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

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.

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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