Communications in Mathematical Sciences

Volume 15 (2017)

Number 6

On the global attractor of the damped Rosenau equation on the whole line

Pages: 1667 – 1684

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n6.a9

Authors

Deqin Zhou (College of Mathematics and Statistics, Chongqing University, Chongqing, China)

Chunlai Mu (College of Mathematics and Statistics, Chongqing University, Chongqing, China)

Ke Lin (College of Mathematics and Statistics, Chongqing University, Chongqing, China)

Abstract

We consider the asymptotic behaviour of the solution for the damped Rosenau equation on $\mathbb{R}^1$. By applying the $I-$ method and a variant form of Riesz-Rellich criteria, we prove that this damped Rosenau equation possesses a global attractor in $H^s (\mathbb{R})$ for any $s \in (\frac{1}{2} , 2)$. Moreover, the global attractor $\mathcal{A}_s$ is contained in $\mathbb{H}^2 (\mathbb{R})$ for any $s \in (\frac{1}{2} , 2)$. Our results establish the lower regularity of the global attractor for the damped Rosenau equation in fractional order Sobolev space and give a partial answer to the open problem in [D. Zhou and C. Mu, Appl. Anal., 1–10, 2016].

Keywords

Rosenau equation, global solution, global attractor

2010 Mathematics Subject Classification

Primary 35B40. Secondary 35B41, 35Q53.

Full Text (PDF format)

This work is in part supported by China Postdoctoral Science Foundation [grant No. 2016M592634], Chongqing Postdoctoral Science Special Foundation [grant No. Xm2016035], and NSFC [grants No. 11371384, 11571244 and 11571062].

Received 9 December 2016

Published 27 June 2017