Recent zbMATH articles in MSC 35K https://www.zbmath.org/atom/cc/35K 2022-06-24T15:10:38.853281Z Werkzeug Analysis on Lie groups. An introduction https://www.zbmath.org/1485.22001 2022-06-24T15:10:38.853281Z "Faraut, Jacques" https://www.zbmath.org/authors/?q=ai:faraut.jacques Publisher's description: Cet ouvrage est issu d'un cours élémentaire en Master 1 de l'université Pierre-et-Marie Curie, cours destiné à initier les étudiants, dès la quatrième année universitaire, aux thèmes et méthodes de l'analyse harmonique non commutative. Partant de connaissances préliminaires réduites à l'algèbre linéaire et au calcul différentiel, l'auteur réussit dans un même texte le pari d'introduire les groupes et algèbres de Lie, de fournir ces outils nécessaires à l'apprentissage de l'analyse que sont la mesure de Haar et l'intégration invariante, mais aussi de traiter de sujets subtils comme la théorie des représentations, les harmoniques sphériques, l'analyse de Fourier et l'équation de la chaleur. Jacques Faraut introduit savamment le lecteur à un territoire mathématique fascinant et à ses méthodes et outils propres. On sait le rôle qu'a joué l'analyse harmonique commutative dans les mathématiques du dix-neuvième siècle et dans la physique classique. C'est à l'analyse harmonique non commutative qu'il est revenu de prendre le relais dans le contexte de la physique moderne, où l'idée de symétrie, incarnée par les groupes de Lie, joue un rôle essentiel. À contexte nouveau, objets nouveaux, mais problématiques traditionnelles -- et, bien sûr, d'autres qui le sont moins : équation de Laplace, fonctions harmoniques, noyau de Poisson, transformation de Fourier, représentations irréductibles, intégrale orbitale. La géométrie des actions de groupes et l'analyse de Fourier se rencontrent dans des développements récents de l'étude des mesures orbitales et des fonctions splines, qui font l'objet du dernier chapitre de cette nouvelle édition. C'est par un choix éclectique de thèmes et dans la convergence des points de vue géométrique, algébrique et infinitésimal que le présent ouvrage offre à un étudiant en Master le loisir de découvrir quelques-uns des plus jolis thèmes et outils de ce territoire mathématique. Ces outils développés élémentairement seront aussi utiles a l'étude des matrices aléatoires et de la statistique multivariée. See the review of the first edition in [Zbl 1096.22001]. Corrigendum and improvements to Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations'' and its consequences https://www.zbmath.org/1485.35001 2022-06-24T15:10:38.853281Z "Fragnelli, Genni" https://www.zbmath.org/authors/?q=ai:fragnelli.genni "Mugnai, Dimitri" https://www.zbmath.org/authors/?q=ai:mugnai.dimitri Summary: This paper is a corrigendum of one hypothesis introduced in [the authors, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1377.93043)] and used again in [\textit{G. Fragnelli}, J. Differ. Equations 260, No. 2, 1314--1371 (2016; Zbl 1331.35199)] and [\textit{G. Fragnelli} and \textit{D. Mugnai}, Adv. Nonlinear Anal. 6, No. 1, 61--84 (2017; Zbl 1358.35219)]. We give here the corrected proofs of the concerned results, improving most of them. On singularities in the quaternionic Burgers equation https://www.zbmath.org/1485.35005 2022-06-24T15:10:38.853281Z "Sverak, Vladimir" https://www.zbmath.org/authors/?q=ai:sverak.vladimir Summary: We consider the equation $$q_t+qq_x=q_{xx}$$ for $$q:{{\mathbb{R}}}\times (0,\infty )\rightarrow{\mathbb{H}}$$ (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small generic'' smooth perturbation of the initial data. Similar results are also established for the same equation in $$\mathbf{S}^1\times (0,\infty)$$, where $$\mathbf{S}^1$$ is the standard one-dimensional circle. Periodic solutions of a phase-field model with hysteresis https://www.zbmath.org/1485.35017 2022-06-24T15:10:38.853281Z "Bin, Chen" https://www.zbmath.org/authors/?q=ai:bin.chen "Timoshin, Sergey A." https://www.zbmath.org/authors/?q=ai:timoshin.sergey-a Summary: In the present paper we consider a partial differential system describing a phase-field model with temperature dependent constraint for the order parameter. The system consists of an energy balance equation with a fairly general nonlinear heat source term and a phase dynamics equation which takes into account the hysteretic character of the process. The existence of a periodic solution for this system is proved under a minimal set of assumptions on the curves defining the corresponding hysteresis region. Convergence of the Allen-Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to $$90^\circ$$ https://www.zbmath.org/1485.35020 2022-06-24T15:10:38.853281Z "Abels, Helmut" https://www.zbmath.org/authors/?q=ai:abels.helmut "Moser, Maximilian" https://www.zbmath.org/authors/?q=ai:moser.maximilian In this paper, the authors study a parabolic Allen-Cahn equation with a nonlinear Robin condition in a bounded smooth domain in $$\mathbb{R}^2$$. The limiting problem is the mean curvature flow with a contact angle $$\alpha$$ on the boundary. When $$\alpha$$ is close to $$\pi/2$$, by assuming the limiting problem has a local smooth solution, they prove that solutions to the Allen-Cahn equation converges to the limit in a smooth way. This is done by taking an expansion of the solution in $$\varepsilon$$ and then estimating the difference. For this purpose, some linearized estimates are developed, and the restriction on $$\alpha$$ arises from the linearization problem at the boundary contact point. Reviewer: Kelei Wang (Wuhan) Fast reaction limit and forward-backward diffusion: a Radon-Nikodym approach https://www.zbmath.org/1485.35024 2022-06-24T15:10:38.853281Z "Skrzeczkowski, Jakub" https://www.zbmath.org/authors/?q=ai:skrzeczkowski.jakub Summary: We consider two singular limits: a fast reaction limit with a non-monotone nonlinearity and a regularization of the forward-backward diffusion equation. We derive pointwise identities satisfied by the Young measure generated by these problems. As a result, we obtain an explicit formula for the Young measure even without the non-degeneracy assumption used in the previous works. The main new idea is an application of the Radon-Nikodym theorem to decompose the Young measure. Diffusion-induced spatio-temporal oscillations in an epidemic model with two delays https://www.zbmath.org/1485.35034 2022-06-24T15:10:38.853281Z "Du, Yan-fei" https://www.zbmath.org/authors/?q=ai:du.yanfei "Niu, Ben" https://www.zbmath.org/authors/?q=ai:niu.ben "Wei, Jun-jie" https://www.zbmath.org/authors/?q=ai:wei.junjie Summary: We investigate a diffusive, stage-structured epidemic model with the maturation delay and freely-moving delay. Choosing delays and diffusive rates as bifurcation parameters, the only possible way to destabilize the endemic equilibrium is through Hopf bifurcation. The normal forms of Hopf bifurcations on the center manifold are calculated, and explicit formulae determining the criticality of bifurcations are derived. There are two different kinds of stable oscillations near the first bifurcation: on one hand, we theoretically prove that when the diffusion rate of infected immature individuals is sufficiently small or sufficiently large, the first branch of Hopf bifurcating solutions is always spatially homogeneous; on the other, xing this diffusion rate at an appropriate size, stable oscillations with different spatial profiles are observed, and the conditions to guarantee the existence of such solutions are given by calculating the corresponding eigenfunction of the Laplacian at the first Hopf bifurcation point. These bifurcation behaviors indicate that spatial di usion in the epidemic model may lead to spatially inhomogeneous distribution of individuals. Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system https://www.zbmath.org/1485.35035 2022-06-24T15:10:38.853281Z "Li, Jing" https://www.zbmath.org/authors/?q=ai:li.jing.13 "Sun, Gui-Quan" https://www.zbmath.org/authors/?q=ai:sun.guiquan "Jin, Zhen" https://www.zbmath.org/authors/?q=ai:jin.zhen Summary: Empirical data exhibit a common phenomenon that vegetation biomass fluctuates periodically over time in ecosystem, but the corresponding internal driving mechanism is still unclear. Simultaneously, considering that the conversion of soil water absorbed by roots of the vegetation into vegetation biomass needs a period time, we thus introduce the conversion time into Klausmeier model, then a spatiotemporal vegetation model with time delay is established. Through theoretical analysis, we not only give the occurence conditions of stability switches for system without and with diffusion at the vegetation-existence equilibrium, but also derive the existence conditions of saddle-node-Hopf bifurcation of non-spatial system and Hopf bifurcation of spatial system at the coincidence equilibrium. Our results reveal that the conversion delay induces the interaction between the vegetation and soil water in the form of periodic oscillation when conversion delay increases to the critical value. By comparing the results of system without and with diffusion, we find that the critical value decreases with the increases of spatial diffusion factors, which is more conducive to emergence of periodic oscillation phenomenon, while spatial diffusion factors have no effects on the amplitude of periodic oscillation. These results provide a theoretical basis for understanding the spatiotemporal evolution behaviors of vegetation system. Fourier transforms and $$L^2$$-stability of diffusion equations https://www.zbmath.org/1485.35039 2022-06-24T15:10:38.853281Z "Kang, Dongseung" https://www.zbmath.org/authors/?q=ai:kang.dongseung "Kim, Hoewoon B." https://www.zbmath.org/authors/?q=ai:kim.hoewoon-b Summary: This paper is concerned with generalized Hyers-Ulam stability of the diffusion equation, $$\displaystyle\frac{\partial u(x,t)}{\partial t} = \triangle u (x, t)$$ with $$u(x,0) = f(x)$$ for $$t > 0$$ and $$x \in \mathbb{R}^n$$. Most of the Hyers-Ulam stability problems of differential equations are involved with $$L^\infty$$-norm or the supremum norm of functions with consideration of either initial conditions or forcing terms. However, an integral method of Fourier transform can be used to obtain the $$L^2$$-estimates for generalized Hyers-Ulam stability of an IVP (initial value problem) of the diffusion equation with a function $$f (x)$$ as an initial condition and we will present the generalized Hyers-Ulam stability of the IVP in the sense of $$L^2$$-norm. On a two-species chemotaxis-competition system with indirect signal consumption https://www.zbmath.org/1485.35041 2022-06-24T15:10:38.853281Z "Xiang, Yuting" https://www.zbmath.org/authors/?q=ai:xiang.yuting "Zheng, Pan" https://www.zbmath.org/authors/?q=ai:zheng.pan Summary: This paper deals with a two-competing-species chemotaxis system with indirect signal consumption \begin{cases} \begin{aligned} &u_t=d_1\Delta u-\chi_1\nabla \cdot (u\nabla w)+\mu_1u(1-u-a_1v),&(x,t)\in \Omega \times (0,\infty ),\\ &v_t=d_2\Delta v-\chi_2\nabla \cdot (v\nabla w)+\mu_2v(1-v-a_2u),&(x,t)\in \Omega \times (0,\infty ),\\ &w_t=\Delta w-wz,&(x,t)\in \Omega \times (0,\infty ),\\ &z_t=\Delta z-z+u+v,&(x,t)\in \Omega \times (0,\infty ),\\ &\frac{{\partial u}}{{\partial \nu }} = \frac{{\partial v}}{{\partial \nu }} = \frac{{\partial w}}{{\partial \nu }} = \frac{{\partial z}}{{\partial \nu }} = 0,&(x,t)\in \partial \Omega \times (0,\infty ),\\ &\left( {u, v, w, z} \right) \left( {x,0} \right) = \left( {{u_0}\left( x \right) , {v_0}\left( x \right) ,{w_0}\left( x \right) , {z_0}\left( x \right) } \right) ,&x\in \Omega ,\ \end{aligned} \end{cases} under homogeneous Neumann boundary conditions in a smooth bounded domain $$\Omega \subset{\mathbb{R}}^n(n\le 2)$$, with the nonnegative initial data $$\left( {u_0, v_0, w_0, z_0} \right) \in{C^0}\left( {{{\bar{\Omega }}} } \right) \times{C^0}\left( {{{\bar{\Omega }}} } \right) \times{W^{1,\infty }}\left( \Omega \right) \times{W^{1,\infty }}\left( \Omega \right)$$, where $$\chi_i>0, d_i>0, a_i>0$$ and $$\mu_i>0 (i=1,2)$$. It is shown that the system has a global bounded classical solution for arbitrary size of $$\mu_1, \mu_2>0$$. Additionally, we consider the asymptotic stabilization of solutions to the above system as follows: \begin{itemize} \item When $$a_1, a_2 \in (0,1)$$, the global bounded classical solution $$(u, v, w, z)$$ exponentially converges to $$\Big (\frac{1-a_1}{1-a_1 a_2}, \frac{1-a_2}{1-a_1 a_2}, 0, \frac{2-a_1-a_2}{1-a_1 a_2}\Big )$$ in the $$L^{\infty }$$-norm as $$t \rightarrow \infty$$; \item When $$a_1>1>a_2>0$$ and $$a_1a_2<1$$, the global bounded classical solution $$(u, v, w, z)$$ exponentially converges to (0, 1, 0, 1) in the $$L^{\infty }$$-norm as $$t \rightarrow \infty$$; \item When $$a_1=1>a_2>0$$, the global bounded classical solution $$(u, v, w, z)$$ polynomially converges to (0, 1, 0, 1) in the $$L^{\infty }$$-norm as $$t \rightarrow \infty$$. \end{itemize} Effect of a membrane on diffusion-driven Turing instability https://www.zbmath.org/1485.35042 2022-06-24T15:10:38.853281Z "Ciavolella, Giorgia" https://www.zbmath.org/authors/?q=ai:ciavolella.giorgia Summary: Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To this aim, we study a reaction-diffusion system with zero-flux boundary conditions on the external boundary and Kedem-Katchalsky membrane conditions on the inner membrane. We use the same approach as in the classical Turing analysis but applied to membrane operators. The introduction of a diagonalization theory for compact and self-adjoint membrane operators is needed. Here, Turing instability is proven with the addition of new constraints, due to the presence of membrane permeability coefficients. We perform an explicit one-dimensional analysis of the eigenvalue problem, combined with numerical simulations, to validate the theoretical results. Finally, we observe the formation of discontinuous patterns in a system which combines diffusion and dissipative membrane conditions, varying both diffusion and membrane permeability coefficients. Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics https://www.zbmath.org/1485.35043 2022-06-24T15:10:38.853281Z "Coelho, Daniel L." https://www.zbmath.org/authors/?q=ai:coelho.daniel-l "Vitral, Eduardo" https://www.zbmath.org/authors/?q=ai:vitral.eduardo "Pontes, José" https://www.zbmath.org/authors/?q=ai:pontes.jose-pedro "Mangiavacchi, Norberto" https://www.zbmath.org/authors/?q=ai:mangiavacchi.norberto Summary: Spatio-temporal pattern formation in complex systems presents rich nonlinear dynamics which leads to the emergence of periodic nonequilibrium structures. One of the most prominent equations for the theoretical and numerical study of the evolution of these textures is the Swift-Hohenberg (SH) equation, which presents a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system, and has been largely employed in phase-field models. Though a large part of the literature on pattern formation addresses uniformly forced systems, nonuniform forcings are also observed in several natural systems, for instance, in developmental biology and in soft matter applications. In these cases, an orientation effect due to forcing gradients is a new factor playing a role in the development of patterns, particularly in the class of stripe patterns, which we investigate through the nonuniformly forced SH dynamics. The present work addresses amplitude instability of stripe textures induced by forcing gradients, and the competition between the orientation effect of the gradient and other bulk, boundary, and geometric effects taking part in the selection of the emerging patterns. A weakly nonlinear analysis suggests that stripes are stable with respect to small amplitude perturbations when aligned with the gradient, and become unstable to such perturbations when when aligned perpendicularly to the gradient. This analysis is vastly complemented by a numerical work that accounts for other effects, confirming that forcing gradients drive stripe alignment, or even reorient them from preexisting conditions. However, we observe that the orientation effect does not always prevail in the face of competing effects, whose hierarchy is suggested to depend on the magnitude of the forcing gradient. Other than the cubic SH equation (SH3), the quadratic-cubic (SH23) and cubic-quintic (SH35) equations are also studied. In the SH23 case, not only do forcing gradients lead to stripe orientation, but also interfere in the transition from hexagonal patterns to stripes. Global solution and spatial patterns for a ratio-dependent predator-prey