GLOBAL SYNCHRONISING BEHAVIOR OF EVOLUTION EQUATIONS WITH EXPONENTIALLY GROWING NONAUTONOMOUS FORCING

. This work is concerned with the following nonautonomous evolutionary system on a Banach space X , where A is a hyperbolic sectorial operator on X , the nonlinearity f ∈ C ( X α × X,X ) is Lipschitz in the ﬁrst variable, the nonautonomous forcing h ∈ C ( R ,X ) is µ -subexponentially growing for some µ > 0 (see (3.4) below for deﬁnition). Under some reasonable assumptions, we ﬁrst establish an existence result for a unique nonautonomous hyperbolic equilibrium for the system in the framework of cocycle semiﬂows. We then demonstrate that the system exhibits a global synchronising behavior with the nonautonomous forcing h as time varies. Fi- nally, we apply the abstract results to stochastic partial diﬀerential equations with additive white noise and obtain stochastic hyperbolic equilibria for the corresponding systems.


1.
Introduction. The notion of equilibria is a fundamental concept in the study of the long time behaviour of dynamical systems. The study of the existence and stability of equilibria is of great interests in both mathematics and physics. In contrast to the autonomous dynamical systems, the existence of equilibria of nonautonomous or random dynamical systems is still a more difficult and subtle problem.
There are many works on the studies of dynamics of evolutionary systems under small bounded perturbations; see e.g. [4, 5, 7, 8, 11, 18, 20-22, 24, 25]. Generally, autonomous hyperbolic equilibria are locally structural stable under small bounded autonomous or nonautonomous perturbations; see e.g. [3][4][5]. This fact usually leads to the structural stability of gradient attractors; see e.g. [4,5,11]. However, it is also of great importance to study the effect of unbounded nonautonomous perturbation on terms of a dynamical system when we consider such systems as models of real phenomena, because there are many examples of unbounded nonautonomous perturbations such as (sub)linear and (sub)exponential ones.
In this paper, we first consider the nonautonomous system (0.1). To have a better understanding of the dynamics of (0.1), as usual we embed the system into the following cocycle system: where H := H[h] is the hull of h (see (3.7) below for definition). Then (1.1) generates a cocycle semiflow ϕ = ϕ(t, p)x on X α with base space H and driving system θ t , where θ t is the shift operator on H for each t ∈ R. Suppose the Lipschitz constant of f and the subexponential growth rate µ of h are sufficiently small. We first prove that the system ϕ possesses a unique nonautonomous equilibrium Γ ∈ C(H, X α ). Next we show the equilibrium is hyperbolic by proving the existences of the section unstable manifold W u (Γ, ·) : H → X α and the section stable manifold W s (Γ, ·) : H → X α of Γ. Meanwhile, we demonstrate that both W u (Γ, ·) : H → X α and W s (Γ, ·) : H → X α are continuous in the sense of Hausdorff distance. In consequence, if h is periodic (resp. pseudo periodic, almost periodic, uniformly almost automorphic), then Γ(θ t h), W u (Γ, θ t h) and W s (Γ, θ t h) are also periodic (resp. pseudo periodic, almost periodic, uniformly almost automorphic). Moreover, we prove that W u (Γ, θ t h) exponentially forward attracts every point in X α through ϕ. Accordingly, the system (0.1) exhibits a global synchronising behavior with the nonautonomous forcing h as time varies.
In the present paper, we also study the existence and asymptotic stability of stochastic hyperbolic equilibrium of the following stochastic equation with additive white noise du + Audt = f (u)dt + dW (t) (1.2) on a Polish space X, where A is also a hyperbolic sectorial operator on X, f : X α → X is Lipschitz continuous, W is a X-valued Wiener process on the classic Wiener space (Ω, F, P). As far as I know, there are few papers to discuss the existence of stochastic hyperbolic equilibria for stochastic systems. When the sectorial operator A is positive, the existence of an exponentially stable non-trivial equilibrium was obtained by Caraballo et al. [6]. Here we treat a general case when the operator A is hyperbolic. We first need to construct a continuous stationary solution Z(ϑ · ω) : R → X α for linear Langevin stochastic partial differential equation (SPDE) in X α : where {ϑ t } t∈R is a family of measure preserving transformations on Ω. It is worth noting that the stationary solution Z(ϑ · ω) : R → X α is subexponentially growing for each ω ∈ Ω. In other words, the random variable Z : Ω → X α is tempered (see Definition 5.5 below). By a transformation v(t) = u(t) − Z(ϑ t ω), then for each fixed ω ∈ Ω, SPDE (1.2) becomes the following nonautonomous equation Then by applying the methods dealing with (0.1), the stochastic system (1.3) is proved to possess a unique tempered stochastic hyperbolic equilibrium Ξ : Ω → X α , whose ω-section Ξ(ω) not only backward attracts every point in its ω-section unstable manifold W u (Ξ, ω) but also forward attracts every point in its ω-section stable manifold W s (Ξ, ω). Other existence results on invariant manifolds for stochastic parabolic and hyperbolic differential equations with additive or multiplicative noise can be refereed to Duan et al. [15,16], Lu and Schmalfuss [23], Brune and Schmalfuss [2]. We also point out that for each ω ∈ Ω, the original system (1.2) exhibits a global synchronising behavior with h(t) = Z(ϑ t ω), t ∈ R as time vaies. This paper is organized as follows. In Section 2 and Section 3, we present basic definitions, and the mathematical setting of the system (1.1), respectively. Section 4 contains the proof of our main abstract results. In Section 5, we apply the abstract results to stochastic partial differential equations with additive white noise.

