Attractors for a plate equation with nonlocal nonlinearities

Abstract

This paper contains results on well‐posedness, stability, and long‐time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of Rn with smooth boundary, γ,ρ⩾0 and frakturl1(u),frakturl2(u) are nonlocal functions. The main result states that the dynamical system {S(t)}t⩾0 associated with this problem has a compact global attractor. In addition, in the limit case γ = 0, it is also shown that {S(t)}t⩾0 has a finite dimensional global attractor by using an approach on quasi‐stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.

Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy

“…The Petrovsky type of equations in the form of u tt þ D 2 u ¼ gðx; t; u; u t Þ originated from the study of beams and plates. In the past decades, many researchers have studied the well-posedness, regularity, asymptotic behavior for such kinds of fourth order wave equations, and for more details we refer the reader to [1][2][3][4][5][6][7][8][9][10] and the references therein.…”

Blow-up of solutions to a viscoelastic wave equation with nonlocal damping

“…. By using the Faedo-Galerkin method, we now state the existence result of problem ( 1)-( 3), which can be established by combining with the argument of [8,15,12,26].…”

“…The above limits (53)-(54) along with (49), combining with the process as the proof of (2.18) and (2.19) in [26], are enough to pass the limit on the system (51)-( 52) and achieve the desired weak solution (see [12,26,8] for details).…”

“…The stability of systems like (4) with degenerate nonlocal damping is also considered by Pucci [28] and Cavalcanti et al [7]. There are many results about the well-posedness and asymptotic stability for the following more general model [15,12,26,20] and reference therein.…”

On a beam model with degenerate nonlocal nonlinear damping

“…The case nonlocal nonlinear damping of the form M ( ∇u(t) 2 )|u t | γ u t was first considered by Narciso in [35] for the plate equation. Well-posedness and longtime behavior were considered.…”