Longtime dynamics of the damped Boussinesq equation

The paper studies the longtime dynamics of the damped Boussinesq equation utt+Δ2u−Δut−Δg(u)=f(x). First, the existence of global solutions to the initial boundary value problem of the equation is obtained provided that the growth exponent of g(u), say p, is either non-supercritical (subcritical and critical) or supercritical, especially, the stability of solutions is established when p is non-supercritical. Second, the existence of a global attractor and an exponential attractor for the related solution semigroup S(t) are respectively established in the non-supercritical case.