Averaging principles for non-autonomous two-time-scale stochastic reaction-diffusion equations with polynomial growth
aa r X i v : . [ m a t h . D S ] A p r Averaging principles for non-autonomous two-time-scale stochasticreaction-diffusion equations with polynomial growth
Ruifang Wang a , Yong Xu a,b, ∗ , Bin Pei c a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China b MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern PolytechnicalUniversity, Xi’an, 710072, China c School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
Abstract
In this paper, we develop the averaging principle for a class of two-time-scale stochasticreaction-diffusion equations driven by Wiener processes and Poisson random measures. Weassume that all coefficients of the equation have polynomial growth, and the drift term ofthe equation is non-Lipschitz. Hence, the classical formulation of the averaging principleunder the Lipschitz condition is no longer available. To prove the validity of the averagingprinciple, the existence and uniqueness of the mild solution are proved firstly. Then, theexistence of time-dependent evolution family of measures associated with the fast equationis studied, by which the averaged coefficient is obtained. Finally, the validity of the averagingprinciple is verified.
Keywords.
Averaging principles, stochastic reaction-diffusion equations, Poisson randommeasures, evolution families of measures, polynomial growth
Mathematics subject classification.
1. Introduction
Multi-scale problems are widely encountered in composites, porous media, finance andother fields [2, 3]. Morever, in practice, the parameters of systems often depend on time,non-autonomous systems are worthy of thorough analysis. For this reason, we are concernedwith the following non-autonomous two-time-scale stochastic partial differential equations(SPDEs) on a bounded domain O of R d ( d ≥ ∂u ǫ ∂t ( t, ξ ) = A ( t ) u ǫ ( t, ξ ) + b ( t, ξ, u ǫ ( t, ξ ) , v ǫ ( t, ξ )) + f ( t, ξ, u ǫ ( t, ξ )) ∂ω Q ∂t ( t, ξ )+ R Z g ( t, ξ, u ǫ ( t, ξ ) , z ) ∂ ˜ N ∂t ( t, ξ, dz ) , ∂v ǫ ∂t ( t, ξ ) = ǫ [( A ( t ) − α ) v ǫ ( t, ξ ) + b ( t, ξ, u ǫ ( t, ξ ) , v ǫ ( t, ξ ))]+ √ ǫ f ( t, ξ, v ǫ ( t, ξ )) ∂ω Q ∂t ( t, ξ ) + R Z g ( t, ξ, v ǫ ( t, ξ ) , z ) ∂ ˜ N ǫ ∂t ( t, ξ, dz ) ,u ǫ (0 , ξ ) = x ( ξ ) , v ǫ (0 , ξ ) = y ( ξ ) , ξ ∈ O , N u ǫ ( t, ξ ) = N v ǫ ( t, ξ ) = 0 , t ≥ , ξ ∈ ∂ O , (1.1) ∗ Corresponding author
Email addresses: [email protected] (Ruifang Wang), [email protected] (Yong Xu), [email protected] (Bin Pei)
Preprint submitted to Elsevier April 25, 2019 here ω Q , ω Q and ˜ N , ˜ N ǫ are mutually independent Wiener processes and Poisson random mea-sures, 0 < ǫ ≪ α is a sufficiently large fixed constant. In addition, N i ( i = 1 ,
2) are the boundary operators, which can be either the identity operator (Dirichletboundary condition) or the first order operator (coefficients satisfying a uniform nontangentialitycondition). The stochastic perturbations of the equations define on the same complete stochasticbasis (cid:16) Ω, F , {F t } t ≥ , P (cid:17) , the specific introduction will be given in Section 2.The averaging principle is an effective method to analysis the slow-fast systems, which cansimplify the system by constructing the averaged equation. In 1961, Bogolyubov and Mitropolskii[4] studied the averaging principle, giving the first rigorous result for the deterministic case. Sincethen, the averaging principle became an active area of research. Khasminskii [5] established theaveraging principle for stochastic differential equations (SDEs) in 1968. Then, Givon [6], Freidlinand Wentzell [7], Duan [8], Xu and his co-workers [9–11] also studied the averaging principle ofSDEs. In addition, many scholars also investigated the averaging principle of SPDEs in recentyears, such as, Cerrai [12, 13], Wang and Roberts [14], Pei and Xu [15–17], Xu and Miao [18]. Itshould be pointed out that most of the current studies about the averaging principle are based onautonomous systems. In practical problems, the parameters of the system often depend on time.Therefore, non-autonomous system can depict some actual models better, which has made itselfattract more and more attention of scholars.In 2017, effetive approximation for non-autonomous slow-fast system has been presented byCerrai [19], and the system of this paper was driven by Gaussian noises. In our previous article[20], we study the non-autonomous slow-fast system driven by Gaussian noises and Poisson randommeasures. An effective approximations for the slow equation of the original system in article [20]was established by using the averaging principle, where the coefficients of the equation satisfy theLipschitz condition and linear growth. But those conditions are too strict to study the validity of theaveraging principle in many other relevant cases, such as, polynomial growth. One of the reaction-diffusion equations for the coefficients satisfy the polynomial growth is the Fitzhugh-Nagumo orGinzburg-Landau type, those systems have appeared in the fields of biology and physics and at-tracted considerable attention. Therefore, we are devoted to developing the averaging principle fornon-autonomous systems of reaction-diffusion equations with polynomial growth.First, with the aid of the Sobolev embedding theorem, fixed point theorem and stopping tech-nique, the existence and uniqueness of the mild solution is proved. That is, for any p ≥ T ≥ s , we prove that system (1.1) admits a unique mild solution depending on the initial datum.Next, as in our previous work [20], assuming the operator A ( t ) is periodic and the functions b , f , g are almost periodic. Analyzing the fast equation with a frozen slow component and usingKunitas first inequality to deal with the Poisson terms, we get that the evolution family of measuresfor the fast equation also exists, and it is almost periodic. Then, the averaged coefficient is definedthrough it, and the following averaged equation is obtained ∂ ¯ u∂t ( t, ξ ) = A ( t ) ¯ u ( t, ξ ) + ¯ b ( ξ, ¯ u ( t, ξ )) + f ( t, ξ, ¯ u ( t, ξ )) ∂ω Q ∂t ( t, ξ )+ R Z g ( t, ξ, ¯ u ( t, ξ ) , z ) ∂ ˜ N ∂t ( t, ξ, dz ) , ¯ u (0 , ξ ) = x ( ξ ) , ξ ∈ O , N ¯ u ( t, ξ ) = 0 , t ≥ , ξ ∈ ∂ O , (1.2)where ¯ b ( ξ, ¯ u ( t, ξ )) is the averaged coefficient, which will be given in Section 5.Finally, the validity of the averaging principle is verified by using the classical Khasminskiimethod. That is, for any T > η >
