Minimal periods for solutions of semilinear wave equations in exterior domains and for solutions of the equations of nonlinear elasticity
type: Item
instance of: scholarly article
title: Minimal periods for solutions of semilinear wave equations in exterior domains and for solutions of the equations of nonlinear elasticity
zbMATH Open document ID: 0676.35063
DOI: 10.1016/0022-247X(88)90155-2
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The author examines i) time-periodic solutions of the semilinear wave equations over \({\mathbb{R}}^ n\), in particular for \(n=3\); ii) the equations of `semilinear elasticity' with spatially periodic elastic coefficients; iii) one-dimensional nonlinear elasticity; iv) nonlinear elasticity. For i), it is shown that radially symmetric time-periodic solutions decaying spatially rapidly enough must be time-independent. A hypothesis on the growth of the nonlinearity at zero is essential to achieve this. In ii), assuming decaying in time of spatial measures of solutions, these are seen to be independent of space. Cases iii) and iv) are similar in spirit to ii). iii) is a special case of iv) about which more is said - an example is given involving the failure of strict hyperbolicity and consequent failure of the above results. A similar point concerns case i) and the sine-Gordon equation.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4107299
zbMATH Keywords: time-periodic solutions
zbMATH Keywords: semilinear
zbMATH Keywords: semilinear elasticity
zbMATH Keywords: periodic elastic coefficients
zbMATH Keywords: nonlinear elasticity
zbMATH Keywords: radially symmetric
zbMATH Keywords: growth
zbMATH Keywords: sine-Gordon equation
author: Howard A. Levine
reviewed by: Ralph Saxton
MaRDI profile type: Publication
cites work: Q4390424
cites work: Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elasto- dynamics
cites work: Q3889412
cites work: Logarithmic convexity and the Cauchy problem for some abstract second order differential inequalities
full work available at URL: 0022-247x(88)90155-2
OpenAlex ID: W2057648680
description: scientific article; zbMATH DE number 4107299