Response solutions for wave equations with variable wave speed and periodic forcing

TL;DR

This paper proves the existence of periodic response solutions in nonlinear wave equations with variable wave speed and periodic forcing, using advanced mathematical techniques under non-resonance conditions.

Contribution

It establishes the existence of response solutions for a class of nonlinear wave equations with variable wave speed and periodic forcing, employing Lyapunov--Schmidt reduction and Nash--Moser iteration.

Findings

Response solutions exist on a Cantor set of asymptotically full measure.

The proof combines Lyapunov--Schmidt reduction with Nash--Moser iteration.

Non-resonance conditions are crucial for the existence results.

Abstract

We consider a model of nonlinear wave equations with periodically varying wave speed and periodic external forcing. By imposing non-resonance conditions on the frequency, we establish the existence of the response solutions (i.e., periodic solutions with the same frequency as the forcing) for such a model in a Cantor set of asymptotically full measure. The proof relies on a Lyapunov--Schmidt reduction together with the Nash--Moser iteration.