Indirect stabilization of nonlinear coupled wave equations in the presence of time delay or indefinite damping

TL;DR

This paper investigates the exponential stability of coupled nonlinear wave equations with time delay and indefinite damping, providing conditions for energy decay using semigroup theory and energy methods.

Contribution

It introduces a novel analysis of coupled wave equations with time delay and indefinite damping, establishing exponential stability under new conditions.

Findings

Exponential stability achieved with damping and delay

Weaker conditions ensure stability with indefinite damping

Well-posedness established for the coupled systems

Abstract

This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other experiences a discrete time delay. By reformulating the problem in an abstract framework, we use semigroup theory and energy methods to establish well-posedness and derive conditions that guarantee exponential energy decay. In the second case, we examine a scenario where a frictional damping term appears in the first equation, while the second equation contains an indefinite damping term, namely with a sign-changing coefficient. Although this setup can be viewed as a special case of the first, we analyze it separately and show that exponential stability still holds under a weaker condition.