Decay bounds for solutions of second order parabolic problems and their derivatives. III.
type: Item
instance of: scholarly article
title: Decay bounds for solutions of second order parabolic problems and their derivatives. III.
zbMATH Open document ID: 1072.35097
DOI: 10.4171/ZAA/1224
author: G. A. Philippin
author: L. E. Payne
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2005-04-11T00:00:00Z
review text: The initial-boundary value problem \[ \nabla (D(x)\nabla u)-u_t'=-f(u),\qquad x\in\Omega,\quad t\in(0,\tau), \] \[ u(x,0)=g(x)\geq0,\quad x\in\Omega, \qquad u(x,t)=0,\quad x\in\partial\Omega \] is considered. It is supposed that \(u(x,t)\) is a classical solution of this problem, where \(\Omega\) is a bounded convex domain in \(\mathbb R^n\), \(n\geq2\), and \(f\), \(g\), \(D\) are given nonnegative functions. Solution \(u(x,t)\) being nonnegative may (without appropriate data restriction) blow up at a finite time. Authors determine conditions on the data and on the geometry of \(\Omega\) sufficient to insure that the solution remains bounded for all time. Moreover, they establish exponential decay (in time) bounds for \(u(x,t)\) and \(| \nabla u| \) under these conditions. The particular case \(D(x)\equiv1\) has already been investigated in part I [Math. Models Meth. Appl. Sci. 5, 95--110 (1995; Zbl 0832.35019)]. For part II see Math. Inequal. Appl. 7, No. 4, 543--549 (2004; Zbl 1069.35506).
reviewed by: Josef Diblík
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2155010
zbMATH Keywords: quasilinear parabolic problem
zbMATH Keywords: maximum principle
zbMATH Keywords: exponential decay bounds
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1224
OpenAlex ID: W2054915322
description: scientific article; zbMATH DE number 2155010