A Liouville theorem for vector-valued nonlinear heat equations and applications
type: Item
instance of: scholarly article
title: A Liouville theorem for vector-valued nonlinear heat equations and applications
zbMATH Open document ID: 0939.35086
author: Frank Merle
author: H. Zaag
published in: Mathematische Annalen
publication date: 2000-06-19T00:00:00Z
review text: The paper deals with the blowing-up solutions to the Dirichlet problem for vector-valued semilinear heat equation \[ u_t=\Delta u+F(|u|) \quad \text{in} \Omega \times [0,T), \qquad u(x,t)=0 \quad \text{on} \partial \Omega\times [0,T), \qquad u(x,0)=u_0(x),\;x\in \Omega, \] where \(\Omega\) is a bounded and convex domain of \({\mathbb R}^N\) or \(\Omega={\mathbb R}^N,\) \(F\in C^1({\mathbb R},{\mathbb R})\) satisfying \(F(|u|)\sim |u|^{p-1}\) as \(|u|\to\infty,\) \(p>1\) and \((N-2)p<N+2.\) By the aid of scalar-case-techniques, the authors extend the parabolic Liouville theorem to the vectorial setting.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 1406573
zbMATH Keywords: vector-valued semilinear heat equation
zbMATH Keywords: blowing-up solutions
MaRDI profile type: Publication
full work available at URL: s002080050006
OpenAlex ID: W2047808905
DOI: 10.1007/S002080050006
description: scientific article; zbMATH DE number 1406573