0, we havelim ǫ → P (cid:16) sup t ∈ [0 ,T ] k u ǫ ( t ) − ¯ u ( t ) k D ( ¯ O ) > η (cid:17) = 0 , (1.3) here ¯ u is the solution of the averaged equation (1.2).We will give a specific definition of the notations in Section 2. In this paper, c >
2. Notations, assumptions and preliminaries
Denote E is the space C ( ¯ O ), endowed with the following sup-norm k x k E = sup ξ ∈ ¯ O | x ( ξ ) | , and the duality h· , ·i E . The norm of the product space E × E denote as k x k E × E = (cid:16) k x k E + k x k E (cid:17) . and the corresponding duality of the product space E × E is h· , ·i E × E .Let X be any space, denote L ( X ) is the space of the bounded linear operators in X . For any0 ≤ s < T and p ≥
1, denote the norm of the space L p (Ω; D ([ s, T ] ; X )) is k u k pL s,T,p ( X ) := E sup t ∈ [ s,T ] k u ( t ) k p X . where D ([ s, T ] ; X ) denotes the space of all c`adl`ag path from [ s, T ] into X .For any p ∈ [1 , ∞ ] with p = 2, denote the norms of the space L p ( O ) and L p ( O ) × L p ( O ) areboth k·k p . When δ > p < ∞ , we denote the norm of the space W δ,p ( O ) is k·k δ,p : k x k δ,p = k x k p + (cid:16) Z D Z D | x ( ξ ) − x ( η ) || ξ − η | δp + d dξdη (cid:17) p . Now, we introduce some notations about subdifferential. The subdifferential of k x k E is definedas ∂ k x k E := { h ∈ E ∗ ; k h k E ∗ = 1 , h h, x i E = k x k E } , where E ∗ is the dual space of E . Due to the characterization of the subdifferential [21, AppendixD], if u : [0 , T ] → E is any differentiable mapping, then ddt − k u ( t ) k E ≤ (cid:10) u ′ ( t ) , δ (cid:11) E , (2.1)for any t ∈ [0 , T ] and δ ∈ ∂ k u ( t ) k E .Now, we assume the space dimension d >
1, the processes ∂ω Q /∂t ( t, ξ ) and ∂ω Q /∂t ( t, ξ ) inthe slow-fast system are the Gaussian noises, assumed it is white in time and colored in space.Here, ω Q i ( t, ξ ) ( i = 1 ,
2) is the cylindrical Wiener processes, and it defined as ω Q i ( t, ξ ) = ∞ X k =1 Q i e k ( ξ ) β k ( t ) , i = 1 , , where { e k } k ∈ N is a complete orthonormal basis in H , { β k ( t ) } k ∈ N is a sequence of mutually indepen-dent standard Brownian motion defined on the same complete stochastic basis (cid:16) Ω, F , {F t } t ≥ , P (cid:17) ,and Q i is a bounded linear operator on H . ext, we give the definitions of Poisson random measures ˜ N ( dt, dz ) and ˜ N ǫ ( dt, dz ). Let( Z , B ( Z )) be a given measurable space and v ( dz ) be a σ -finite measure on it. D p it , i = 1 , R + . Moreover, let p t , t ∈ D p t be a stationary F t -adapted Poisson pointprocess on Z with characteristic v , and p t , t ∈ D p t be the other stationary F t -adapted Poisson pointprocess on Z with characteristic v/ǫ . Denote by N i ( dt, dz ) , i = 1 , p it , i.e., N i ( t, Λ ) := X s ∈ D pit ,s ≤ t I Λ (cid:0) p it (cid:1) , i = 1 , . Let us denote the two independent compensated Poisson measures˜ N ( dt, dz ) := N ( dt, dz ) − v ( dz ) dt and ˜ N ǫ ( dt, dz ) := N ( dt, dz ) − ǫ v ( dz ) dt, where v ( dz ) dt and ǫ v ( dz ) dt are the compensators.In this paper, for any t ∈ R , the operators A ( t ) and the operators A ( t ) are the second orderuniformly elliptic operators with continuous coefficients on ¯ O . As in our previous work [20], weassume that the operator A i ( t ) has the following form A i ( t ) = γ i ( t ) A i + L i ( t ) , t ∈ R , i = 1 , , (2.2)where A i independent of t is a second order uniformly elliptic operator, having continuous coeffi-cients on ¯ O . And the operator L i ( t ) is a first order differential operator has the following form L i ( t, ξ ) u ( ξ ) = h l i ( t, ξ ) , ∇ u ( ξ ) i R d , t ∈ R , ξ ∈ ¯ O . (2.3)Finally, for i = 1 ,
2, denote the realization of the operators A i and L i in E are A i and L i , andthe operator A i generates an analytic semigroup e tA i .Now, we give the following assumptions about the operators A i and Q i as in [20] and [13].(A1) (a) For i = 1 ,
2, the function γ i : R → R is continuous, and there exist γ , γ > γ ≤ γ i ( t ) ≤ γ, t ∈ R . (2.4)(b) For i = 1 ,
2, the function l i : R × ¯ O → R d is continuous and bounded.(A2) For i = 1 ,
2, there exist a complete orthonormal system { e i,k } k ∈ N of E , and two sequences ofnonnegative real numbers { α i,k } k ∈ N and { λ i,k } k ∈ N such that A i e i,k = − α i,k e i,k , Q i e i,k = λ i,k e i,k , k ≥ , (2.5)and κ i := ∞ X k =1 λ ρ i i,k k e i,k k ∞ < ∞ , ζ i := ∞ X k =1 α − β i i,k k e i,k k ∞ < ∞ , (2.6)for some constants ρ i ∈ (2 , + ∞ ] and β i ∈ (0 , + ∞ ) such that[ β i ( ρ i − /ρ i < . (2.7) bout the coefficients of the system (1.1), we assume it satisfy the following conditions.(A3) (a) The mappings b : R × ¯ O × R → R is continuous and there exists m ≥ ( t,ξ ) ∈ R × ¯ O | b ( t, ξ, x, y ) | ≤ c (1 + | x | m + | y | ) , ( x, y ) ∈ R . (2.8)(b) There exists c > x, h ∈ R ,sup ( t,ξ ) ∈ R × ¯ O ( b ( t, ξ, x + h ) − b ( t, ξ, x )) h ≤ c | h | (1 + | x | + | h | ) . (2.9)(c) There exists θ > ( t,ξ ) ∈ R × ¯ O | b ( t, ξ, x ) − b ( t, ξ, y ) | ≤ c (cid:16) | x | θ + | y | θ (cid:17) | x − y | , x, y ∈ R . (2.10)(A4) (a) The mappings b : R × ¯ O × R → R is continuous and there exists m ≥ ( t,ξ ) ∈ R × ¯ O | b ( t, ξ, x, y ) | ≤ c (1 + | x | + | y | m ) , ( x, y ) ∈ R . (2.11)(b) There exists c > x, h ∈ R ,sup ( t,ξ ) ∈ R × ¯ O ( b ( t, ξ, x + h ) − b ( t, ξ, x )) h ≤ c | h | (1 + | x | + | h | ) . (2.12)(c) The mapping b ( t, ξ, · ) : R → R is locally Lipschitz-continuous, uniformly with respectto ( t, ξ ) ∈ R × ¯ O .(d) For all x, y , y ∈ R , we have b ( t, ξ, x, y ) − b ( t, ξ, x, y ) = − τ ( t, ξ, x, y , y ) ( y − y ) . (2.13)for some measurable function τ : R × ¯ O × R → [0 , ∞ ).(A5) The mappings f : R × ¯ O × R → R , g : R × ¯ O × R × Z → R , f : R × ¯ O × R → R , g : R × ¯ O × R × Z → R are continuous, and the mappings f ( t, ξ, · ) : R → R , g ( t, ξ, · , z ) : R → R , f ( t, ξ, · ) : R → R , g ( t, ξ, · , z ) : R → R are Lipschitz-continuous, uniformly with respectto ( t, ξ, z ) ∈ R × ¯ O × Z . Moreover, for all p ≥
