Impulsive evolution processes: abstract results and an application to a coupled wave equations

Abstract

The aim of this paper is to study the long-time behavior of impulsive evolution processes. We obtain qualitative properties for impulsive evolution processes, and we prove an existence result of impulsive pullback attractors. Additionally, we provide sufficient conditions to obtain the upper semicontinuity at zero for a family of impulsive pullback attractors. As an application, we study the asymptotic dynamics of the following non-autonomous coupled wave system subject to impulsive effects at variable times, given by the following evolution system $$ \begin{cases} u_{tt} - \Delta u + u + \eta(-\Delta)^{\frac{1}{2}}u_t + a_{\epsilon}(t)(-\Delta)^{\frac{1}{2}}v_t = f(u), & (x, t) \in \Omega \times (\tau, \infty), \\ v_{tt} - \Delta v + \eta(-\Delta)^{\frac{1}{2}}v_t - a_{\epsilon}(t)(-\Delta)^{\frac{1}{2}}u_t = 0, & (x, t) \in \Omega \times (\tau, \infty), \\ u = v = 0, & (x, t) \in \partial\Omega\times (\tau, \infty), \\ \left\{I_t\colon M(t)\subset Y_0\to Y_0\right\}_{t \in \mathbb R}, \end{cases} $$ with 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 boundary $\partial\Omega$ assumed to be regular enough, $\eta > 0$ is constant, $a_{\epsilon}$ and $f$ are suitable functions, the family $\hat{M} = \{M(t)\}_{t \in \mathbb R}$ is an impulsive family, $\hat{I} = \{I_t\colon M(t)\subset Y_0\to Y_0\}_{t \in \mathbb R}$ is an impulse function and $Y_0$ is a Hilbert space.