On the global attractor for the damped Benjamin-Bona-Mahony equation

Milena Stanislavova

doi:10.3934/proc.2005.2005.824

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2005, Volume 2005: 824-832. Doi: 10.3934/proc.2005.2005.824 open access

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On the global attractor for the damped Benjamin-Bona-Mahony equation

Milena Stanislavova^1,

1.
Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523

Received: August 31, 2004

Revised: March 31, 2005

Published: August 2005

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Abstract

We present a new necessary and sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood-Paley projection operators. We then use this condition to prove the existence of an attractor for the damped \bbme in the phase space $H^1({\bf R})$ by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to $H^{3/2-\ve}$ for every $\ve>0$.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41.

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