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].