Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well‐posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system associated with the problem in the space and , respectively. After that, the asymptotic smoothness of the dynamical system and the further quasi‐stability are demonstrated by the energy reconstruction method. Finally, we show not only the existence of the finite global attractor but also the existence of the generalized exponential attractor.