A nonlinear degenerate diffusion equation not in divergence form
type: Item
instance of: scholarly article
title: A nonlinear degenerate diffusion equation not in divergence form
zbMATH Open document ID: 0961.35077
DOI: 10.1007/PL00001503
author: Shu Wang
author: Mingxin Wang
author: Chunhong Xie
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2001-05-15T00:00:00Z
review text: The interesting paper under review studies positive solutions of the nonlinear degenerate diffusion problem: \[ u_t=u^p(\Delta u+u), \quad x\in \Omega,\;t>0,\qquad u=0, \quad x\in \partial\Omega,\;t>0,\qquad u(x,0)=\phi(x), \quad x\in \Omega, \] where \(\Omega\subset {\mathbb R}^N\) is a bounded domain and \(p>1.\) The authors prove that all positive classical solutions to the above problem exist globally if and only if \(\lambda_1\geq 1\) where \(\lambda_1\) is the first eigenvalue of \(-\Delta \) in \(\Omega\) with homogeneous Dirichlet boundary data.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1437372
zbMATH Keywords: positive classical solutions
zbMATH Keywords: global solution
zbMATH Keywords: blow up
zbMATH Keywords: upper and lower solutions method
MaRDI profile type: Publication
full work available at URL: pl00001503
OpenAlex ID: W2314721840
description: scientific article; zbMATH DE number 1437372