학술논문
Impulsive evolution processes: abstract results and an application to a coupled wave equations.
Document Type
Journal
Author
Bonotto, Everaldo M. (BR-SPL3-CMC) AMS Author Profile; Nascimento, Marcelo J. D. (BR-SACA) AMS Author Profile; Santiago, Eric B. (BR-SPL3-CMC) AMS Author Profile
Source
Subject
35 Partial differential equations -- 35B Qualitative properties of solutions
35B40Asymptotic behavior of solutions
35Partial differential equations -- 35K Parabolic equations and systems
35K40Second-order parabolic systems
37Dynamical systems and ergodic theory -- 37B Topological dynamics
37B55Nonautonomous dynamical systems
35B40
35
35K40
37
37B55
Language
English
Abstract
Summary: ``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 $$ \cases u_{tt} - \Delta u + u + \eta(-\Delta)^{\frac 12} u_t + a_\varepsilon(t)(-\Delta)^{\frac 12} v_t = f (u), & (x, t) \in\Omega \times (\tau, \infty),\\ v_{tt} - \Delta v + \eta(-\Delta)^{\frac 12} v_t - a_\varepsilon(t)(-\Delta)^{\frac 12} u_t = 0, & (x, t) \in\Omega\times (\tau, \infty),\\ u=v=0 &(x,t)\in\partial\Omega\times(\tau,\infty),\\ \{I_t : M (t) \subset Y_0 \to Y_0\}_{t\in \Bbb R}, \endcases $$ with initial conditions $u(\tau, x) = u_0(x), u_t(\tau, x) = u_1(x), v(\tau, x) = v_0(x), v_t(\tau, x) = v_1(x), x \in\Omega, \tau \in \Bbb R$, 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 function and $Y_0$ is a Hilbert space.''