Well-posedness and stability for semilinear wave-type equations with time delay

TL;DR

This paper investigates the well-posedness and exponential stability of semilinear damped wave equations with time delay, demonstrating results under variable delay feedback and small initial data through energy estimates.

Contribution

It introduces new conditions for stability and well-posedness of wave equations with variable time delay feedback, extending existing theories.

Findings

Proves well-posedness under certain assumptions.

Establishes exponential stability for small initial data.

Provides examples illustrating the abstract results.

Abstract

In this paper we analyze a semilinear abstract damped wave-type equation with time delay. We assume that the delay feedback coefficient is variable in time and belonging to $L^1_{loc}([0, +\infty)).$ Under suitable assumptions, we show well-posedness and exponential stability for small initial data. Our strategy combines careful energy estimates and continuity arguments. Some examples illustrate the abstract results.