Quasi-stability and Upper Semicontinuity for Coupled Wave Equations with Fractional Damping

• Published in: Applied mathematics & optimization Vol. 89; no. 1; p. 26
• Main Authors: Qin, Yuming, Han, Xiaoyue
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2024; Springer Nature B.V
• Subjects: Applied mathematics; Attractors (mathematics); Calculus of Variations and Optimal Control; Optimization; Control; Damping; Dynamic stability; Dynamical systems; Hilbert space; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Metric space; Nonlinear dynamics; Nonlinear systems; Numerical and Computational Physics; Optimization; Simulation; Systems Theory; Theoretical; Wave equations; Upper semicontinuity; Quasi-stability; 37L05; Coupled wave equation; 35L05; 35B41; 26A15; 35B40; Fractional damping; Global attractors
• DOI: 10.1007/s00245-023-10072-8
• ISSN: 0095-4616

This paper deals with the long-time dynamics of a nonlinear system of coupled wave equations with fractional damping term and subjected to small perturbations of autonomous external forces. Inspired by the works of Chueshov and Lasiecka (J Dyn Differ Equ 16:469–512, 2004; Appl Math Optim Equ 58:195–241, 2008; Mem AMS 912:1–167, 2008; Von Karman evolution equations. Well-posedness and long time dynamics, Springer, New York, 2010) on the property of quasi-stability of dynamic systems, we show first the quasi-stability property holds for the problem considered here and then prove the existence of global and exponential attractors. We also prove the upper semicontinuity of global attractors with respect to the fractional exponent. Finally, we show the continuity of global attractors with respect to a pair of parameters in a residual dense set and their upper semicontinuities in a complete metric space.