On nonexistence of similarity solutions
type: Item
instance of: scholarly article
title: On nonexistence of similarity solutions
zbMATH Open document ID: 0673.35045
DOI: 10.1016/0022-247X(88)90364-2
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: Consider the semilinear parabolic equation \[ (1)\quad u_ t=\Delta u- u^ p,\quad x\in R^ N,\quad t>0, \] where \(N\geq 1\), \(0<p<1\). Let u(x,t) be a nonnegative solution of (1) for which the initial value u(x,0) is uniformly bounded in \(R^ N\) by a constant M. Let \(T=\sup \{t>0:\quad u(x,t)=0\}\leq \{M/(1-p)\}^{1-p}.\) Then (1) may be changed into the form \[ (2)\quad \Delta v-\eta \cdot \nabla v+(v/(1-p))-v^ p=0. \] This paper proves that (2) has no radially symmetric solutions in \(R^ N\) which are nonnegative and bounded, except \(v(\eta)=0\), \(v(\eta)=k=(1-p)^{1/(1-p)}.\)
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 4101714
zbMATH Keywords: nonexistence
zbMATH Keywords: similarity solutions
zbMATH Keywords: semilinear
zbMATH Keywords: nonnegative solution
zbMATH Keywords: initial value
zbMATH Keywords: radially symmetric solutions
author: Lambertus A. Peletier
author: W. C. Troy
MaRDI profile type: Publication
cites work: Q4077008
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full work available at URL: 0022-247x(88)90364-2
OpenAlex ID: W1978391055
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description: scientific article; zbMATH DE number 4101714