2.
Preliminaries. In this section we introduce some basic definitions and notions [9,10].
Let X be a complete metric space with metric d(·, ·). Given M ⊂ X, we denote M , int M , ∂M and M c the closure, interior, boundary and complement of M of X, The Hausdorff semidistance and the Hausdorff distance in X are defined, respectively, as 2.1. Cocycle semiflows. A nonautonomous system consists of a "base flow" and a "cocycle semiflow" that is in some sense driven by the base flow. A base flow {θ t } t∈R is a group of continuous transformations from a metric space Σ into itself such that Definition 2.1. A cocycle semiflow ϕ on the phase space X over θ is a continuous mapping ϕ : • ϕ(t + s, σ, x) = ϕ(t, θ s σ, ϕ(s, σ, x)) (cocycle property).
For convenience in statement, a family of subsets B = {B σ } σ∈Σ of X is called a nonautonomous set in X.
Let B = {B σ } σ∈Σ be a nonautonomous set. For convenience, we will rewrite B σ as B(σ), called the σ-section of B. We also denote P(B) the union of the sets B(σ) × {σ} (σ ∈ Σ), i.e., Note that P(B) is a subset of X × Σ.
A nonautonomous set B is said to be closed (resp. open, compact), if P(B) is closed (resp. open, compact) in X × Σ.
Let B and C be two nonautonomous subsets of X. We say that B pullback (resp. forward) attracts C under ϕ if for any σ ∈ Σ, A solution on J = R is called a full solution.
Remark 1. We will also call a solution γ defined as above a σ-solution of ϕ to emphasize the dependence of γ on σ.
Let Γ be a nonautonomous equilibrium of ϕ.
3. Mathematical setting. Let X be a Banach space with norm · , and let A be a sectorial operator in X. Pick a number a > 0 sufficiently large so that Re σ(A + aI) > 0.
Note that the definition of X α is independent of the choice of the number a.
A sectorial operator A is said to be hyperbolic if its spectrum σ(A) has a decomposition σ(A) = σ u ∪ σ s with Accordingly, the space X has a direct sum decomposition: X = X u X s . Let be the projection from X to X i . Denote A u = A| Xu and A s = A| Xs . By the basic knowledge on sectorial operators (see Henry [19]), we know that there exist M ≥ 1, β > 0 such that 2) In the present paper, we will study the qualitative behaviour of a nonautonomous equation on X which have the form: where the nonautonomous forcing h ∈ C(R, X) is µ-subexponentially growing for some µ > 0, namely, lim t→±∞ h(t) = ∞ and lim sup Note that it also covers the case when p is a periodic function, quasiperiodic function, almost periodic function or local almost periodic function [9,21].
Suppose the nonlinearity f ∈ C(X α × X, X) satisfies (1) Lipschitz condition: (3.5) (2) Linear growth condition: there exists a constant C > 0 such that Denote by C(R, X) the set of continuous functions from R to X. Equip with C(R, X) the compact-open topology generated by the metric: , h 1 , h 2 ∈ C(R, X).
Then C(R, X) is a complete metric space. Define the hull of the nonautonomous forcing h as follows and define the shift operator on H: It is clear that θ t : H → H is continuous. Instead of (3.3), we will consider the more general cocycle system in X α (where α ∈ [0, 1)): For convenience, we always assume that the unique solution (3.9) is globally defined. Define Then ϕ is a cocycle semiflow on X α driven by the base flow θ on H.

