- 15-10 Renato C. Calleja, Alessandra Celletti, Livia Corsi, Rafael de la Llave
- Response solutions for quasi-periodically forced, dissipative wave equations
Jan 23, 15
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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 $arepsilon$ (where $arepsilon$ is the inverse of the coefficient
multiplying the damping) for $arepsilon$ 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.