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

Abstract

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.