Solvability conditions for some semi-linear parabolic equations
type: Item
instance of: scholarly article
title: Solvability conditions for some semi-linear parabolic equations
zbMATH Open document ID: 1186.35104
author: James M. Wrkich
author: Qi S. Zhang
published in: Journal of Differential Equations
publication date: 2009-11-05T00:00:00Z
review text: The authors deal with the Cauchy problem for a semi-linear parabolic equation of the following form: \[ \begin{cases} \Delta u-\partial _tu+Vu^p+w =0,\;p>1 \\ u(x,0) =u_0(x)\end{cases}\tag{1} \] where \(u_0(x)\geq 0\), \(w=w(x)\), \(V=V(x)\), \(u_0,w,V\in L_{\text{loc}}^1(\mathbb R^N)\), \(x\in \mathbb R^N\), \(t>0\). Starting from the fact that the solvability of problem (1) was for a long time a open problem, the authors establish necessary conditions on the potential \(V\) so that problem (1) has a positive solution for some \(u_0=u_0(x)\geq 0\), \(w=w(x)\geq 0\). Finally the authors give an application in the case when \(V\) is the important inverse square potential.
reviewed by: Cristian Chifu
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5625448
zbMATH Keywords: semi-linear parabolic equations
zbMATH Keywords: solvability conditions
zbMATH Keywords: positive solutions
zbMATH Keywords: inverse square potential
MaRDI profile type: Publication
full work available at URL: j.jde.2009.07.028
OpenAlex ID: W2063237287
cites work: Q3842692
cites work: The Heat Equation with a Singular Potential
cites work: Critère d'existence de solutions positives pour des équations semi- linéaires non monotones. (Existence condition of positive solutions of non-monotonic semilinear equations)
cites work: Nonlinear equations and weighted norm inequalities
cites work: An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci \(\geq 0\)
DOI: 10.1016/J.JDE.2009.07.028
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description: scientific article; zbMATH DE number 5625448