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

### Claudia Valls

Instituto Superior Técnico, Lisboa, Portugal

## 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.

## Cite this article

Claudia Valls, Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain. Comment. Math. Helv. 81 (2006), no. 4, pp. 783–800

DOI 10.4171/CMH/73