On the Dynamics of Abstract Retarded Evolution Equations

Abstract

This paper is concerned with the dynamics of the following abstract retarded evolution equation: (/) () + () = ((− 1),. .. , (−)) + () in a Hilbert space , where : () ⊂ → is a self-adjoint positive-definite operator with compact resolvent and : () → (∈ [0, 1/2]) is a locally Lipschitz continuous mapping. The dissipativity and pullback attractors are investigated, and the existence of locally almost periodic solutions is established.

Recurrent Solutions of a Nonautonomous Modified Swift-Hohenberg Equation

“…and Theorem 16 in [14], we know that A is compact and invariant. Therefore we only need to show the minimality of A, i.e., there are no proper compact invariant sets in A.…”

“…We apply the main theorem to the study of recurrent solutions of nonautonomous modified Swift-Hohenberg equations (1.1). Different with the linear growth assumed in [14], the nonlinear term of (1.1) has a super-linear growth and a modified term related to the gradient of u, which increases the difficulty of analysis. However, the topological tool, Conley index can help to overcome these difficulties.…”

The Existence and Stability of Synchronizing Solution of Non-Autonomous Equations with Multiple Delays

“…Proof. The proof can be obtained by Theorem 5 in [3]. Remark 2.4 u satisfies the following integral equation: Then Φ is a cocycle system on C with the base space Σ and driving system θ .…”

“…In this paper, we consider the following non-autonomous evolution equation with multiple delays in a Hilbert space H: ∈  is bounded. This partial differential equations with delays (1.1) has extensive physical background and realistic mathematical model, hence it has been considerably developed and the numerous properties of their solutions have been studied, see [3]- [5] and references therein. Ref.…”

“…Ref. [3] is concerned with the existence of locally almost periodic solutions of (1.1) by pullback attractor theory.…”