1, there exist positive constants c , c , suchthat for all x, y ∈ R , havesup ( t,ξ ) ∈ R × ¯ O Z Z | g i ( t, ξ, x, z ) − g i ( t, ξ, y, z ) | p υ i ( dz ) ≤ c i | x − y | p , i = 1 , . (A6) For any x, y ∈ R , it hold thatsup ( t,ξ ) ∈ R × ¯ O (cid:16) | f i ( t, ξ, x ) | p + Z Z | g i ( t, ξ, x, z ) | p v i ( dz ) (cid:17) ≤ c (cid:16) | x | pmi (cid:17) , i = 1 , , (2.14)where m and m are the constants introduced in (2.8) and (2.11). emark 2.1. For any ( t, ξ ) ∈ R × ¯ O and x, y, h ∈ E , z ∈ Z , we shall set B ( t, x, y ) ( ξ ) := b ( t, ξ, x ( ξ ) , y ( ξ )) , B ( t, x, y ) ( ξ ) := b ( t, ξ, x ( ξ ) , y ( ξ )) , [ F ( t, x ) h ] ( ξ ) := f ( t, ξ, x ( ξ )) h ( ξ ) , [ F ( t, x ) h ] ( ξ ) := f ( t, ξ, x ( ξ )) h ( ξ ) , [ G ( t, x, z ) h ] ( ξ ) := g ( t, ξ, x ( ξ ) , z ) h ( ξ ) , [ G ( t, x, z ) h ] ( ξ ) := g ( t, ξ, x ( ξ ) , z ) h ( ξ ) . Due to the assumption (A3) and (A4), we know the mappings B : R × E × E → E and B : R × E × E → E are well defined and continuous. According to (2.8) and (2.11), for any x, y ∈ E and t ∈ R , we have k B ( t, x, y ) k E ≤ c (1 + k x k m E + k y k E ) , k B ( t, x, y ) k E ≤ c (1 + k x k E + k y k m E ) . (2.15)As a consequence of (2.9) and (2.12), it is immediate to check that, for any x, y, h, k ∈ E , any t ∈ R ,and any δ ∈ ∂ k h k E , h B i ( t, x + h, y + k ) − B i ( t, x, y ) , δ i E ≤ c (1 + k x k E + k y k E + k h k E + k k k E ) (2.16)In view of (2.10), for any x , y , x , y ∈ E , we have k B ( t, x , y ) − B ( t, x , y ) k E ≤ c (cid:0) k ( x , y ) k θ E × E + k ( x , y ) k θ E × E (cid:1) ( k x − x k E + k y − y k E ) . (2.17)In addition, from the equation (2.13), for every δ ∈ ∂ k k k E , we have h B ( t, x, y + k ) − B ( t, x, y ) , δ i E ≤ t, z ) ∈ ( R , Z ) , the mappings F i ( t, · ) : E → L ( E ) , G i ( t, · , z ) : E → L ( E ) , i = 1 , , are Lipschitz-continuous.Now, for i = 1 ,
2, we define γ i ( t, s ) := Z ts γ i ( r ) dr, s < t, and let γ ( t, s ) := ( γ ( t, s ) , γ ( t, s )). For any ǫ > β ≥
0, set U β,ǫ,i ( t, s ) = e ǫ γ i ( t,s ) A i − βǫ ( t − s ) , s < t, in the case ǫ = 1, we write U β,i ( t, s ), and in the case ǫ = 1 and β = 0, we write U i ( t, s ).Next, for any ǫ > , β ≥ u ∈ D ([ s, t ] ; E ) , r ∈ [ s, t ] , we define ψ β,ǫ,i ( u ; s ) ( r ) = 1 ǫ Z rs U β,ǫ,i ( r, ρ ) L i ( ρ ) u ( ρ ) dρ, s < r < t, in the case ǫ = 1, we write ψ β,i ( u ; s ) ( r ), and in the case ǫ = 1 and β = 0, we write ψ i ( u ; s ) ( r ).We can easily get that ψ β,ǫ,i ( u ; s ) ( t ) is the solution of du ( t ) = 1 ǫ ( A i ( t ) − β ) u ( t ) dt, t > s, u ( s ) = 0 . . Existence, uniqueness of the solutions More general, in this section, we mainly study the existence and uniqueness of solutions for thefollowing problems du ( t ) = [ A ( t ) u ( t ) + B ( t, u ( t ))] dt + F ( t, u ( t )) dw Q ( t ) + Z Z G ( t, u ( t ) , z ) ˜ N ( dt, dz ) , (3.1)where u ( t ) := (cid:18) u ( t ) u ( t ) (cid:19) , A ( t ) := (cid:18) A ( t ) 00 A ( t ) (cid:19) , B ( t, u ( t )) = (cid:18) B ( t, u ( t ) , u ( t )) B ( t, u ( t ) , u ( t )) (cid:19) , and F ( t, u ( t )) := (cid:18) F ( t, u ( t )) 00 F ( t, u ( t )) (cid:19) , G ( t, u ( t ) , z ) := (cid:18) G ( t, u ( t ) , z ) 00 G ( t, u ( t ) , z ) (cid:19) , and w Q ( t ) := (cid:18) w Q ( t ) w Q ( t ) (cid:19) , ˜ N ( dt, dz ) := (cid:18) ˜ N ( dt, dz )˜ N ( dt, dz ) (cid:19) . According to the assumption (A5), it is easy to know that for any fixed ( t, z ) ∈ ( R , Z ) , the mappings F ( t, · ) : E × E → L ( E × E ) , G ( t, · , z ) : E × E → L ( E × E ) , are Lipschitz-continuous. Definition 3.1.
For any fix ( x , x ) ∈ E × E , a process u ( t ) is a mild solution of the equation(3.1), if u ( t ) = U ( t, s ) x + ψ ( u ; s ) ( t ) + Z ts U ( t, r ) B ( r, u ( r )) dr + Z ts U ( t, r ) F ( r, u ( r )) dw Q ( r )+ Z ts Z Z U ( t, r ) G ( r, u ( r ) , z ) ˜ N ( dr, dz ) , (3.2)where U ( t, s ) = (cid:18) U ( t, s ) 00 U ( t, s ) (cid:19) , ψ ( u ; s ) ( t ) = (cid:18) ψ ( u ; s ) ( t ) ψ ( u ; s ) ( t ) (cid:19) , x = (cid:18) x x (cid:19) . Now, we denote Φ( u )( t ) := Z ts U ( t, r ) B ( r, u ( r )) dr,Γ ( u )( t ) := Z ts U ( t, r ) F ( r, u ( r )) dw Q ( r ) , and Ψ ( u )( t ) := Z ts Z Z U ( t, r ) G ( r, u ( r ) , z ) ˜ N ( dr, dz ) . First, we prove that the mapping Ψ ( u )( t ) is a contraction in L p ( Ω ; D ([ s, T ] ; E )). emma 3.2. Under the assumptions (A1)-(A6), for any u, v ∈ L p ( Ω ; D ([ s, T ] ; E )) with p ≥ ,the mapping Ψ maps L p ( Ω ; D ([ s, T ] ; E )) into itself, and we have k Ψ ( u ) − Ψ ( v ) k L s,T,p ( E ) ≤ c Ψs,p ( T ) k u − v k L s,T,p ( E ) , (3.3) where c Ψs,p is a continuous increasing function with c Ψs,p ( s ) = 0 . Proof:
By using a factorization argument [21, Theorem 8.3], we have Ψ ( u ) ( t ) − Ψ ( v ) ( t ) = sinπλπ Z ts ( t − r ) λ − U ( t, r ) φ λ ( u, v ) ( r ) dr where φ λ ( u, v ) ( r ) := Z rs Z Z ( r − σ ) − λ U ( r, σ ) [ G ( σ, u ( σ ) , z ) − G ( σ, v ( σ ) , z )] ˜ N ( dσ, dz ) , and λ ∈ (0 , / t, ǫ > p ≥
1, the semigroup e tA maps L p ( O ; R ) into W ǫ,p ( O ; R ) and by usingthe semigroup law, we can obtain (cid:13)(cid:13) e tA x (cid:13)(cid:13) ǫ,p ≤ c ( t ∧ − ǫ k x k p , x ∈ L p (cid:0) O ; R (cid:1) , (3.4)for some constant c independent of p . Then, according to (3.4), using the H¨older inequality, forany ǫ < λ , we have k Ψ ( u ) ( t ) − Ψ ( v ) ( t ) k E ≤ k Ψ ( u ) ( t ) − Ψ ( v ) ( t ) k ǫ,p ≤ c λ Z ts (( t − r ) ∧ λ − ǫ − k φ λ ( u, v ) ( r ) k p dr ≤ c λ sup r ∈ [ s,T ] k φ λ ( u, v ) ( r ) k p · Z t − s ( r ∧ λ − ǫ − dr, (3.5)so, if we show that φ λ ( u, v ) ( r ) ∈ L p (cid:0) O ; R (cid:1) , we can get Ψ ( u ) − Ψ ( v ) ∈ D (cid:0) [ s, T ] ; W ǫ,p (cid:0) O ; R (cid:1)(cid:1) ,P − a.s. Using Kunita’s first inequality [22, Theorem 4.4.23] and the H¨older inequality, because G is Lipschitz-continuous, for any p ≥
