This paper is concerned with the long-time dynamics of a semilinear wave equation with degenerate viscoelasticity
defined in a bounded domain 惟 of R3, with Dirichlet boundary condition and nonlinear forcing f(u) with critical growth. The problem is degenerate in the sense that the function a(x)≥0 in the memory term is allowed to vanish in a part of 658c">. When 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)ut, which is in a certain sense arbitrarily small, such that a+b>0 in 658c">. 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.