An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source
type: Item
instance of: scholarly article
title: An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source
zbMATH Open document ID: 1172.35038
author: Philippe Souplet
published in: Journal of Differential Equations
publication date: 2009-05-19T00:00:00Z
review text: The author considers nonnegative weak solutions of the porous medium equation with source. Assume \(m>1\) and \(1 < p/m < p_s\), where \(p_s\) is the Sobolev exponent defined by \(p_s = (n+1)/(n-2)\) if \(n >2\) and \(\infty\) if \(n \leq 2\). The author proves that the equation \[ u_t - \Delta u^m = u^p,\quad x \in {\mathbb R}^n, t \in {\mathbb R} \] has no nontrivial, bounded radial solutions \(u \geq 0\). In one space-dimensional, the conclusion of the result mentioned above remains true without the assumption of radial symmetry. In fact the conclusions of the results remain true without the boundedness assumption on \(u\). The proof is based on intersection-comparison arguments and a key step is to show the positivity of possible bounded radial entire solutions. As an auxiliary result, the author establishes pointwise gradient estimates for bounded, radial nonnegative solutions of the equation \(u_t - \Delta u^m = u^p\), which may be of independent interest.
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 5555288
zbMATH Keywords: porous medium equation
zbMATH Keywords: bounded radial solutions
zbMATH Keywords: Bernstein-type estimates
zbMATH Keywords: parabolic Liouville-type theorem
zbMATH Keywords: nonnegative weak solutions
MaRDI profile type: Publication
full work available at URL: j.jde.2008.10.018
OpenAlex ID: W2058929262
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DOI: 10.1016/J.JDE.2008.10.018
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description: scientific article; zbMATH DE number 5555288