model with predator-taxis https://www.zbmath.org/1485.35044 2022-06-24T15:10:38.853281Z "Gao, Xiaoyan" https://www.zbmath.org/authors/?q=ai:gao.xiaoyan Summary: This paper analyzes the dynamic behavior of a ratio-dependent predator-prey model with predator-taxis, which the prey can move in the opposite direction of predator gradient. The first purpose is to prove rigorously the global existence and boundedness of the classical solution for the model based on the heat operator semigroup theory and some priori estimates. The another purpose is to analyze the stability of positive equilibrium, which the results will be extended to the case that the derivative of prey's functional response with prey is positive, and it will be found that large predator-taxis can stabilize equilibrium even diffusion-driven instability has occurred. Finally, the numerical simulations present that the pattern formation may arise and predator-taxis is the driving factor for the evolution of spatial inhomogeneity into homogeneity. Oscillations and bifurcation structure of reaction-diffusion model for cell polarity formation https://www.zbmath.org/1485.35045 2022-06-24T15:10:38.853281Z "Kuwamura, Masataka" https://www.zbmath.org/authors/?q=ai:kuwamura.masataka "Izuhara, Hirofumi" https://www.zbmath.org/authors/?q=ai:izuhara.hirofumi "Ei, Shin-ichiro" https://www.zbmath.org/authors/?q=ai:ei.shin-ichiro Summary: We investigate the oscillatory dynamics and bifurcation structure of a reaction-diffusion system with bistable nonlinearity and mass conservation, which was proposed by \textit{M. Otsuji} et al. [A mass conserved reaction-diffusion system captures properties of cell polarity'', PLoS Comput. Biol. 3, No. 6, e108, 15 p. (2007; \url{doi:10.1371/journal.pcbi.0030108})]. The system is a useful model for understanding cell polarity formation. We show that this model exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusion-driven (Turing-like) instability. Moreover, we investigate the effects of extracellular signals on the cell polarity oscillations. Quantitative dynamics of irreversible enzyme reaction-diffusion systems https://www.zbmath.org/1485.35047 2022-06-24T15:10:38.853281Z "Braukhoff, Marcel" https://www.zbmath.org/authors/?q=ai:braukhoff.marcel "Einav, Amit" https://www.zbmath.org/authors/?q=ai:einav.amit "Tang, Bao Quoc" https://www.zbmath.org/authors/?q=ai:tang.bao-quoc Relation between solutions and initial values for double-nonlinear diffusion equation https://www.zbmath.org/1485.35051 2022-06-24T15:10:38.853281Z "Deng, Liwei" https://www.zbmath.org/authors/?q=ai:deng.liwei "Wang, Liangwei" https://www.zbmath.org/authors/?q=ai:wang.liangwei "Li, Min" https://www.zbmath.org/authors/?q=ai:li.min.10|li.min.8|li.min.1|li.min.7|li.min.6|li.min.2|li.min.3|li.min.9|li.min.5|li.min.4|li.min "Yin, Jingxue" https://www.zbmath.org/authors/?q=ai:yin.jingxue Summary: In this paper, we consider the Cauchy problem of the double-nonlinear diffusion equation. We establish the propagation speed estimates and space-time decay estimates for the solutions and study the equivalent relation between the solutions and the initial values. As an application of this relationship, we prove two different asymptotic behaviors for the solutions in the last of this paper. Boundedness and stabilization in the 3D minimal attraction-repulsion chemotaxis model with logistic source https://www.zbmath.org/1485.35060 2022-06-24T15:10:38.853281Z "Ren, Guoqiang" https://www.zbmath.org/authors/?q=ai:ren.guoqiang "Liu, Bin" https://www.zbmath.org/authors/?q=ai:liu.bin.5|liu.bin.4|liu.bin.8|liu.bin.1|liu.bin.2|liu.bin.6|liu.bin|liu.bin.7|liu.bin.3|liu.bin.9 Summary: In this paper, we consider the fully parabolic attraction-repulsion chemotaxis system with logistic source in a three-dimensional bounded domain with smooth boundary. We first derive an explicit formula $$\mu_*=\mu_*(3,d_1,d_2,d_3,\beta_1,\beta_2,\chi ,\xi )$$ for the logistic damping rate $$\mu$$ such that the system has no blowups whenever $$\mu >\mu_*$$. In addition, the asymptotic behavior of the solutions is discussed; we obtain the other explicit formula $$\mu^*=\mu^*(d_1,d_2,d_3,\alpha_1,\alpha_2,\beta_1,\beta_2,\chi ,\xi ,\lambda )$$ for the logistic damping rate so that the convergence rate is expressed explicitly whenever $$\mu >\mu^*$$. Our results generalize and improve partial previously known ones. Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $$\mathbb{R}^N$$. III: Transition fronts https://www.zbmath.org/1485.35061 2022-06-24T15:10:38.853281Z "Salako, Rachidi B." https://www.zbmath.org/authors/?q=ai:salako.rachidi-bolaji "Shen, Wenxian" https://www.zbmath.org/authors/?q=ai:shen.wenxian Summary: The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic-elliptic chemotaxis system with space and time dependent logistic source, $\begin{cases} \partial_tu=\Delta u -\chi \nabla \cdot (u\nabla v)+u(a(x,t)-b(x,t)u),&{}\quad x\in{\mathbb{R}}^N,\\ 0=\Delta v-\lambda v+\mu u ,&{}\quad x\in{\mathbb{R}}^N, \end{cases}\tag{1}$ where $$N\ge 1$$ is a positive integer, $$\chi$$, $$\lambda$$ and $$\mu$$ are positive constants, and the functions $$a(x, t)$$ and $$b(x, t)$$ are positive and bounded. In the first of the series [the authors, Math. Models Methods Appl. Sci. 28, No. 11, 2237--2273 (2018; Zbl 1426.35034)], we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initial data, and the asymptotic spreading for solutions with compactly supported or front like initial data. In the second of the series [the authors, J. Math. Anal. Appl. 464, No. 1, 883--910 (2018; Zbl 1390.35379)], we investigate the existence, uniqueness and stability of strictly positive entire solutions of (1). In particular, in the case of space homogeneous logistic source (i.e. $$a(x,t)\equiv a(t)$$ and $$b(x,t)\equiv b(t))$$, we proved in [Zbl 1390.35379, loc. cit.] that the unique spatially homogeneous strictly positive entire solution $$(u^*(t),v^*(t))$$ of (1) is uniformly and exponentially stable with respect to strictly positive perturbations when $$0<2\chi \mu <\inf_{t\in{\mathbb{R}}}b(t)$$. In the current part of the series, we discuss the existence of transition front solutions of (1) connecting (0, 0) and $$(u^*(t),v^*(t))$$ in the case of space homogeneous logistic source. We show that for every $$\chi >0$$ with $$\chi \mu \big (1+\frac{\sup_{t\in{\mathbb{R}}}a(t)}{\inf_{t\in{\mathbb{R}}}a(t)}\big )<\inf_{t\in{\mathbb{R}}}b(t)$$, there is a positive constant $${c}^*_\chi$$ such that for every $$\underline{c}> {c}^*_{\chi }$$ and every unit vector $$\xi$$, (1) has a transition front solution of the form $$(u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))$$ satisfying that $$C'(t)=\frac{a(t)+\kappa^2}{\kappa }$$ for some positive number $$\kappa , \liminf_{t-s\rightarrow \infty }\frac{C(t)-C(s)}{t-s}=\underline{c}$$, and $\lim_{x\rightarrow -\infty }\sup_{t\in{\mathbb{R}}}|U(x,t)-u^*(t)|=0 \quad \text{and}\quad \lim_{x\rightarrow \infty }\sup_{t\in{\mathbb{R}}}|\frac{U(x,t)}{e^{-\kappa x}}-1|=0.$ Furthermore, we prove that there is no transition front solution $$(u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))$$ of (1) connecting (0, 0) and $$(u^*(t),v^*(t))$$ with least mean speed less than $$2\sqrt{\underline{a}}$$, where $$\underline{a}=\liminf_{t-s\rightarrow \infty }\frac{1}{t-s}\int_s^ta(\tau )d\tau$$. Large-time behavior of solutions to the Cauchy problem for degenerate parabolic system https://www.zbmath.org/1485.35062 2022-06-24T15:10:38.853281Z "Tedeev, Anatoli F." https://www.zbmath.org/authors/?q=ai:tedeev.anatolii-fedorovich Summary: We consider nonnegative solutions to the Cauchy problem for a degenerate parabolic system of the form \begin{aligned} u_t &=\mathrm{div}(v^{\alpha_1} \nabla u^{m_1}, \quad (x,t) \in S_T = \mathbb{R}^N \times (0,T)\\ v_t &=\mathrm{div}(u^{\alpha_2} \nabla v^{m_2}, \quad (x,t) \in S_T\\ u(x,0) &=u_0(x) \geq 0, \quad v(x,0)=v_0(x) \geq 0, \quad x \in \mathbb{R}^N, \quad N \geq 1. \end{aligned} Under the suitable assumptions on the parameters of nonlinearities and initial data, we obtained optimal decay estimates of a solution for a large time. Moreover, the phenomena of extinction in finite time was established. Provided that initial data are compactly supported, we proved the property of finite speed of propagation. Asymptotic behavior for solutions to an oncolytic virotherapy model involving triply haptotactic terms https://www.zbmath.org/1485.35064 2022-06-24T15:10:38.853281Z "Wei, Ya-nan" https://www.zbmath.org/authors/?q=ai:wei.yanan "Wang, Yifu" https://www.zbmath.org/authors/?q=ai:wang.yifu "Li, Jing" https://www.zbmath.org/authors/?q=ai:li.jing.13 Summary: In this paper, based on $$L^p-L^q$$ estimate for the Neumann heat semigroup, we investigate the asymptotic behavior for solutions to an oncolytic virotherapy model given by $\begin{cases} u_t=\Delta u-\xi_u\nabla \cdot \left( u\nabla v \right) -\rho_uuz, &x\in \Omega ,t> 0,\\ w_t=\Delta w-\xi_w\nabla \cdot \left( w\nabla v \right) -\delta_ww+\rho_wuz, &x\in \Omega ,t> 0,\\ v_t=-(\alpha_uu+\alpha_ww)v-\delta_v v, &x\in \Omega ,t> 0,\\ z_t=\Delta z-\xi_z\nabla \cdot \left( z\nabla v \right) -\delta_zz-\rho_zuz+\beta w, &x\in \Omega ,t> 0, \end{cases} \tag{0.1}$ where $$u, w, v$$ and $$z$$ denote the density of uninfected cancer cells, oncolytic viruses infected cancer cells, extracellular matrix and oncolytic virus particles, respectively. It is showed that when suitably regular initial data satisfy a certain small condition, infected cancer cells and virus particle populations will both become extinct asymptotically. The rates of convergence for the chemotaxis-Navier-Stokes equations in a strip domain https://www.zbmath.org/1485.35065 2022-06-24T15:10:38.853281Z "Wu, Jie" https://www.zbmath.org/authors/?q=ai:wu.jie.4|wu.jie.2|wu.jie.3|wu.jie|wu.jie.6|wu.jie.1|wu.jie.5 "Lin, Hongxia" https://www.zbmath.org/authors/?q=ai:lin.hongxia Summary: In this paper, we study the long-time behavior of the chemotaxis-Navier-Stokes system \begin{aligned} &\partial_t n + \boldsymbol{u} \cdot \nabla n = \lambda \Delta n - \nabla \cdot (\chi (c)n \nabla c),\\ &\partial_t c + \boldsymbol{u} \cdot \nabla c = \mu \Delta c - f(c)n,\\ &\partial_t \boldsymbol{u} + \boldsymbol{u} \cdot \nabla \boldsymbol{u} = \zeta \Delta \boldsymbol{u} - n\nabla \phi,\\ &\nabla \cdot \boldsymbol{u} =0, \quad t>0, x \in \Omega \end{aligned} posed in a strip domain $$\Omega := \mathbb{R}^2 \times [0,1] \subset \mathbb{R}^3$$. In [\textit{Y. Peng} and \textit{Z. Xiang}, Math. Models Methods Appl. Sci. 28, No. 5, 869--920 (2018; Zbl 1391.35206)], the authors have established the global existence of strong solutions to this system with non-flux boundary conditions for $$n$$ and $$c$$ and non-slip boundary conditions for $$\boldsymbol{u}$$. Our main purpose is to establish the time-decay rates for such solutions. This will be done by using the anisotropic $$L^p$$ interpolation and the iterative techniques. Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux https://www.zbmath.org/1485.35067 2022-06-24T15:10:38.853281Z "Zheng, Jiashan" https://www.zbmath.org/authors/?q=ai:zheng.jiashan Summary: We consider the spatially 3-D version of the following Keller-Segel-Navier-Stokes system with rotational flux $\begin{cases} n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x, n, c)\nabla c),\quad x\in \Omega,\; t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+n,\quad x\in \Omega,\; t>0,\\ u_t+\kappa (u \cdot \nabla )u+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega,\; t>0,\\ \nabla \cdot u=0,\quad x\in \Omega,\; t>0 \end{cases}\tag{$$\ast$$}$ under no-flux boundary conditions in a bounded domain $$\Omega \subseteq \mathbb{R}^3$$ with smooth boundary, where $$\phi \in W^{2, \infty}(\Omega)$$ and $$\kappa \in \mathbb{R}$$ represent the prescribed gravitational potential and the strength of nonlinear fluid convection, respectively. Here the matrix-valued function $$S(x, n, c)\in C^2(\bar{\Omega}\times [0, \infty)^2; \mathbb{R}^{3\times 3})$$ denotes the rotational effect which satisfies $$|S(x, n, c)|\le C_S(1 + n)^{-\alpha}$$ with some $$C_S > 0$$ and $$\alpha \ge 0$$. Compared with the signal consumption case as in chemotaxis-(Navier-)Stokes system, the quantity $$c$$ of system ($$\ast$$) is no longer a priori bounded by its initial norm in $$L^\infty$$, which means that we have less regularity information on $$c$$. Moreover, the tensor-valued sensitivity functions result in new mathematical difficulties, mainly linked to the fact that a chemotaxis system with such rotational fluxes thereby loses an energy-like structure. In this paper, under an explicit condition on the size of $$C_S$$ relative to $$C_N$$, we can prove that the \textit{weak} solution $$(n, c, u)$$ becomes smooth ultimately, and that it approaches the unique spatially homogeneous steady state $$(\bar{n}_0, \bar{n}_0, 0)$$, where $$\bar{n}_0=\frac{1}{|\Omega|}\int_\Omega n_0$$ and $$C_N$$ is the best Poincaré constant. To the best of our knowledge, there are the first results on asymptotic behavior of the system. Existence and characterization of attractors for a nonlocal reaction-diffusion equation with an energy functional https://www.zbmath.org/1485.35069 2022-06-24T15:10:38.853281Z "Caballero, R." https://www.zbmath.org/authors/?q=ai:caballero.ruben "Marín-Rubio, P." https://www.zbmath.org/authors/?q=ai:marin-rubio.pedro "Valero, José" https://www.zbmath.org/authors/?q=ai:valero.jose Summary: In this paper we study a nonlocal reaction-diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories. Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system https://www.zbmath.org/1485.35072 2022-06-24T15:10:38.853281Z "Chiyo, Yutaro" https://www.zbmath.org/authors/?q=ai:chiyo.yutaro "Yokota, Tomomi" https://www.zbmath.org/authors/?q=ai:yokota.tomomi Summary: This paper deals with the quasilinear attraction-repulsion chemotaxis system $\begin{cases} u_t=\nabla \cdot \big ((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big ) +f(u), \\ 0=\Delta v+\alpha u-\beta v, \\ 0=\Delta w+\gamma u-\delta w \end{cases}$ in a bounded domain $$\Omega \subset{\mathbb{R}}^n (n \in{\mathbb{N}})$$ with smooth boundary $$\partial \Omega$$, where $$m, p, q \in{\mathbb{R}}, \chi , \xi , \alpha , \beta , \gamma , \delta >0$$ are constants, and $$f$$ is a function of logistic type such as $$f(u)=\lambda u-\mu u^{\kappa }$$ with $$\lambda , \mu >0$$ and $$\kappa \ge 1$$, provided that the case $$f(u) \equiv 0$$ is included in the study of boundedness, whereas $$\kappa$$ is sufficiently close to 1 in considering blow-up in the radially symmetric setting. In the case that $$\xi =0$$ and $$f(u) \equiv 0$$, global existence and boundedness have already been proved under the condition $$p<m+\frac{2}{n}$$. Also, in the case that $$m=1, p=q=2$$ and $$f$$ is a function of logistic type, finite-time blow-up has already been established by assuming $$\chi \alpha -\xi \gamma >0$$. This paper classifies boundedness and blow-up into the cases $$p<q$$ and $$p>q$$ without any condition for the sign of $$\chi \alpha -\xi \gamma$$ and the case $$p=q$$ with $$\chi \alpha -\xi \gamma <0$$ or $$\chi \alpha -\xi \gamma >0$$. Blowup time estimates for the heat equation with a nonlocal boundary condition https://www.zbmath.org/1485.35074 2022-06-24T15:10:38.853281Z "Lu, Heqian" https://www.zbmath.org/authors/?q=ai:lu.heqian "Hu, Bei" https://www.zbmath.org/authors/?q=ai:hu.bei "Zhang, Zhengce" https://www.zbmath.org/authors/?q=ai:zhang.zhengce Summary: We study the blowup time for the heat equation $$u_t=\Delta u$$ in a bounded domain $$\Omega \subset{\mathbb{R}}^n(n\geqslant 2)$$ with the nonlocal boundary condition, where the normal derivative $$\partial u/\partial \vec{\eta}=\int \limits_{\Omega }u^p\text{d}z$$ on one part of boundary $$\Gamma_1\subseteq \partial \Omega$$ for some $$p>1$$, while $$\partial u/\partial \vec{\eta}=0$$ on the rest part of the boundary. By constructing suitable auxiliary functions and analyzing the representation formula of $$u$$, we establish the finite time blowup of the solution and get both upper and lower bounds for the blowup time in terms of the parameter $$p$$, the initial value $$u_0(x)$$ and the volume of $$\Gamma_1$$. In many other studies, they require the convexity of the domain $$\Omega$$ and only deal with the case $$\Gamma_1=\partial \Omega$$. In this article, we remove the convexity assumption and consider the problem with $$\Gamma_1\subseteq \partial \Omega$$. Gradient estimates for singular parabolic $$p$$-Laplace type equations with measure data https://www.zbmath.org/1485.35081 2022-06-24T15:10:38.853281Z "Dong, Hongjie" https://www.zbmath.org/authors/?q=ai:dong.hongjie "Zhu, Hanye" https://www.zbmath.org/authors/?q=ai:zhu.hanye Summary: We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic $$p$$-Laplace equation $$u_t-\Delta_p u=\mu$$ with $$p\in (1, 2)$$. The case when $$p\in\big (2-\frac{1}{n+1},2\big)$$ were studied in [\textit{T. Kuusi} and \textit{G. Mingione}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 12, No. 4, 755--822 (2013; Zbl 1288.35145)]. In this paper, we extend the results in Kuusi and Mingione [loc. cit.] to the open case when $$p\in\big(\frac{2n}{n+1},2-\frac{1}{n+1}\big]$$ if $$n\ge 2$$ and $$p\in (\frac{5}{4}, \frac{3}{2}]$$ if $$n=1$$. More specifically, in a more singular range of $$p$$ as above, we establish pointwise gradient estimates via linear parabolic Riesz potential and gradient continuity results via certain assumptions on parabolic Riesz potential. Stability results for backward heat equations with time-dependent coefficient in the Banach space $$L_p (\mathbb{R})$$ https://www.zbmath.org/1485.35082 2022-06-24T15:10:38.853281Z "Duc, Nguyen Van" https://www.zbmath.org/authors/?q=ai:duc.nguyen-van "Muoi, Pham Quy" https://www.zbmath.org/authors/?q=ai:muoi.pham-quy "Anh, Nguyen Thi Van" https://www.zbmath.org/authors/?q=ai:anh.nguyen-thi-van Summary: In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space $$L_p (\mathbb{R}),\; (1 < p < \infty)$$. For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme. Harnack estimate for positive solutions to a nonlinear equation under geometric flow https://www.zbmath.org/1485.35083 2022-06-24T15:10:38.853281Z "Fasihi-Ramandi, Ghodratallah" https://www.zbmath.org/authors/?q=ai:ramandi.ghodratallah-fasihi "Azami, Shahroud" https://www.zbmath.org/authors/?q=ai:azami.shahroud Summary: In the present paper, we obtain gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds $\frac{ \partial u}{ \partial t} = \triangle u + a (x, t) u^p + b (x, t) u^q$ where, $$0 < p, q < 1$$ are real constants and $$a(x, t)$$ and $$b(x, t)$$ are functions which are $$C^2$$ in the $$x$$-variable and $$C^1$$ in the $$t$$-variable. We shall get an interesting Harnack inequality as an application. Carleson estimates for the singular parabolic $$p$$-Laplacian in time-dependent domains https://www.zbmath.org/1485.35084 2022-06-24T15:10:38.853281Z "Gianazza, Ugo" https://www.zbmath.org/authors/?q=ai:gianazza.ugo-pietro Summary: We deal with the parabolic $$p$$-Laplacian in the so-called singular super-critical range $$\frac{2N}{N+1}< p < 2$$, and we prove Carleson estimates for non-negative solutions in suitable non-cylindrical domains $$\Omega\subset\mathbb{R}^{N+1}$$. The sets $$\Omega$$ satisfy a proper NTA condition, tailored on the parabolic $$p$$-Laplacian. As an intermediate step, we show that in these domains non-negative solutions which vanish at the boundary, are Hölder continuous up to the same boundary. Nonlinear parabolic equations with Robin boundary conditions and Hardy-Leray type inequalities https://www.zbmath.org/1485.35085 2022-06-24T15:10:38.853281Z "Goldstein, Gisèle Ruiz" https://www.zbmath.org/authors/?q=ai:ruiz-goldstein.gisele "Goldstein, Jerome A." https://www.zbmath.org/authors/?q=ai:goldstein.jerome-a "Kömbe, Ismail" https://www.zbmath.org/authors/?q=ai:kombe.ismail "Tellioğlu Balekoğlu, Reyhan" https://www.zbmath.org/authors/?q=ai:tellioglu-balekoglu.reyhan Summary: We are primarily concerned with the absence of positive solutions of the following problem, $\begin{cases} \frac{\partial u}{\partial t}=\Delta(u^m)+V(x)u^m+\lambda u^q & \text{ in }\Omega\times (0, T),\\ u(x,0)=u_0(x)\geq 0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial\nu}=\beta(x)u & \text{ on }\partial\Omega\times (0,T), \end{cases}$ where $$0<m<1$$, $$V\in L_{\mathrm{loc}}^1(\Omega)$$, $$\beta\in L_{\mathrm{loc}}^1(\partial\Omega)$$, $$\lambda\in\mathbb{R}$$, $$q>0$$, $$\Omega\subset\mathbb{R}^N$$ is a bounded open subset of $$\mathbb{R}^N$$ with smooth boundary $$\partial\Omega$$, and $$\frac{\partial u}{\partial\nu}$$ is the outer normal derivative of $$u$$ on $$\partial\Omega$$. Moreover, we also present some new sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation of our interest. For the entire collection see [Zbl 07455846]. Estimates on fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic equations with nonnegative potentials and their applications https://www.zbmath.org/1485.35087 2022-06-24T15:10:38.853281Z "Tang, Lin" https://www.zbmath.org/authors/?q=ai:tang.lin "Zhao, Yuan" https://www.zbmath.org/authors/?q=ai:zhao.yuan Summary: We study the fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic operators with nonnegative potentials in the reverse Hölder class. The main aim of the paper is to give pointwise estimates of the heat kernel of the operators above, which improve and generalize the main results by \textit{K. Kurata} [J. Lond. Math. Soc., II. Ser. 62, No. 3, 885--903 (2000; Zbl 1013.35020)]. On a comparison theorem for parabolic equations with nonlinear boundary conditions https://www.zbmath.org/1485.35088 2022-06-24T15:10:38.853281Z "Kita, Kosuke" https://www.zbmath.org/authors/?q=ai:kita.kosuke "Ôtani, Mitsuharu" https://www.zbmath.org/authors/?q=ai:otani.mitsuharu Summary: In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions. Regularity of solutions to Kolmogorov equation with Gilbarg-Serrin matrix https://www.zbmath.org/1485.35095 2022-06-24T15:10:38.853281Z "Kinzebulatov, D." https://www.zbmath.org/authors/?q=ai:kinzebulatov.damir "Semënov, Yu. A." https://www.zbmath.org/authors/?q=ai:semenov.yuliy-a Summary: In $${\mathbb{R}}^d$$, $$d \ge 3$$, consider the divergence and the non-divergence form operators \begin{aligned} & -\Delta - \nabla \cdot (a-I) \cdot \nabla + b \cdot \nabla, \\ & - \Delta - (a-I) \cdot \nabla^2 + b \cdot \nabla, \end{aligned} where the second-order perturbations are given by the matrix $a-I=c|x|^{-2}x \otimes x, \quad c>-1.$ The vector field $$b:{\mathbb{R}}^d \rightarrow{\mathbb{R}}^d$$ is form-bounded with form-bound $$\delta >0$$. (This includes vector fields with entries in $$L^d$$, as well as vector fields having critical-order singularities.) We characterize quantitative dependence on $$c$$ and $$\delta$$ of the $$L^q \rightarrow W^{1,qd/(d-2)}$$ regularity of solutions of the corresponding elliptic and parabolic equations in $$L^q$$, $$q \ge 2 \vee ( d-2)$$. Gaussian bounds of fundamental matrix and maximal $$L^1$$ regularity for Lamé system with rough coefficients https://www.zbmath.org/1485.35096 2022-06-24T15:10:38.853281Z "Xu, Huan" https://www.zbmath.org/authors/?q=ai:xu.huan Summary: The purpose of this paper is twofold. First, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lamé system with only bounded and measurable coefficients. Second, we derive a maximal $$L^1$$ regularity result for the abstract Cauchy problem associated with a composite operator. In a concrete example, we also obtain maximal $$L^1$$ regularity for the Lamé system, from which it follows that the Lipschitz seminorm of the solutions to the Lamé system is globally $$L^1$$-in-time integrable. As an application, we use a Lagrangian approach to prove a global-in-time well-posedness result for a viscous pressureless flow in a perturbation framework, but with possibly discontinuous densities. The method established in this paper might be a useful tool for studying many issues arising from viscous fluids with truly variable densities. Exact results on some nonlinear Laguerre-type diffusion equations https://www.zbmath.org/1485.35098 2022-06-24T15:10:38.853281Z "Garra, Roberto" https://www.zbmath.org/authors/?q=ai:garra.roberto "Tomovski, Zivorad" https://www.zbmath.org/authors/?q=ai:tomovski.zivorad Summary: In this paper we obtain some new explicit results for nonlinear equations involving Laguerre derivatives in space and/or in time. In particular, by using the invariant subspace method, we have many interesting cases admitting exact solutions in terms of Laguerre functions. Nonlinear diffusive-type and telegraph-type equations are considered and also the space and time-fractional counterpart are analyzed, involving Caputo or Prabhakar-type derivatives. The main aim of this paper is to point out that it is possible to construct many new interesting examples of nonlinear diffusive equations with variable coefficients admitting exact solutions in terms of Laguerre and Mittag-Leffler functions. Propagation dynamics of a nonlocal periodic organism model with non-monotone birth rates https://www.zbmath.org/1485.35102 2022-06-24T15:10:38.853281Z "Bai, Zhenguo" https://www.zbmath.org/authors/?q=ai:bai.zhenguo "Zhang, Liang" https://www.zbmath.org/authors/?q=ai:zhang.liang|zhang.liang.3|zhang.liang.2|zhang.liang.1 Summary: This work is concerned with a nonlocal reaction-diffusion system modeling the propagation dynamics of organisms owning mobile and stationary states in periodic environments. We establish the existence of the asymptotic speed of spreading for the model system with monotone birth function via asymptotic propagation theory of monotone semiflow, and then discuss the case for non-monotone birth function by using the squeezing technique. In terms of the truncated problem on a finite interval, we apply the method of super- and sub-solutions and the fixed point theorem combined with regularity estimation and limit arguments to obtain the existence of time periodic traveling waves for the model system without quasi-monotonicity. The non-existence proof is to use the results of the spreading speed. Finally, as an application, we study the spatial dynamics of the model with the birth rate function of Ricker type and numerically demonstrate analytic results. Speed-up of reaction-diffusion fronts by a line of fast diffusion https://www.zbmath.org/1485.35103 2022-06-24T15:10:38.853281Z "Berestycki, Henri" https://www.zbmath.org/authors/?q=ai:berestycki.henri "Coulon, Anne-Charline" https://www.zbmath.org/authors/?q=ai:coulon.anne-charline "Roquejoffre, Jean-Michel" https://www.zbmath.org/authors/?q=ai:roquejoffre.jean-michel "Rossi, Luca" https://www.zbmath.org/authors/?q=ai:rossi.luca Summary: In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction of the line, but also in the plane. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. When the diffusion is the Laplacian, the global asymptotic speed of spreading on the line grows as the square root of the diffusion. In the other directions, the line of strong diffusion influences the spreading up to a critical angle, from which one recovers the classical spreading velocity. When the diffusion is the fractional Laplacian, the spreading on the line is exponential in time, and propagation in the plane is equivalent to that of a one-dimensional infinite planar front parallel to the line. Propagation phenomena for time-space periodic monotone semiflows and applications to cooperative systems in multi-dimensional media https://www.zbmath.org/1485.35106 2022-06-24T15:10:38.853281Z "Du, Li-Jun" https://www.zbmath.org/authors/?q=ai:du.lijun "Li, Wan-Tong" https://www.zbmath.org/authors/?q=ai:li.wan-tong "Shen, Wenxian" https://www.zbmath.org/authors/?q=ai:shen.wenxian Summary: The current paper is concerned with propagation phenomena for time-space periodic monotone semiflows and applications to time-space periodic cooperative systems in multi-dimensional media. We first establish some abstract theory on spreading speeds and traveling waves for time-space periodic monotone semiflows in the space of vector-valued functions on $$\mathbb{R}^N$$. Among others, we prove the equivalence of spreading speeds adopted by two different approaches, several spreading properties in terms of the spreading speeds, and the existence of periodic traveling waves which extends several known results in various special cases. By applying the abstract theory, we study the spreading speeds and traveling waves of time-space periodic cooperative systems in multi-dimensional media. It is proved that such a system admits a single spreading speed (resp. asymptotic spreading ray speed and asymptotic spreading set) under certain conditions. A set of sufficient conditions are also given for the single spreading speed to be linearly determinate. Furthermore, we show that the spreading speed can be characterized as the minimal wave speed of periodic traveling waves. The obtained results are then applied to the two-species periodic competition system in multi-dimensional media. Piecewise linear model of phytoplankton wave propagation in periodical vortex flow https://www.zbmath.org/1485.35108 2022-06-24T15:10:38.853281Z "Miroshnichenko, Taisia" https://www.zbmath.org/authors/?q=ai:miroshnichenko.taisia-p "Gubernov, Vladimir" https://www.zbmath.org/authors/?q=ai:gubernov.vladimir-vladimirovich "Minaev, Sergey" https://www.zbmath.org/authors/?q=ai:minaev.sergey-s "Mislavskii, Vladimir" https://www.zbmath.org/authors/?q=ai:mislavskii.vladimir "Okajima, Junnosuke" https://www.zbmath.org/authors/?q=ai:okajima.junnosuke Propagation dynamics for an age-structured population model in time-space periodic habitat https://www.zbmath.org/1485.35109 2022-06-24T15:10:38.853281Z "Pan, Yingli" https://www.zbmath.org/authors/?q=ai:pan.yingli Summary: How do environmental heterogeneity influence propagation dynamics of the age-structured invasive species? We investigate this problem by considering a yearly generation invasive species in time-space periodic habitat. Starting from an age-structured population growth law, we formulate a reaction-diffusion model with time-space periodic dispersal, mortality and recruitment. Thanks to the fundamental solution for linear part of the model, we reduce to study the dynamics of a time-space periodic semiflow which is defined by the solution map. By the recent developed dynamical theory in [\textit{J. Fang} et al., J. Funct. Anal. 272, No. 10, 4222--4262 (2017; Zbl 1398.35116)], we obtained the spreading speed and its coincidence with the minimal wave speed of time-space periodic traveling waves, as well as the variational characterization of spreading speed in terms of a principal eigenvalue problem. Such results are also proved back to the reaction-diffusion model. Spreading speeds and traveling waves for a time periodic DS-I-A epidemic model https://www.zbmath.org/1485.35110 2022-06-24T15:10:38.853281Z "Yang, Xiying" https://www.zbmath.org/authors/?q=ai:yang.xiying "Lin, Guo" https://www.zbmath.org/authors/?q=ai:lin.guo Summary: This paper is devoted to studying the speed of asymptotic spreading and minimal wave speed of traveling wave solutions for a time periodic and diffusive DS-I-A epidemic model, which describes the propagation threshold of disease spreading. The main feature of this model is the possible deficiency of the classical comparison principle such that many known results do not directly work. The speed of asymptotic spreading is estimated by constructing auxiliary equations and applying the classical theory of asymptotic spreading for Fisher type equation. The minimal wave speed is established by proving the existence and nonexistence of the nonconstant traveling wave solutions. Moreover, some numerical examples are presented to model the propagation dynamics of this system. Global well-posedness for the 2D chemotaxis-fluid system with logistic source https://www.zbmath.org/1485.35118 2022-06-24T15:10:38.853281Z "Lin, Yina" https://www.zbmath.org/authors/?q=ai:lin.yina "Zhang, Qian" https://www.zbmath.org/authors/?q=ai:zhang.qian Summary: In this paper, the two-dimensional incompressible chemotaxis fluid with logical source is studied as following: \begin{aligned} &n_t + u \cdot \nabla n = \Delta n - \nabla \cdot (n \nabla c) + n - n^2,\\ &c_t + u \cdot \nabla c = \Delta c - nc,\\ &u_t + u \cdot \nabla u + \nabla P = \Delta u - n \nabla \phi,\\ &\nabla \cdot u = 0. \end{aligned} By taking advantage of a coupling structure of the equations and using a scale decomposition technique, the global existence and uniqueness of weak solutions to the above system for a large class of initial data is obtained. Global existence and boundedness in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity https://www.zbmath.org/1485.35119 2022-06-24T15:10:38.853281Z "Zheng, Jiashan" https://www.zbmath.org/authors/?q=ai:zheng.jiashan Summary: In this paper, we consider the following chemotaxis-Stokes system with nonlinear diffusion and rotational flux \begin{aligned} \left\{ \begin{array}{l} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),\quad x\in \Omega , t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,\quad x\in \Omega , t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega , t>0,\\ \nabla \cdot u=0,\quad x\in \Omega , t>0\ \end{array} \right. \end{aligned} \tag{CNF} in a bounded domain $$\Omega \subseteq \mathbb{R}^3$$ with smooth boundary, which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. Here the matrix-valued function $$S\in C^2(\bar{\Omega }\times [0,\infty )^2;\mathbb{R}^{3\times 3})$$ fulfills $$|S(x,n,c)| \le S_0(c)$$ for all $$(x,n,c)\in \bar{\Omega } \times [0, \infty )\times [0, \infty )$$ with $$S_0(c)$$ nondecreasing on $$[0,\infty )$$. With developing some new methods (see Sect. 4 and Sect. 5), it is proved that under the condition $$m>\frac{10}{9}$$ and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global weak solution, which is uniformly bounded. In view of $$S$$ is a tensor-valued chemotactic sensitivity, it is easy to see that the restriction on $$m$$ here is optimal (see Remark 3.1) and thus answer the open problem left in [\textit{N. Bellomo} et al., Math. Models Methods Appl. Sci. 25, No. 9, 1663--1763 (2015; Zbl 1326.35397)] and [\textit{Y. Tao} and \textit{M. Winkler}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 1, 157--178 (2013; Zbl 1283.35154)]. This result significantly improves or extends previous results of several authors (see Remark 1.1). Riemann integral operator for stationary and non-stationary processes https://www.zbmath.org/1485.35151 2022-06-24T15:10:38.853281Z "Alexandrovich, I. M." https://www.zbmath.org/authors/?q=ai:alexandrovich.i-m "Lyashko, S. I." https://www.zbmath.org/authors/?q=ai:lyashko.sergei-ivanovich "Sydorov, M. V.