4.
Nonautonomous hyperbolic equilibrium and global synchronising behavior of the forced nonautonomous system with h.

4.1.
Basics. Let us first introduce several Banach spaces that will used throughout the paper. For µ ≥ 0, define Replacing R ± with R in the above definition, one immediately obtains the definition of the space X µ . Clearly The following lemma will play a basic role in the proof of our main theorem in the next section.
(a) Let x ∈ X − µ . Then x is the solution of (3.8) if and only if it solves the following integral equation Then x is a solution of (3.8) if and only if it solves the following integral equation Then x is the solution of (3.8) if and only if it solves the following integral equation for any t 0 ≤ t. Since Consequently This is precisely what we desired in (4.1).
Conversely if x satisfies (4.1), then one can easily see that it solves (3.8) on R − . Thus This is precisely what we desired in (4.2). If x satisfies (4.2), then it clearly solves (3.8) on R + . The proof of (b) is complete.
(c) Let x ∈ X µ be a solution of (3.8). Write Similar to (a) and (b), we have It follows that Conversely, if x satisfies (4.3), it is trivial to check that x is a full solution of (3.8).

Main results.
Denote Our main results in the section can be summarized as Let h ∈ C(R, X) be µ-subexponentially growing for some µ ∈ (0, β). Suppose the Lipschitz constant L f of f is sufficiently small. Then (a) The cocycle semiflow ϕ has a unique µ-subexponential equilibrium Γ ∈ C(H, X α ), namely, the full solution Γ(θ · p) : R → X α is µ-subexponentially growing for each p ∈ H. (b) The equilibrium Γ is hyperbolic. Specifically, for each p ∈ H, there exist two family of Lipschitz continuous mappings such that the p-section unstable and stable manifold of Γ are represented as (1) The continuous dependence of ξ p and ζ p on p imply that and respectively. (2) It is clear that if h is periodic (resp. pseudo periodic, almost periodic, uniformly almost automorphic), then θ t h is periodic (resp. pseudo periodic, almost periodic, uniformly almost automorphic). In consequence, the continuity of Γ : H → X α , (4.7) and (4.8) manifest that if h is periodic (resp. pseudo periodic, almost periodic, uniformly almost automorphic), then Γ(θ t h), W u (Γ, θ t h) and W s (Γ, θ t h) are also periodic (resp. pseudo periodic, almost periodic, uniformly almost automorphic). This means Γ(θ t h), W u (Γ, θ t h) and W s (Γ, θ t h) exhibit a synchronising behavior with the nonautonomous forcing h as time varies.
where β = β − µ. For p ∈ H, one can use the righthand side of equation (4.3) to define a contraction mapping T on X µ as follows: We first verify that T maps X µ into itself. Let x be in X µ . Then by (3.1), (3.2) and (3.6), One observes that (4.10) We have Notice from (3.4) that p ∞,µ < ∞. Hence T x Xµ < ∞, i.e. T x ∈ X µ . Next, we check that T is a contraction mapping. Indeed, in a quite similar fashion as above, it can be shown that for any x, x ∈ X µ , M β L f < 1 by (4.9), it follows that T is contracting on X µ . Now thanks to the Banach fixed-point theorem, T has a unique fixed point x p ∈ X µ which is precisely a full solution of (3.8) satisfying (4.3).
Define Γ : H → X α by It is easy to verify that Γ(θ t p) = x p (t), t ∈ R, hence Γ is an equilibrium of ϕ. In what follows we show that Γ is continuous.
(b) For each p ∈ H and y ∈ X α u , the righthand side of (4.1) can define a mapping T − = T − p,y on X − µ as follows: We first show that T − maps X − µ into itself. Let x ∈ X − µ . For any t ≤ 0, µ . By virtue of the Banach fixed-point theorem, T − has a unique fixed point x p,y in X − µ . So x p,y (t) is precisely a solution of (3.8) on R − with Π u x p,y (0) = y, which equivalently solves the following integral equation  We claim that x p,y (0) is Lipschitz continuous in y uniformly on p ∈ H. Indeed, for y, z ∈ X α u and t ≤ 0, and thus x p,y (0) is Lipschitz continuous in y uniformly on p ∈ H. Define Setting t = 0 in (4.15) leads to x p,y (0) = y + ξ p (y), y ∈ X α u . (4.17) The Lipschitz continuity of x p,y (0) in y then implies ξ p : X α u → X α s is a Lipschitz continuous mapping. Set Then M u (p) is homeomorphic to X α u . We claim that In what follows we check that Define a Banach space Let x p (t) be the full solution of ϕ satisfying (4.11). For each fixed y ∈ X α u , define a mapping G − = G − y : W − µ → W − µ as follows: (It is trivial to see that z α < ∞.) We first check that G − is well defined. Indeed, let w ∈ W − µ . Then for any t ≤ 0, µ . Therefore G − has a unique fixed point w in W − µ , which solves the following integral equation on R − : (Note that w(t) α tends to zero exponentially fast for t → −∞.) Adding (4.11) to (4.19), we find (4.20) Setx p,y (t) = x p (t) + w(t), t ≤ 0. The equation (4.20) means thatx p,y (t), t ≤ 0 satisfies (4.15). By the uniqueness of the backward solution x p,y (t), t ≤ 0 with Π u x p,y (0) = y, one knows that We conclude that the backward solution x p,y (t) tends to the full solution x p (t) exponentially as t → −∞. Since y ∈ X α u is arbitrary, we prove (4.18) and complete the assertion.
(c) Let y ∈ X α u , and p, q ∈ H. Then similar computations as in (4.12) and (4.13) show that This together with (4.17) shows that Since p → q (in H) implies that p − q ∞,µ → 0, we conclude that lim p→q sup y∈X α u ξ p (y) − ξ q (y) α = 0, which verifies the Lipschitz continuity property of ξ p in p.
The corresponding argument for stable manifold of Γ is completely similar to that for the unstable manifold, so is omitted.

