Long-time behavior of a quasilinear viscoelastic equation with past history

This paper is concerned with a class of quasilinear wave equations with memory|ut|ρutt−αΔu−Δutt+∫τtμ(t−s)Δu(s)ds−γΔut+f(u)=h,ρ>0, which was considered by several authors, with τ=0τ=0, since 2001. Existing results are mainly devoted to global existence, energy decay, existence with small data and blow-up of solutions. However uniqueness seems to be an open problem and existence of attractors was no yet considered. The objective of the present paper is to provide some results on the well-posedness and long-time behavior to this equation in a more general setting which includes past history, that is, by taking τ=−∞τ=−∞ in the memory term.