Global existence of the solutions to nonlinear hyperbolic equations in exterior domains
type: Item
instance of: scholarly article
title: Global existence of the solutions to nonlinear hyperbolic equations in exterior domains
zbMATH Open document ID: 0726.35084
author: Yunmei Chen
author: Jianmin Gao
published in: Chinese Annals of Mathematics. Series B
publication date: 1990-01-01T00:00:00Z
review text: The authors prove global existence and uniqueness results concerning smooth solutions of initial boundary value problems for the nonlinear wave equation: \[ \partial_ t^ 2u- \sum^{n}_{i,j=1}a_{ij}(u,Du)\partial_{x_ i}\partial_{x_ j}u=b(u,Du);\quad t>0,\quad x\in \Omega \] \[ u(0,x)=u_ 0(x),\quad \partial_ tu(0,x)=u_ 1(x);\quad x\in \Omega;\quad u(t,x)=0;\quad t>0,\quad x\in \partial \Omega \] in an exterior domain \(\Omega\) of a compact set in \({\mathbb{R}}^ n\) and \(Du:=(\partial_ tu,\partial_{x_ 1}u,...,\partial_{x_ n}u)\). Under that assumption that all \(a_{ij}\) and b are infinitely differentiable functions of their \(n+2\) arguments and if there is a \(k\in {\mathbb{N}}\) such that \(| a_{ij}(y)- \delta_{ij}| ={\mathcal O}(| y|^ k)\) and \(| b(y)| ={\mathcal O}(| y|^{k+1})\) where \(\delta_{ij}\) denotes the Kronecker symbol, the existence and uniqueness of a global smooth solution of the afore-mentioned problem is shown for the two cases \(n\geq 7\), \(k=1\) and \(n\geq 4\), \(k\geq 2\). Furthermore, decay estimates for the problem \(\partial_ t^ 2u-\nabla u=f\) with initial and boundary data as in the nonlinear case are given.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4198624
zbMATH Keywords: global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: smooth solutions
zbMATH Keywords: nonlinear wave equation
zbMATH Keywords: exterior domain
zbMATH Keywords: decay estimates
reviewed by: Thomas Sonar
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4198624