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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

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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