Existence and Stability of Pullback Exponential Attractors for a Nonautonomous Semilinear Evolution Equation of Second Order

In this work we consider a family of nonautonomous semilinear evolution equations of second order given by \(\begin{aligned} \left\{ \begin{array}{l} u_{tt} - \Delta u - \eta _\epsilon (t) \Delta u_t - \Delta u_{tt} = f(u), \; t > s, \; x \in \Omega , \\ u=0, \; t \ge s, x \in \partial \Omega , \\ u(s,x) = u_{0}(x), \; \; u_{t}(s,x) = v_{0}(x), \; x \in \Omega , \end{array} \right. \end{aligned}\) where \(\Omega \) is a bounded smooth domain in \(\mathbb {R}^{N}\) with \(N\ge 3\) , \(\epsilon \in [0,1]\) is a parameter, \(\eta _\epsilon : \mathbb {R} \longrightarrow (0,\infty )\) is a continuously differentiable function satisfying \(0 < a_{1} \le \eta _\epsilon (t) \le a_{2} < \infty \) for all \(t \in \mathbb {R}\) with \(\lim _{\epsilon \rightarrow 0}\Vert \eta _\epsilon -\eta _0\Vert _{L^{\infty }(\mathbb R)}=0\) and it has a uniformly (with respect to \(t\in \mathbb R\) and \(\epsilon \in [0,1]\) ) bounded derivative, and \(f:\mathbb {R} \longrightarrow \mathbb {R}\) is a locally Lipschitz function satisfying suitable growth and dissipativeness conditions. For that problem we will present results on the global well-posedness of solutions for the nonautonomous semilinear evolution equation, existence and stability of pullback exponential attractors in space \(H^{1}_0(\Omega ) \times H^{1}_0(\Omega )\) . This material can be seen in more details in the article [1].