1, we have E | φ λ ( u, v ) ( t, ξ ) | p ≤ c p E (cid:16) Z ts Z Z ( t − σ ) − λ | U ( r, σ ) [ G ( σ, u ( σ ) , z ) ( ξ ) − G ( σ, v ( σ ) , z ) ( ξ )] | v ( dz ) dσ (cid:17) p + c p E Z ts Z Z ( t − σ ) − pλ | U ( r, σ ) [ G ( σ, u ( σ ) , z ) ( ξ ) − G ( σ, v ( σ ) , z ) ( ξ )] | p v ( dz ) dσ ≤ c p E (cid:16) Z ts Z Z ( t − σ ) − λ k U ( r, σ ) [ G ( σ, u ( σ ) , z ) − G ( σ, v ( σ ) , z )] k E v ( dz ) dσ (cid:17) p + c p E Z ts Z Z ( t − σ ) − pλ k U ( r, σ ) [ G ( σ, u ( σ ) , z ) − G ( σ, v ( σ ) , z )] k p E v ( dz ) dσ c p E (cid:16) Z ts ( t − σ ) − λ k u ( σ ) − v ( σ ) k E dσ (cid:17) p + c p E Z ts ( t − σ ) − pλ k u ( σ ) − v ( σ ) k p E dσ ≤ c p,T sup r ∈ [ s,T ] E k u ( r ) − v ( r ) k p E · h(cid:16) Z t − s σ − pλp − dσ (cid:17) p − + Z t − s σ − pλ dσ i , so E k φ λ ( u, v ) ( t ) k p = E (cid:16) Z D | φ λ ( u, v ) ( t, ξ ) | p dξ (cid:17) p ≤ c p,T |O| p E k u − v k L s,T,p ( E ) h(cid:16) Z t − s σ − pλp − dσ (cid:17) p − + Z t − s σ − pλ dσ i p . (3.6)where |O| is Lebesgue measure of the bounded domain O . Because u, v ∈ L p ( Ω ; D ([ s, T ] ; E )), sowe know that Ψ ( u ) − Ψ ( v ) ∈ D (cid:0) [ s, T ] ; W ǫ,p (cid:0) O ; R (cid:1)(cid:1) , P − a.s. for any λ < min( p − p , p ). In addition,we know k Ψ ( u ) − Ψ ( v ) k L s,T,p ( E ) = h sup t ∈ [ s,T ] k Ψ ( u ) ( t ) − Ψ ( v ) ( t ) k p E i p (3.7)according to the equation (3.5) and (3.6), we can get that Ψ maps the space L p ( Ω ; D ([ s, T ] ; E ))into itself, and (3.3) holds with c Ψs,p ( t ) = Z t − s ( r ∧ λ − ǫ − dr · h(cid:16) Z t − s σ − pλp − dσ (cid:17) p − + Z t − s σ − pλ dσ i p . (cid:3) Remark 3.3.
For any x := ( x , x ) ∈ E × E , according to the assumption (A6), we know thatthere exists m := ( m , m ) ∈ R × R and positive constants c , such that sup ξ ∈ ¯ O Z Z | G ( t, x, z ) | v ( dz ) ≤ c (cid:0) | x | m (cid:1) , ( t, z ) ∈ ( R , Z ) . For any p ≥ , if u ∈ L p ( Ω ; D ([ s, T ] ; E )) , by proceeding as Lemma 3.2, we can get that Ψ ( u ) ∈ D (cid:0) [ s, T ] ; W ǫ,p (cid:0) O ; R (cid:1)(cid:1) , and it is easy to prove that there exists some continuous increasing function c Ψs,p ( t ) with c Ψs,p ( s ) = 0 , such that k Ψ ( u ) k pL s,T,p ( E ) ≤ c Ψs,p ( T ) (cid:0) k u k pm L s,T,p ( E ) (cid:1) . (3.8) Moreover, as the space W ǫ,p (cid:0) ¯ O ; R (cid:1) continuously into C θ ( ¯ O ) for any θ < ǫ − d/p , so we have that Ψ ( u ) ∈ C θ ( ¯ O ) , and E sup t ∈ [ s,T ] k Ψ ( u ) ( t ) k pC θ ( ¯ O ) ≤ c Ψs,p ( T ) (cid:16) k u k pm L s,T,p ( E ) (cid:17) . (3.9)Now, for any α > u ∈ L p ( Ω ; D ([ s, T ] ; E )), we define Ψ α ( u ) ( t ) := Z ts Z Z U α ( t, r ) G ( r, u ( r ) , z ) ˜ N ( dr, dz ) . e also can prove that Ψ α maps L p ( Ω ; D ([ s, T ] ; E )) into itself for any p ≥ k Ψ α ( u ) k pL s,T,p ( E ) ≤ c Ψ,αs,p ( T ) (cid:0) k u k pm L s,T,p ( E ) (cid:1) , (3.10)for some continuous increasing function c Φ,αs,p ( t ) and c Φ,αs,p ( s ) = 0.Now, we prove the existence and uniqueness of the solution for system (3.1). Theorem 3.4.
Under the assumptions (A1)-(A6), for any x ∈ E and p ≥ , there exists a uniquemild solution u xs ∈ L p (Ω; D ([ s, T ] ; E )) for equation (3.1). Moreover, there have k u k L s,T,p ( E ) ≤ c s,p ( T ) (1 + k x k E ) , (3.11) for some continuous increasing function c s,p . Proof:
In order to prove the existence of the solution for system (3.1), we construct the followingequations. For any n ∈ N , i = 1 , t, ξ ) ∈ [0 , ∞ ) × ¯ O , we define b i,n ( t, ξ, σ ) := (cid:26) b i ( t, ξ, σ ) b i ( t, ξ, nσ/ | σ | ) if | σ | ≤ n,if | σ | > n. For any n ∈ N , we can easily know that b i,n ( t, ξ, · ) : R → R is Lipschitz-continuous uniformly withrespect to ξ ∈ ¯ O and t ∈ [ s, T ]. For any x ∈ E , define the corresponding composition operator B n associated with b n = ( b ,n , b ,n ) is B n ( t, x )( ξ ) := b n ( t, ξ, x ( ξ )) , ξ ∈ ¯ O . It is easy to get that B n ( t, · ) is Lipschitz-continuous. Moreover, if m < n , we have k x k E ≤ m ⇒ B m ( t, x ) = B n ( t, x ) = B ( t, x ) . (3.12)Next, we give the following problem du ( t ) = [ A ( t ) u ( t ) + B n ( t, u ( t ))] dt + F ( t, u ( t )) dw Q ( t ) + Z Z G ( t, u ( t ) , z ) ˜ N ( dt, dz ) . (3.13)Because B n ( t, · ) is Lipschitz-continuous, so the mapping Φ n Φ n ( u ) ( t ) := Z ts U ( t, r ) B n ( r, u ( r )) dr is Lipschitz-continuous in L p ( Ω ; D ([ s, T ] ; E )). By proceeding as [23, Lemma 6.1.2], we can provethat for any t ∈ [ s, T ] , ǫ ∈ (0 ,
1] and u, v ∈ L p ( Ω ; D ([ s, T ]; E )), it yield k ψ ( u ; s ) ( t ) k E ≤ c Z ts (( t − r ) ∧ − ǫ k u ( r ) k E dr ≤ c Z t − s ( r ∧ − ǫ dr sup r ∈ [ s,t ] k u ( r ) k E , (3.14)so, for any p ≥ , we can get k ψ ( u ) − ψ ( v ) k L s,T,p ( E ) ≤ k ψ ( u − v ) k L s,T,p ( E ) ≤ c ψs,p ( T ) k u − v k L s,T,p ( E ) . (3.15) here c ψs,p is a continuous increasing function with c ψs,p ( s ) = 0. In addition, according to [24,Theorem 4.2, Remark 4.3], we know that there exists a constant p ∗ ≥
1, such that for any p ≥ p ∗ ,we have k Γ ( u ) − Γ ( v ) k pL s,T,p ( E ) ≤ c Γs,p ( T ) k u − v k L s,T,p ( E ) , (3.16)and k Γ ( u ) k pL s,T,p ( E ) ≤ c Γs,p ( T ) (cid:0) k u k pm L s,T,p ( E ) (cid:1) . (3.17)where c Ψs,p is a continuous increasing function with c Ψs,p ( s ) = 0.Due to Lemma 3.2, we have get that the mapping Ψ is a contraction in L p ( Ω ; D ([ s, T ] ; E )).Moreover, because Φ n ( u ) is Lipschitz-continuous and according to the equation (3.15) and (3.16),we can know that the mapping Φ n , ψ and Γ are contraction in L p ( Ω ; D ([ s, T ] ; E )). So, we can getthat the mild solution u n of the equation (3.13) is the unique fixed point of the following mapping u ( t ) U ( t, s ) x + ψ ( u ; s ) ( t ) + Φ n ( u ) ( t ) + Γ ( u ) ( t ) + Ψ ( u ) ( t ) . Next, we prove that the sequence { u n } is bounded in L p (Ω; D ([ s, T ] ; E )) . Lemma 3.5.
