Exponential and polynomial decay for a quasilinear viscoelastic equation

We consider the following nonlinear viscoelastic equation: |ut|ρutt−Δu−Δutt+∫0tg(t−s)Δu(s)ds=0, in a bounded domain ΩΩ for ρ>0ρ>0. We show that the dissipation induced by the integral term is strong enough to stabilize the solution. This result improves an earlier one given by Cavalcanti et al. in [M.M. Cavalcanti, V.N.D. Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043–1053].