Infinitely many periodic solutions for a semilinear wave equation with $x$-dependent coefficients

TL;DR

This paper proves the existence of infinitely many periodic solutions for a one-dimensional semilinear wave equation with spatially varying coefficients, relevant to nonhomogeneous string vibrations and seismic wave propagation.

Contribution

It introduces a novel combination of variational methods and approximation techniques to establish infinite periodic solutions for wave equations with variable coefficients.

Findings

Existence of infinitely many periodic solutions when the period is a rational multiple of the spatial interval.

Solutions are obtained under various homogeneous boundary conditions.

Spectral analysis of the wave operator with x-dependent coefficients underpins the results.

Abstract

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.