Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

Abstract

Abstract. This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodicsolutions. An application of the result to the Chafee-Infante equation is discussed.

A global solution for a reaction-diffusion equation on bounded domains

“…Also, [3] has obtained trivial attractor {0} for the unperturbed equation when λ < 1, while here the perturbed version has a global solution {u(t)} as its pullback global attractor when λ < 1 which generalizes that result. Moreover, this result is consistent with the result about this problem in [12], Section 4, that is, if h(t) = εh 1 (t)r(x) where h 1 : R → R is an almost periodic function and r(x) ∈ L 2 (0, π) then λ ∈ (0, 1) implies n = 0, and all solutions converge to its unique almost periodic solution u(t), so we have the trivial pullback global attractor as {u(t)} which its sections are one-point sets [ [12], Theorem 4.1]. In fact, our result shows that not only for almost periodic perturbations but also for all bounded perturbations, the pullback global attractor only contains a global solution which, in some sense, improves the result of [12] for λ ∈ (0, 1).…”

“…Hence, in fulfilling the requirements of Corollary 1 to obtain a pullback exponential attractor, for meeting (H 1 ), we can replace the assumption (6) with the assumption (12). Now, we state the main result in L 2 (Ω):…”

“…This is our new assumption which takes the place of the Lipschitz condition in dealing with (10). According to (12), there exists η 0 ≥ 0 such that | f u (t, z)| ≤ η 0 a.e. for t ≤ t 0 .…”

“…2 ) −1 and for a convex Ω with larger diameters, we must check (13) to be held for assuring zero dimensionality of sections of pullback global attractor of reaction-diffusion equation defined on it. In what follows, As an application to Theorem 3, we examine the asymptotic behavior of a new version of non-autonomous Chafee-Infante equation respecting the new assumption (12). Some other versions of this equation in stronger spaces have arisen in [5], [3], [2] and [7].…”

“…with the initial condition u(s, x) = u 0 (x), x ∈ (0, π) and 0 < b 0 ≤ β (t) ≤ B 0 is C 1 [3] and h(t) ≤ M. Obviously, this equation meets the conditions (2)-(5) with f (t, u) = β (t)u 3 − λ u , α = 0 and c 5 = λ . For checking (12), We have f u (t, u) = 3β (t)u 2 − λ and so obviously, g(t) is bounded in the past. Hence, Corollary 5 shows that its pullback global attractor in L 2 (0, π) consists only of one global solution if λ < 1.…”