July/August 2023 Impulsive evolution processes: abstract results and an application to a coupled wave equations
Everaldo M. Bonotto, Marcelo J.D. Nascimento, Eric B. Santiago
Adv. Differential Equations 28(7/8): 569-612 (July/August 2023). DOI: 10.57262/ade028-0708-569

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.

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Everaldo M. Bonotto. Marcelo J.D. Nascimento. Eric B. Santiago. "Impulsive evolution processes: abstract results and an application to a coupled wave equations." Adv. Differential Equations 28 (7/8) 569 - 612, July/August 2023. https://doi.org/10.57262/ade028-0708-569

Information

Published: July/August 2023
First available in Project Euclid: 10 April 2023

Digital Object Identifier: 10.57262/ade028-0708-569

Subjects:
Primary: 35B40 , 35B41 , 35K40 , 37B55

Rights: Copyright © 2023 Khayyam Publishing, Inc.

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Vol.28 • No. 7/8 • July/August 2023
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