Longtime Dynamics Of The Boussinesq Type Equation With Fractional Damping

The paper is concerned with long dynamics of the Boussinesq equation with fractional damping:where ??(0,1),?(?)RN is a bounded domain with smooth boundary (?)?, g(u) is the nonlinear term, and f is external force. Under the dissipativeness condition and the growth condition on nonlinear term, this paper proves the well-posedness of the above mentioned problem in the space X = H01(?)×H-1(?), and both the energy identity and strong continuity of solutions with to time are valid. We prove that the related dynamical system possesses a global attractor, and we establish exponential attractor by virtue of weak quasi-stability estimates.