| Issue |
ESAIM: COCV
Volume 26, 2020
|
|
|---|---|---|
| Article Number | 7 | |
| Number of page(s) | 20 | |
| DOI | https://doi.org/10.1051/cocv/2019007 | |
| Published online | 10 February 2020 | |
Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients★
1
School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University,
Changchun
130024, P.R. China.
2
School of Mathematics, Jilin University,
Changchun
130012, P.R. China.
** Corresponding author: jishuguan@hotmail.com
Received:
18
May
2018
Accepted:
8
February
2019
This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. 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 an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.
Mathematics Subject Classification: 35L71 / 35B10
Key words: Periodic solutions / wave equation / variational methods / ℤ2-index.
© EDP Sciences, SMAI 2020
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