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Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

Part of: Partial differential equations Hyperbolic equations and systems

Published online by Cambridge University Press:  04 June 2019

Hui Wei  and
Shuguan Ji
Hui Wei
Affiliation:
School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun130024, P.R. China (weihui01@163.com)
Shuguan Ji*
Affiliation:
School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun130024, P.R. China and School of Mathematics, Jilin University, Changchun130012, P.R. China (jishuguan@hotmail.com)
*
*Corresponding author:
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Abstract

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

Keywords

existenceperiodic solutionswave equation

MSC classification

Primary: 35L71: Semilinear second-order hyperbolic equations
Secondary: 35B10: Periodic solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2586 - 2606
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Copyright
Copyright © Royal Society of Edinburgh 2019

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