Time-dependent attractor  for the Oscillon equation

Francesco Di Plinio; Gregory S. Duane; Roger Temam

doi:10.3934/dcds.2011.29.141

Article Contents

2011, Volume 29, Issue 1: 141-167. Doi: 10.3934/dcds.2011.29.141

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Time-dependent attractor for the Oscillon equation

1.
Indiana University Mathematics Department, Bloomington, IN 47405
2.
Rosenstiel School of Marine and Atmospheric Sciences, University of Miami, Miami, FL 33149
3.
The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405

Received: January 2010

Revised: May 2010

Published: January 2011

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Abstract

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation

∂ _tt $u(x,t) +H $ ∂ _t$ u(x,t) -\e^{-2Ht}$ ∂ _xx $ u(x,t) + V'(u(x,t)) =0, \quad (x,t)\in (0,1) \times \R,$

with periodic boundary conditions, where $H>0$ is the Hubble constant and $V$ is a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular global attractor $\A=\A(t)$. The kernel sections $\A(t)$ have finite fractal dimension.

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Mathematics Subject Classification: Primary: 37L30, 35B41; Secondary: 83D05.

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