On the regularity of the global attractor for a damped Rosenau equation on R

Abstract

In this paper, we consider the asymptotic behaviour of the solution for a damped Rosenau equation on . We apply a variant of Riesz–Rellich criteria, which involves the Littlewood–Paley projection operators, to prove that the damped Rosenau equation possesses a global attractor in for any . Moreover, the global attractor is contained in , if the time-independent source term is in and the initial data are in . Our results establish the regularity of the global attractor for the damped Rosenau equation in fractional order Sobolev space, which is a new ingredient in this paper.

Keywords:

• Rosenau equation
• global attractor
• global solution

AMS Subject Classifications:

• 35B40
• 35B41
• 35Q53

Acknowledgements

The authors would like to thank two anonymous referees for their valuable comments and suggestions which help improve the quality of this paper. Especially, the authors want to thank one reviewer for pointing out a mistake in the proof of Theorem 1.1, and the other reviewer’s suggestion to Corollary 1.4 which did not appear in the first version of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by China Postdoctoral Science Foundation [grant number 2016M592634]", The Fundamental Research Funds for the Central Universities [grant number 106112016CDJCR101207] and NSFC [grant number 11571244]; the third author is supported by NSFC [grant number 11371384], [grant number 11571062] and the Basic and Advanced Research Project of CQC-STC [grant number cstc2015jcyjBX0007].