Lower semicontinuity of pullback attractors for a non-autonomous coupled system of strongly damped wave equations
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 $$\left\{ \begin{array}{lr} u_{tt} - Δu + u + η(-Δ)^{1/2}u_t + a_ε(t)(-Δ)^{1/2}v_t = f(u), &(x, t) \inΩ\times (τ, \infty),\\ v_{tt} - Δv + η(-Δ)^{1/2}v_t - a_ε(t)(-Δ)^{1/2}u_t = 0, &(x, t) \inΩ\times (τ, \infty),\end{array}\right.$$ subject to boundary conditions $$u = v = 0, \; (x, t) \in\partialΩ\times (τ, \infty),$$ and initial conditions $$u(τ, x) = u_0(x), \ u_t(τ, x) = u_1(x), \ v(τ, x) = v_0(x), \ v_t(τ, x) = v_1(x), \ x \in Ω, \ τ\in\mathbb{R},$$ where $Ω$ is a bounded smooth domain in $\mathbb{R}^n$, $n \geq 3$, with the boundary $\partialΩ$ assumed to be regular enough, $η> 0$ is a constant, $a_ε$ is a Hölder 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 $ε= 0$.