-S." https://www.zbmath.org/authors/?q=ai:sydorov.mykola-v-s "Lyashko, N. I." https://www.zbmath.org/authors/?q=ai:lyashko.n-i "Bondar, O. S." https://www.zbmath.org/authors/?q=ai:bondar.o-s Summary: Integral operators based on the Riemann function, which transform arbitrary analytical functions into regular solutions of equations of elliptic, parabolic, and hyperbolic types of second order, are constructed. The Riemann operator method is generalized for the biaxisymmetric Helmholtz equation. A method for finding solutions to the above equations in analytical form is developed. In some cases, formulas for inverting integral representations of solutions are constructed. The conditions for solving the Cauchy problem for the axisymmetric Helmholtz equation are formulated. Heat kernel of supercritical nonlocal operators with unbounded drifts https://www.zbmath.org/1485.35256 2022-06-24T15:10:38.853281Z "Menozzi, Stéphane" https://www.zbmath.org/authors/?q=ai:menozzi.stephane "Zhang, Xicheng" https://www.zbmath.org/authors/?q=ai:zhang.xicheng.1|zhang.xicheng Summary: Let $$\alpha\in (0,2)$$ and $$d\in\mathbb{N}$$. Consider the following stochastic differential equation (SDE) in $$\mathbb{R}^d$$: $\mathrm{d}X_t=b(t,X_t)\,\mathrm{d}t+a(t,X_{t-})\,\mathrm{d} L^{(\alpha )}_t,\quad X_0=x,$ where $$L^{(\alpha)}$$ is a $$d$$-dimensional rotationally invariant $$\alpha$$-stable process, $$b:\mathbb{R}_+\times \mathbb{R}^d\rightarrow\mathbb{R}^d$$ and $$a:\mathbb{R}_+\times \mathbb{R}^d\rightarrow\mathbb{R}^d\otimes\mathbb{R}^d$$ are Hölder continuous functions in space, with respective order $$\beta,\gamma\in (0,1)$$ such that $$(\beta\wedge\gamma)+\alpha>1$$, uniformly in $$t$$. Here $$b$$ may be unbounded. When $$a$$ is bounded and uniformly elliptic, we show that the unique solution $$X_t(x)$$ of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole \textit{supercritical} range $$\alpha\in (0,1)$$. Our proof is based on ad hoc parametrix expansions and probabilistic techniques. On backward uniqueness for parabolic equations when Osgood continuity of the coefficients fails at one point https://www.zbmath.org/1485.35257 2022-06-24T15:10:38.853281Z "Del Santo, Daniele" https://www.zbmath.org/authors/?q=ai:del-santo.daniele "Prizzi, Martino" https://www.zbmath.org/authors/?q=ai:prizzi.martino Summary: We prove the uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for $$t>0$$ but not at $$t=0$$. On a generalized diffusion problem: a complex network approach https://www.zbmath.org/1485.35258 2022-06-24T15:10:38.853281Z "Cantin, Guillaume" https://www.zbmath.org/authors/?q=ai:cantin.guillaume "Thorel, Alexandre" https://www.zbmath.org/authors/?q=ai:thorel.alexandre Summary: In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution \textit{via} operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors. Lie symmetries of the Shigesada-Kawasaki-Teramoto system https://www.zbmath.org/1485.35259 2022-06-24T15:10:38.853281Z "Cherniha, Roman" https://www.zbmath.org/authors/?q=ai:cherniha.roman-m "Davydovych, Vasyl'" https://www.zbmath.org/authors/?q=ai:davydovych.vasyl "Muzyka, Liliia" https://www.zbmath.org/authors/?q=ai:muzyka.liliia Summary: The Shigesada-Kawasaki-Teramoto system, which consists of two reaction-diffusion equations with variable cross-diffusion and quadratic nonlinearities, is considered. The system is the most important case of the biologically motivated model proposed by \textit{N. Shigesada} et al. [Spatial segregation of interacting species'', J. Theor. Biol. 79, No. 1, 83--99 (1979; \url{doi:10.1016/0022-5193(79)90258-3})]. A complete description of Lie symmetries for this system is derived. It is proved that the Shigesada-Kawasaki-Teramoto system admits a wide range of different Lie symmetries depending on coefficient values. In particular, the Lie symmetry operators with highly unusual structure are unveiled and applied for finding exact solutions of the relevant nonlinear system with cross-diffusion. Properties of generalized degenerate parabolic systems https://www.zbmath.org/1485.35260 2022-06-24T15:10:38.853281Z "Kim, Sunghoon" https://www.zbmath.org/authors/?q=ai:kim.sunghoon "Lee, Ki-Ahm" https://www.zbmath.org/authors/?q=ai:lee.ki-ahm Summary: In this article, we consider the parabolic system $(u^i )_t =\nabla \cdot (mU^{m-1} \mathcal{A} (\nabla u^i, u^i, x,t)+ \mathcal{B} (u^i, x,t)), \quad (1\leq i\leq k)$ in the range of exponents $$m>\frac{n-2}{n}$$ where the diffusion coefficient $$U$$ depends on the components of the solution $$\mathbf{u} =(u^1, \ldots, u^k)$$. Under suitable structure conditions on the vector fields $$\mathcal{A}$$ and $$\mathcal{B}$$, we first showed the uniform $$L^{\infty}$$ boundedness of the function $$U$$ for $$t\geq \tau > 0$$. We also proved the law of $$L^1$$ mass conservation and the local continuity of solution $$\mathbf{u}$$. In the last result, all components of the solution $$\mathbf{u}$$ have the same modulus of continuity if the ratio between $$U$$ and $$u^i, (1\leq i\leq k)$$, is uniformly bounded above and below. On the first initial-boundary value problem for a model parabolic system in a domain with curvilinear lateral boundaries https://www.zbmath.org/1485.35261 2022-06-24T15:10:38.853281Z "Fedorov, K. D." https://www.zbmath.org/authors/?q=ai:fedorov.k-d Summary: We consider the first initial-boundary value problem for a second-order Petrovskii parabolic homogeneous system with constant coefficients in a bounded domain $$\Omega$$ on the plane with curvilinear lateral boundaries nonsmooth at $$t=0$$. The existence of a solution of this problem in the class $$C^{2,1}_{x,t}(\overline{\Omega })$$ is proved by the method of boundary integral equations. An initial-boundary value problem for systems of linear partial differential equations with a differential operator of Gegenbauer type https://www.zbmath.org/1485.35262 2022-06-24T15:10:38.853281Z "Gadjiev, Tahir S." https://www.zbmath.org/authors/?q=ai:gadjiev.tahir-s "Guliyev, Vagif S." https://www.zbmath.org/authors/?q=ai:guliyev.vagif-sabir "Ibragimov, Elman J." https://www.zbmath.org/authors/?q=ai:ibragimov.elman-j Summary: This article discusses the initial boundary value problem for the systems of linear parabolic equations. The system is written in a matrix form. Its elements are polynomials with the Gegenbauer operator having the same order. The class of functions is located in which a way that the problem under consideration is correct, i.e., there is only a unique solution that depends on the initial function. Explicit formulas for solving the problem are given, while the method of generalized functions is developed by Gelfand and Shilov. Global existence for a singular phase field system related to a sliding mode control problem https://www.zbmath.org/1485.35263 2022-06-24T15:10:38.853281Z "Colli, Pierluigi" https://www.zbmath.org/authors/?q=ai:colli.pierluigi "Colturato, Michele" https://www.zbmath.org/authors/?q=ai:colturato.michele Summary: In the present contribution we consider a singular phase field system located in a smooth and bounded three-dimensional domain. The entropy balance equation is perturbed by a logarithmic nonlinearity and by the presence of an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. The second equation of the system accounts for the phase dynamics, and it is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. The resulting system is highly nonlinear; the main difficulties lie in the contemporary presence of two nonlinearities, one of which under time derivative, in the entropy balance equation. Consequently, we are able to prove only the existence of solutions. To this aim, we will introduce a backward finite differences scheme and argue on this by proving uniform estimates and passing to the limit on the time step. The weak solutions of a doubly nonlinear parabolic equation related to the $$p(x)$$-Laplacian https://www.zbmath.org/1485.35264 2022-06-24T15:10:38.853281Z "Zhan, Huashui" https://www.zbmath.org/authors/?q=ai:zhan.huashui Summary: A nonlinear degenerate parabolic equation related to the $$p(x)$$-Laplacian ${u_t}= \operatorname{div} \bigl({b(x)} {\bigl\vert{\nabla a(u)} \bigr\vert^{p(x) - 2}}\nabla a(u) \bigr)+\sum_{i=1}^N\frac{\partial b_i(u)}{\partial x_i}+c(x,t) -b_0a(u)$ is considered in this paper, where $$b(x)|_{x\in \varOmega}>0, b(x)|_{x \in \partial \varOmega}=0, a(s)\geq 0$$ is a strictly increasing function with $$a(0)=0$$, $$c(x,t)\geq 0$$ and $$b_0>0$$. If $$\int_{\varOmega}b(x)^{-\frac{1}{p^--1}}\,dx\leq c$$ and $$\vert \sum_{i=1}^Nb_i'(s) \vert \leq c a'(s)$$, then the solutions of the initial-boundary value problem is well-posedness. When $$\int_{\varOmega}b(x)^{-(p(x)-1)} dx<\infty$$, without the boundary value condition, the stability of weak solutions can be proved. On a predator-prey reaction-diffusion model with nonlocal effects https://www.zbmath.org/1485.35265 2022-06-24T15:10:38.853281Z "Han, Bang-Sheng" https://www.zbmath.org/authors/?q=ai:han.bang-sheng "Yang, Ying-Hui" https://www.zbmath.org/authors/?q=ai:yang.yinghui Summary: In this paper, we consider an initial value problem of a predator-prey system with integral term. By establishing comparison principle and constructing monotone sequences, the existence and uniqueness for the solution of that problem is proved. Then we further show the uniform boundedness. Finally, some conditions of Turing bifurcation occurring are given and illustrated by numerical simulations. Global existence in reaction-diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects https://www.zbmath.org/1485.35266 2022-06-24T15:10:38.853281Z "Lankeit, Johannes" https://www.zbmath.org/authors/?q=ai:lankeit.johannes "Winkler, Michael" https://www.zbmath.org/authors/?q=ai:winkler.michael Summary: We introduce a generalized concept of solutions for reaction-diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate. Traveling wave solutions for a neutral reaction-diffusion equation with non-monotone reaction https://www.zbmath.org/1485.35267 2022-06-24T15:10:38.853281Z "Liu, Yubin" https://www.zbmath.org/authors/?q=ai:liu.yubin Summary: In the present paper, we firstly improve the results on traveling wave solution that were established in [the author and \textit{P. Weng}, J. Differ. Equations 258, No. 11, 3688--3741 (2015; Zbl 1345.35127)] for a neutral reaction-diffusion equation with quasi-monotone reaction. Secondly, by constructing two auxiliary equations and using Schauder's Fixed Point Theorem, we further establish the existence and the asymptotic properties of the traveling wave solution for the equation with non-monotone reaction. Two examples are also given as the application of our results. Pullback attractors of nonautonomous discrete $$p$$-Laplacian complex Ginzburg-Landau equations with fast-varying delays https://www.zbmath.org/1485.35268 2022-06-24T15:10:38.853281Z "Pu, Xiaoqin" https://www.zbmath.org/authors/?q=ai:pu.xiaoqin "Wang, Xuemin" https://www.zbmath.org/authors/?q=ai:wang.xuemin "Li, Dingshi" https://www.zbmath.org/authors/?q=ai:li.dingshi Summary: In this paper, we consider a class of nonautonomous discrete $$p$$-Laplacian complex Ginzburg-Landau equations with time-varying delays. We prove the existence and uniqueness of pullback attractor for these equations. The existing results of studying attractors for time-varying delay equations require that the derivative of the delay term should be less than 1 (called slow-varying delay). By using differential inequality technique, our results remove the constraints on the delay derivative. So, we can deal with the equations with fast-varying delays (without any constraints on the delay derivative). Entire solutions for a delayed lattice competitive system https://www.zbmath.org/1485.35269 2022-06-24T15:10:38.853281Z "Yan, Rui" https://www.zbmath.org/authors/?q=ai:yan.rui "Wang, Yang" https://www.zbmath.org/authors/?q=ai:wang.yang "Yao, Meiping" https://www.zbmath.org/authors/?q=ai:yao.meiping Summary: In this paper, we investigate the existence of entire solutions for a delayed lattice competitive system. Here the entire solutions are the solutions that exist for all $$(n,t)\in \mathbb{Z}\times \mathbb{R}$$. In order to prove the existence, we firstly embed the delayed lattice system into the corresponding larger system, of which the traveling front solutions are identical to those of the delayed lattice system. Then based on the comparison theorem and the sup-sub solutions method, we construct entire solutions which behave as two opposite traveling front solutions moving towards each other from both sides of $$x$$-axis and then annihilating. Moreover, our conclusions extend the invading way, which the superior species invade the inferior ones from both sides of $$x$$-axis and then the inferior ones extinct, into the lattice and delay case. Optimal control of semilinear parabolic equations with non-smooth pointwise-integral control constraints in time-space https://www.zbmath.org/1485.35270 2022-06-24T15:10:38.853281Z "Casas, Eduardo" https://www.zbmath.org/authors/?q=ai:casas.eduardo "Kunisch, Karl" https://www.zbmath.org/authors/?q=ai:kunisch.karl Summary: This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form $$\| u(t)\|_{L^1(\varOmega)} \leq \gamma$$ for $$t \in (0,T)$$. This limits the total control that can be applied to the system at any instant of time. The $$L^1$$-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to $$\gamma$$ is investigated on the basis of different second order conditions. Second-order Lagrange multiplier rules in multiobjective optimal control of semilinear parabolic equations https://www.zbmath.org/1485.35271 2022-06-24T15:10:38.853281Z "Dinh, Tuan Nguyen" https://www.zbmath.org/authors/?q=ai:nguyen-dinh.tuan Summary: We consider multiobjective optimal control problems for semilinear parabolic systems subject to pointwise state constraints, integral state-control constraints and pointwise state-control constraints. In addition, the data of the problems need not be twice Fréchet differentiable. Employing the second-order directional derivative (in the sense of Demyanov-Pevnyi) for the involved functions, we establish necessary optimality conditions, via second-order Lagrange multiplier rules of Fritz-John type, for local weak Pareto solutions of the problems. An existence result for a strongly nonlinear parabolic equations with variable nonlinearity https://www.zbmath.org/1485.35272 2022-06-24T15:10:38.853281Z "Ait Hammou, Mustapha" https://www.zbmath.org/authors/?q=ai:ait-hammou.mustapha "Azroul, Elhoussine" https://www.zbmath.org/authors/?q=ai:azroul.elhoussine "Lahmi, Badr" https://www.zbmath.org/authors/?q=ai:lahmi.badr Summary: We prove the existence of a solution for the strongly nonlinear parabolic initial boundary value problem associated to the equation $ut- \text{ div } a(x, t, \nabla u) + g(x, t, u, \nabla u) = f,$ where the vector field $$a(x, t, \xi)$$ exhibits non-standard growth conditions. Evolution equations with nonlocal initial conditions