Exponential decay estimates for semilinear wave-type equations with time-dependent time delay

TL;DR

This paper investigates the exponential decay and stability of semilinear wave equations with time-dependent delays, establishing conditions for well-posedness and demonstrating the effectiveness of Lyapunov methods.

Contribution

It introduces new exponential decay estimates for wave equations with time-varying delays, extending existing stability results to more general delay feedback scenarios.

Findings

Proved well-posedness for small initial data.

Established exponential stability under certain conditions.

Applied results to specific damped wave equations with delay feedback.

Abstract

In this paper, we analyze a semilinear damped second order evolution equation with time-dependent time delay and time-dependent delay feedback coefficient. The nonlinear term satisfies a local Lipschitz continuity assumption. Under appropriate conditions, we prove well-posedness and exponential stability of our model for small initial data. Our arguments combine a Lyapunov functional approach with some continuity arguments. Moreover, as an application of our abstract results, the damped wave equation with a source term and delay feedback is analyzed.