Higher regularity of global attractor for a damped Benjamin–Bona–Mahony equation on R

Abstract

For one-dimensional damped Benjamin–Bona–Mahony equation, a recent result claimed that there exists a global attractor in , in fact the attractor is smoother and it belongs to for every . In this paper, by some techniques of harmonic analysis, we first show that the solution operator of one-dimensional damped BBM equation is asymptotically compact, then we give the existence of global attractor in . Finally, we obtain the higher regularity of the global attractor, it is in fact bounded in .

Keywords:

• Benjamin–Bona–Mahony equation
• global attractor
• regularity

AMS Subject Classifications:

• 35K57
• 35C07
• 35R60
• 35B40

Acknowledgements

The authors are grateful to the anonymous referee for helpful comments and suggestions that greatly improved the presentation of this paper. For example, the referee gave us a new idea to establish the estimate (3.123.12 ) and so on.

Notes

This work is supported by NSF of China [grant number 11272277], [grant number 61304175]; FRF for the Central Universities of China [grant number 2013ZZGH027].