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

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 ] [k,Z] - multiplier method, we establish the multi-linear estimates, including the bilinear and trilinear estimates on the Bourgain space X s , b . X_{s,b}. Then, combining the multi-linear estimates with the contraction mapping principle as well as L ~ 2 \widetilde {L}^{2} energy method, we establish the global well-posedness and existence of the bounded absorbing sets in L ~ 2 . \widetilde {L}^{2}. Finally, we show the existence of global attractor in L ~ 2 \widetilde {L}^{2} and its compactness in H ~ 5 \widetilde {H}^{5} by means of the high-low frequency decomposition method, cut-off function, tail estimate together with Kuratowski α \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].