For any n ∈ N and t ∈ [ s, T ] , there exists a continuous increaing function c s,p ( t ) such that k u n k L s,T,p ( E ) ≤ c s,p ( T ) (1 + k x k E ) , (3.18) Proof:
Denote Λ( u n ) is the solution of dv ( t ) = A ( t ) v ( t ) dt + F ( t, u n ( t )) dw Q ( t ) + Z Z G ( t, u n ( t ) , z ) ˜ N ( dt, dz ) , v ( s ) = 0 , (3.19)we can get that Λ( u n ) ∈ L p ( Ω ; D ([ s, T ] ; E )) is the unique fixed point of the mapping v ( t ) ψ ( v ; s ) ( t ) + Γ ( u n ) ( t ) + Ψ ( u n ) ( t ) . So, we have k Λ ( u n ) ( t ) k E ≤ k ψ ( Λ ( u n ) ; s ) ( t ) k E + k Γ ( u n ) ( t ) k E + k Ψ ( u n ) ( t ) k E . (3.20)According to the equation (3.14) and (3.20), using the Gronwall inequality, we can get k Λ ( u n ) ( t ) k E ≤ c Z ts (( t − r ) ∧ − ǫ [ k Γ ( u n ) ( r ) k E + k Ψ ( u n ) ( r ) k E ] e c R tr (( t − σ ) ∧ − ǫ dσ dr + k Γ ( u n ) ( t ) k E + k Ψ ( u n ) ( t ) k E ≤ (cid:16) sup r ∈ [ s,T ] k Γ ( u n ) ( r ) k E + sup r ∈ [ s,T ] k Ψ ( u n ) ( r ) k E (cid:17) × (cid:16) e c R ts (( t − σ ) ∧ − ǫ dσ − (cid:17) + (cid:16) sup r ∈ [ s,T ] k Γ ( u n ) ( r ) k E + sup r ∈ [ s,T ] k Ψ ( u n ) ( r ) k E (cid:17) ≤ c s ( t ) (cid:16) sup r ∈ [ s,T ] k Γ ( u n ) ( r ) k E + sup r ∈ [ s,T ] k Ψ ( u n ) ( r ) k E (cid:17) . (3.21)If we set v n ( t ) := u n ( t ) − Λ ( u n ) ( t ), we know that v n is the solution of the problem dv n dt ( t ) = A ( t ) v n ( t ) dt + B n ( t, v n ( t ) + Λ ( u n ) ( t )) , v n ( s ) = x. (3.22) ccording to the assumptions (A3) and (A4), we know that there exists m := ( m , m ) ∈ R × R ,such that for any δ v n ∈ ∂ k v n ( t ) k E , we yield ddt − k v n ( t ) k E ≤ h A ( t ) v n ( t ) , δ v n i E + h B n ( t, v n ( t ) + Λ ( u n ) ( t )) , δ v n i E ≤ h A ( t ) v n ( t ) , δ v n i E + h B n ( t, v n ( t ) + Λ ( u n ) ( t )) − B n ( t, Λ ( u n ) ( t )) , δ v n i E + h B n ( t, Λ ( u n ) ( t )) , δ v n i E ≤ c k v n ( t ) k E + c (1 + k Λ ( u n ) ( t ) k m E ) , so k v n ( t ) k E ≤ e c ( t − s ) k x k E + c Z ts e c ( t − r ) (1 + k Λ ( u n ) ( r ) k m E ) dr ≤ c s ( t ) (cid:0) k x k E + sup r ∈ [ s,T ] k Γ ( u n ) ( r ) k m E + sup r ∈ [ s,T ] k Ψ ( u n ) ( r ) k m E (cid:1) . (3.23)Due to the definition of u n ( t ) and the equation (3.21) and (3.23), we can get that k u n ( t ) k E ≤ c s ( t ) (cid:16) k x k E + sup r ∈ [ s,T ] k Γ ( u n ) ( r ) k m E + sup r ∈ [ s,T ] k Ψ ( u n ) ( r ) k m E (cid:17) . So, due to (3.17) and Remark 3.3, we can get that there exists a constant p ∗ ≥
1, such that, forany p ≥ p ∗ , we have E sup r ∈ [ s,t ] k u n ( r ) k p E ≤ c s,p ( t ) (cid:16) k x k p E + (cid:0) c Γs,p ( t ) + c Ψs,p ( t ) (cid:1) E sup r ∈ [ s,t ] k u n ( r ) k p E (cid:17) , because c Γs,p ( s ) = c Ψs,p ( s ) = 0 and c s,p , c Γs,p , c
Ψs,p are continuous, there exists t , such that c s,p ( s + t ) · [ c Γs,p ( s + t ) + c Ψs,p ( s + t )] ≤ /
2. For any t ∈ [ s, s + t ] we have E sup t ∈ [ s,s + t ] k u n ( t ) k p E ≤ c s,p ( t ) (cid:0) k x k p E (cid:1) . (3.24)By proceeding it in the intervals [ s + t , s + 2 t ] , [ s + 2 t , s + 3 t ] etc., we get that for any T > s and p ≥ p ∗ , (3.18) holds. If p < p ∗ , using the H¨older inequality, we can get (3.18) also holds. (cid:3) Finally, through the sequence { u xn } , we can prove that Theorem 3.4 holds. For any n ∈ N and x ∈ E , we define τ n := inf { t ≥ s : k u n ( t ) k E ≥ n } , and let τ := sup n ∈ N τ n . We can prove that the sequence of stopping times { τ n } is non-decreasing, and thanks to (3.18), wecan get that P ( τ = + ∞ ) = 1.Therefore, for any t ≥ s and w ∈ { τ = + ∞} , there exists m ∈ N such that for any t ∈ [ s, T ],have t ≤ τ m ( w ), and then we define u ( t )( w ) := u m ( t )( w ) . Set η := τ n ∧ τ m , due to (3.12), we can get k u m ( t ∧ η ) − u n ( t ∧ η ) k E = k ψ ( u m − u n ; s ) ( t ∧ η ) k E (cid:13)(cid:13)(cid:13) Z t ∧ ηs U ( t ∧ η, r ) [ B m ( r, u m ( r )) − B n ( r, u n ( r ))] dr (cid:13)(cid:13)(cid:13) E + k Γ ( u m ) ( t ∧ η ) − Γ ( u n ) ( t ∧ η ) k E + k Ψ ( u m ) ( t ∧ η ) − Ψ ( u n ) ( t ∧ η ) k E = k ψ ( u m − u n ; s ) ( t ∧ η ) k E + (cid:13)(cid:13)(cid:13) Z ts I { r ≤ η } U ( t ∧ η, r ) × [ B m ∨ n ( r ∧ η, u m ( r ∧ η )) − B m ∨ n ( r ∧ η, u n ( r ∧ η ))] dr (cid:13)(cid:13)(cid:13) E + k Γ ( u m ) ( t ∧ η ) − Γ ( u n ) ( t ∧ η ) k E + k Ψ ( u m ) ( t ∧ η ) − Ψ ( u n ) ( t ∧ η ) k E ≤ sup r ∈ [ s,t ] k ψ ( u m − u n ; s ) ( r ∧ η ) k E + c Z ts sup σ ∈ [ s,r ] k u m ( σ ∧ η ) − u n ( σ ∧ η ) k E dr + sup r ∈ [ s,t ] k Γ ( u m ) ( r ∧ η ) − Γ ( u n ) ( r ∧ η ) k E + sup r ∈ [ s,t ] k Ψ ( u m ) ( r ∧ η ) − Ψ ( u n ) ( r ∧ η ) k E . (3.25)By proceeding as the proof of Lemma 3.2, using the factorization arguement for Ψ ( u m ) ( r ∧ η ) − Ψ ( u n ) ( r ∧ η ), we can obtain E sup r ∈ [ s,t ] k Ψ ( u m ) ( r ∧ η ) − Ψ ( u n ) ( r ∧ η ) k E ≤ c Ψs, ( t ) E sup r ∈ [ s,t ] k u m ( r ∧ η ) − u n ( r ∧ η ) k E . (3.26)Then, substitue (3.15), (3.16) and (3.26) into (3.25), we havesup r ∈ [ s,t ] k u m ( r ∧ η ) − u n ( r ∧ η ) k E ≤ (cid:0) c ψs, ( t ) + c Γs, ( t ) + c Ψs, ( t ) (cid:1) sup r ∈ [ s,t ] k u m ( r ∧ η ) − u n ( r ∧ η ) k E + c Z ts sup σ ∈ [ s,r ] k u m ( r ∧ η ) − u n ( r ∧ η ) k E dr. Fix t >
0, such that c ψs, ( t ) + c Γs, ( t ) + c Ψs, ( t ) ≤ /
2, we can get E sup r ∈ [ s,s + t ] k u m ( r ∧ η ) − u n ( r ∧ η ) k E ≤ c Z s + t s E sup σ ∈ [ s,r ] k u m ( r ∧ η ) − u n ( r ∧ η ) k E dr. According to the Gronwall lemma, we have E sup r ∈ [ s,s + t ] k u m ( r ∧ η ) − u n ( r ∧ η ) k E = 0, that is, for any t ∈ [ s, s + t ], we have u m ( t ∧ η ) = u n ( t ∧ η ) . Repeat it in the interval [ s + t , s + 2 t ] , [ s + 2 t , s + 3 t ] , etc., we obtain u m ( t ) = u n ( t ) , s ≤ t ≤ τ m ∧ τ n , (3.27)for any n ∈ N . Because when w ∈ { τ = + ∞} and t ≤ τ m , we have denote u ( t ) = u m ( t ), thanks to(3.12), this yields u ( t ) = U ( t, s ) x + ψ ( u ; s ) ( t ) + Z ts U ( t, r ) B ( r, u ( r )) dr + Z ts U ( t, r ) F ( r, u ( r )) dw Q ( r )+ Z ts Z Z U ( t, r ) G ( r, u ( r ) , z ) ˜ N ( dr, dz ) ,P − a.s. , that is, u ( t ) is the mild solution of the system (3.1). ow, we prove the solution of system (3.1) is unique. Denote another solution of system (3.1)is v , by proceeding as the equation (3.27), we can get that for any n ∈ N u ( t ) = v ( t ) , s ≤ t ≤ τ n . For any T ≥ s , we know { τ n ≤ T } ↓ { τ ≤ T } , we get that u = v .Finally, for any p ≥ T > s , we havesup t ∈ [ s,T ] k u ( t ) k p E = lim n → + ∞ sup t ∈ [ s,T ] k u ( t ) k p E I { T ≤ τ n } = lim n → + ∞ sup t ∈ [ s,T ] k u n ( t ) k p E I { T ≤ τ n } , according to the estimate (3.18) and the Fatou lemma, we can get (3.11).
4. The slow-fast system
According to the introduced in Section 2, system (1.1) can be rewritten as: du ǫ ( t ) = [ A ( t ) u ǫ ( t ) + B ( t, u ǫ ( t ) , v ǫ ( t ))] dt + F ( t, u ǫ ( t )) dω Q ( t )+ R Z G ( t, u ǫ ( t ) , z ) ˜ N ( dt, dz ) ,dv ǫ ( t ) = ǫ [( A ( t ) − α ) v ǫ ( t ) + B ( t, u ǫ ( t ) , v ǫ ( t ))] dt + √ ǫ F ( t, v ǫ ( t )) dω Q ( t ) + R Z G ( t, v ǫ ( t ) , z ) ˜ N ǫ ( dt, dz ) ,u ǫ ( s ) = x, v ǫ ( s ) = y. (4.1)Since the coefficients under the assumptions (A1)-(A6) are uniform with respect to t ∈ R ,according the prove in Section 3, we can get that there exist two unique adapted u ǫ and v ǫ in L p (Ω; D ([ s, T ] ; E )), such that u ǫ ( t ) = U ( t, s ) x + ψ ( u ǫ ; s ) ( t ) + R ts U ( t, r ) B ( r, u ǫ ( r ) , v ǫ ( r )) dr + R ts U ( t, r ) F ( r, u ǫ ( r )) dw Q ( r )+ R ts R Z U ( t, r ) G ( r, u ǫ ( r ) , z ) ˜ N ( dr, dz ) ,v ǫ ( t ) = U α,ǫ, ( t, s ) y + ψ α,ǫ, ( v ǫ ; s ) ( t ) + ǫ R ts U α,ǫ, ( t, r ) B ( r, u ǫ ( r ) , v ǫ ( r )) dr + √ ǫ R ts U α,ǫ, ( t, r ) F ( r, v ǫ ( r )) dw Q ( r )+ R ts R Z U α,ǫ, ( t, r ) G ( r, v ǫ ( r ) , z ) ˜ N ǫ ( dr, dz ) . (4.2)Under the assumptions (A1)-(A6), by proceeding as [20, Lemma 5.1] and [13, Lemma 3.1], wecan get that for any p ≥ T >
0, there exists a positive constant c p,T , such that for any x, y ∈ E and ǫ ∈ (0 , E sup t ∈ [ s,T ] k u ǫ ( t ) k p E ≤ c p,T (cid:0) k x k p E + k y k p E (cid:1) , (4.3)and Z Ts E k v ǫ ( t ) k p E dt ≤ c p,T (cid:0) k x k p E + k y k p E (cid:1) . (4.4)Then, due to the equation (3.9) and the estimates (4.3) and (4.4), using the proof of [13,Proposition 3.2] to the present situation, we can prove that there exists ¯ θ >
0, such that for any
T > s, x ∈ C θ ( ¯ O ) with θ ∈ [0 , ¯ θ ) and y ∈ E , we havesup ǫ ∈ (0 , E k u ǫ ( t ) k L ∞ ( s,T ; C θ ( ¯ O )) ≤ c θ,T (cid:16) k x k m C θ ( ¯ O ) + k y k m E (cid:17) , (4.5) here c θ,T > θ ∈ [0 , ¯ θ ),there also exists β ( θ ) >
0, such that, for any
T > , p ≥ , x ∈ C θ ( ¯ O ) , y ∈ E and s, t ∈ [0 , T ], wehave sup ǫ ∈ (0 , E k u ǫ ( t ) − u ǫ ( s ) k p E ≤ c p,θ,T (cid:0) | t − s | β ( θ ) p + | t − s | (cid:1) (cid:16) k x k m pC θ ( ¯ O ) + k y k m p E (cid:17) . (4.6)According to the equation (4.5) and (4.6), using the Arzel`a-Ascoli theorem, we know that thefamily { L ( u ǫ ) } ǫ ∈ (0 , is tight.
