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 $$\{\begin{array}{lr}{u}_{tt}-\mathrm{\Delta }u+u+\eta (-\mathrm{\Delta }{)}^{1\mathrm{/}2}{u}_t+a_{\epsilon }(t)(-\mathrm{\Delta }{)}^{1\mathrm{/}2}{v}_t=f(u),& (x,t)\in \mathrm{\Omega }\times (\tau ,\mathrm{\infty }),\\ v_{tt}-\mathrm{\Delta }v+\eta (-\mathrm{\Delta }{)}^{1\mathrm{/}2}{v}_t-a_{\epsilon }(t)(-\mathrm{\Delta }{)}^{1\mathrm{/}2}{u}_t=0,& (x,t)\in \mathrm{\Omega }\times (\tau ,\mathrm{\infty }),\end{array}$$ subject to boundary conditions $$u=v=0,(x,t)\in \mathrm{\partial }\mathrm{\Omega }\times (\tau ,\mathrm{\infty }),$$ and initial conditions $$u(\tau ,x)=u_0(x),u_t(\tau ,x)=u_1(x),v(\tau ,x)=v_0(x),v_t(\tau ,x)=v_1(x),x\in \mathrm{\Omega },\tau \in \mathbb{R},$$ where $\mathrm{\Omega }$ is a bounded smooth domain in ${\mathbb{R}}^n$, $n\ge 3$, with the boundary $\mathrm{\partial }\mathrm{\Omega }$ assumed to be regular enough, $\eta >0$ is a constant, $a_{\epsilon }$ 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 $\epsilon =0$.