Long-time behaviour for a non-autonomous Klein-Gordon-Zakharov system

TL;DR

This paper investigates the long-term behavior of solutions to a non-autonomous Klein-Gordon-Zakharov system, establishing well-posedness and the existence of pullback attractors in a bounded domain.

Contribution

It introduces a non-autonomous Klein-Gordon-Zakharov system, proving well-posedness and the existence of pullback attractors using sectorial operators theory.

Findings

Established local and global well-posedness in relevant Sobolev spaces.

Proved existence and regularity of pullback attractors.

Showed upper semicontinuity of attractors with respect to parameters.

Abstract

The aim of this paper is to study the long-time dynamics of solutions of the evolution system \[ \begin{cases} u_{tt} - \Delta u + u + \eta(-\Delta)^{\frac{1}{2}}u_t + a_{\epsilon}(t)(-\Delta)^{\frac{1}{2}}v_t = f(u), & \; (x, t) \in \Omega \times (\tau, \infty), \\ v_{tt} - \Delta v + \eta(-\Delta)^{\frac{1}{2}}v_t - a_{\epsilon}(t)(-\Delta)^{\frac{1}{2}}u_t = 0, & \; (x, t) \in \Omega \times (\tau, \infty), \end{cases} \] subject to boundary conditions \[ u = v = 0, \;\; (x, t)\in \partial\Omega\times (\tau, \infty), \] where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$, $n \geq 3$, with the boundary $\partial\Omega$ assumed to be regular enough, $\eta > 0$ is constant, $a_{\epsilon}$ is a H\"older continuous function and $f$ is a dissipative nonlinearity. This problem is a non-autonomous version of the well known Klein-Gordon-Zakharov system. Using the uniform sectorial operators theory, we will show the local and global well-posedness of this problem in $H_0^1(\Omega) \times L^2(\Omega) \times H_0^1(\Omega) \times L^2(\Omega)$. Additionally, we prove existence, regularity and upper semicontinuity of pullback attractors.