Asymptotics of viscoelastic materials with nonlinear density and memory effects

This paper is concerned with a nonlinear viscoelastic equation with past history. This class of equations was studied by many authors, but well-posedness in the whole admissible range of the involved parameters and for a nonlinearity growing up to the critical exponent were established only recently. The existence of global attractors was proved in presence of an additional viscous or frictional damping. In the present work we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor and for critical nonlinearities. Moreover, for a large class of dissipative nonlinearities, we prove the exponential decay of the associated energy.