Families of periodic orbits for some PDE’s in higher dimensions

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June 2002, 1(2): 269-279. doi: 10.3934/cpaa.2002.1.269

Families of periodic orbits for some PDE’s in higher dimensions

Dario Bambusi ^1, and Simone Paleari ^1,

1.	Dipartimento di Matematica “F. Enriques”, Universita di Milano, Via Saldini 50, 20133 Milano, Italy, Italy

Received July 2001 Revised November 2001 Published March 2002

Abstract
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In this paper we consider the following nonlinear plate equations:

$u_{t t} +\Delta \Delta u + m u = \psi(x, u) ,$

$\psi(x, u) = \pm u^3 + O(u^5),\quad \psi(-x,-u)=-\psi(x,u), \qquad\qquad\qquad $(1)

with Navier boundary conditions in a $n$–dimensional cube, here $\psi$ is a $C^\infty$ function, and $m$ is a positive parameter. For this equation we construct some Cantor families of periodic orbits. Our proof is very simple and is based on contraction mapping principle and on a suitable correspondence between Lyapunov Schmidt decomposition and averaging theory.

Keywords: Lyapunov Schmidt decomposition., Periodic solutions, higher dimensional PDEs.

Mathematics Subject Classification: 37K55, 35B10, 35B1.

Citation: Dario Bambusi, Simone Paleari. Families of periodic orbits for some PDE’s in higher dimensions. Communications on Pure & Applied Analysis, 2002, 1 (2) : 269-279. doi: 10.3934/cpaa.2002.1.269

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