Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain

Abstract

We study the long-time behavior of the solutions to a nonlinear damped driven Schrödinger type equation with quadratic potential on a strip. We prove that this behavior is described by a regular compact global attractor with finite fractal dimension.2000 Mathematics Subject Classification. Primary: 35L05, 35Q55; Secondary: 76B03.

Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential

“…which is none other then (1.1) with α = 2, was studied first of all in [1] with critical nonlinearity (cubic nonlinearity) in a thin strip. Then, with subcritical nonlinearities in [3] and [2] where the authors have prove in both cases the existence of a compact global attractor with finite fractal dimension. Finally, we point out that the case in which α = 1 in (1.1) is very interesting and rises many questions which will be the subject of a forthcoming paper.…”

“…Under the same heading, in addition of proving the existence of standing waves, numerical results about the dynamics of (1.3) with harmonic potential were given in [17]. In [29], K. Kirkpatrick where α ∈ (1,2). They prove in [23] the existence of a regular compact global attractor in H 2α (R) with finite fractal dimension (under suitable assumption on f ).…”

“…Lemma 2.3. Let α ∈ (1, 2) and q ∈ [2,4]. Then there exists a reel constant C α > 0 that depends on α and q such that for all…”

Global attractor for a one dimensional weakly damped half-wave equation

“…Now consider the problem (28) in Ω = R × [0, 1] supplemented with initial data u 0 ∈ X s , s > 1 2 , and assuming suitable assumptions on the growth of the nonlinearity g one can prove, as in [4], the existence of a continuous semi-group of operators that maps X s into itself, through vector valued Strichartz type estimates as established in [1] and [2].…”

“…where D denotes for the sake of simplicity − ∂ 2 ∂x 2 . The unknown u = u(t, x) maps R + × R into C. The equation (1) is supplemented with initial data at t = 0 u(0) = u 0 ,…”