4.3.
Dynamical completeness of hyperbolic equilibrium. Here we show that the hyperbolic equilibrium Γ obtained in Theorem 4.2 will completely characterizes the dynamics of the cocycle semiflow ϕ by proving its unstable manifold is globally forward stable. Consequently, the original system (3.3) exhibits a global synchronising behavior (in the sense of Remark 2) with the forcing h.
Remark 3. Theorem 4.3 and Remark 2 indicate that the system ϕ exhibits a global synchronising behavior with the nonautonomous forcing h as time varies.
Then for t ≥ 0, and thus
5. Application to stochastic partial differential equations with additive white noise. In the section, we apply the main results obtained in Section 4 to stochastic partial equation with additive white noise. Let X be a Polish space with norm · . We first recall some basic concepts in random dynamical systems (RDSs).

5.1.
RDSs. An RDS consists of a "metric dynamical system" and a "cocycle semiflow" that is in some sense driven by the metric dynamical system. Definition 5.1. Let (Ω, F, P) be a probability space and ϑ := {ϑ t } t∈R be a family of measure preserving transformations on Ω such that (t, ω) → ϑ t ω is measurable, ϑ 0 = id Ω , and ϑ t+s = ϑ t • ϑ s for all t, s ∈ R. Then the quadruple (Ω, F, P, ϑ) is called a metric dynamical system.

