Hyperbolic trapped rays and global existence of quasilinear wave equations
type: Item
instance of: scholarly article
title: Hyperbolic trapped rays and global existence of quasilinear wave equations
zbMATH Open document ID: 1077.35083
author: Jason Metcalfe
author: Christopher D. Sogge
published in: Inventiones Mathematicae
publication date: 2005-02-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0301290
review text: The authors considered the initial boundary value problem of the system of wave equations: \[ \square u= Q(du, d^2u),\;(t,x)\in\mathbb{R}_+\times \mathbb{R}^3\setminus{\mathcal K},\;u(t,\cdot)|_{\partial{\mathcal K}}= 0,\;u(0,\cdot)= f,\;\partial_t u(0,\cdot)= g,\tag{1} \] where \(\square= (\square_{c_1},\dots, \square_{c_D})\) with \(\square_{c_I}= \partial^2_t- c^2_I\Delta\) and \(c_I> 0\), \(I= 1,\dots, D\), \(Q(du, d^2u)\) are quadratic quasilinear nonlinearities satisfy a form of the null condition, the obstacle \({\mathcal K}\) is smooth and compact. The main purpose of this paper is to prove the global solvability to (1) under the smallness of initial data in a weighted Sobolev space. In particular, they can drop the star-shaped hypothesis and handle nontrapping obstacle as well as any obstacle that has exponential local decay rate of energy for \(H^2\) data for the linear equation.
reviewed by: Fumihiko Hirosawa
Mathematics Subject Classification ID: 35L55
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 58J45
zbMATH DE Number: 2132026
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: non star-shaped obstacle
zbMATH Keywords: weighted Sobolev space
MaRDI profile type: Publication
OpenAlex ID: W3102291120
arXiv ID: math/0301290
cites work: Local energy decay of the wave equation in an exterior problem and without resonance in the neighborhood of the real line
cites work: Global solutions of nonlinear hyperbolic equations for small initial data
cites work: An elementary proof of global or almost global existence for quasi-linear wave equations
cites work: Lectures on nonlinear hyperbolic differential equations
cites work: Decay of solutions of the wave equation in the exterior of two convex obstacles
cites work: Decay of solutions of the wave equation in the exterior of several convex bodies
cites work: Global existence for a quasilinear wave equation outside of star-shaped domains
cites work: Almost global existence for some semilinear wave equations
cites work: Almost global existence for quasilinear wave equations in three space dimensions
cites work: Uniform decay estimates and the lorentz invariance of the classical wave equation
cites work: Q3733338
cites work: Q4884687
cites work: Exponential decay of solutions of the wave equation in the exterior of a star‐shaped obstacle
cites work: Q3996331
cites work: Singularities and energy decay in acoustical scattering
cites work: The decay of solutions of the exterior initial-boundary value problem for the wave equation
cites work: Decay of solutions of the wave equation outside nontrapping obstacles
cites work: Solutions of the wave equation with localized energy
cites work: On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain
cites work: Nonresonance and global existence of prestressed nonlinear elastic waves
cites work: The null condition and global existence of nonlinear elastic waves
cites work: Global existence for systems of nonlinear wave equations in 3D with multiple speeds
cites work: Global strichartz estimates for nonthapping perturbations of the laplacian
cites work: Q5475904
cites work: Q4462264
cites work: Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions
DOI: 10.1007/S00222-004-0383-2
description: scientific article