Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 49, Issue 4, Pages 3161-3207Publisher
SIAM PUBLICATIONS
DOI: 10.1137/151005397
Keywords
dissipative wave equation; quasi-periodic solution; response solution
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Funding
- NSF [DMS-1162544, DMS-1500943]
- CONACYT [133036]
- PRIN-MIUR [2010JJ4KPA_009]
- GNFM/INdAM
- ERC project Hamiltonian PDEs and Small Divisor Problems: A Dynamical Systems Approach
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1500943] Funding Source: National Science Foundation
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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 E (where E is the inverse of the coefficient multiplying the damping) for E 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.
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