Well-posedness and general decay of solution for a transmission problem with viscoelastic term and delay

TL;DR

This paper investigates the well-posedness and decay behavior of solutions to a transmission problem involving viscoelastic and delay effects, establishing general decay results that encompass exponential and polynomial decay.

Contribution

It proves well-posedness using Faedo-Galerkin method and introduces Lyaponov functionals to establish a unified decay result including exponential and polynomial cases.

Findings

Proved well-posedness of the transmission problem

Established general decay results for solutions

Unified decay behavior including exponential and polynomial types

Abstract

In this paper, we consider a transmission problem in a bounded domain with a viscoelastic term and a delay term. Under appropriate hypothesis on the relaxation function and the relationship between the weight of the damping and the weight of the delay, we prove the well-posedness result by using Faedo-Galerkin method. By introducing suitable Lyaponov functionals, we establish a general decay result, from which the exponential and polynomial types of decay are only special cases.