Asymptotics of viscoelastic materials with nonlinear density and memory effects

Abstract

This paper is concerned with the nonlinear viscoelastic equation

$$|{\partial }_{t}u{|}^{\rho }{\partial }_{tt}u-\mathrm{\Delta }{\partial }_{tt}u-\mathrm{\Delta }u+\underset{0}{\overset{\infty }{\int }}\mu (s)\mathrm{\Delta }u(t-s)\mathrm{d}s+f(u)=h,$$

suitable to modeling extensional vibrations of thin rods with nonlinear material density $\varrho ({\partial }_{t}u)=|{\partial }_{t}u{|}^{\rho }$, and presence of memory effects. This class of equations was studied by many authors, but well-posedness in the whole admissible range $\rho \in [0,4]$ and for f 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 ${\mathcal{A}}_{\rho }$ for $\rho <4$ and critical nonlinearity f.