Smooth attractors for strongly damped wave equations

Abstract

This paper is concerned with the semilinear strongly damped wave equationThe existence of compact global attractors of optimal regularity is proved for nonlinearities ϕ of critical and supercritical growth.

Asymptotic behavior for a semilinear second order evolution equation

“…We study the long-time behavior of the following semilinear evolution equation of second order in time: When ε = 0, (E 0 ) is the usual strongly damped wave equation, and its asymptotic behavior has been studied extensively in terms of attractors; see [4,5,7,13,16,23,25,32,35,36].…”

Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity

“…7) ρ(x) = (1 + η|x| 2 ) −l , with l > N 2 , η > 0, where, obviously, one can obtain the estimates that |∇ρ| ≤ C √ ηρ and |∆ρ| ≤ Cη ρ. Problem (1.1)-(1.2) has many relevant physical applications (e.g., see [28] for a summary), and its dynamics have been studied extensively by many authors; see [5,6,7,13,27,28] and the reference therein. The existence and smoothness of the global attractor for the case where the spacial domain is bounded have been achieved recently, e.g., see [6,7,27,28].…”

“…Furthermore, when the spacial domain is a bounded domain Ω, for the system (1.1)-(1.2) with homogeneous Dirichlet boundary conditions, the authors in [27] and [28] have proven that the corresponding (H (Ω) for subcritical and critical cases respectively. Under the framework of locally uniform spaces, the authors in [10,16,31,34] have considered the weak dissipative wave equations, i.e., α = 0 and β > 0.…”

“…This is an essential difficulty in studying the asymptotic behavior, even for the case that the spacial domain is bounded, e.g., see [6,27,28].…”

Asymptotic Regularity and Uniform Attractor for Non-autonomous Viscoelastic Equations with Memory

“…In [10,11] etc, authors studied this equations with Dirichlet boundary conditions as 0   . Recently, Ara jo et al [5] and M. Conti [4], H. Yassine and A. Abbas [9] studied the well posedness for this equations.…”