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Nonlinear vibrations of completely resonant wave equations. (English) Zbl 1121.35083

Jachymski, Jacek (ed.) et al., Fixed point theory and its applications. Proceedings of the international conference, Bȩdlewo, Poland, August 1–5, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 77, 49-60 (2007).
This paper surveys results obtained by the author and his collaborators concerning the existence of small amplitude periodic and quasi periodic solutions of completely resonant nonlinear wave equations on a finite interval with Dirichlet boundary conditions. Both the autonomous and the periodically-forced cases are discussed. The paper describes the difficulties arising in treating these problems – small divisors and inifinite-dimensional bifurcations, presents the results obtained by the author and co-workers, and outlines the methods used: bifurcation theory, Nash-Moser implicit function theorems, dynamical systems techniques, and variational methods. For the detailed proofs of the results one is refered to the several recent papers.
For the entire collection see [Zbl 1112.47302].

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B15 Almost and pseudo-almost periodic solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
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