Response Solutions for Quasi-Periodically Forced, Dissipative Wave Equations

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 or it creates multiple time scales. 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 in appropriate spaces of smooth functions, constructing an approximate solution, and studying the properties of iterations that converge to the solutions of the fixed point problem. In particular we do not use dynamical properties of the models, so the method applies even to ill-posed equations.