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

TL;DR

This paper proves the existence of multiple 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 saddle point reduction to establish multiple solutions based on spectral analysis of the wave operator.

Findings

At least three periodic solutions exist for certain periods.

Spectral properties of the wave operator are crucial for the proof.

The method applies to equations with $x$-dependent coefficients.

Abstract

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficient. 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.