Recent zbMATH articles in MSC 35B51https://www.zbmath.org/atom/cc/35B512021-04-16T16:22:00+00:00WerkzeugViscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term.https://www.zbmath.org/1456.350802021-04-16T16:22:00+00:00"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Požár, Norbert"https://www.zbmath.org/authors/?q=ai:pozar.norbertSummary: A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.Qualitative properties of weak solutions for \(p\)-Laplacian equations with nonlocal source and gradient absorption.https://www.zbmath.org/1456.351182021-04-16T16:22:00+00:00"Chaouai, Zakariya"https://www.zbmath.org/authors/?q=ai:chaouai.zakariya"El Hachimi, Abderrahmane"https://www.zbmath.org/authors/?q=ai:el-hachimi.abderrahmaneSummary: We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source \[\dfrac{\partial u}{\partial t} -\operatorname{div}(|\nabla u|^{p-2}\nabla u) =\lambda u^k\int_{\Omega}u^sdx- \mu u^l|\nabla u|^q\] in a bounded domain \(\Omega\subset\mathbb{R}^N \), where \(p>1\), the parameters \(k,s,l,q,\lambda>0\) and \(\mu\geq 0\). Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on \(|\nabla u|\). Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.An order approach to SPDEs with antimonotone terms.https://www.zbmath.org/1456.352452021-04-16T16:22:00+00:00"Scarpa, Luca"https://www.zbmath.org/authors/?q=ai:scarpa.luca"Stefanelli, Ulisse"https://www.zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle.