Existence and asymptotic behavior of solutions for some hyperbolic equations with time delay

Abstract

The thesis aims to provide the reader with how to use the most popular method for studying the existence and uniqueness of the solution and the general energy decay of some wave problems with strong delay and distributed delay, similar to the Kirchhoff system and the Lam´e system. The first chapter deals with introducing some basic notions in bounded and unbounded operators and some main theorems in functional analysis. In second chapter, we proved the well-posedness and an exponential decay result under a suitable assumptions on the weight of the damping and the weight of the delay for a wave equation with a strong damping and a strong constant (respectively, distributed) delay. Finally "third and forth chapters", we proved the global existence of Kirchhoff’s coupled system and general decay for the coupled system of Kirchhoff and Lam´e with a distributed term delay.