Quasi-stability and continuity of attractors for nonlinear system of wave equations

Abstract

In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

Long-time dynamics for a piezoelectric system with time-varying delay and memory

“…Proof. Follow the same arguments as in [17] and take Λ = [0, 1]. Then the assumption (H1) holds by Theorem 5.1.…”

“…In section 5, we apply the infinite dynamical system theory to establish the existence of global attractors and generalized exponential attractors. In section 6, the upper semicontinuity of global attractors is obtained by the similar argument as in [17]. In section 7, the exponential decay of system (1.1)-(1.3) under the condition (7.1) is proved by constructing some differential inequalities.…”

“…Moreover, Ma et al [35] and Freitas et al [17,18] investigated a system with external forces with the parameter ε ∈ [0, 1], they mainly obtained the properties and upper-semicontinuity of the attractors with respect to the perturbations.…”

Long-time dynamical behavior for a piezoelectric system with magnetic effect and nonlinear dampings

“…In this paper, they mainly proved the existence and properties of global attractors and attractors are continuous under autonomous perturbations. Freitas et al [16] investigated the following wave equations:…”