Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain

• Autores: Clàudia Valls Anglès
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 81, Nº 4, 2006, págs. 783-800
• Idioma: inglés
• DOI: 10.4171/cmh/73
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• Resumen

  • The elliptic equation $\partial_{tt} u= -\partial_{xx} u - \alpha u - g(u)$, $\alpha >0$ is ill-posed and "most'' initial conditions lead to no solutions. Nevertheless, we show that for almost every $\alpha$ there exist smooth solutions which are quasi-periodic. These solutions are anti-symmetric in space, and hence they are not traveling waves. Our approach uses the existence of an invariant center manifold, and the solutions are obtained from a KAM-type theorem for the restriction of the equation to that manifold.