Global existence of the solutions of nonlinear parabolic equations in exterior domains
type: Item
instance of: scholarly article
title: Global existence of the solutions of nonlinear parabolic equations in exterior domains
zbMATH Open document ID: 0663.35037
author: Yunmei Chen
published in: Chinese Annals of Mathematics. Series B
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with the following initial boundary value problem of the nonlinear parabolic equation: \[ u_ t=\Delta u+F(u,D_ xu,D^ 2_ xu),\quad (t,x)\in R^+\times \Omega,\quad u(0,x)=\phi (x),\quad x\in \Omega,\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is the exterior domain of a compact set in \(R^ n\) with smooth boundary and F satisfies \(| F(\lambda)| =o(| \lambda |^ 2)\), near \(\lambda =0\). It is proved that when \(n\geq 3\), under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, it is also proved that the solution has the decay property \(\| u(t)\|_{L^{\infty}(\Omega)}=o(t^{-n/2}),\) as \(t\to +\infty\).
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4084166
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: nonlinear parabolic equation
zbMATH Keywords: exterior domain
zbMATH Keywords: smoothness
zbMATH Keywords: global smooth solution
zbMATH Keywords: small initial data
zbMATH Keywords: decay property
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4084166