Nonresonance and global existence of nonlinear elastic wave equations with external forces
type: Item
instance of: scholarly article
title: Nonresonance and global existence of nonlinear elastic wave equations with external forces
zbMATH Open document ID: 1258.35140
author: Yi Zhou
author: Wei Xu
published in: Chinese Annals of Mathematics. Series B
publication date: 2012-12-06T00:00:00Z
review text: The nonlinear dynamical system \[ \partial _t^2u^i-c_2^2\Delta u^i-(c_1^2-c_2^2)\partial _i(\nabla \cdot u)=B_{lmn}^{ijk}\partial _l(\partial _mu^j\partial _nu^k)+h^i(x), \; i=1,2,3 \] is treated. The authors generalize the results of \textit{T. C. Sideris} [Ann. Math. (2) 151, No. 2, 849--874 (2000; Zbl 0957.35126)] (for \(h^i(x)\equiv 0\)) to the case of time-dependent external forces. First, they prove the existence and uniqueness of the stationary solution. Then they prove the global existence of the solution to the initial value problem by use of estimates, the invariance of the above system under translations, simultaneous rotations and changes of scaling.
Mathematics Subject Classification ID: 35L52
Mathematics Subject Classification ID: 35L72
Mathematics Subject Classification ID: 35Q74
zbMATH DE Number: 6113074
zbMATH Keywords: small data
zbMATH Keywords: null condition
zbMATH Keywords: external force
reviewed by: Marie Kopáčková
MaRDI profile type: Publication
full work available at URL: s11401-012-0707-2
OpenAlex ID: W2060572262
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DOI: 10.1007/S11401-012-0707-2
description: scientific article; zbMATH DE number 6113074