Non-autonomous Klein-Gordon-Zakharov system: pullback dynamics in the continuous and impulsive approaches

Abstract

This work is dedicated to study a non-autonomous formulation of the Klein-Gordon-Zakharov system, which is a coupled system consisting of two non-autonomous evolution equations, where each one is of second order in time. This model is closely related to the interaction of waves and it appears frequently in thermoelasticity, mechanics, plasma physics, and other areas alike.

The present work is divided into two main parts. In a first moment, using the uniform sectorial operators theory, we will show that our formulation has parabolic structure and then, making use of the natural energy associated to the system, we will obtain its global well-posedness. With the global solution in hands, we can define a nonlinear evolution process. Thus, in order to study the long-time dynamics of solutions, we shall use the abstract evolution processes theory to prove existence, regularity and upper semicontinuity of pullback attractors.

In the second main moment of this work, we are going to investigate the asymptotic dynamics of solutions of the non-autonomous Klein-Gordon-Zakharov system when they are subject to the action of impulses. To do that, we will study the qualitative properties of evolution processes under conditions of impulses and present sufficient conditions for the existence of pullback attractors for evolution processes in the impulsive scenario. Finally, we apply the abstract results in order to ensure the existence of an impulsive pullback attractor for the impulsive evolution process associated with the non-autonomous Klein-Gordon-Zakharov system with impulsive action.