai"https://www.zbmath.org/authors/?q=ai:wang.shuai"Yuan, Guangwei"https://www.zbmath.org/authors/?q=ai:yuan.guangweiSummary: A nonlinear correction technique for finite element methods to anisotropic diffusion problems on general triangular and quadrilateral meshes are introduced. The classic linear or bi-linear finite element methods are modified, and then the resulting schemes satisfy the discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well-known restrictions on diffusion coefficients and geometry of mesh-cell (e.g. ``acute angle'' condition). Convergence rate for smooth and piecewise smooth solutions and the discrete extremum principle property are verified by numerical examples.Locking-free enriched Galerkin method for linear elasticityhttps://www.zbmath.org/1483.651952022-05-16T20:40:13.078697Z"Yi, Son-Young"https://www.zbmath.org/authors/?q=ai:yi.son-young-jun|yi.son-young"Lee, Sanghyun"https://www.zbmath.org/authors/?q=ai:lee.sanghyun"Zikatanov, Ludmil"https://www.zbmath.org/authors/?q=ai:zikatanov.ludmil-tGeneralized Swift-Hohenberg and phase-field-crystal equations based on a second-gradient phase-field theoryhttps://www.zbmath.org/1483.740052022-05-16T20:40:13.078697Z"Espath, Luis"https://www.zbmath.org/authors/?q=ai:espath.luis-f-r"Calo, Victor M."https://www.zbmath.org/authors/?q=ai:calo.victor-manuel"Fried, Eliot"https://www.zbmath.org/authors/?q=ai:fried.eliotSummary: The principle of virtual power is used derive a microforce balance for a second-gradient phase-field theory. In conjunction with constitutive relations consistent with a free-energy imbalance, this balance yields a broad generalization of the Swift-Hohenberg equation. When the phase field is identified with the volume fraction of a conserved constituent, a suitably augmented version of the free-energy imbalance yields constitutive relations which, in conjunction with the microforce balance and the constituent content balance, delivers a broad generalization of the phase-field-crystal equation. Thermodynamically consistent boundary conditions for situations in which the interface between the system and its environment is structureless and cannot support constituent transport are also developed, as are energy decay relations that ensue naturally from the thermodynamic structure of the theory.Motions of a charged particle in the electromagnetic field induced by a non-stationary currenthttps://www.zbmath.org/1483.780012022-05-16T20:40:13.078697Z"Garzón, Manuel"https://www.zbmath.org/authors/?q=ai:garzon.manuel"Marò, Stefano"https://www.zbmath.org/authors/?q=ai:maro.stefanoSummary: In this paper we study the non-relativistic dynamic of a charged particle in the electromagnetic field induced by a periodically time dependent current \(J\) along an infinitely long and infinitely thin straight wire. The motions are described by the Lorentz-Newton equation, in which the electromagnetic field is obtained by solving the Maxwell's equations with the current distribution \(\vec{J}\) as data. We prove that many features of the integrable time independent case are preserved. More precisely, introducing cylindrical coordinates, we prove the existence of (non-resonant) radially periodic motions that are also of twist type. In particular, these solutions are Lyapunov stable and accumulated by subharmonic and quasiperiodic motions.Spinning-off stringy electro-magnetic memorieshttps://www.zbmath.org/1483.811162022-05-16T20:40:13.078697Z"Aldi, Alice"https://www.zbmath.org/authors/?q=ai:aldi.alice"Bianchi, Massimo"https://www.zbmath.org/authors/?q=ai:bianchi.massimo"Firrotta, Maurizio"https://www.zbmath.org/authors/?q=ai:firrotta.maurizioSummary: We extend and generalise the string corrections to the EM memory to the Type I superstring including spin effects. Very much as in the simpler bosonic string context, the relevant corrections are non-perturbative in \(\alpha^\prime\), slowly decaying (as \(1/R\)) at large distances and modulated in retarded time \(u = t-R\). For spin \(N\) states in the first Regge trajectory they entail a sequence of \(N\) derivatives wrt \(u\) on the `parent' \(N = 0\) amplitude. We also briefly discuss how to include loop effects, that broaden and shift the string resonances, and how to modify our analysis for macroscopic semi-classical quasi-BPS coherent states, whose collisions may lead to detectable string memory signals in viable Type I models.Reprint of: Marginal CFT perturbations at the integer quantum Hall transitionhttps://www.zbmath.org/1483.811282022-05-16T20:40:13.078697Z"Zirnbauer, Martin R."https://www.zbmath.org/authors/?q=ai:zirnbauer.martin-rSummary: According to recent arguments by the author, the conformal field theory (CFT) describing the scaling limit of the integer quantum Hall plateau transition is a deformed level-4 Wess-Zumino-Novikov-Witten model with Riemannian target space inside a complex Lie supergroup \(\operatorname{GL}\). After a summary of that proposal and some of its predictions, the leading irrelevant and relevant perturbations of the proposed CFT are discussed.
