Energy decay for semilinear evolution equations with memory and time-dependent time delay feedback

TL;DR

This paper investigates the stability and decay of solutions to semilinear evolution equations with memory effects and time-dependent delays, establishing conditions for exponential decay of small initial data.

Contribution

It introduces new conditions on delay feedback that ensure global existence and exponential decay for semilinear equations with memory and delays.

Findings

Solutions with small initial data are globally well-posed.

Under certain conditions, solutions exhibit exponential decay.

The delay feedback's properties are crucial for stability.

Abstract

In this paper, we study well-posedness and exponential stability for semilinear second order evolution equations with memory and time-varying delay feedback. The time delay function is assumed to be continuous and bounded. Under a suitable assumption on the delay feedback, we are able to prove that solutions corresponding to small initial data are globally defined and satisfy an exponential decay estimate.