Existence and stability for abstract linear and nonlinear evolution problems with single or multiple delay terms

Abstract

In this work, we were interested in studying the existence and stability of a general model of evolution equations with a delay term. By the theory of semi-groups and under appropriate conditions on the delayed damping term, we established the global existence and the exponential stability of linear and non-linear evolution systems with a constant delay first and then with a variable and multiple delay term considering a local Lipshitz source term. The exponential stability of the solution was obtained directly from the Duhamel formula of the solution in the three cases cited. The research aims to understand how delays affect the behaviour and stability of these systems. We terminated this work by illustrating the results obtained with some applications.