Argued to be marginal, these result in a non-standard renormalization group (RG) flow near criticality, which calls for modified finite-size scaling analysis and may explain the long-standing inability of numerical work to reach agreement on the values of critical exponents. The technique of operator product expansion is used to compute the RG-beta functions up to cubic order in the couplings. The mean value of the dissipative conductance at the RG-fixed point is calculated for a cylinder geometry with any aspect ratio.Green's functions on a renormalized lattice: an improved method for the integer quantum Hall transitionhttps://www.zbmath.org/1483.811692022-05-16T20:40:13.078697Z"Puschmann, Martin"https://www.zbmath.org/authors/?q=ai:puschmann.martin"Vojta, Thomas"https://www.zbmath.org/authors/?q=ai:vojta.thomasSummary: We introduce a performance-optimized method to simulate localization problems on bipartite tight-binding lattices. It combines an exact renormalization group step to reduce the sparseness of the original problem with the recursive Green's function method. We apply this framework to investigate the critical behavior of the integer quantum Hall transition of a tight-binding Hamiltonian defined on a simple square lattice. In addition, we employ an improved scaling analysis that includes two irrelevant exponents to characterize the shift of the critical energy as well as the corrections to the dimensionless Lyapunov exponent. We compare our findings with the results of a conventional implementation of the recursive Green's function method, and we put them into broader perspective in view of recent development in this field.Quasinormal resonances of rapidly-spinning Kerr black holes and the universal relaxation boundhttps://www.zbmath.org/1483.830102022-05-16T20:40:13.078697Z"Hod, Shahar"https://www.zbmath.org/authors/?q=ai:hod.shaharSummary: The universal relaxation bound suggests that the relaxation times of perturbed thermodynamical systems is bounded from below by the simple time-times-temperature (TTT) quantum relation \(\tau \times T \geq \frac{\hbar}{\pi}\). It is known that some perturbation modes of near-extremal Kerr black holes in the regime \(M T_{\mathrm{BH}}/\hbar \ll m^{-2}\) are characterized by normalized relaxation times \(\pi \tau \times T_{\mathrm{BH}}/\hbar\) which, in the approach to the limit \(M T_{\mathrm{BH}}/\hbar\to 0\), make infinitely many oscillations with a tiny constant amplitude around 1 and therefore cannot be used directly to verify the validity of the TTT bound in the entire parameter space of the black-hole spacetime (Here \(\{T_{\mathrm{BH}}, M\}\) are respectively the Bekenstein-Hawking temperature and the mass of the black hole, and \(m\) is the azimuthal harmonic index of the linearized perturbation mode). In the present compact paper we explicitly prove that all rapidly-spinning Kerr black holes respect the TTT relaxation bound. In particular, using analytical techniques, it is proved that all black-hole perturbation modes in the complementary regime \(m^{-1} \ll M T_{\mathrm{BH}}/\hbar\ll 1\) are characterized by relaxation times with the simple dimensionless property \(\pi \tau \times T_{\mathrm{BH}}/\hbar\geq 1\).Phase space analysis of Tsallis agegraphic dark energyhttps://www.zbmath.org/1483.830302022-05-16T20:40:13.078697Z"Huang, Hai"https://www.zbmath.org/authors/?q=ai:huang.hai"Huang, Qihong"https://www.zbmath.org/authors/?q=ai:huang.qihong"Zhang, Ruanjing"https://www.zbmath.org/authors/?q=ai:zhang.ruanjingSummary: Based on the generalized Tsallis entropy and holographic hypothesis, the Tsallis agegraphic dark energy (TADE) was proposed by introducing the timescale as infrared cutoff. In this paper, we analyze the evolution of the universe in the TADE model and the new Tsallis agegraphic dark energy (NTADE) model by considering an interaction between dark matter and dark energy as \(Q=H(\alpha\rho_m+\beta\rho_D)\). Through the phase space and stability analysis, we find an attractor which represents a late-time accelerated expansion phase can exist only in NTADE model. When \(0\leq\alpha<1\) and \(\beta=0\), this attractor becomes a dark energy dominated de Sitter solution and the universe can eventually evolve into an accelerated expansion era which is depicted by the \(\Lambda\) cold dark matter model. Thus, the expansion history of the universe can be depicted by the NTADE model.QNMs of branes, BHs and fuzzballs from quantum SW