Global attractors for the plate equation with nonlocal nonlinearity in unbounded domains

Abstract

We consider the initial value problem for the semilinear plate equation with nonlocal nonlinearity. We prove the existence of global attractor and then establish the regularity and finite dimensionality of this attractor.Moreover, if (u 0 , u 1 ) ∈ H 4 (R n )×H 2 (R n ), then u is a strong solution from the class C [0, ∞); H 4 (R n ). Therefore, the problem (1.1)-(1.2) generates a strongly continuous semigroup {S (t)} t≥0 in H 2 (R n ) × L 2 (R n ) by the formula (u (t) , u t (t)) = S (t) (u 0 , u 1 ),2000 Mathematics Subject Classification. 35B41, 35G20, 37L30, 74K20.

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

“…Applying the semigroup theory (see [4, p.56-58]) and repeating the arguments done in the introduction of [1], one can prove the following well-posedness result.…”

“…This situation does not allow us to apply the standard splitting method and the energy method devised in [2]. Recently in [1], the obstacle mentioned above is handled for the nonlinearity f ( ∇u L 2 (R n ) )∆u by using compensated compactness method introduced in [11]. In that paper, the strictly positivity condition on the damping coefficient α (•) is critically used.…”

Dynamics of Plate Equations with Memory Driven by Multiplicative Noise on Bounded Domains

“…In this paper, we investigate the existence of a random attractor for the following stochastic plate equations with linear memory and multiplicative noise on bounded domain: U is an open bounded set of 5  with smooth boundary U ∂ , ( )…”

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]). On the other hand, in the case when α ≡ 0 and even β ≡ 1, the semigroup {S (t)} t≥0 generated by (1.1)-(1.2) does not possess a global attractor in H 2 (R n ) × L 2 (R n ).…”