Asymptotic properties of a semilinear heat equation with strong absorption and small diffusion
type: Item
instance of: scholarly article
title: Asymptotic properties of a semilinear heat equation with strong absorption and small diffusion
zbMATH Open document ID: 0702.35013
DOI: 10.1007/BF01444558
author: J. J. L. Velázques
author: Miguel A. Herrero
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164760
review text: We consider the following family of Cauchy problems \[ (1)\quad u_ t- \epsilon \Delta u+u^ p=0,\quad x\in {\mathbb{R}}^ N,\quad t>0;\quad u(x,0)=u_ 0(x),\quad x\in {\mathbb{R}}^ N \] where \(N\geq 1\), \(0<p<1\), \(u_ 0(x)\) is continuous, nonnegative and bounded, and \(0<\epsilon \ll 1\). We discuss the behaviour of the corresponding solutions \(u_{\epsilon}(x,t)\) as \(\epsilon\downarrow 0\). In particular, we derive an asymptotic expansion for the difference \((u_{\epsilon}(x,t)-\bar u(x,t)),\) where \(\bar u(x,t)\) solves \[ (2)\quad u_ t+u^ p=0,\quad x\in {\mathbb{R}}^ N,\quad t>0;\quad u(x,0)=u_ 0(x),\quad x\in {\mathbb{R}}^ N. \] Solutions of (1) and (2) vanish in a finite time. We use the previous asymptotic result to study the convergence as \(\epsilon\downarrow 0\) of the local and global extinction times of (1) to the corresponding quantities for (2), these last being explicitly written in terms of \(u_ 0(x)\) and p.
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 4150752
zbMATH Keywords: semilinear heat equation
zbMATH Keywords: strong absorption
zbMATH Keywords: small diffusion
zbMATH Keywords: Cauchy problems
zbMATH Keywords: convergence
zbMATH Keywords: extinction times
MaRDI profile type: Publication
cites work: Q3207312
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description: scientific article; zbMATH DE number 4150752