The Existence Of The Global Attractor For A Strongly Damped Viscoelastic Wave Equation

Let Ω be a bounded domain in□3with smooth boundary. Given ε>0, we consider the following strongly damped viscoelastic wave equationWe prove the existence of the universal attractor for the above equation in the presence of a quite general nonlinearity of critical growth on the energy space Xo=D(A2)×L(Ω)×M1with M1={η|∫0∞μ(τ)‖A2η’(τ)‖2dτ<∞}. While in the subcritical case, we demonstrate the existence of an exponential attractor of optimal regularity, the basin of attraction of the exponential attraction is the whole phase-space. We also prove the boundedness of fractal dimension of a universal attractor. In the paper we pursue a detailed analysis of the longtime behavior of solutions in dependence of the damping coefficient co.