geometrieshttps://www.zbmath.org/1483.830352022-05-16T20:40:13.078697Z"Bianchi, Massimo"https://www.zbmath.org/authors/?q=ai:bianchi.massimo"Consoli, Dario"https://www.zbmath.org/authors/?q=ai:consoli.dario"Grillo, Alfredo"https://www.zbmath.org/authors/?q=ai:grillo.alfredo"Morales, Francisco"https://www.zbmath.org/authors/?q=ai:morales.franciscoSummary: QNMs govern the linear response to perturbations of BHs, D-branes and fuzzballs and the gravitational wave signals in the ring-down phase of binary mergers. A remarkable connection between QNMs of neutral BHs in 4d and quantum SW geometries describing the dynamics of \(\mathcal{N} = 2\) SYM theories has been recently put forward. We extend the gauge/gravity dictionary to a large class of gravity backgrounds including charged and rotating BHs of Einstein-Maxwell theory in \(d = 4\), 5 dimensions, D3-branes, D1D5 `circular' fuzzballs and smooth horizonless geometries; all related to \(\mathcal{N} = 2\) SYM with a single \(SU(2)\) gauge group and fundamental matter. We find that photon-spheres, a common feature of all examples, are associated to degenerations of the classical elliptic SW geometry whereby a cycle pinches to zero size. Quantum effects resolve the singular geometry and lead to a spectrum of quantized energies, labelled by the overtone number \(n\). We compute the spectrum of QNMs using exact WKB quantization, geodetic motion and numerical simulations and show excellent agreement between the three methods. We explicitly illustrate our findings for the case D3-brane QNMs.Higher-dimensional non-extremal Reissner-Nordström black holes, scalar perturbation and superradiance: an analytical studyhttps://www.zbmath.org/1483.830482022-05-16T20:40:13.078697Z"Huang, Jia-Hui"https://www.zbmath.org/authors/?q=ai:huang.jiahui"Zhao, Run-Dong"https://www.zbmath.org/authors/?q=ai:zhao.rundong"Zou, Yi-Feng"https://www.zbmath.org/authors/?q=ai:zou.yi-fengSummary: The superradiant stability of asymptotically flat higher dimensional non-extremal Reissner-Nordstrom black holes under charged massive scalar perturbation is analytically studied. We extend an analytical method developed by one of the authors in the extremal Reissner-Nordstrom black hole cases to non-extremal cases. Using the new analytical method, we revisit four-dimensional Reissner-Nordstrom black hole case and obtain that four-dimensional Reissner-Nordstrom black hole is superradiantly stable, which is consistent with results in previous works. We then analytically prove that the five-dimensional Reissner-Nordstrom black holes are also superradiantly stable under charged massive scalar perturbation. Our result implies that all higher dimensional non-extremal Reissner-Nordstrom black holes may be superradiantly stable under charged massive scalar perturbation.Oscillons during Dirac-Born-Infeld preheatinghttps://www.zbmath.org/1483.830952022-05-16T20:40:13.078697Z"Sang, Yu"https://www.zbmath.org/authors/?q=ai:sang.yu"Huang, Qing-Guo"https://www.zbmath.org/authors/?q=ai:huang.qingguoSummary: Oscillons are long-lived, localized, oscillating nonlinear excitations of a real scalar field which can be abundantly produced during preheating after inflation. We give the first \((3 + 1)\)-dimensional simulation for the oscillon formation during preheating with noncanonical kinetic terms, e.g. the Dirac-Born-Infeld form, and find that the formation of oscillons is significantly influenced by the noncanonical effect.Aspects of GRMHD in high-energy astrophysics: geometrically thick disks and tori agglomerates around spinning black holeshttps://www.zbmath.org/1483.850102022-05-16T20:40:13.078697Z"Pugliese, D."https://www.zbmath.org/authors/?q=ai:pugliese.daniela"Montani, G."https://www.zbmath.org/authors/?q=ai:montani.giovanniSummary: This work focuses on some key aspects of the general relativistic (GR) -- magneto-hydrodynamic (MHD) applications in high-energy astrophysics. We discuss the relevance of the GRHD counterparts formulation exploring the geometrically thick disk models and constraints of the GRMHD shaping the physics of accreting configurations. Models of clusters of tori orbiting a central super-massive black hole (\textbf{SMBH}) are described. These orbiting tori aggregates form sets of geometrically thick, pressure supported, perfect fluid tori, associated to complex instability processes including tori collision emergence and
empowering a wide range of activities related expectantly to the embedding matter environment of Active Galaxy Nuclei. Some notes are included on aggregates combined with proto-jets, represented by open cusped solutions associated to the geometrically thick tori.
