Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$

Abstract

We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.2000 Mathematics Subject Classification. 35B41, 35G20, 74K20.

Long-time dynamics of the strongly damped semilinear plate equation inRN

“…The situation becomes more difficult when the equation contains localized damping terms and nonlocal nonlinearities. Recently, in [12] and [13], the plate equation with localized weak damping (the case β ≡ 0 in (1.1)) and involving nonlocal nonlinearities as −f ( ∇u L 2 (R n ) )∆u and f ( u L p (R n ) ) |u| p−2 u have been considered.…”

“…In these articles, the existence of global attractors has been proved when the coefficient α(·) of the weak damping term is strictly positive (see [12]) or, in addition to (1.3), is positive (see [13]) almost everywhere in R n . However, in the case when α(·) vanishes in a set of positive measure, the existence of the global attractor for (1.1) with β ≡ 0 remained as an open question (see [12,Remark 1.2]).…”

Attractors for a plate equation with nonlocal nonlinearities

“…We note that nonlocal sources have been recently approached in the theory of plates. Indeed, in Khanmamedov and Simsek it is considered the following plate equation The authors prove, under proper conditions on the damping coefficient, that the dynamical system associated with possesses a global attractor. In particular, the nonlocal source with ρ = 0 arises in Kerr‐like medium models (e.g., ).…”

“…We note that nonlocal sources have been recently approached in the theory of plates. Indeed, in Khanmamedov and Simsek [6] it is considered the following plate equation…”

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

“…For instance, in the case when σ is identically zero, equation (1.1) becomes an equation with polynomial nonlinearity which arises in aeroelasticity modeling (see, for example, [14,15]), and the problem with (or without) damping, memory, timedelay etc. were studied extensively (see [17,21,25,26,27,33,34,37,40,55,44] and references therein for the topics on well-posedness, global existence, finite time blow-up, global attractor etc. ).…”