A nonlinear heat equation with singular initial data
type: Item
instance of: scholarly article
title: A nonlinear heat equation with singular initial data
zbMATH Open document ID: 0868.35058
DOI: 10.1007/BF02790212
author: Thierry Cazenave
author: Haïm Brezis
published in: Journal d'Analyse Mathématique
publication date: 1996-10-27T00:00:00Z
review text: The authors take up the model problem of a semilinear heat equation with a powertype source on a bounded domain \(u_t-\Delta u=|u|^{p-1}u\) with Dirichlet conditions and an initial value \(u_0\in L_q\). They show that the number \(q^*= N(p-1)/2\) is critical in the sense that for \(q>q^*\) and \(q\geq 1\) (or \(q=q^*>1\)) a unique local classical solution exists, belonging to \(C([0,T],L_q)\); and this solution fulfills additionally \(\lim_{t\to0}t^{n/2q}|u(t)|_\infty=0\). For \(q>q^*\) and \(q\geq p\), the same applies to weak solutions (of the corresponding integral equation). In this context, remarkable results were proven by \textit{F. B. Weißler} [see, e.g., Indiana Univ. Math. J. 29, 79-102 (1980; Zbl 0443.35034) and Proc. Symp. Pure Math. 45/2, 545-551 (1986; Zbl 0631.35049)]. The authors present a lot of open and interesting problems, which are worth to study.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35A07
zbMATH DE Number: 927203
zbMATH Keywords: semilinear heat equation
zbMATH Keywords: powertype source
zbMATH Keywords: unique local classical solution
reviewed by: Michael Wiegner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02790212
OpenAlex ID: W2008427960
description: scientific article; zbMATH DE number 927203