Global attractors for nonlinear viscoelastic equations with memory

We study the asymptotic properties of the semigroup $S(t)$ arising from thenonlinear viscoelastic equation with hereditary memoryon a bounded three-dimensional domain{eqnarray}|∂_t u|^ρ ∂t t u-Δ ∂t t u-Δ ∂_t u\\ -(1+∫_0^∞ μ(s)Δ s )Δ u+∫_0^∞ μ(s)Δ u(t-s)Δ s +f(u)=h{eqnarray}written in the past history framework of Dafermos [10].We establish the existence of the global attractor of optimal regularity for $S(t)$when ρ∈ [0,4)and f has polynomial growth of (at most) critical order 5.