and superlinear growth https://www.zbmath.org/1485.35273 2022-06-24T15:10:38.853281Z "Benedetti, Irene" https://www.zbmath.org/authors/?q=ai:benedetti.irene "Ciani, Simone" https://www.zbmath.org/authors/?q=ai:ciani.simone Summary: We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case, we are able to show the existence of at least one periodic solution on the half line. Strong traces to degenerate parabolic equations https://www.zbmath.org/1485.35274 2022-06-24T15:10:38.853281Z "Erceg, Marko" https://www.zbmath.org/authors/?q=ai:erceg.marko "Mitrović, Darko" https://www.zbmath.org/authors/?q=ai:mitrovic.darko The null controllability for a singular heat equation with variable coefficients https://www.zbmath.org/1485.35275 2022-06-24T15:10:38.853281Z "Qin, Xue" https://www.zbmath.org/authors/?q=ai:qin.xue "Li, Shumin" https://www.zbmath.org/authors/?q=ai:li.shumin Summary: The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential: $$\partial_t u(x,t) - \mathrm{div}(p(x) \nabla u (x,t)) - (\mu / |x|^2)u(x,t) = f(x,t)$$. Here $$\mu$$ is a real constant. It was proved in the paper of Goldstein and Zhang (8) that the equation is well-posedness when $$0 \leq \mu \leq p_1(n-2)^2 / 4$$, and in this paper, we mainly consider the case $$0 \leq \mu < (p_1^2 / p_2) (n-2)^2 / 4$$, where $$p_1, p_2$$ are two positive constants which satisfy: $$0 < p_1 \leq p(x) \leq p_2$$, $$\forall \; x \in \Omega$$. We extend the specific Carleman estimates in the paper of Ervedoza [Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun Partial Differ Equ. 2008;33:1996--2019] and Vancostenoble [Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287--1317] to the system. We obtain that we can control the equation from any non-empty open subset as for the heat equation. Moreover, we will study the case $$\mu > p_2(n-2)^2 /4$$. We consider a sequence of regularized potentials $$\mu / (|x|^2 + \epsilon^2)$$ and prove that we cannot stabilize the corresponding systems uniformly with respect to $$\epsilon > 0$$. A matrix Harnack estimate for a Kolmogorov type equation https://www.zbmath.org/1485.35276 2022-06-24T15:10:38.853281Z "Jiang, Feida" https://www.zbmath.org/authors/?q=ai:jiang.feida "Shen, Xinyi" https://www.zbmath.org/authors/?q=ai:shen.xinyi The authors study the properties of the solutions $$f$$ of equations of Kolmogorov type that are generalizations of the ultra-parabolic equation $$f_t=f_{xx}-xf_y$$ to more dimensions. More precisely: $f_t=\sum_{i=1}^n\alpha_if_{x_ix_i}+\sum_{i=1}^k\beta_ix_i\left(\sum_{j=1}^{m_i} f_{x_{i_j}}\right) \quad \text{on} \; \mathbb R^N\times (0,T),$ where $$\alpha_i>0$$ and $$\beta_i\neq 0$$; $$1 \le k \le n$$, $$\sum_{i=1}^km_i=N-n$$ and $$i_j=n+m_1+\dots +m_{i-1}+j$$. They obtain a matrix Harnack inequality via nontrivial technical calculations. From that, by tracing or integrating on suitable paths, they deduce a number of scalar Harnack inequalities. For the entire collection see [Zbl 1455.58001]. Reviewer: Antonio Vitolo (Fisciano) Staffans-Weiss perturbations for maximal $$L^p$$-regularity in Banach spaces https://www.zbmath.org/1485.35277 2022-06-24T15:10:38.853281Z "Amansag, Ahmed" https://www.zbmath.org/authors/?q=ai:amansag.ahmed "Bounit, Hamid" https://www.zbmath.org/authors/?q=ai:bounit.hamid "Driouich, Abderrahim" https://www.zbmath.org/authors/?q=ai:driouich.abderrahim "Hadd, Said" https://www.zbmath.org/authors/?q=ai:hadd.said Summary: In this paper, we show that the concept of maximal $$L^p$$-regularity is stable under a large class of unbounded perturbations, namely Staffans-Weiss perturbations. To that purpose, we first prove that the analyticity of semigroups is preserved under this class of perturbations, which is a necessary condition for the maximal regularity. In UMD spaces, $${\mathcal{R}}$$-boundedness is exploited to give conditions guaranteeing the maximal regularity. For Banach spaces, a condition is imposed to prove maximal regularity. Moreover, we apply the obtained results to perturbed boundary value problems. Strong solutions for semilinear problems with almost sectorial operators https://www.zbmath.org/1485.35278 2022-06-24T15:10:38.853281Z "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.karina Summary: In this paper, we study a semilinear parabolic problem $u_t +A u = f(t,u), \;\, t>\tau; \quad u (\tau) = u_0 \in X,$ in a Banach space $$X$$, where $$A:D(A) \subset X \rightarrow X$$ is an almost sectorial operator. This problem is locally well-posed in the sense of mild solutions. By exploring properties of the semigroup of growth $$1-\alpha$$ generated by $$-A$$, we prove that the local mild solution is actually strong solution for the equation. This is done without requiring any extra regularity for the initial condition $$u_0\in X$$ and under suitable assumptions on the nonlinearity $$f$$. We apply the results for a reaction-diffusion equation in a domain with handle where the nonlinearity $$f$$ satisfies a polynomial growth $|f(t,u)-f(t,v)| \le C|u-v| (1+|u|^{\rho -1}+|v|^{\rho -1}),$ and we establish values of $$\rho$$ for which the problem still have strong solution. $$\Gamma$$-stability of maximal monotone processes https://www.zbmath.org/1485.35279 2022-06-24T15:10:38.853281Z "Visintin, Augusto" https://www.zbmath.org/authors/?q=ai:visintin.augusto Summary: The flow of noncyclic maximal monotone operators is formulated variationally as \textit{null-minimization} problems, via results of Brezis, Ekeland, Nayroles and Fitzpatrick. By means of De Giorgi's theory of $$\Gamma$$-convergence, the compactness and the stability of these flows are derived under nonparametric perturbations of the operator. These results are here reviewed, and are applied to quasilinear equations arising in electromagnetism. Decay estimate and non-extinction of solutions of $$p$$-Laplacian nonlocal heat equations https://www.zbmath.org/1485.35280 2022-06-24T15:10:38.853281Z "Toualbia, Sarra" https://www.zbmath.org/authors/?q=ai:toualbia.sarra "Zaraï, Abderrahmane" https://www.zbmath.org/authors/?q=ai:zarai.abderrahmane "Boulaaras, Salah" https://www.zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud Summary: The main goal of this work is to study the initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity in a bounded domain. By using the logarithmic Sobolev inequality and potential wells method, we obtain the decay, blow-up and non-extinction of solutions under some conditions, and the results extend the results of a recent paper [\textit{L. Yan} and \textit{Z. Yang}, Bound. Value Probl. 2018, Paper No. 121, 11 p. (2018; Zbl 07509577)]. Existence of solutions to the Poisson-Nernst-Planck system with singular permanent charges in $$\mathbb{R}^2$$ https://www.zbmath.org/1485.35353 2022-06-24T15:10:38.853281Z "Hsieh, Chia-Yu" https://www.zbmath.org/authors/?q=ai:hsieh.chia-yu "Yu, Yong" https://www.zbmath.org/authors/?q=ai:yu.yong The drift-diffusion system of Nernst-Planck type for two species of charge carriers is studied in the presence of fixed point charges in two-dimensional domains. The authors study solvability and well-posedness of the initial-boundary value problem (with no-flux boundary conditions and the Dirichlet condition for the electric potential), under suitable smallness assumptions on each singular charge. They apply a weighted $$L^2$$-spaces framework to deal with singularities in the equations, a completely different approach than in [\textit{P. Biler} and \textit{T. Nadzieja}, Math. Methods Appl. Sci. 20, No. 9, 767--782 (1997; Zbl 0885.35051)]. Reviewer: Piotr Biler (Wrocław) Collective dynamics in science and society https://www.zbmath.org/1485.35357 2022-06-24T15:10:38.853281Z "Bellomo, N." https://www.zbmath.org/authors/?q=ai:bellomo.nicola "Brezzi, F." https://www.zbmath.org/authors/?q=ai:brezzi.franco This is an editorial statement by the two authors. It presents articles published in a special issue devoted to active particle methods applied to modeling, qualitative analysis, and simulation of the collective dynamics of large systems of interacting living entities in science and society. Reviewer: Adrian Muntean (Karlstad) Local vs nonlocal models for mitochondria swelling https://www.zbmath.org/1485.35358 2022-06-24T15:10:38.853281Z "Efendiev, Messoud A." https://www.zbmath.org/authors/?q=ai:efendiev.messoud "Muradova, Antiga" https://www.zbmath.org/authors/?q=ai:muradova.antiga "Muradov, Nijat" https://www.zbmath.org/authors/?q=ai:muradov.nijat "Zischka, Hans" https://www.zbmath.org/authors/?q=ai:zischka.hans Summary: In this paper, we consider deterministic, continuous, nonlocal models for the mitochondrial permeability transition, i.e. mitochondrial swelling. Based on seminal papers [the first author et al., SIAM J. Math. Anal. 52, No. 1, 543--569 (2020; Zbl 1429.35139); Discrete Contin. Dyn. Syst. 37, No. 7, 4131--4158 (2017; Zbl 1360.35024); Math. Methods Appl. Sci. 41, No. 5, 2162--2177 (2018; Zbl 1387.35335)], \textit{S. Eisenhofer} et al. [Discrete Contin. Dyn. Syst., Ser. B 20, No. 4, 1031--1057 (2015; Zbl 1307.35045)] and the book [the first author, Mathematical modeling of mitochondrial swelling. Cham: Springer (2018; Zbl 1406.92003)], in which ODE-PDE and PDE-PDE local models for the swelling of mitochondria have been considered, we suggest here new nonlocal models for this process. This new nonlocal deterministic continuous model for mitochondrial swelling scenario contains nonlocal diffusion, nonlocal chemotaxis, as well as nonlocal source term. We would like to especially emphasize that some of the new nonlocal models that we consider in this paper do not have local counterparts in the literature. Ill-posedness issue for a multidimensional hyperbolic-parabolic model of chemotaxis in critical Besov spaces $$\dot{B}_{2 d , 1}^{- \frac{ 3}{ 2}} \times ( \dot{B}_{2 d , 1}^{- \frac{ 1}{ 2}} )^d$$ https://www.zbmath.org/1485.35359 2022-06-24T15:10:38.853281Z "Nie, Yao" https://www.zbmath.org/authors/?q=ai:nie.yao "Yuan, Jia" https://www.zbmath.org/authors/?q=ai:yuan.jia The authors continue their studies on ill-posedness of the parabolic-hyperbolic system of chemotaxis $u_t-\Delta u=\nabla\cdot(uv),$ $v_t-\nabla v=0,$ begun in [\textit{Y. Nie} and \textit{J. Yuan}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 196, Article ID 111782, 17 p. (2020; Zbl 1442.35006)]. They prove that there exists a sequence of initial data $$(u_{0,N},v_{0,N})$$ converging to zero in the homogeneous Besov space $$\dot B^{-3/2}_{2d,1}(\mathbb R^d)\times \dot B^{-1/2}_{2d,1}(\mathbb R^d)$$ while the norms of solutions $$u_N(t_N)$$ at some moments $$t_N\to 0$$ are unbounded. Reviewer: Piotr Biler (Wrocław) Two flow approaches to the Loewner-Nirenberg problem on manifolds https://www.zbmath.org/1485.35362 2022-06-24T15:10:38.853281Z "Li, Gang" https://www.zbmath.org/authors/?q=ai:li.gang.11|li.gang.1|li.gang.6|li.gang.4|li.gang.8|li.gang.10|li.gang.2|li.gang.9 The flow approach is applied to study the generalized Loewner-Nirenberg problem on a Riemannian manifold $$(M,g)$$ with boundary: \begin{align*} \begin{cases} \frac{4(n-1)}{n-2}\Delta u - R_g u - n(n-1) u^{\frac{n+2}{n-2}} = 0, & \text{in}\ \mathring{M}, \\ u(p) \to \infty, & \text{as}\ p\to \partial M. \end{cases} \end{align*} The first main result is Theorem 1.1. The author stated that the Cauchy-Dirichlet problem \begin{align*} \begin{cases} u_t = \frac{4(n-1)}{n-2}\Delta u - R_g u - n(n-1) u^{\frac{n+2}{n-2}} & \text{in}\ \mathring{M}, \\ u(p,0) u_0(p), & p\to \partial M, \\ u(q,t)=\phi(q,t), & q\in \partial M \end{cases} \end{align*} has a solution $$u(t,p)$$ under certain mild technical assumptions. Moreover, as $$t\to\infty$$, the solution $$u(t,p)$$ converges to a limit $$u_{\infty}$$ which solves the above Loewner-Nirenberg problem. In the second main result, i.e., Theorem 1.2, the author used limiting solutions of the Cauchy-Dirichlet problem regarding the Yamabe flow to solve the Loewner-Nirenberg problem. Reviewer: Ruobing Zhang (Stony Brook) Controllability of impulsive neutral stochastic integro-differential systems driven by fractional Brownian motion with delay and Poisson jumps https://www.zbmath.org/1485.35370 2022-06-24T15:10:38.853281Z "Youssef, Benkabdi" https://www.zbmath.org/authors/?q=ai:youssef.benkabdi "El Hassan, Lakhel" https://www.zbmath.org/authors/?q=ai:el-hassan.lakhel Summary: In this paper the controllability of a class of impulsive neutral stochastic integro-differential systems driven by fractional Brownian motion and Poisson process in a separable Hilbert space with infinite delay is studied. The controllability result is obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result. Analytic solutions of fractal and fractional time derivative-Burgers-Nagumo equation https://www.zbmath.org/1485.35371 2022-06-24T15:10:38.853281Z "Abdel-Gawad, H. I." https://www.zbmath.org/authors/?q=ai:abdel-gawad.hamdy-ibrahim "Tantawy, M." https://www.zbmath.org/authors/?q=ai:tantawy.mohammad "Abdel-Aziz, B." https://www.zbmath.org/authors/?q=ai:abdelaziz.batoul|abdelaziz.benallegue "Bekir, Ahmet" https://www.zbmath.org/authors/?q=ai:bekir.ahmet Summary: The Nagumo equation describes a reaction-diffusion system in biology. Here, it is coupled to Burgers equation, via including convection, which is the Burgers-Nagumo equation (BNE). The first objective of this work is to present a theorem to reduce, approximately, the different versions of the fractional time derivatives (FTD) to an ordinary derivative (OD) with time dependent coefficients (non autonomous). The second objective is to find the exact solutions of the fractal and FTD-BNE is reduced to BNE with time dependent coefficient. Further similarity transformations are introduced. The unified and extended unified method are used to find the exact traveling waves solutions. Also, self-similar solutions are obtained. The novelties in this work are (i) reducing, via an analytic approximation, the different versions of FTD to non autonomous OD. (ii) Traveling and self-similar waves solutions of the FTD-BNE are derived. (iii) The effect of the order of fractional and fractal derivatives, on waves structure, are investigated. It is found that significant fractal effects hold for smaller order derivatives. While significant fractional effects hold for higher-order derivatives. It is found that, the solutions obtained show solitary wave, wrinkle soliton, solitons with double kinks or with spikes and undulated wave. Further It is shown that wrinkle soliton, with double kink configuration holds for smaller fractal order. While in the case of fractional derivative, this holds for higher orders. We mention that the results found here are completely new. Symbolic computations are carried by using Mathematica. Novel investigation of fractional-order Cauchy-reaction diffusion equation involving Caputo-Fabrizio operator https://www.zbmath.org/1485.35372 2022-06-24T15:10:38.853281Z "Alesemi, Meshari" https://www.zbmath.org/authors/?q=ai:alesemi.meshari "Iqbal, Naveed" https://www.zbmath.org/authors/?q=ai:iqbal.naveed-h "Abdo, Mohammed S." https://www.zbmath.org/authors/?q=ai:abdo.mohammed-salem Summary: In this article, the new iterative transform technique and homotopy perturbation transform method are applied to calculate the fractional-order Cauchy-reaction diffusion equation solution. Yang transformation is mixed with the new iteration method and homotopy perturbation method in these methods. The fractional derivative is considered in the sense of Caputo-Fabrizio operator. The convection-diffusion models arise in physical phenomena in which energy, particles, or other physical properties are transferred within a physical process via two processes: diffusion and convection. Four problems are evaluated to demonstrate, show, and verify the present methods' efficiency. The analytically obtained results by the present method suggest that the method is accurate and simple to implement. Direct and inverse Cauchy problems for generalized space-time fractional differential equations https://www.zbmath.org/1485.35404 2022-06-24T15:10:38.853281Z "Restrepo, Joel E." https://www.zbmath.org/authors/?q=ai:restrepo.joel-esteban "Suragan, Durvudkhan" https://www.zbmath.org/authors/?q=ai:suragan.durvudkhan Summary: In this paper, explicit solutions of a class of generalized space-time fractional Cauchy problems with time-variable coefficients are given. The representation of a solution involves kernels given by convergent infinite series of fractional integro-differential operators, which can be extensively and efficiently applied for analytic and computational goals. Time-fractional operators