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 ψ , namely ψ ′ ( t ) ≤ − μ ( t ) G ( ψ ( t ) ) , where G is a convex and increasing function near the origin, and μ is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions μ and G , as well as the function F defined by f 0 , which characterizes the growth behavior of f 1 at the origin.