On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay

TL;DR

This paper investigates a viscoelastic wave equation with a strong delay in a Kelvin-Voigt material, establishing well-posedness, stability, and decay properties, and explores the singular limit with applications in biomechanics.

Contribution

It introduces a novel analysis of a delayed Kelvin-Voigt wave equation, including well-posedness, exponential stability, and the singular limit, with a real-world biomechanics application.

Findings

Global well-posedness in Sobolev and BV spaces

Exponential decay rate under certain conditions

Numerical example demonstrating biomechanics application

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. 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 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 the Lyapunov's indirect method. The singular limit $\tau \to 0$ is further studied with the aid of the energy method. Finally, a numerical example from a real-world application in biomechanics is presented.