ASYMPTOTIC PROPERTIES FOR SOME DELAYED DIFFERENTIAL EQUATIONS

The aim of this thesis is to analyze the asymptotic behavior of solutions to some delayed differential equations. It's well-known that the presence of a delay term can induce instability. Here, we will study Hegselmann-Krause opinion formation models and abstract evolution equations. We will present the Hegselmann-Krause model with different kind of time delays. Afterwards, we will study a Hegselmann-Krause-type control system with leadership. We will show the controllability result under a smallness assumption on the delay and on the other parameters appearing in the model and some simulations will be given. We will focus our attention also to the energy decay of abstract evolution equations. These systems are fundamental for the study of semilinear wave-type equations with delay feedback and viscoelastic/frictional damping. We will show the stability property under some assumptions on the parameters and small initial data. We extend our results also to abstract evolution equations with an infinite number of constant time delays and Lipschitz continuous source terms. Finally, we will present a bifurcation theory for a simplified model for the blood flow control in the kidney. Via semigroup theory, we will study existence and uniqueness of solutions and we will give results of bifurcation depending on the coefficients appearing in the model. Some numerical examples will illustrate the validity of the theory presented.