of complex orders with respect to a given function are used. Further, we study inverse Cauchy problems of finding time dependent coefficients for fractional wave and heat type equations, which involve the explicit representation of the solution of the direct Cauchy problem and a recent method to recover variable coefficients for the considered inverse problems. Concrete examples and particular cases of the obtained results are discussed. The fractional porous medium equation on manifolds with conical singularities. I https://www.zbmath.org/1485.35405 2022-06-24T15:10:38.853281Z "Roidos, Nikolaos" https://www.zbmath.org/authors/?q=ai:roidos.nikolaos "Shao, Yuanzhen" https://www.zbmath.org/authors/?q=ai:shao.yuanzhen Summary: This is the first of a series of two papers which studies the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. In this article, we show $$R$$-sectoriality for the fractional powers of possibly non-invertible $$R$$-sectorial operators. Applications concern existence, uniqueness and maximal $$L^q$$-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided, and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator. Mild solutions to a time-fractional Cauchy problem with nonlocal nonlinearity in Besov spaces https://www.zbmath.org/1485.35408 2022-06-24T15:10:38.853281Z "Tuan, Nguyen Huy" https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan. "Au, Vo Van" https://www.zbmath.org/authors/?q=ai:au.vo-van "Nguyen, Anh Tuan" https://www.zbmath.org/authors/?q=ai:nguyen.anh-tuan Summary: In this paper, we aim to study a time-fractional Cauchy problem for a heat equation with a nonlocal nonlinearity driven by simulation problems arising in populations, and biological mathematics. Using the Banach fixed-point argument, we investigate the existence and uniqueness of mild solutions in Besov spaces defined on an open subset of $$\mathbb{R}^N$$. The key tools of our method are some linear estimates of the heat semigroup generated by the Dirichlet Laplacian and techniques of the M-Wright function. Some embeddings are also used for our proofs. Identification of source term for the ill-posed Rayleigh-Stokes problem by Tikhonov regularization method https://www.zbmath.org/1485.35416 2022-06-24T15:10:38.853281Z "Binh, Tran Thanh" https://www.zbmath.org/authors/?q=ai:binh.tran-thanh "Nashine, Hemant Kumar" https://www.zbmath.org/authors/?q=ai:nashine.hemant-kumar "Long, Le Dinh" https://www.zbmath.org/authors/?q=ai:long.le-dinh "Luc, Nguyen Hoang" https://www.zbmath.org/authors/?q=ai:luc.nguyen-hoang "Nguyen, Can" https://www.zbmath.org/authors/?q=ai:nguyen.can-huu Summary: In this paper, we study an inverse source problem for the Rayleigh-Stokes problem for a generalized second-grade fluid with a fractional derivative model. The problem is severely ill-posed in the sense of Hadamard. To regularize the unstable solution, we apply the Tikhonov method regularization solution and obtain an a priori error estimate between the exact solution and regularized solutions. We also propose methods for both a priori and a posteriori parameter choice rules. In addition, we verify the proposed regularized methods by numerical experiments to estimate the errors between the regularized and exact solutions. On an inverse problem for a parabolic equation in a domain with moving boundaries https://www.zbmath.org/1485.35417 2022-06-24T15:10:38.853281Z "Akhundov, Adalat Ya." https://www.zbmath.org/authors/?q=ai:akhundov.adalat-ya "Habibova, Arasta Sh." https://www.zbmath.org/authors/?q=ai:habibova.arasta-sh Summary: This paper considers the inverse problem of determining the unknown coefficient on the right-hand side of a parabolic equation in a domain with moving boundaries. An additional condition for finding the unknown coefficient, which depends on the variable time, is given in integral form. A theorem on uniqueness and conditional'' stability of the solution is proved. Inverse problem for a Cahn-Hilliard type system modeling tumor growth https://www.zbmath.org/1485.35426 2022-06-24T15:10:38.853281Z "Sakthivel, K." https://www.zbmath.org/authors/?q=ai:sakthivel.kumarasamy "Arivazhagan, A." https://www.zbmath.org/authors/?q=ai:arivazhagan.anbu "Barani Balan, N." https://www.zbmath.org/authors/?q=ai:baranibalan.n|barani-balan.natesan Summary: In this paper, we address an inverse problem of reconstructing a space-dependent semilinear coefficient in the tumor growth model described by a system of semilinear partial differential equations (PDEs) with Dirichlet boundary condition using boundary-type measurement. We establish a new higher order weighted Carleman estimate for the given system with the help of Dirichlet boundary conditions. By deriving a suitable regularity of solutions for this nonlinear system of PDEs and the new Carleman estimate, we prove Lipschitz-type stability for the tumor growth model. Simultaneous identification of initial field and spatial heat source for heat conduction process by optimizations https://www.zbmath.org/1485.35429 2022-06-24T15:10:38.853281Z "Wang, Bingxian" https://www.zbmath.org/authors/?q=ai:wang.bingxian "Yang, Bin" https://www.zbmath.org/authors/?q=ai:yang.bin "Xu, Mei" https://www.zbmath.org/authors/?q=ai:xu.mei Summary: Consider the simultaneous identification of the initial field and spatial heat source for heat conduction process from extra measurements with the two additional measurement data at different times. The uniqueness and conditional stability for this inverse problem are established by using the properties of a parabolic equation and the representation of solution after reforming the equation. By combining the least squares method with the regularization technique, the inverse problem is transformed into an optimal control problem. Based on the existence and uniqueness of the minimizer of the cost functional, an alternative iteration process is built to solve this optimizing problem by the variational adjoint method. The negative gradient direction is selected as the first search direction. For further iterations, the alternative iteration algorithm is used for the initial field and heat source identification. The efficiency of the proposed scheme is tested by the numerical simulation experiments. Classification of the spreading behaviors of a two-species diffusion-competition system with free boundaries https://www.zbmath.org/1485.35432 2022-06-24T15:10:38.853281Z "Du, Yihong" https://www.zbmath.org/authors/?q=ai:du.yihong "Wu, Chang-Hong" https://www.zbmath.org/authors/?q=ai:wu.changhong Summary: In this paper, we revisit the spreading behavior of two invasive species modelled by a diffusion-competition system with two free boundaries in a radially symmetric setting, where the reaction terms depict a weak-strong competition scenario. Our previous work [the authors, ibid. 57, No. 2, Paper No. 52, 36 p. (2018; Zbl 1396.35028)] proves that from certain initial states, the two species develop into a chase-and-run coexistence'' state, namely the front of the weak species $$v$$ propagates at a fast speed and that of the strong species $$u$$ propagates at a slow speed, with their population masses largely segregated. Subsequent numerical simulations in [\textit{K. Khan} et al., J. Math. Biol. 83, No. 3, Paper No. 23, 43 p. (2021; Zbl 1477.35285)] suggest that for all possible initial states, only four different types of long-time dynamical behaviours can be observed: (1) chase-and-run coexistence, (2) vanishing of $$u$$ with $$v$$ spreading successfully, (3) vanishing of $$v$$ with $$u$$ spreading successfully, and (4) vanishing of both species. In this paper, we rigorously prove that, as the initial states vary, there are exactly five types of long-time dynamical behaviors: apart from the four mentioned above, there exists a fifth case, where both species spread successfully and their spreading fronts are kept within a finite distance to each other all the time. We conjecture that this new case can happen only when a parameter takes an exceptional value, which is why it has eluded the numerical observations of Khan et al. [loc. cit.]. Projection method for droplet dynamics on groove-textured surface with merging and splitting https://www.zbmath.org/1485.35433 2022-06-24T15:10:38.853281Z "Gao, Yuan" https://www.zbmath.org/authors/?q=ai:gao.yuan "Liu, Jian-Guo" https://www.zbmath.org/authors/?q=ai:liu.jian-guo A free boundary problem of some modified Leslie-gower predator-prey model with nonlocal diffusion term https://www.zbmath.org/1485.35435 2022-06-24T15:10:38.853281Z "Niu, Shiwen" https://www.zbmath.org/authors/?q=ai:niu.shiwen "Cheng, Hongmei" https://www.zbmath.org/authors/?q=ai:cheng.hongmei "Yuan, Rong" https://www.zbmath.org/authors/?q=ai:yuan.rong Summary: This paper is mainly considered a Leslie-Gower predator-prey model with nonlocal diffusion term and a free boundary condition. The model describes the evolution of the two species when they initially occupy the bounded region $$[0,h_0]$$. We first show that the problem has a unique solution defined for all $$t>0$$. Then, we establish the long-time dynamical behavior, including Spreading-vanishing dichotomy and Spreading-vanishing criteria. The dynamics of partially degenerate nonlocal diffusion systems with free boundaries https://www.zbmath.org/1485.35436 2022-06-24T15:10:38.853281Z "Zhang, Heting" https://www.zbmath.org/authors/?q=ai:zhang.heting "Li, Lei" https://www.zbmath.org/authors/?q=ai:li.lei.4|li.lei.3|li.lei.6|li.lei.5|li.lei.1|li.lei|li.lei.7|li.lei.2 "Wang, Mingxin" https://www.zbmath.org/authors/?q=ai:wang.mingxin|wang.mingxin.1 Summary: We consider a class of partially degenerate nonlocal diffusion systems with free boundaries. Such problems can describe the evolution of one species with nonlocal diffusion and the other without diffusion or with much slower diffusion. The existence, uniqueness, and regularity of global solutions are first proven. The criteria of spreading and vanishing are also established for the Lotka-Volterra type competition and prey-predator growth terms. Moreover, we investigate long-time behaviors of the solution and propose estimates of spreading speeds when spreading occurs. Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain https://www.zbmath.org/1485.35438 2022-06-24T15:10:38.853281Z "Allwright, Jane" https://www.zbmath.org/authors/?q=ai:allwright.jane Summary: A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a critical' case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius. Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations https://www.zbmath.org/1485.35441 2022-06-24T15:10:38.853281Z "Hernández-Santamaría, Víctor" https://www.zbmath.org/authors/?q=ai:hernandez-santamaria.victor "Peralta, Liliana" https://www.zbmath.org/authors/?q=ai:peralta.liliana Summary: In this paper, we study some controllability and observability problems for stochastic systems coupling fourth- and second-order parabolic equations. The main goal is to control both equations with only one controller localized on the drift of the fourth-order equation. We analyze two cases: on the one hand, we study the controllability of a linear backward system where the couplings are made only through first-order terms. The key point is to use suitable Carleman estimates for the heat equation and the fourth-order operator with the same weight to deduce an observability inequality for the adjoint system. On the other hand, we study the controllability of a simplified nonlinear coupled model of forward equations. This case, which is well known to be harder to solve, follows a methodology that has been introduced recently and relies on an adaptation of the well-known source term method in the stochastic setting together with a truncation procedure. This approach gives a new concept of controllability for stochastic systems. Removable singularities for Lipschitz caloric functions in time varying domains https://www.zbmath.org/1485.42030 2022-06-24T15:10:38.853281Z "Mateu, Joan" https://www.zbmath.org/authors/?q=ai:mateu.joan "Prat, Laura" https://www.zbmath.org/authors/?q=ai:prat.laura "Tolsa, Xavier" https://www.zbmath.org/authors/?q=ai:tolsa.xavier Summary: In this paper we study removable singularities for regular $$(1,1/2)$$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $$L^2$$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation. Correction to: A weak reverse Hölder inequality for caloric measure'' https://www.zbmath.org/1485.42035 2022-06-24T15:10:38.853281Z "Genschaw, Alyssa" https://www.zbmath.org/authors/?q=ai:genschaw.alyssa "Hofmann, Steve" https://www.zbmath.org/authors/?q=ai:hofmann.steve From the text: In our paper [ibid. 30, No. 2, 1530--1564 (2020; Zbl 1436.42028)], in which we presented a parabolic version of results of [\textit{B. Bennewitz} and \textit{J. L. Lewis}, Complex Variables, Theory Appl. 49, No. 7--9, 571--582 (2004; Zbl 1068.31001)], the proof of Lemma 2.7 (the main lemma) was divided into several cases, one of which we had inadvertently failed to note and treat. We now rectify this omission. Fortunately, this will be a simple matter. On the equivalence of certain seminorms on some weighted Hölder spaces https://www.zbmath.org/1485.46025 2022-06-24T15:10:38.853281Z "Degtyarev, S. P." https://www.zbmath.org/authors/?q=ai:degtyarev.sergei-p Summary: The present paper is devoted to studying of some weighted Hölder spaces. These spaces are designed in the way to serve as a framework for studying different statements for the thin film equations in weighted classes of smooth functions in the multidimensional setting. These spaces can serve also for considering of other equations with the degeneration on the boundary of the domain of definition. We prove the equivalence of certain metrics on these spaces. Optimal control for a phase field model of melting arising from inductive heating https://www.zbmath.org/1485.49012 2022-06-24T15:10:38.853281Z "Xiong, Zonghong" https://www.zbmath.org/authors/?q=ai:xiong.zonghong "Wei, Wei" https://www.zbmath.org/authors/?q=ai:wei.wei.6|wei.wei.2|wei.wei.3|wei.wei.7|wei.wei.5|wei.wei.4 "Zhou, Ying" https://www.zbmath.org/authors/?q=ai:zhou.ying "Wang, Yue" https://www.zbmath.org/authors/?q=ai:wang.yue.6 "Liao, Yumei" https://www.zbmath.org/authors/?q=ai:liao.yumei Summary: Due to its unique performance of high efficiency, fast heating speed and low power consumption, induction heating is widely and commonly used in many applications. In this paper, we study an optimal control problem arising from a metal melting process by using a induction heating method. Metal melting phenomena can be modeled by phase field equations. The aim of optimization is to approximate a desired temperature evolution and melting process. The controlled system is obtained by coupling Maxwell's equations, heat equation and phase field equation. The control variable of the system is the external electric field on the local boundary. The existence and uniqueness of the solution of the controlled system are showed by using Galerkin's method and Leray-Schauder's fixed point theorem. By proving that the control-to-state operator $$P$$ is weakly sequentially continuous and Fréchet differentiable, we establish an existence result of optimal control and derive the first-order necessary optimality conditions. This work improves the limitation of the previous control system which only contains heat equation and phase field equation. Mean curvature flow with boundary https://www.zbmath.org/1485.53109 2022-06-24T15:10:38.853281Z "White, Brian" https://www.zbmath.org/authors/?q=ai:white.brian-cabell The author develop a theory of surfaces with boundary moving by the mean curvature flow. As a consequence, he proves a general existence theorem using elliptic regularization. He also proves boundary regularity at all positive times under very mild hypotheses. Reviewer: Shu-Yu Hsu (Chiayi) Loss of initial data under limits of Ricci flows https://www.zbmath.org/1485.53112 2022-06-24T15:10:38.853281Z "Topping, Peter M." https://www.zbmath.org/authors/?q=ai:topping.peter-miles Summary: We construct a sequence of smooth Ricci flows on $$T^2$$, with standard uniform $$C/t$$ curvature decay, and with initial metrics converging to the standard flat unit-area square torus $$g_0$$ in the Gromov-Hausdorff sense, with the property that the flows themselves converge not to the static Ricci flow $$g(t)\equiv g_0$$, but to the static Ricci flow $$g(t)\equiv 2g_0$$ of twice the area. For the entire collection see [Zbl 1473.53006]. The Calabi flow with rough initial data https://www.zbmath.org/1485.53113 2022-06-24T15:10:38.853281Z "He, Weiyong" https://www.zbmath.org/authors/?q=ai:he.weiyong "Zeng, Yu" https://www.zbmath.org/authors/?q=ai:zeng.yu.1 The author proves that there exists a constant $$\delta>0$$ such that for any given background Kähler metric $$\omega$$ there exists a unique short-time smooth solution to the Calabi flow with initial data $$u_0$$ satisfying $\partial\overline{\partial}u_0\in L^{\infty}(M)\quad\mbox{ and }\quad (1-\delta)\omega<\omega_{u_0}<(1+\delta)\omega$ where $$\omega_{u_0}=\omega+\sqrt{-1}\partial\overline{\partial}u_0$$. As a corollary the Calabi flow has short-time existence for any initial data satisfying $\partial\overline{\partial}u_0\in C^0(M)\quad\mbox{ and }\quad \omega_{u_0}>0.