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

Clàudia Valls Anglès

• 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.