Asymptotic behavior for the damped Boussinesq equation with critical nonlinearity

This work is focused on the long-time behavior of solutions to the damped Boussinesq equationutt+Δ2u−Δut−Δf(u)=g(x),inΩ⊂R3, which describes the small nonlinear oscillations of elastic membranes in the presence of viscosity. The existence of the global attractor of optimal regularity and an exponential attractor are respectively established in the critical case.