$ The main technical ingredient is a new Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time-weighted Hölder norms. Reviewer: Shu-Yu Hsu (Chiayi) Ancient finite entropy flows by powers of curvature in $$\mathbb{R}^2$$ https://www.zbmath.org/1485.53114 2022-06-24T15:10:38.853281Z "Choi, Kyeongsu" https://www.zbmath.org/authors/?q=ai:choi.kyeongsu "Sun, Liming" https://www.zbmath.org/authors/?q=ai:sun.liming Summary: We show the existence of non-homothetic ancient flows by powers of curvature embedded in $$\mathbb{R}^2$$ whose entropy is finite. We determine the Morse indices and kernels of the linearized operator of shrinkers to the flows, and construct ancient flows by using unstable eigenfunctions of the linearized operator. Asymptotics and zeta functions on compact nilmanifolds https://www.zbmath.org/1485.58023 2022-06-24T15:10:38.853281Z "Fischer, Véronique" https://www.zbmath.org/authors/?q=ai:fischer.veronique The author obtains asymptotic formulae on nilmanifolds $$\Gamma\setminus G$$ where $$G$$ is any stratified nilpotent Lie group equipped with a co-compact discrete subgroup $$\Gamma$$. He also studies the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group $$G$$. The author shows that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. The author also studies the associated zeta functions. Reviewer: Shu-Yu Hsu (Chiayi) Weak time discretization for slow-fast stochastic reaction-diffusion equations https://www.zbmath.org/1485.60032 2022-06-24T15:10:38.853281Z "Shi, Chungang" https://www.zbmath.org/authors/?q=ai:shi.chungang "Wang, Wei" https://www.zbmath.org/authors/?q=ai:wang.wei.15|wang.wei.3|wang.wei.16|wang.wei.25|wang.wei.2|wang.wei.26|wang.wei.21|wang.wei.8|wang.wei.23|wang.wei.30|wang.wei.1|wang.wei.27|wang.wei.24|wang.wei.9|wang.wei.17|wang.wei.29|wang.wei.19|wang.wei.28|wang.wei.5|wang.wei.13|wang.wei.12|wang.wei.20 "Chen, Dafeng" https://www.zbmath.org/authors/?q=ai:chen.dafeng In this article, the authors construct a time discretization for the slow-fast stochastic partial differential equations (SPDEs) on the interval $$D = [0,1]$$ $\partial_t u^{\varepsilon}(t,x) = \partial_{xx} u^{\varepsilon}(t,x) + f(u^{\varepsilon}(t,x),v^{\varepsilon}(t,x))$ $\partial_t v^{\varepsilon}(t,x) = \frac{1}{\varepsilon} \big[ \partial_{xx} v^{\varepsilon}(t,x) + g(u^{\varepsilon}(t,x),v^{\varepsilon}(t,x)) \big] + \frac{1}{\sqrt{\varepsilon}} \partial_t W(t)$ $u^{\varepsilon}(0,x) = u_0(x), \,\,\,\,\,\, v^{\varepsilon}(0,x) = v_0(x), \,\,\,\,\,\, x \in D,$ $u^{\varepsilon}(t,0) = u^{\varepsilon}(t,1) = 0, \,\,\,\,\,\, v^{\varepsilon}(t,0) = v^{\varepsilon}(t,1) = 0, \,\,\,\,\,\, t > 0,$ where $$f(u,v) = u - u^3 + uv$$, the function $$g$$ is a Lipschitz nonlinearity, and $$\{ W(t) : t \geq 0 \}$$ is an $$L^2(D)$$-valued Wiener process. For this purpose, the authors show that the SPDEs above can be written in abstract form as $d u^{\varepsilon}(t) = [ A u^{\varepsilon}(t) + f(u^{\varepsilon}(t),v^{\varepsilon}(t)) ] dt$ $d v^{\varepsilon}(t) = \frac{1}{\varepsilon} \big[ A v^{\varepsilon}(t) + g(u^{\varepsilon}(t),v^{\varepsilon}(t)) \big] dt + \frac{1}{\sqrt{\varepsilon}} dW(t)$ $u^{\varepsilon}(0) = u_0, \,\,\,\,\,\, v^{\varepsilon}(0) = v_0,$ where $$A = \Delta$$ on $$D$$ with Dirichlet boundary conditions. Then the idea is to consider the averaging equation $d \bar{u}(t) = [A \bar{u}(t) + \bar{f}(\bar{u}(t))] dt.$ Here the function $$\bar{f}$$ is given by $\bar{f}(u) = \int_{L^2(D)} f(u,v) \mu^u(dv),$ where $$\mu^u$$ denotes the unique invariant measure for the fast component with frozen slow component $$u$$. Afterwards, they introduce the implicit Euler scheme $u_{n+1} = S_{\Delta t} u_n + \Delta t S_{\Delta t} \bar{f}(u_n),$ where $$S_{\Delta t} = (I - A \Delta t)^{-1}$$. In their main result (Theorem 3.1) the authors provide an estimate for the weak error $| \mathbb{E} \phi(u^{\varepsilon}(T)) - \mathbb{E} \phi(u_n) |$ for any $$T > 0$$ and $$\phi \in C_b^3$$. The authors also present a result about the discretization approximation of the solution of the averaging equation (see Theorem 4.5) and a result about asymptotic expansions (see Theorem 5.1). Reviewer: Stefan Tappe (Freiburg) Burgers equation in the adhesion model https://www.zbmath.org/1485.60060 2022-06-24T15:10:38.853281Z "Gliklikh, Yuri" https://www.zbmath.org/authors/?q=ai:gliklikh.yuri-e "Shamarova, Evelina" https://www.zbmath.org/authors/?q=ai:shamarova.evelina Summary: We prove the existence and uniqueness of a classical solution to a multidimensional non-potential stochastic Burgers equation with Hölder continuous initial data. Our motivation is the adhesion model in the theory of formation of the large-scale structure of the universe. Importantly, we drop the assumption on the potentiality of the velocity flow that has been questioned in physics literature. Discrete monotone method for space-fractional nonlinear reaction-diffusion equations https://www.zbmath.org/1485.65093 2022-06-24T15:10:38.853281Z "Flores, Salvador" https://www.zbmath.org/authors/?q=ai:flores.salvador "Macías-Díaz, Jorge E." https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardo "Hendy, Ahmed S." https://www.zbmath.org/authors/?q=ai:hendy.ahmed-s Summary: A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion-reaction equation. More precisely, we propose a Crank-Nicolson discretization of a reaction-diffusion system with fractional spatial derivative of the Riesz-type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher's equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method. Error estimates and blow-up analysis of a finite-element approximation for the parabolic-elliptic Keller-Segel system https://www.zbmath.org/1485.65102 2022-06-24T15:10:38.853281Z "Chen, Wenbin" https://www.zbmath.org/authors/?q=ai:chen.wenbin "Liu, Qianqian" https://www.zbmath.org/authors/?q=ai:liu.qianqian "Shen, Jie" https://www.zbmath.org/authors/?q=ai:shen.jie.4|shen.jie.1|shen.jie.5|shen.jie.2|shen.jie|shen.jie.3 Summary: The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, a new fully discrete scheme for this model was proposed in [\textit{J. Shen} and \textit{J. Xu}, SIAM J. Numer. Anal. 58, No. 3, 1674--1695 (2020; Zbl 1440.65101)], mass conservation, positivity and energy decay were proved for the proposed scheme, which are important properties of the original system. In this paper, we establish the error estimates of this scheme. Then, based on the error estimates, we derive the finite-time blowup of nonradial numerical solutions under some conditions on the mass and the moment of the initial data. Numerical analysis of high order time stepping schemes for a predator-prey system https://www.zbmath.org/1485.65103 2022-06-24T15:10:38.853281Z "Chrysafinos, Konstantinos" https://www.zbmath.org/authors/?q=ai:chrysafinos.konstantinos "Kostas, Dimitrios" https://www.zbmath.org/authors/?q=ai:kostas.dimitrios (no abstract) Correction to: A B-spline finite element method for solving a class of nonlinear parabolic equations modeling epitaxial thin-film growth with variable coefficient'' https://www.zbmath.org/1485.65106 2022-06-24T15:10:38.853281Z "Qin, Dandan" https://www.zbmath.org/authors/?q=ai:qin.dandan "Tan, Jiawei" https://www.zbmath.org/authors/?q=ai:tan.jiawei "Liu, Bo" https://www.zbmath.org/authors/?q=ai:liu.bo.3|liu.bo.2|liu.bo.4|liu.bo|liu.bo.1 "Huang, Wenzhu" https://www.zbmath.org/authors/?q=ai:huang.wenzhu From the text: In the original publication of the authors' article [ibid. 2020, Paper No. 172, 26 p. (2020; Zbl 1482.65187)] the name of the second author is incorrect. The correct name of the second author is Jiawei Tan rather than John Jiawei Tan. The error in this correction has been updated in the original article. Hexagonal grid approximation of the solution of the heat equation on special polygons https://www.zbmath.org/1485.65112 2022-06-24T15:10:38.853281Z "Buranay, Suzan C." https://www.zbmath.org/authors/?q=ai:buranay.suzan-c "Arshad, Nouman" https://www.zbmath.org/authors/?q=ai:arshad.nouman Summary: We consider the first type boundary value problem of the heat equation in two space dimensions on special polygons with interior angles $$\alpha_j\pi$$, $$j=1,2,\dots,M$$, where $$\alpha_j\in \{ \frac{1}{2},\frac{1}{3},\frac{2}{3}\}$$. To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points. It is proved that the given implicit schemes in both difference problems are unconditionally stable. It is also shown that the solutions of the constructed Difference Problem 1 and Difference Problem 2 converge to the exact solution on the grids of order $$O (h^2+\tau^2)$$ and $$O (h^4+\tau)$$ respectively, where $$h$$ and $$\frac{\sqrt{3}}{2}h$$ are the step sizes in space variables $$x_1$$ and $$x_2$$ respectively and $$\tau$$ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram. Non-perturbative quantum Galileon in the exact renormalization group https://www.zbmath.org/1485.83183 2022-06-24T15:10:38.853281Z "Steinwachs, Christian F." https://www.zbmath.org/authors/?q=ai:steinwachs.christian-friedrich Cosmic ray propagation in the universe in presence of a random magnetic field https://www.zbmath.org/1485.85029 2022-06-24T15:10:38.853281Z "Supanitsky, A. D." https://www.zbmath.org/authors/?q=ai:supanitsky.a-d Global existence of solutions without Dirac-type singularity to a chemotaxis-fluid system with arbitrary superlinear degradation https://www.zbmath.org/1485.92020 2022-06-24T15:10:38.853281Z "Ding, Mengyao" https://www.zbmath.org/authors/?q=ai:ding.mengyao "Lyu, Wenbin" https://www.zbmath.org/authors/?q=ai:lyu.wenbin Summary: In this paper, we study a chemotaxis-fluid model in a two-dimensional setting as below, $\begin{cases} n_t +\mathbf{u}\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)+f(n), & x\in \Omega,\, t>0, \\ c_t +\mathbf{u}\cdot \nabla c=\Delta c - c+ n, & x\in \Omega,\, t>0, \\ \mathbf{u}_t =\Delta\mathbf{u} +\nabla P+ n\nabla \phi, & x\in \Omega, \, t>0, \\ \nabla \cdot\mathbf{u}=0, & x\in \Omega, \, t>0. \end{cases}$ The global solvability of the system in a generalized sense is obtained under the hypothesis that the logistic source function $$f\in C^1 ([0,\infty))$$ satisfies a very mild condition: $$f(0)\geq 0$$ and $$\frac{f(s)}{s}\rightarrow -\infty$$ as $$s\rightarrow \infty$$. This result exhibits that without any critical superlinear exponent restriction on $$f$$, persistent Dirac-type singularities can be ruled out in our model. This work can be regarded as a natural follow-up to the recent paper due to [\textit{M. Winkler}, $$L^1$$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation'', Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) (to appear); \url{doi:10.2422/2036-2145.202005_016}]. Global classical solution to a chemotaxis consumption model with singular sensitivity https://www.zbmath.org/1485.92021 2022-06-24T15:10:38.853281Z "Liu, Dongmei" https://www.zbmath.org/authors/?q=ai:liu.dongmei Summary: In this paper, we are concerned with the chemotaxis consumption model with singular sensitivity $\begin{cases} u_t=\Delta u-\nabla \cdot \left( \frac{f(u)}{v} \nabla v \right),\quad & x\in \Omega,\; t > 0, \\ v_t = \Delta v - u v,\quad & x \in \Omega,\; t > 0, \end{cases}$ under homogeneous Neumann boundary conditions in a smooth bounded domain $$\Omega \subset \mathbb{R}^n$$ $$(n \geq 1)$$, for $$0 < f(u) \leq K(u + 1)^\alpha$$ with some $$K > 0$$ and $$\alpha < 2$$. It is shown that for any sufficiently smooth initial data, the above system admits a global classical solution when either $$n = 1$$ and $$\alpha < 2$$, or $$n \geq 2$$ and $$\alpha<1-\frac{n}{4}$$. Predator-prey pattern formation driven by population diffusion based on Moore neighborhood structure https://www.zbmath.org/1485.92090 2022-06-24T15:10:38.853281Z "Huang, Tousheng" https://www.zbmath.org/authors/?q=ai:huang.tousheng "Zhang, Huayong" https://www.zbmath.org/authors/?q=ai:zhang.huayong "Hu, Zhengran" https://www.zbmath.org/authors/?q=ai:hu.zhengran "Pan, Ge" https://www.zbmath.org/authors/?q=ai:pan.ge "Ma, Shengnan" https://www.zbmath.org/authors/?q=ai:ma.shengnan "Zhang, Xiumin" https://www.zbmath.org/authors/?q=ai:zhang.xiumin "Gao, Zichun" https://www.zbmath.org/authors/?q=ai:gao.zichun Summary: Diffusion-driven instability is a basic nonlinear mechanism for pattern formation. Therefore, the way of population diffusion may play a determinative role in the spatiotemporal dynamics of biological systems. In this research, we launch an investigation on the pattern formation of a discrete predator-prey system where the population diffusion is based on the Moore neighborhood structure instead of the von Neumann neighborhood structure widely applied previously. Under pattern formation conditions which are determined by Turing instability analysis, numerical simulations are performed to reveal the spatiotemporal complexity of the system. A pure Turing instability can induce the self-organization of many basic types of patterns as described in the literature, as well as new spiral-spot and labyrinth patterns which show the temporally oscillating and chaotic property. Neimark-Sacker-Turing and flip-Turing instability can lead to the formation of circle, spiral and much more complex patterns, which are self-organized via spatial symmetry breaking on the states that are homogeneous in space and non-periodic in time. Especially, the emergence of spiral pattern suggests that spatial order can generate from temporal disorder, implying that even when the predator-prey dynamics in one site is chaotic, the spatially global dynamics may still be predictable. The results obtained in this research suggest that when the way of population diffusion changes, the pattern formation in the predator-prey systems demonstrates great differences. This may provide realistic significance to explain more general predator-prey coexistence. Design and analysis of a discrete method for a time-delayed reaction-diffusion epidemic model https://www.zbmath.org/1485.92143 2022-06-24T15:10:38.853281Z "Macías-Díaz, Jorge E." https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardo "Ahmed, Nauman" https://www.zbmath.org/authors/?q=ai:ahmed.nauman "Jawaz, Muhammad" https://www.zbmath.org/authors/?q=ai:jawaz.muhammad "Rafiq, Muhammad" https://www.zbmath.org/authors/?q=ai:rafiq.muhammad-h "Rehman, Muhammad Aziz Ur" https://www.zbmath.org/authors/?q=ai:rehman.muhammad-aziz-ur In this paper, the authors first proposed a two-dimensional and time-delayed reaction-diffusion model to describe the propagation of infectious viral diseases like COVID-19, and then established the existence and stability of the equilibrium points. Moreover, the authors examined the bifurcation of this system in terms of one of the parameters. In particular, they developed a time-splitting nonlocal finite-difference scheme to simulate numerically this mathematical model. Reviewer: Jia-Bing Wang (Wuhan) A note on advection-diffusion cholera model with bacterial hyperinfectivity https://www.zbmath.org/1485.92169 2022-06-24T15:10:38.853281Z "Wu, Xiaoqing" https://www.zbmath.org/authors/?q=ai:wu.xiaoqing "Shan, Yinghui" https://www.zbmath.org/authors/?q=ai:shan.yinghui "Gao, Jianguo" https://www.zbmath.org/authors/?q=ai:gao.jianguo In the recent paper by \textit{X. Wang} and \textit{F.-B. Wang} [J. Math. Anal. Appl. 480, No. 2, Article ID 123407, 29 p. (2019; Zbl 1423.92241)] a system of advection-diffusion equations was suggested to model the transmission of cholera. The authors complement these results proving two new theorems in the paper under review. Namely, Theorem 1.2 establishes local asymptotic stability and global attractivity of the cholera-free equilibrium $$E_{0}$$ for the case when the basic reproduction number $$\mathcal{R}_{0}=1.$$ For $$\mathcal{R}_{0}>1,$$ Theorem 1.3 furnishes sufficient conditions for global asymptotic stability of the positive equilibrium $$E^{\ast}.