Attractors for wave equations with degenerate memory

This paper is concerned with the long-time dynamics of a semilinear wave equation with degenerate viscoelasticityutt−Δu+∫0∞g(s)div[a(x)∇u(t−s)]ds+f(u)=h(x), defined in a bounded domain Ω of R3R3, with Dirichlet boundary condition and nonlinear forcing f(u)f(u) with critical growth. The problem is degenerate in the sense that the function a(x)≥0a(x)≥0 in the memory term is allowed to vanish in a part of Ω‾. When a(x)a(x) does not degenerate and g   decays exponentially it is well-known that the corresponding dynamical system has a global attractor without any extra dissipation. In the present work we consider the degenerate case by adding a complementary frictional damping b(x)utb(x)ut, which is in a certain sense arbitrarily small, such that a+b>0a+b>0 in Ω‾. Despite that the dissipation is given by two partial damping terms of different nature, none of them necessarily satisfying a geometric control condition, we establish the existence of a global attractor with finite-fractal dimension.