Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values
type: Item
instance of: scholarly article
title: Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values
zbMATH Open document ID: 0936.35034
DOI: 10.1016/S0362-546X(98)00161-8
author: Kentaro Mukai
author: K. Mochizuki
author: Qing Huang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2000-05-09T00:00:00Z
review text: The following initial value problem is considered \[ \partial_t u=\Delta u^m+u^p,\quad x\in \mathbb{R}^N,\;t>0 \qquad u(x,0)=u_0(x),\quad x\in\mathbb{R}^N, \] where \(1<m<p\), \(N\geq 1\), and \(u_0(x)\) is a nonnegative bounded and continuous function. The problem describes a combustion process in a stationary medium, where \(u\) represents the temperature, and it is assumed that thermal conductivity and volume heat source depend on some powers of \(u\). It is well known that this problem has a unique, nonnegative and bounded solution in some weak sense at least locally in time. The paper establishes some sufficient conditions implying that the considered solution exists only on a finite time interval (blow up) or that it has an infinite life span. It concentrates on the case of initial values \(u_0\) having slow decay \(u_0\sim\lambda|x|^a\), \(\lambda>0\), \(a\geq 0\), near \(x=\infty\). The problems of global existence and nonexistence, large time behavior or life span are investigated in terms of \(\lambda\) and \(a\).
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 80A25
zbMATH DE Number: 1383357
zbMATH Keywords: blow up
zbMATH Keywords: global existence
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00161-8
OpenAlex ID: W2056864792
description: scientific article; zbMATH DE number 1383357