Long time behavior of solutions to the damped forced generalized Ostrovsky equation below the energy space

Zhang, Zaiyun; Liu, Zhenhai; Deng, Youjun; Huang, Jianhua; Huang, Chuangxia

doi:10.1090/proc/15322

Long time behavior of solutions to the damped forced generalized Ostrovsky equation below the energy space
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by Zaiyun Zhang, Zhenhai Liu, Youjun Deng, Jianhua Huang and Chuangxia Huang PDF

Proc. Amer. Math. Soc. 149 (2021), 1527-1542 Request permission

Abstract:

In this paper, we investigate the long time behavior of the damped forced generalized Ostrovsky equation below the energy space. First, by using Fourier restriction norm method and Tao’s $[k,Z]$- multiplier method, we establish the multi-linear estimates, including the bilinear and trilinear estimates on the Bourgain space $X_{s,b}.$ Then, combining the multi-linear estimates with the contraction mapping principle as well as $\widetilde {L}^{2}$ energy method, we establish the global well-posedness and existence of the bounded absorbing sets in $\widetilde {L}^{2}.$ Finally, we show the existence of global attractor in $\widetilde {L}^{2}$ and its compactness in $\widetilde {H}^{5}$ by means of the high-low frequency decomposition method, cut-off function, tail estimate together with Kuratowski $\alpha$-measure in order to overcome the non-compactness of the classical Sobolev embedding. This result improves earlier ones in the literatures, such as Goubet and Rosa [J. Differential Equations 185 (2002), no. 1, 25–53], Moise and Rosa [Adv. Differential Equations 2 (1997), no. 2, 251–296], Wang et al. [J. Math. Anal. Appl. 390 (2012), no. 1, 136–150], Wang [Discrete Contin. Dyn. Syst. 35 (2015), no. 8, 3799–3825], and Guo and Huo [J. Math. Anal. App. 329 (2007), no. 1, 392–407].

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