General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions

This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition a(2)>0 and without imposing any restrictive growth assumption on the damping term f(1), using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function psi, namely psi'(t)<=-mu(t)G(psi(t)), where G is a convex and increasing function near the origin, and mu is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions mu and G, as well as the function F defined by f(0), which characterizes the growth behavior of f1 at the origin.