Exponential attractors for the strongly damped wave equations

For the strongly damped wave equation with critical nonlinearity, we first show the existence of a (H01(Ω)×L2(Ω),H01(Ω)×H01(Ω))-global attractor when the external forcing g∈H−1g∈H−1; then we prove that for each T>0T>0 fixed, there is a bounded (in H2×H1H2×H1) set which attracts exponentially every H1×L2H1×L2-bounded set w.r.t. the stronger H1×H1H1×H1-norm for all t⩾Tt⩾T and has finite fractal dimension in H01(Ω)×H01(Ω) for the case g∈L2(Ω)g∈L2(Ω).