Critical exponents for the decay rate of solutions in a semilinear parabolic equation
type: Item
instance of: scholarly article
title: Critical exponents for the decay rate of solutions in a semilinear parabolic equation
zbMATH Open document ID: 0937.35087
author: Noriko Mizoguchi
author: Eiji Yanagida
published in: Archive for Rational Mechanics and Analysis
publication date: 2000-06-07T00:00:00Z
review text: The authors study the 1-D Cauchy problem for the heat equation with power-type absorption \[ u_t=u_{xx}-|u|^{p-1}u \] with \(u(x,0)= u_0(x)\) uniformly bounded. Then \(\max_{x\in \mathbb{R}}|u(x,t) |t^{1/(p-1)}\) is uniformly bounded, and a solution is called to decay fast, if this quantity tends to zero. The authors show, that this cannot happen, if \(u_0\) changes sign exactly \(k\)-times and \(1<p<p_k =1+2/(k+1)\), while for \(p\geq p_k\) such an initial value leading to a fast decaying solution will exist. Earlier, the authors had proved, that the same exponents occur in the case of finite time blow-up if dealing with a source term ``\(+|u|^{p-1}u\)''.
reviewed by: Michael Wiegner
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1287909
zbMATH Keywords: power-type absorption
zbMATH Keywords: fast decaying solution
MaRDI profile type: Publication
full work available at URL: s002050050132
OpenAlex ID: W1986536080
DOI: 10.1007/S002050050132
description: scientific article; zbMATH DE number 1287909