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

Hui Wei, Shuguan Ji

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.

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

Abstract

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with -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 -dependent coefficients, and the spectral properties play an essential role in the proof.

Paper Structure

This paper contains 6 sections, 12 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Variational problem and its reduction
  4. Verification the $(PS)_c$ condition
  5. Bounds of the reduction functional
  6. Proof of Theorem \ref{['th:2.1']}

Key Result

Lemma 2.1

Assume that $\rho(x)$ satisfies (A1), then the eigenvalues of problem eqa:2.2 have the form where and $\rho_1 = \frac{2}{\pi} \int^{\pi}_0 \eta_{\rho}(x) \textrm{d}x$, $\rho_2 = \sqrt{\rho_0 +1} -1$.

Theorems & Definitions (23)

  • Definition 2.1
  • Lemma 2.1: Barbu.(1997)a
  • Lemma 2.2
  • Theorem \oldthetheorem
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 13 more