Stochastic hyperbolic equilibrium and synchronising behavior.
In what follows we suppose that (Ω, F, P) is the classic Wiener space, i.e., Ω = {ω : ω(·) ∈ C(R, X), ω(0) = 0} endowed with the open compact topology, F is the associated Borel-σ-algebra, P is the Wiener measure and the σ-fields {F t } t∈R , called a filtration, given by Since ω ∈ Ω if and only if ω − ∈ Ω, we may assume without loss of generality that each ω ∈ Ω is an even function, i.e., We can define a measurable dynamical system ϑ := {ϑ t } t∈R on (Ω, F, P) by ϑ t ω(·) = ω(· + t) − ω(t). It is well-known that P is invariant and ergodic under ϑ. Then (Ω, F, P, ϑ) is a metric dynamical system. A Wiener process {W (t)} t∈R defined on (Ω, F, P, ϑ) is given by We now consider stochastic partial differential equations (SPDEs) with additive noise on X which have the form: where A is a hyperbolic sectorial operator in X, f : X α → H is a Lipshitz continuous mapping with Lipschitz constant L f , W is a X valued Wiener process on (Ω, F, P) with covariance operator Q. Let B : X α → X be a Lipschitz continuous Hilbert-Schmidt operator. When the noise term dW in (5.1) is replaced by B(u)dW , we will obtain a more general SPDE. The existence theory for such general equations is formulated as in Da Prato and Zabczyk [14]. Specifically, for any initial data u 0 ∈ X α , there exists a unique mild solution given by The above mild solution is only defined almost surely where the exceptional set may depend on the initial data u 0 . Such a dependence contradicts the definition of an RDS. In fact, it is still an open problem to interpret general SPDEs as RDSs. However, for the special additive noise in (5.1), using a perfection procedure, the equation (5.1) indeed generates an RDS (see [1,17]). The key idea is to transform this SPDE into a random evolutionary equation with random coefficients. For this purpose, we first need to construct a stationary solution of linear Langevin SPDE in X α : dz + Azdt = dW (t). Lemma 5.6. Suppose the Wiener process W (t), t ∈ R has a covariance operator Q such that Tr(Q(−A u ) 2α−1+ε ) < ∞ and Tr(QA 2α−1+ε s ) < ∞ for some ε > 0. Then the equation (5.2) possesses a stationary solution R × Ω (t, ω) → Z(ϑ t ω) ∈ X α , which is {F |t| } t∈R -adapted and is given by a tempered and F 0 -measurable random variable Z ∈ X α . Moreover, t → Z(ϑ t ω) is continuous.
Proof. Split (5.2) into two linear equations Since the real part of the spectrum Re σ(A s ) > 0 and Tr(QA 2α−1+ε s ) < ∞, from [2, Lemma 4.2], there is a tempered and F 0 -measurable random variable Z s ∈ X α s such that the stochastic process Z s (ϑ t ω) ∈ X α s is an {F t } t∈R -adapted and continuous (in t) stationary solution for (5.3), where Meanwhile, we use a coordinate transform y(t) = z u (−t) for (5.4), then y solves dy − A u ydt = Π u dW (−t), (5.5) where −A u is a bounded linear operator on X α with Re σ(−A u ) > 0. By Remark 4, we can rewrite (5.5) as Since Tr(Q(−A u ) 2α−1+ε ) < ∞, repeating the argument above, there is a tempered and F 0 -measurable random variable Y u ∈ X α u such that the stochastic process Y u (ϑ t ω) ∈ X α u is an {F t } t∈R -adapted and continuous (in t) stationary solution for is an {F −t } t∈R -adapted stationary solution for (5.4). Consequently, is the desired stationary solution for (5.2), and Z : Ω → X α is tempered. Let Then v(t) satisfies the following random evolution (5.6) Let x(t, 0; v 0 , ω) denote the unique globally defined solution of (5.6) for the initial value v 0 ∈ X α . Define Suppose φ is an RDS on X α driven by the base flow ϑ on Ω. We summarise our conclusions in the following theorem.
Theorem 5.7. Suppose the assumptions of Lemma 5.6 hold and the Lipschitz constant L f of f is sufficiently small such that Then (a) the RDS φ has a unique tempered stochastic hyperbolic equilibrium Ξ : Ω → X α , with its the ω-section unstable and stable manifolds being represented as W u (Ξ, ω) = {y + ξ ω (y) : y ∈ X α u } W s (Ξ, ω) = {ζ ω (y) + y : y ∈ X α s } respectively, where ξ ω : X α u → X α s and ζ ω : X α s → X α u are the graphs for the two manifolds. Moreover, Proof. For each ω ∈ Ω, let h(t) = Z(ϑ t ω) and v(t) = u(t) − h(t). Then v(t) satisfies the following evolution equation v t + Av = f v + h(t) . The unique solution of (5.9) will generate a cocycle semiflow ϕ on X α driven by the base flow θ on H, namely, ϕ(t, p)x 0 := x(t, 0; x 0 , p), x 0 ∈ X α .
The proof is complete.