Estabilidade assintótica de uma classe de equações quasilineares viscoelásticas com história

Abstract

This work is concerned with the long-time behaviour of a class nonlinear viscoelastic equations of the form |ut|utt −∆u−∆utt + ∫ t −τ μ(t− s)∆u(s)ds = h, ρ > 0, defined in a bounded domain of R . Such class of problems was studied by several authors since 2001, with τ = 0. Existing results are mainly devoted to global existence, energy decay, with or without additional dampings, existence with small data, among others. However, uniqueness and existence of global attractors were not considered previously. In the present work, we establish some results on the uniqueness of solutions and existence of global attractors in a more general setting, including τ = −∞. In addition, we have added a second problem concerned with a fourth order equation where we study the existence of exponential attractors.