Commentarii Mathematici Helvetici


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Volume 81, Issue 4, 2006, pp. 783–800
DOI: 10.4171/CMH/73

Published online: 2006-12-31

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

Raquel M. Gaspar[1]

(1) Universidade T├ęcnica de Lisboa, Portugal

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.

Keywords: Center manifolds, elliptic equations, quasi-periodic solutions

Gaspar Raquel: Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain. Comment. Math. Helv. 81 (2006), 783-800. doi: 10.4171/CMH/73