This exploration of some key concepts of the GRMHD formulation in its applications to High-Energy Astrophysics starts with the discussion of the initial data problem for a most general Einstein-Euler-Maxwell system addressing the problem with a relativistic geometric background. The system is then set in quasi linear hyperbolic form, and the reduction procedure is argumented. Then, considerations follow on the analysis of the stability problem
for self-gravitating systems with determined symmetries considering the perturbations also of the geometry part on the quasi linear hyperbolic onset. Thus we focus on the ideal GRMHD and self-gravitating plasma ball. We conclude with the models of geometrically thick GRHD disks gravitating around a Kerr \textbf{SMBH} in their GRHD formulation and including in the force balance equation of the disks the influence of a toroidal magnetic field, determining its impact in tori topology and stability.Local asymptotic stability of a system of integro-differential equations describing clonal evolution of a self-renewing cell population under mutationhttps://www.zbmath.org/1483.920582022-05-16T20:40:13.078697Z"Busse, Jan-Erik"https://www.zbmath.org/authors/?q=ai:busse.jan-erik"Cuadrado, Sílvia"https://www.zbmath.org/authors/?q=ai:cuadrado.silvia"Marciniak-Czochra, Anna"https://www.zbmath.org/authors/?q=ai:marciniak-czochra.anna-kSummary: In this paper we consider a system of non-linear integro-differential equations (IDEs) describing evolution of a clonally heterogeneous population of malignant white blood cells (leukemic cells) undergoing mutation and clonal selection. We prove existence and uniqueness of non-trivial steady states and study their asymptotic stability. The results are compared to those of the system without mutation. Existence of equilibria is proved by formulating the steady state problem as an eigenvalue problem and applying a version of the Krein-Rutmann theorem for Banach lattices. The stability at equilibrium is analysed using linearisation and the Weinstein-Aronszajn determinant which allows to conclude local asymptotic stability.Optimal control of pattern formations for an SIR reaction-diffusion epidemic modelhttps://www.zbmath.org/1483.921262022-05-16T20:40:13.078697Z"Chang, Lili"https://www.zbmath.org/authors/?q=ai:chang.lili"Gao, Shupeng"https://www.zbmath.org/authors/?q=ai:gao.shupeng"Wang, Zhen"https://www.zbmath.org/authors/?q=ai:wang.zhen.1Summary: Patterns arising from the reaction-diffusion epidemic model provide insightful aspects into the transmission of infectious diseases. For a classic SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates. A quantitative indicator, ``normal serious prevalent area (\textit{NSPA})'', is introduced to characterize the relationship between patterns and the extent of the epidemic. The extent of epidemic is positively correlated to \textit{NSPA}. To effectively reduce \textit{NSPA} of patterns under the large transmission rates, taken removed (recovery or isolation) rate as a control parameter, we consider the mathematical formulation and numerical solution of an optimal control problem for the SIR reaction-diffusion model. Numerical experiments demonstrate the effectiveness of our method in terms of control effect, control precision and control cost.Numerical bifurcation analysis and pattern formation in a minimal reaction-diffusion model for vegetationhttps://www.zbmath.org/1483.921552022-05-16T20:40:13.078697Z"Kabir, M. Humayun"https://www.zbmath.org/authors/?q=ai:kabir.md-humayun"Gani, M. Osman"https://www.zbmath.org/authors/?q=ai:gani.m-osmanSummary: Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by \textit{C. A. Klausmeier} [``Regular and irregular patterns in semiarid vegetation'', Science 284, No. 5421, 1826--1828 (1999; \url{doi:10.1126/science.284.5421.1826})] that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction-diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction-diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction-diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction-diffusion system.