JournalscmhVol. 81, No. 4pp. 783–800

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

  • Claudia Valls

    Instituto Superior Técnico, Lisboa, Portugal
Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain cover
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Abstract

The elliptic equation ttu=xxuαug(u)\partial_{tt} u= -\partial_{xx} u - \alpha u - g(u), α>0\alpha >0 is ill-posed and "most'' initial conditions lead to no solutions. Nevertheless, we show that for almost every α\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