Response solutions for quasi-periodically forced, dissipative wave equations

TL;DR

This paper proves the existence and analyticity of response solutions for nonlinear wave equations with strong damping and quasi-periodic forcing, using fixed point methods and asymptotic expansions.

Contribution

It establishes the existence, asymptotic expansions, and analyticity of response solutions under general non-resonance conditions, extending previous results in the literature.

Findings

Response solutions exist under broad non-resonance conditions.

Response solutions depend analytically on the inverse damping coefficient.

Asymptotic expansions of solutions are rigorously justified.

Abstract

We consider several models of nonlinear wave equations subject to very strong damping and quasi-periodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under very general non-resonance conditions on the frequency, we show the existence of asymptotic expansions of the response solution; moreover, we prove that the response solution indeed exists and depends analytically on $\varepsilon$ (where $\varepsilon$ is the inverse of the coefficient multiplying the damping) for $\varepsilon$ in a complex domain, which in some cases includes disks tangent to the imaginary axis at the origin. In other models, we prove analyticity in cones of aperture $\pi/2$ and we conjecture it is optimal. These results have consequences for the asymptotic expansions of the response solutions considered in the literature. The proof of our results relies on reformulating the problem as a fixed point problem, constructing an approximate solution and studying the properties of iterations that converge to the solutions of the fixed point problem.