Attractors for semilinear wave equations with localized damping and external forces

Abstract

This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of R 3 with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global attractors established by Chueshov, Lasiecka and Toundykov (2008) reflects a good deal of the current state of the art on this matter. Our contribution is threefold. First, we prove uniform boundedness of attractors with respect to a forcing parameter. Then, we study the continuity of attractors with respect to the parameter in a residual dense set. Finally, we show the existence of generalized exponential attractors. These aspects were not previously considered for wave equations with localized damping.

Tools Used from Mathematics in The Study of a Riemannian Wave Equation to Prevent the Collapse of Bridges

“…The mathematical results seen in (Cavalcanti, Ma, Marín-Rubio, & Seminario-Huertas, 2021), (Lasiecka, Triggiani, & Zhang, 2000), (Ma & Seminario-Huertas, 2020), (Rauch & Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, 1975), (Triggiani & Yao, 2002), and (Yao, 2011) (2)…”

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

“…Definition 6.1. [35] Let Λ be a complete metric space and S σ (t) be a family of semigroups on H, where σ ∈ Λ. The global attractors A σ is called upper semi-continuous on σ ∈ Λ if…”

“…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.…”

Global attractor of the Euler-Bernoulli equations with a localized nonlinear damping

“…Then the quasi-stability property with multiplier methods as well as some estimates of Carleman type were established, which entail the existence of a finite dimensional global attractor. Based on these results, the authors [25] further proved the residual semi-continuity of the global attractor with respect to a small parameter and the existence of a generalized exponential attractor. Moreover, by developing a special version of Carleman's estimates and some special tangential estimates based on microlocal analysis, the existence of a finite dimensional global attractor and a generalized exponential attractor has also been investigated in [9] for the wave equation with nonlinear localized boundary damping and a source term of critical exponent.…”

“…For the wave equations with localized interior/boundary damping, there are some results concerning the existence of global attractors in [1,8,9,10,21,25]. In particular, the authors [10] considered the long-time behavior of solutions for the wave equation with localized nonlinear damping and critical nonlinearity for the first time, and proved the existence of a global attractor by using the multiplier methods, a unique continuation property of wave equation and the composition of the critical nonlinearity as in [1].…”