부산대학교 도서관

Summary: ``The aim of this paper is to study the long-time behavior ofimpulsive evolution processes. We obtain qualitative properties forimpulsive evolution processes, and we prove an existence result ofimpulsive pullback attractors. Additionally, we provide sufficientconditions to obtain the upper semicontinuity at zero for a family ofimpulsive pullback attractors. As an application, we study theasymptotic dynamics of the following non-autonomous coupled wave systemsubject to impulsive effects at variable times, given by the followingevolution system$$\cases u_{tt} - \Delta u + u + \eta(-\Delta)^{\frac 12} u_t +a_\varepsilon(t)(-\Delta)^{\frac 12} v_t = f (u), & (x, t) \in\Omega\times (\tau, \infty),\\ v_{tt} - \Delta v + \eta(-\Delta)^{\frac 12} v_t -a_\varepsilon(t)(-\Delta)^{\frac 12} u_t = 0, & (x, t) \in\Omega\times(\tau, \infty),\\ u=v=0 &(x,t)\in\partial\Omega\times(\tau,\infty),\\ \{I_t : M(t) \subset Y_0 \to Y_0\}_{t\in \Bbb R},\endcases$$with initial conditions $u(\tau, x) = u_0(x), u_t(\tau, x) = u_1(x),v(\tau, x) = v_0(x), v_t(\tau, x) = v_1(x), x \in\Omega, \tau \in \BbbR$, where $\Omega$ is a bounded smooth domain in $\Bbb R^n (n \geq 3)$with boundary $\partial\Omega$ assumed to be regular enough, $\eta > 0$is constant, $a_\varepsilon$ and $f$ are suitable functions, the family$\hat M = \{M (t)\}_{t\in\Bbb R}$ is an impulsive family, $\hat I =\{I_t : M (t) \subset Y_0 to Y_0\}_{t\in\Bbb R}$ is an impulse functionand $Y_0$ is a Hilbert space.''