On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain
type: Item
instance of: scholarly article
title: On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain
zbMATH Open document ID: 0592.35028
DOI: 10.1007/BF01164023
author: Yoshihiro Shibata
author: Yoshio Tsutsumi
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173664
review text: Let \(\Omega\) be an unbounded domain in \({\mathbb R}^ n\) with compact \(C^{\infty}\) boundary \(\partial \Omega\). Let \(t\) or \(x_ 0\) denote the time variable and \(x=(x_ 1,...,x_ n)\) denote the space variables. Let \(\partial_ t=\partial /\partial t\), \(\partial_ x=(\partial_ 1,...,\partial_ n)\) with \(\partial_ j=\partial /\partial x_ j\) \((j=1,...,n)\) and \(\underline{D}^ 1 D^ 1u=(\partial^ j_ t\partial_ x^{\alpha}u;\,1\leq j+| \alpha | \leq 2)\). The mixed problem \[ (\partial_ t^ 2-\Delta)u+F(t,x,\underline D^ 1D^ 1u)=f(t,x)\text{ in } [0,\infty)\times \Omega, \] \[ u=0\text{ on } [0,\infty)\times \partial \Omega,\tag{*} \] \[ u(0,x)=u_ 0(x), \quad (\partial_ tu)(0,x)=u_ 1(x) \text{ in }\Omega \] is considered. The authors prove that there exists a unique, global classical solution of (*) if the data \(u_ 0\), \(u_ 1\), and \(f\) are small and sooth in some sense. They extend results due to \textit{J. Shatah} [J. Differ. Equations 46, 409--425 (1982; Zbl 0518.35046)] to the mixed problem (*). The proof is divided into two main steps. The first is to obtain an existence theorem for time-local solutions. As \(F\) is fully nonlinear, one encounters loss of derivatives if the contraction mapping principle is used. One of the methods used to overcome this difficulty is the Nash-Moser technique. But for the regularity of solutions, such results seem somewhat rough. Using a method of \textit{P. A. Dionne} [J. Anal. Math. 10, 1--90 (1962; Zbl 0112.32301)] one can overcome this difficulty for the Cauchy problem, but not for (*). However, by differentiating the differential equation in (*) with respect to \(t\) and setting \(v=\partial_ t u\) one obtains two equations for \(u\) and \(v\). The first, original equation is regarded as a fully nonlinear elliptic equation for \(u\) and the second as a quasi-linear equation for \(v\). The authors thus obtain an existence theorem via the usual contraction principle. In particular, if the data are sufficiently smooth,…
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 3951081
zbMATH Keywords: mixed problem
zbMATH Keywords: unique, global classical solution
zbMATH Keywords: existence theorem
zbMATH Keywords: Nash-Moser technique
zbMATH Keywords: regularity of solutions
zbMATH Keywords: contraction principle
zbMATH Keywords: uniform decay estimate
zbMATH Keywords: scattering obstacle
MaRDI profile type: Publication
cites work: Q5651602
cites work: Sur les problèmes de Cauchy hyperboliques bien posés
cites work: Mixed problems for hyperbolic equations of second order
cites work: Blow-up for quasi-linear wave equations in three space dimensions
cites work: Global, small amplitude solutions to nonlinear evolution equations
cites work: Q5616408
cites work: Exponential decay of solutions of the wave equation in the exterior of a star‐shaped obstacle
cites work: The initial value problem for the equations of motion of viscous and heat-conductive gases
cites work: Singularities and energy decay in acoustical scattering
cites work: Q5679204
cites work: Periodic solutions of nonlinear hyperbolic partial differential equations. II
cites work: Non-linear semi-groups
cites work: Global existence of small solutions to nonlinear evolution equations
cites work: Q3037925
cites work: On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain
cites work: Local existence of solution for the initial boundary value problem of fully nonlinear wave equation
cites work: Global solutions of the nonlinear schrödinger equation in exterior domains
cites work: Q3344580
cites work: ON THE SHORT WAVE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF STATIONARY PROBLEMS AND THE ASYMPTOTIC BEHAVIOUR ASt→ ∞ OF SOLUTIONS OF NON-STATIONARY PROBLEMS
cites work: \(L^ p\)-decay rates for homogeneous wave-equations
cites work: Characterization of a convex obstacle by singularities of the scattering kernel
OpenAlex ID: W2000720828
description: scientific article; zbMATH DE number 3951081