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Forced vibrations via Nash-Moser iterations

In this thesis, we prove the existence of large frequency periodic solutions for the nonlinear wave equations utt − uxx − v(x)u = u3 + [fnof]([Omega]t, x) (1) with Dirichlet boundary conditions. Here, [Omega] represents the frequency of the solution. The method we use to find the periodic solutions u([Omega]) for large [Omega] originates in the work of Craig and Wayne [10] where they constructed solutions for free vibrations, i.e., for [fnof] = 0. Here we construct smooth solutions for forced vibrations ([fnof] [not equal to] 0). Given an x-dependent analytic potential v(x) previous works on (1) either assume a smallness condition on [fnof] or yields a weak solution. The study of equations like (1) goes back at least to Rabinowitz in the sixties [25]. The main difficulty in finding periodic solutions of an equation like (1), is the appearance of small denominators in the linearized operator stemming from the left hand side. To overcome this difficulty, we used a Nash-Moser scheme introduced by Craig and Wayne in [10]. / text

Identiferoai:union.ndltd.org:UTEXAS/oai:repositories.lib.utexas.edu:2152/23960
Date11 April 2014
CreatorsFokam, Jean-Marcel
Source SetsUniversity of Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis
Formatelectronic
RightsCopyright is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works.

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