Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions
Articolo
Data di Pubblicazione:
2005
Citazione:
Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions / G. Gentile, V. Mastropietro, M. Procesi. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 256:2(2005 Jun), pp. 437-490.
Abstract:
We consider the nonlinear string equation with Dirichlet boundary conditions u(tt) - u(xx) = phi(u), with phi(u) = Phi u(3) + O(u(5)) odd and analytic, Phi not equal 0, and we construct small amplitude periodic solutions with frequency omega for a large Lebesgue measure set of omega 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 non-resonant 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 to the nonlinear wave equations u(tt) - u(xx) + Mu = phi(u), M not equal 0, is that not only the P equation but also the Q equation is infinite-dimensional.
Tipologia IRIS:
01 - Articolo su periodico
Elenco autori:
G. Gentile, V. Mastropietro, M. Procesi
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