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 $$\{\begin{array}{lr}{u}_{tt}-\Delta u+u+\eta (-\Delta {)}^{1/2}{u}_t+a_{\epsilon }(t)(-\Delta {)}^{1/2}{v}_t=f(u),& (x,t)\in \Omega \times (\tau ,\infty ),\\ v_{tt}-\Delta v+\eta (-\Delta {)}^{1/2}{v}_t-a_{\epsilon }(t)(-\Delta {)}^{1/2}{u}_t=0,& (x,t)\in \Omega \times (\tau ,\infty ),\end{array}$$ 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 ${\mathbb{R}}^n$, $n\ge 3$, with the boundary $\partial \Omega$ 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 $\epsilon =0$.

This new version of the paper contains several improvements and corrections in the results