Global attractor for the damped Benjamin–Bona–Mahony equations on R¹

Abstract

We utilize a new necessary and sufficient condition to verity 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 Benjamin–Bona–Mahony equation in the phase space H ¹(R ¹) by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to H ^3/2−ϵ for every ϵ>0.

Keywords:

• Damped Benjamin–Bona–Mahony equation
• Global attractor
• Asymptotic behavior

Keywords:

• 2000 Mathematics Subject Classifications: 35B40
• 35B41

Acknowledgment

This work is supported in part by program for New Century Excellent Talents in University and in part by NNSF of China (10571126).