Periodic solutions for completely resonant nonlinear wave equations


We consider the nonlinear string equation with Dirichlet boundary conditions uxx−utt = φ(u), with φ(u) = Φu+O(u) odd and analytic, Φ 6= 0, and we construct small amplitude periodic solutions with frequency ω for a large Lebesgue measure set of ω close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and nonresonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations uxx−utt+Mu = φ(u), M 6= 0, is that not only the P equation but also the Q equation is infinite-dimensional.

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@inproceedings{Gentile2004PeriodicSF, title={Periodic solutions for completely resonant nonlinear wave equations}, author={Guido Gentile and Vieri Mastropietro and Michela Procesi}, year={2004} }