$$ Reviewer: Svitlana P. Rogovchenko (Kristiansand) A Fisher-KPP model with a nonlocal weighted free boundary: analysis of how habitat boundaries expand, balance or shrink https://www.zbmath.org/1485.92175 2022-06-24T15:10:38.853281Z "Feng, Chunxi" https://www.zbmath.org/authors/?q=ai:feng.chunxi "Lewis, Mark A." https://www.zbmath.org/authors/?q=ai:lewis.mark-a "Wang, Chuncheng" https://www.zbmath.org/authors/?q=ai:wang.chuncheng "Wang, Hao" https://www.zbmath.org/authors/?q=ai:wang.hao.4 Summary: In this paper, we propose a novel free boundary problem to model the movement of single species with a range boundary. The spatial movement and birth/death processes of the species found within the range boundary are assumed to be governed by the classic Fisher-KPP reaction-diffusion equation, while the movement of a free boundary describing the range limit is assumed to be influenced by the weighted total population inside the range boundary and is described by an integro-differential equation. Our free boundary equation is a generalization of the classical Stefan problem that allows for nonlocal influences on the boundary movement so that range expansion and shrinkage are both possible. In this paper, we prove that the new model is well-posed and possesses steady state. We show that the spreading speed of the range boundary is smaller than that for the equivalent problem with a Stefan condition. This implies that the nonlocal effect of the weighted total population on the boundary movement slows down the spreading speed of the population. While the classical Stefan condition categorizes asymptotic behavior via a spreading-vanishing dichotomy, the new model extends this dichotomy to a spreading-balancing-vanishing trichotomy. We specifically analyze how habitat boundaries expand, balance or shrink. When the model is extended to have two free boundaries, we observe the steady state scenario, asymmetric shifts, or even boundaries moving synchronously in the same direction. These are newly discovered phenomena in the free boundary problems for animal movement. The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation https://www.zbmath.org/1485.92177 2022-06-24T15:10:38.853281Z "Jaïbi, Olfa" https://www.zbmath.org/authors/?q=ai:jaibi.olfa "Doelman, Arjen" https://www.zbmath.org/authors/?q=ai:doelman.arjen "Chirilus-Bruckner, Martina" https://www.zbmath.org/authors/?q=ai:chirilus-bruckner.martina "Meron, Ehud" https://www.zbmath.org/authors/?q=ai:meron.ehud Summary: In this paper we consider the 2-component reaction-diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics \textit{E. Gilad} et al. [Ecosystem engineers: from pattern formation to habitat creation'', Phys. Rev. Lett. 93, No. 9, Article ID 098105, 4 p. (2004; \url{doi:10.1103/PhysRevLett.93.098105})]. The nonlinear structure of this model is more involved than other more conceptual models, such as the extended \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})] model, and the analysis a priori is more complicated. However, the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations. Moreover, we find that the model exhibits a richness of analytically tractable patterns that exceeds that of Klausmeier-type models. Our study focuses on the 4-dimensional dynamical system associated with the reaction-diffusion model by considering traveling waves in 1 spatial dimension. We use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic/homoclinic/periodic orbits that jump' between (normally hyperbolic) slow manifolds, representing various kinds of localized vegetation patterns. The basic 1-front invasion patterns and 2-front spot/gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model. By exploiting the rich nonlinear structure of the model, we construct many multi-front patterns that are novel, both from the ecological and the mathematical point of view. In fact, we argue that these orbits/patterns are not specific for the model considered here, but will also occur in a much more general (singularly perturbed reaction-diffusion) setting. We conclude with a discussion of the ecological and mathematical implications of our findings. Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method https://www.zbmath.org/1485.93057 2022-06-24T15:10:38.853281Z "Allonsius, Damien" https://www.zbmath.org/authors/?q=ai:allonsius.damien "Boyer, Franck" https://www.zbmath.org/authors/?q=ai:boyer.franck "Morancey, Morgan" https://www.zbmath.org/authors/?q=ai:morancey.morgan In this paper, authors computed the exact value of the minimal null control time for the Grushin equation controlled on a strip based on the moments method. The approach here is involved in a careful spectral analysis of a truncated harmonic oscillator. The main results also extended some known results on biorthogonal families to real exponentials in the absence of a gap condition to get uniform estimates with respect to the asymptotic behavior of the associated counting function. Reviewer: Yong-Kui Chang (Xi'an) Boundary null controllability of degenerate heat equation as the limit of internal controllability https://www.zbmath.org/1485.93058 2022-06-24T15:10:38.853281Z "Araújo, B. S. V." https://www.zbmath.org/authors/?q=ai:araujo.b-s-v "Demarque, R." https://www.zbmath.org/authors/?q=ai:demarque.reginaldo "Viana, L." https://www.zbmath.org/authors/?q=ai:viana.laura|viana.leonardo-p|viana.luiz Summary: In this paper, we recover the boundary null controllability for the degenerate heat equation by analyzing the asymptotic behavior of an eligible family of state-control pairs $$((u_\varepsilon, h_\varepsilon))_{\varepsilon > 0}$$ solving corresponding singularly perturbed internal null controllability problems. As in other situations studied in the literature, our approach relies on Carleman estimates and meticulous weak convergence results. However, for the degenerate parabolic case, some specific trace operator inequalities must be obtained, in order to justify correctly the passage to the limit argument. Nonnegative multiplicative controllability for semilinear multidimensional reaction-diffusion equations https://www.zbmath.org/1485.93065 2022-06-24T15:10:38.853281Z "Floridia, Giuseppe" https://www.zbmath.org/authors/?q=ai:floridia.giuseppe The paper is well written and deserves for reading by reserachers, who are interested in mathematical control theory, especially for systems governed by pde. It would be interesting to see some results for practical applications of the received theory. Reviewer: Krzysztof Gałkowski (Zielona Gora) Null controllability of a semilinear degenerate parabolic equation with a gradient term https://www.zbmath.org/1485.93082 2022-06-24T15:10:38.853281Z "Xu, Fengdan" https://www.zbmath.org/authors/?q=ai:xu.fengdan "Zhou, Qian" https://www.zbmath.org/authors/?q=ai:zhou.qian "Nie, Yuanyuan" https://www.zbmath.org/authors/?q=ai:nie.yuanyuan Summary: This paper concerns the null controllability of a semilinear control system governed by degenerate parabolic equation with a gradient term, where the nonlinearity of the problem is involved with the first derivative. We first establish the well-posedness and prove the approximate null controllability of the linearized system, then we can get the approximate null controllability of the semilinear control system by a fixed point argument. Finally, the semilinear control system with a gradient term is shown to be null controllable. Output regulation for a heat equation with unknown exosystem https://www.zbmath.org/1485.93252 2022-06-24T15:10:38.853281Z "Guo, Bao-Zhu" https://www.zbmath.org/authors/?q=ai:guo.baozhu "Zhao, Ren-Xi" https://www.zbmath.org/authors/?q=ai:zhao.ren-xi Summary: In this paper, we consider output regulation for a $$1$$-d heat equation where the disturbances generated from an unknown finite-dimensional exosystem enter all possible channels. We adopt adaptive observer internal model approach which has been well developed for lumped parameter systems over two decades to estimate all possible unknown frequencies that have entered into a transformed system. By the estimates of the unknown frequencies, we are able to design a tracking error based feedback control to achieve output regulation and disturbance rejection for this PDE. A significance of the problem lies in the fact that both the control and observation operators are unbounded. The proposed approach is potentially applicable to other PDEs. Performance output tracking for a one-dimensional heat equation with input delay https://www.zbmath.org/1485.93260 2022-06-24T15:10:38.853281Z "Wang, Li" https://www.zbmath.org/authors/?q=ai:wang.li.1|wang.li.4|wang.li|wang.li.3|wang.li.2|wang.li.6|wang.li.5 Summary: In this paper, we focus on the performance output tracking for a one-dimensional heat partial differential equation with unknown external disturbance and non-collocated configuration, in which the control input is subject to a time delay. By writing the time delay as a first-order hyperbolic equation, the control system can be modeled as a cascade construction in which the transport equation can be viewed as the actuator dynamics of the heat partial differential system. With the help of the actuator dynamics compensation, the difficulties caused by input delay are addressed. The full state feedback law is constructed to achieve the performance output tracking and an error based observer is designed successfully. The exponential stability of the closed-loop system and the well-posedness of the observer are obtained. Performance output tracking for a one-dimensional unstable heat equation with input delay https://www.zbmath.org/1485.93261 2022-06-24T15:10:38.853281Z "Wang, Li" https://www.zbmath.org/authors/?q=ai:wang.li.1|wang.li.4|wang.li|wang.li.5|wang.li.3|wang.li.2|wang.li.6 "Feng, Hongyinping" https://www.zbmath.org/authors/?q=ai:feng.hongyinping Summary: In this paper, we investigate the performance output tracking for a one-dimensional unstable heat equation with input delay and disturbance. Both the performance output and the disturbance are non-collocated to the controller. By writing the time delay as a transport equation, the control plant becomes a cascade system in which the transport equation can be regarded as the actuator dynamics of the unstable heat equation. As a result, the method of actuator dynamics compensation can be used. The difficulties caused by the non-collocated configuration are figured out in the method of trajectory planning. The full state feedback is presented to realize the output tracking, and an error based observer is constructed successfully. The exponential stability of the closed-loop system and the well-posedness of the observer are proved. Guaranteed cost design for controlling semilinear parabolic PDE systems with mobile collocated actuators and sensors https://www.zbmath.org/1485.93263 2022-06-24T15:10:38.853281Z "Wu, Huai-Ning" https://www.zbmath.org/authors/?q=ai:wu.huaining "Zhang, Xiao-Wei" https://www.zbmath.org/authors/?q=ai:zhang.xiaowei Summary: This paper investigates the guaranteed cost design problem for controlling a class of semilinear parabolic partial differential equation (PDE) systems using mobile collocated actuators and sensors. Initially, a mode indicator function is employed to indicate the different modes for all actuator/sensor pairs according to whether each actuator/sensor pair is static or mobile. Subsequently, a mode-dependent switching control scheme is proposed and the well-posedness of the closed-loop PDE system is also analysed. Then, based on Lyapunov direct method, an integrated design of switching controllers and mobile actuator/sensor guidance laws is developed in the form of linear matrix inequalities, such that the closed-loop PDE system is exponentially stable while providing an upper bound for the prescribed quadratic cost function. Moreover, a suboptimal guaranteed cost design problem is also addressed to make the cost bound as small as possible. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method. Feedback stabilization of parabolic systems with input delay https://www.zbmath.org/1485.93445 2022-06-24T15:10:38.853281Z "Djebour, Imene Aicha" https://www.zbmath.org/authors/?q=ai:djebour.imene-aicha "Takahashi, Takéo" https://www.zbmath.org/authors/?q=ai:takahashi.takeo "Valein, Julie" https://www.zbmath.org/authors/?q=ai:valein.julie Summary: This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the $$N$$-dimensional linear reaction-convection-diffusion equation with $$N\geq 1$$ and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state. Feedback stabilization of Cahn-Hilliard phase-field systems https://www.zbmath.org/1485.93452 2022-06-24T15:10:38.853281Z "Marinoschi, Gabriela" https://www.zbmath.org/authors/?q=ai:marinoschi.gabriela Summary: We provide a compact presentation, relying on the papers [\textit{V. Barbu} et al., J. Differ. Equations 262, No. 3, 2286--2334 (2017; Zbl 1351.93113); the author, Springer INdAM Ser. 22, 357--377 (2017; Zbl 1382.35128); Pure Appl. Funct. Anal. 3, No. 1, 107--135 (2018; Zbl 1474.93183)], of the internal feedback stabilization of phase-field systems of Cahn-Hilliard type by using a feedback controller with support in a subset of the flow domain. Here the feedback stabilization technique is based on the design of the controller as a linear combination of the unstable modes of the corresponding linearized system, followed by its representation in a feedback form by means of an optimization method. Results are provided both for a regular potential involved in the phase field equation (the doublewell potential) and for a singular case represented by a logarithmic-type potential. The feedback stabilization is studied in the case with viscosity effects and in the limit case as the viscosity tends to zero. For the entire collection see [Zbl 07438181]. Adaptive stabilization for an uncertain reaction-diffusion equation with dynamic boundary condition at control end https://www.zbmath.org/1485.93479 2022-06-24T15:10:38.853281Z "Li, Jian" https://www.zbmath.org/authors/?q=ai:li.jian.1|li.jian.2|li.jian|li.jian.3 "Wu, Zhaojing" https://www.zbmath.org/authors/?q=ai:wu.zhaojing "Liu, Yungang" https://www.zbmath.org/authors/?q=ai:liu.yungang Summary: This paper is devoted to the stabilization of a class of uncertain reaction-diffusion equations with dynamic boundary condition. Remarkably, the dynamics of boundary condition at control end are considered while unknown parameters are contained in both the intra-domain and the dynamic boundary. This greatly relaxes the restrictions on boundary dynamics and system uncertainties of the related literature where dynamic boundary condition is neglected or considered but unknown parameters are only contained in the dynamic boundary or even completely excluded from the whole system. To solve the control problem, a novel control framework is established by infinite-dimensional backstepping method combining with adaptive compensation technique based on passive identifier. Then, an adaptive state-feedback controller is explicitly constructed which guarantees that all the states of the resulting closed-loop system are bounded while those of the original system converge to zero. A simulation example is provided to validate the effectiveness of the proposed theoretical results. Boundary stabilization and observation of a weak unstable heat equation in a general multi-dimensional domain https://www.zbmath.org/1485.93484 2022-06-24T15:10:38.853281Z "Feng, Hongyinping" https://www.zbmath.org/authors/?q=ai:feng.hongyinping "Lang, Pei-Hua" https://www.zbmath.org/authors/?q=ai:lang.pei-hua "Liu, Jiankang" https://www.zbmath.org/authors/?q=ai:liu.jiankang.1|liu.jiankang Summary: In this paper, we consider the exponential stabilization and observation of an unstable heat equation in a general multi-dimensional domain by combining the finite-dimensional spectral truncation technique and the recently developed dynamics compensation approach. In contrast to the unstable one-dimensional partial differential equation (PDE), such as the transport equation, wave equation and the heat equation, stabilization of unstable PDE in a general multi-dimensional domain by using the backstepping approach is still a challenging problem. We treat the stabilization and observation problems separately. A dynamical state feedback law is proposed firstly to stabilize the unstable heat equation exponentially and then a state observer is designed via a boundary measurement. Both the stability of the closed-loop system and the well-posedness of the observer are proved. Some of the theoretical results are validated by the numerical simulations.