Existence Of Global Solutions And Attractor To A Class Of Wave Equations With Nonlinear Damping And Linear Memory Term

LetΩbe a bounded open domain in R³ with smooth boundary. We consider a class of semilinear wave equations with nonlinear damping and linear memory term u_tt+g(u_t)-K(0)Δu-∫₀^∞K'(s)Δu(t-s)ds+f(u)=h(x),(?)(x,t)∈Ω×R⁺. Where f is supposed to satisfying the growth condition |f′(z)|≤c₄ (1+|z|^p), e p≤2.Based on the Faedo-Galerkin approach method, we proved the global existence of the strong and weak solutions and study long time dynamics. It is well known that the Poincaréimbedding is not compact when p = 2, which is an essential difficulty in proving the existence of global attractor. In this paper, based on a new a priori estimate method given in [14,15,16], we prove the asymptotic compactness of the semigroup in phase space V¹Ã—L² (Ω)×M_μ,1, for the above equations, hence the existence of the global attractor is obtained.