Longtime dynamics of the damped Boussinesq equation

The paper studies the longtime dynamics of the damped Boussinesq equation u t t + Δ 2 u − Δ u t − Δ g ( 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.