On a Kelvin--Voigt Viscoelastic Wave Equation with Strong Delay

Abstract

An initial-boundary value problem for a viscoelastic wave equation subject to a strong time-localized delay in a Kelvin--Voigt-type material law is considered. After transforming the equation to an abstract Cauchy problem on the extended phase space, a global well-posedness theory is established using the operator semigroup theory both in Sobolev-valued $C^{0}$- and $\mathrm{BV}$-spaces. Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained and the stability region in the parameter space is further explored using Lyapunov's indirect method. The singular limit $\tau \to 0$ is studied with the aid of the energy method. Finally, a numerical example from a real-world application in biomechanics is presented.