Well-posedness and asymptotic behavior of some evolution problems with delay

Abstract

In this thesis, we will study some evolution problems that represent some physical phenomena (Piezoelectric beam, Kirchhoff beam) with some types of delay (for example, distributed delay, neutral delay) acting on linear or nonlinear internal feedbacks. We will prove the well-posedness (existence and uniqueness) of solutions to these systems by semigroup theory or by Faedo--Galerkin method. With regard to the asymptotic behavior of the solutions, we will get the exponential decay of solutions, which represents the rapid decrease of energy, by constructing a Lyapunov functional using the multiplication method. Or we get the blow-up of solutions by using Georgiev and Todorova's method