Lower semicontinuity of pullback attractors for a non-autonomous coupled system of strongly damped wave equations

Abstract

The aim of this paper is to study the robustness of the family of pullback attractors associated to a non-autonomous coupled system of strongly damped wave equations, given by the following evolution system $$\left\{ \begin{array}{lr} u_{tt} - \Delta u + u + \eta(-\Delta)^{1/2}u_t + a_{\epsilon}(t)(-\Delta)^{1/2}v_t = f(u), &(x, t) \in\Omega\times (\tau, \infty),\\ v_{tt} - \Delta v + \eta(-\Delta)^{1/2}v_t - a_{\epsilon}(t)(-\Delta)^{1/2}u_t = 0, &(x, t) \in\Omega\times (\tau, \infty),\end{array}\right.$$ subject to boundary conditions $$u = v = 0, \; (x, t) \in\partial\Omega\times (\tau, \infty),$$ and initial conditions $$u(\tau, x) = u_0(x), \ u_t(\tau, x) = u_1(x), \ v(\tau, x) = v_0(x), \ v_t(\tau, x) = v_1(x), \ x \in \Omega, \ \tau\in\mathbb{R},$$ 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 a constant, $a_{\epsilon}$ is a H\"{o}lder continuous function satisfying uniform boundedness conditions, and $f\in C^1(\mathbb{R})$ is a dissipative nonlinearity with subcritical growth. This problem is a modified version of the well known Klein-Gordon-Zakharov system. Under suitable hyperbolicity conditions, we obtain the gradient-like structure of the limit pullback attractor associated with this evolution system, and we prove the continuity of the family of pullback attractors at $\epsilon = 0$.