General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback

TL;DR

This paper investigates the energy decay behavior of a viscoelastic wave equation with a time-varying delay, establishing a general decay result that encompasses exponential and polynomial decay as special cases.

Contribution

The paper introduces a novel approach using energy and Lyapunov functionals to analyze decay rates in viscoelastic wave equations with non-positive delay coefficients.

Findings

Established a general energy decay result for the equation.

Derived conditions under which exponential and polynomial decay occur.

Provided a unified framework for decay analysis in delayed viscoelastic systems.

Abstract

In this paper we consider a viscoelastic wave equation with a time-varying delay term, the coefficient of which is not necessarily positive. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.