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

In this paper, we consider an abstract non-autonomous evolution equation with multiple delays in a Hilbert space H: ′ 1 ( ) ( ) ( ( ), , ( )) ( ), n u t Au t F u t r u t r g t + = − ... − + where : ( ) A D A H H ⊂ → is a positive definite selfadjoint operator, : n F H H → α is a nonlinear mapping, 1 , , n r r ... are nonnegative constants, and ( ) ( ; ) g t C H ∈  is bounded. Motivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition. By this result, we provide a general approach for guaranteeing the existence and stability of periodic, quasiperiodic or almost periodic solution of the equation.

is bounded.Motivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition.By this result, we provide a general approach for guaranteeing the existence and stability of periodic, quasiperiodic or almost periodic solution of the equation.

Introduction
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. [4] and [5] mainly discussed the existence and stability of periodic solutions of (1.1).Ref. [3] is concerned with the existence of locally almost periodic solutions of (1.1) by pullback attractor theory.
In this paper, our aim is to study the existence and stability of synchronizing solution of Equation (1.1).Mo-tivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition.The result be of most interest when we choose ( ) ( , ) b g t C H ∈  be translation compact (resp.recurrent or almost periodic or quasiperiodic or periosdic), then we can obtain the synchronizing solution of Equation (1.1) is also translation compact (resp.recurrent or almost periodic or quasiperiodic or periosdic).This result provides a general approach for guaranteeing the existence and stability of periodic, quasiperiodic, almost periodic or recurrent solution of the equation.
The rest of the paper is organized as follows.In Section 2, we provide some preliminaries.In Section 3, we establish the existence and stability of synchronizing solutions under some convergence condition.

Preliminaries
This section consists of some preliminary work.

Analytic Semigroups
Let H be a Hilbert space with the inner product ( , ) ⋅ ⋅ .We Be the eigenvalues of A (counting with multiplicity) with the corresponding eigenvectors α ∈ , define the powers A α as follows: Then, H α is a Hilbert space with the inner product ( , ) α ⋅ ⋅ and norm | | α ⋅ defined as respectively.We also know that for any 0 1

Pullback Attractors
We recall some basic definitions and facts in the theory of non-autonomous dynamical systems for skew-product flows on complete metric spaces.Let ( , ) X d be a complete metric space, ( , ) ρ Σ be a metric space which will be called the base space (or symbol space).: and is called a skew-product flow.

Definition 2.2 A family { ( )} A
and pullback attracting, that is, for any bounded subset B of X, lim ( ( , , ), ) 0, , and is the minimal family of compact sets that is both invariant and pullback attracting.

Global Pullback Attractor of (1.1)
We present essential conditions on the nonlinearity F to guarantee the dissipation and the existence of pullback attractor of (1.1).
We first discuss the well-posedness of the initial value problem of the equation.
. C is endowed with the norm Consider the initial value problem of the evolution equation with delays , . where is continuous and there exist positive constants 1 , , n β β … and 1 k such that F satisfies the following conditions: (H1) For all 1 ( , , ) ;

Synchronizing Solutions
In this section, we establish some results on synchronizing solutions for (1.1), by developing some techniques inspired by works [2] and [1].It is known that if g has some special structure, i.e., periodic, quasiperiodic, almost periodic etc., then we can obtain a compact base space with same structure.Combined with the theory of uniform pullback attractors for dynamical systems in [6], we will prove that under some convergence condition, Equation (1.1) have some entire solution ( ) t γ that synchronize with the motion of the driving system.We call ( ) t γ synchronizing solutions for (1.1).Now, we consider that , and we know that ⊂  U is bounded.So  is given as the union all bounded entire solution.
One can also write the non-autonomous invariance property as , , .
In what follows we show that for each σ ∈ Σ , ( ) A σ is in fact a singleton, i.e., for some a σ ∈C .

Let , ( ) A ϕ ψ σ ∈
. By invariance property (3.1), for any 0 is the solution of (2.2) with initial value 1 ϕ , and where . We obain the unique bounded entire solution of (1.1) on  ;2) For any ϕ ∈C , there exists a unique solution ( ; , ) u t σ ϕ of (1.1) on +  with initial value 0 By Theorem 2.5, we have proved that the cocycle mapping Φ has a pullback attractor {

2
and (H4), we have ˆ0 η > .Then by Gronwall's lemma we have We infer from Corollary 2.8 in[6] that ( ) A σ is upper semi-continuous in σ .This reduces to the continuity of a σ in σ when the { ( )} A σ are single point sets.Hence, Γ is continuous.For each σ ∈ Σ

C
of  one trivially checks that σ γ is precisely the unique solution of (1.1) on  .Since Theorem 4.3 in [5],  is a uniform pullback attractor.That is, for any ϕ ∈C , we have * lim sup ( ( , , ), ( )) 0, denotes the semi-Hausdorff distance in C .Then, it is uniformly forwards attracting,