5. The averaged equation
In this section, we research the fast equation with frozen slow component x ∈ E , we main provethat there also exists an evolution family of measures for this fast equation and define the averagedequation through it.First, for any s ∈ R , any frozen slow component x ∈ E and initial condition y ∈ E , we introducethe following problem dv ( t ) = [( A ( t ) − α ) v ( t ) + B ( t, x, v ( t ))] dt + F ( t, v ( t )) d ¯ ω Q ( t )+ Z Z G ( t, v ( t ) , z ) ˜ N ′ ( dt, dz ) , v ( s ) = y, (5.1)where ¯ w Q ( t ) = ( w Q ( t ) ,w Q ( − t ) , if t ≥ ,if t < , ˜ N ′ ( t, z ) = (cid:26) ˜ N ′ ( t, z ) , ˜ N ′ ( − t, z ) , if t ≥ ,if t < , where ˜ N ′ ( dt, dz ) and ˜ N ′ ( dt, dz ) has the same L´evy measure. The process w Q ( t ), w Q ( t ),˜ N ′ ( dt, dz ) and ˜ N ′ ( dt, dz ) are independent and the definition of which is given in Section 2.According to the prove in Section 3, we can get that for any x, y ∈ E , p ≥ s < T , thereexists a unique mild solution v x ( · ; s, y ) ∈ L p (Ω; D ([ s, T ] ; E )). And using the same argument as ourprevious work [20], we can get that there also exists δ >
0, such that for any x, y ∈ E and p ≥ E k v x ( t ; s, y ) k p E ≤ c p (cid:16) k x k p E + e − δp ( t − s ) k y k p E (cid:17) , s < t. (5.2)Next, same as our previous work [20], if t ∈ R , we also giving the following problem dv ( t ) = [( A ( t ) − α ) v ( t ) + B ( t, x, v ( t ))] dt + F ( t, v ( t )) d ¯ ω Q ( t )+ Z Z G ( t, v ( t ) , z ) ˜ N ′ ( dt, dz ) , (5.3)for every s < t .By proceeding as [19] and using the conclusion we have proved in [20], it is easy to prove thatfor any t ∈ R and p ≥
1, there exists η x ( t ) ∈ L p ( Ω ; E ) such that for all x, y ∈ E , we havelim s →−∞ E k v x ( t ; s, y ) − η x ( t ) k p E = 0 . (5.4) nd we can get that η x is a mild solution in R of equation (5.3). Moreover, for any R >
0, therealso exists c R > x , x ∈ E x , x ∈ B E ( R ) ⇒ sup t ∈ R E k η x ( t ) − η x ( t ) k E ≤ c R k x − x k E . (5.5)Then, for any t ∈ R and x ∈ E , we denote that the law of the random variable η x ( t ) is µ xt . Asthe prove of our previous work [20], we also can get that { µ xt } t ∈ R defines an evolution family ofmeasures on E for equation (5.1).Now, we give the following assumption.(A7) (a) The functions γ : R → (0 , ∞ ) and l : R × O → R d are periodic, with the same period.(b) The families of functions B ,R := { b ( · , ξ, σ ) : ξ ∈ O , σ ∈ B R ( R ) } , B ,R := { b ( · , ξ, σ ) : ξ ∈ O , σ ∈ B R ( R ) } , F R := { f ( · , ξ, σ ) : ξ ∈ O , σ ∈ B R ( R ) } , G R := { g ( · , ξ, σ, z ) : ξ ∈ O , σ ∈ B R ( R ) , z ∈ Z } , are uniformly almost periodic for any R > Remark 5.1.
Similar with the proof of [19, Lemma 6.2], we get that under the assumption (A7),for any
R >
0, the families of functions { B ( · , x, y ) : ( x, y ) ∈ B E × E ( R ) } , { B ( · , x, y ) : ( x, y ) ∈ B E × E ( R ) } , { F ( · , y ) : y ∈ B E ( R ) } , { G ( · , y, z ) : ( y, z ) ∈ B E ( R ) × Z } , are uniformly almost periodic.As in [20] and [19], we can prove that under the assumptions (A1)-(A7), the mapping t ∈ R µ xt ∈ P ( E ) is almost periodic. Then, due to (5.5), we also can get that for every compact set K ⊂ E , the family of functions (cid:26) t ∈ R Z E B ( t, x, y ) µ xt ( dy ) : x ∈ K (cid:27) (5.6)is uniformly almost periodic. So, we define¯ B ( x ) := lim T →∞ T Z T Z E B ( t, x, y ) µ t ( dy ) dt, x ∈ E , (5.7)we can get that the mapping ¯ B : E → E is locally Lipschitz-continuous. Similar with the prove of[20, Lemma 4.2] and [19, Lemma 7.2], we can conclude that the following crucial results are alsoestablished in this paper. Lemma 5.2.
Under the assumptions (A1)-(A7) , for any
T > , s ∈ R and x, y ∈ E , there existsome constants κ , κ ≥ , we have E (cid:12)(cid:12)(cid:12) T Z t + Tt B ( t, x, v x ( t ; s, y )) dr − ¯ B ( x ) (cid:12)(cid:12)(cid:12) ≤ cT (1 + k x k κ E + k y k κ E ) + α ( T, x ) (5.8) or some mapping α : [0 , ∞ ) × E → [0 , ∞ ) such that sup T > α ( T, x ) ≤ c (1 + k x k m E ) , x ∈ E , (5.9) and for any compact set K ⊂ E , have lim T →∞ sup x ∈ E α ( T, x ) = 0 . (5.10)We introduce the following averaged equation du ( t ) = (cid:2) A ( t ) u ( t ) + ¯ B ( u ( t )) (cid:3) dt + F ( t, u ( t )) dw Q ( t )+ Z Z G ( t, u ( t ) , z ) ˜ N ( dt, dz ) , u (0) = x ∈ E . (5.11)Due to Theorem 3.4, we can prove that for any T > , p ≥ x ∈ E , equation (5.11) admits aunique mild solution ¯ u .
6. Averaging principles
In this section, we will show that the validity of the averaging principle. That is, the slowmotion u ǫ will converges to the averaged motion ¯ u , as ǫ → Theorem 6.1.
Under the assumptions (A1)-(A7), fix x ∈ C θ ( ¯ O ) with θ ∈ [0 , ¯ θ ) , and y ∈ E , forany T > and η > , we have lim ǫ → P (cid:16) sup t ∈ [0 ,T ] k u ǫ ( t ) − ¯ u ( t ) k E > η (cid:17) = 0 , (6.1) where ¯ u is the solution of the averaged equation (5.11). Proof:
For any h ∈ D ( A ), we have Z O u ǫ ( t, ξ ) h ( ξ ) dξ = Z O x ( ξ ) h ( ξ ) dξ + Z t Z O u ǫ ( r, ξ ) A ( r ) h ( ξ ) dξdr + Z t Z O ¯ B ( u ǫ ( r )) ( ξ ) h ( ξ ) dξdr + Z t Z O [ F ( r, u ǫ ( r )) h ] ( ξ ) dw Q ( r, ξ )+ Z t Z Z Z O [ G ( r, u ǫ ( r ) , z ) h ] ( ξ ) dξ ˜ N ( dr, dz ) + R ǫ ( t ) , where R ǫ ( t ) := Z t Z O (cid:0) B ( r, u ǫ ( r ) , v ǫ ( r )) ( ξ ) − ¯ B ( u ǫ ( r )) ( ξ ) (cid:1) h ( ξ ) dξdr. As the proof in [20], because we have get that the family { L ( u ǫ ) } ǫ ∈ (0 , is tight in Section4. If we want to prove Theorem 6.1, it is sufficient to prove that for any T >
0, we havelim ǫ → E sup t ∈ [0 ,T ] k R ǫ ( t ) k E = 0.First, for any n ∈ N , we define b i,n ( t, ξ, σ , σ ) := (cid:26) b i ( t, ξ, σ , σ ) ,b i ( t, ξ, σ n/ | σ | , σ ) , if | σ | ≤ n,if | σ | > n. (6.2) or each b i,n , denote the corresponding composition operator is B i,n , and we have x ∈ B E ( n ) ⇒ B i,n ( t, x, y ) = B i ( t, x, y ) , t ∈ R , y ∈ E . (6.3)It is easy to get that the mapping b ,n and b ,n satisfy all conditions in (A3) and (A4), respectively.And for any fixed ( t, ξ ) ∈ R × ¯ O and σ ∈ R , the mapping b i,n ( t, ξ, · , σ ) are Lipschitz-continuous.In addition, for any n ∈ N , we define f ,n ( t, ξ, σ ) := (cid:26) f ( t, ξ, σ ) ,f ( t, ξ, σn/ | σ | ) , if | σ | ≤ n,if | σ | > n. , and g ,n ( t, ξ, σ ) := (cid:26) g ( t, ξ, σ, z ) ,g ( t, ξ, σn/ | σ | , z ) , if | σ | ≤ n,if | σ | > n. , where ( t, ξ ) ∈ R × ¯ O and z ∈ Z . The corresponding composition operator of f ,n and g ,n aredenoted by F ,n and G ,n , respectively.Now, for any n ∈ N , we introduce the following system du ( t ) = [ A ( t ) u ( t ) + B ,n ( t, u ( t ) , v ( t ))] dt + F ,n ( t, u ( t )) dω Q ( t )+ R Z G ,n ( t, u ( t ) , z ) ˜ N ( dt, dz ) ,dv ( t ) = ǫ [( A ( t ) − α ) v ( t ) + B ,n ( t, u ( t ) , v ( t ))] dt + √ ǫ F ( t, v ( t )) dω Q ( t ) + R Z G ( t, v ( t ) , z ) ˜ N ǫ ( dt, dz ) ,u ( s ) = x, v ( s ) = y, (6.4)we denote the solution of (6.4) is ( u ǫ,n , v ǫ,n ).Then, for any n ∈ N and any frozen slow component x ∈ E , we introduce the following problem dv ( t ) = [( A ( t ) − α ) v ( t ) + B ,n ( t, x, v ( t ))] dt + F ( t, v ( t )) dω Q ( t )+ Z Z G ( t, v ( t ) , z ) ˜ N ( dt, dz ) , v ( s ) = y, (6.5)and denote its solution is v xn ( t ; s, y ). Thanks to (6.3), for any t ≥ x ∈ E , we have v xn ( t ; s, y ) = (cid:26) v x ( t ; s, y ) ,v x n ( t ; s, y ) , if | x ( ξ ) | ≤ n,if | x ( ξ ) | > n, where x n = nsignx ( ξ ).Due to the coefficients of equation (6.5) satisfy the same conditions as the equation (5.3), foreach x ∈ E , there exists an evolution of measures family { µ x,nt } t ∈ R for equation (6.5) µ x,nt = (cid:26) µ xt ,µ x n t , if | x ( ξ ) | ≤ n,if | x ( ξ ) | > n. As proof of (5.2), for any
T > , x ∈ E and p ≥
1, we can get that there also exists δ >
0, such that E k v xn ( t ; s, y ) k p E ≤ c p,n (cid:16) e − δp ( t − s ) k y k p E (cid:17) , s < t. (6.6)Similarly, we can define¯ B ,n ( x ) := lim T →∞ T Z T Z E B ,n ( t, x, y ) µ x,nt ( dy ) dt, x ∈ E, (6.7) nd, we have k x k E ≤ n ⇒ ¯ B ,n ( x ) = ¯ B ( x ) . Moreover, it is easy to prove that the mapping ¯ B ,n : E → E is Lipschitz-continuous, and the resultssimilar with (5.8)-(5.10) also can be established.Next, we prove that the validity of the averaging principle by using the classical Khasminskiimethod as in [20]. For any ǫ >
0, we divide the interval [0 , T ] in subintervals of the size δ ǫ , where δ ǫ > v ǫ in eachtime interval [ kδ ǫ , ( k + 1) δ ǫ ] , k = 0 , , · · · , ⌊ T /δ ǫ ⌋ d ˆ v ǫ,n ( t ) = ǫ [( A ( t ) − α ) ˆ v ǫ,n ( t ) + B ,n ( t, u ǫ,n ( kδ ǫ ) , ˆ v ǫ,n ( t ))] dt + √ ǫ F ( t, ˆ v ǫ,n ( t )) dω Q ( t ) + R Z G ( t, ˆ v ǫ,n ( t ) , z ) ˜ N ǫ ( dt, dz ) , ˆ v ǫ,n ( kδ ǫ ) = v ǫ,n ( kδ ǫ ) . (6.8)Like the equation (4.4), we also can prove that for any p ≥
1, we have Z T E k ˆ v ǫ,n ( t ) k p E dt ≤ c p,T (cid:0) k x k p E + k y k p E (cid:1) . (6.9) Lemma 6.2.
Under the assumptions (A1)-(A7), fix x ∈ C θ ( ¯ O ) with θ ∈ [0 , ¯ θ ) , and y ∈ E , thereexists a constant κ > , such that if δ ǫ = ǫ ln ǫ − κ , and, for any fixed n ∈ N , we have lim ǫ → sup t ∈ [0 ,T ] E k ˆ v ǫ,n ( t ) − v ǫ,n ( t ) k p E = 0 . (6.10) Proof:
Fixed ǫ > n ∈ N . For any t ∈ [ kδ ǫ , ( k + 1) δ ǫ ] , k = 0 , , · · · , ⌊ T /δ ǫ ⌋ , let ρ ǫ,n ( t ) be thesolution of the following problem dρ ǫ,n ( t ) = 1 ǫ ( A ( t ) − α ) ρ ǫ,n ( t ) dt + 1 √ ǫ K ǫ,n ( t ) dω Q ( t )+ Z Z H ǫ,n ( t, z ) ˜ N ǫ ( dt, dz ) , ρ ǫ ( kδ ǫ ) = 0 , where K ǫ,n ( t ) := F ( t, ˆ v ǫ,n ( t )) − F ( t, v ǫ,n ( t )) ,H ǫ,n ( t, z ) := G ( t, ˆ v ǫ,n ( t ) , z ) − G ( t, v ǫ,n ( t ) , z ) . We have ρ ǫ,n ( t ) = ψ α,ǫ, ( ρ ǫ,n ; kδ ǫ ) ( t ) + Γ ǫ,n ( t ) + Ψ ǫ,n ( t ) , t ∈ [ kδ ǫ , ( k + 1) δ ǫ ] , where Γ ǫ,n ( t ) = 1 √ ǫ Z tkδ ǫ U α,ǫ, ( t, r ) K ǫ,n ( r ) dw Q ( r ) ,Ψ ǫ,n ( t ) = Z tkδ ǫ Z Z U α,ǫ, ( t, r ) H ǫ,n ( r, z ) ˜ N ǫ ( dr, dz ) . sing the same arguement as [13, Lemma 6.3], we yield E k Γ ǫ,n ( t ) k p E ≤ c p,n ǫ Z tkδ ǫ E k ˆ v ǫ,n ( r ) − v ǫ,n ( r ) k p E dr. (6.11)For Ψ ǫ,n ( t ), using Kunita’s first inequality and the H¨older inequality, we can get E k Ψ ǫ,n ( t ) k p E ≤ c p,n E (cid:16) ǫ Z tkδ ǫ Z Z (cid:13)(cid:13)(cid:13) e − αǫ ( t − r ) e γ t,r ) ǫ A H ǫ,n ( r, z ) (cid:13)(cid:13)(cid:13) E v ( dz ) dr (cid:17) p + c p,n ǫ E Z tkδ ǫ Z Z (cid:13)(cid:13)(cid:13) e − αǫ ( t − r ) e γ t,r ) ǫ A H ǫ,n ( r, z ) (cid:13)(cid:13)(cid:13) p E v ( dz ) dr ≤ c p,n ǫ p E (cid:16) Z tkδ ǫ Z Z k H ǫ,n ( r, z ) k E v ( dz ) dr (cid:17) p + c p,n ǫ E Z tkδ ǫ Z Z k H ǫ,n ( r, z ) k p E v ( dz ) dr ≤ c p,n (cid:0) δ p − ǫ /ǫ p + 1 /ǫ (cid:1) Z tkδ ǫ E k ˆ v ǫ,n ( r ) − v ǫ,n ( r ) k p E dr. (6.12)Thanks to α > k ρ ǫ,n ( t ) k E ≤ k ψ α,ǫ, ( ρ ǫ,n ; kδ ǫ ) ( t ) k E + k Γ ǫ,n ( t ) k E + k Ψ ǫ,n ( t ) k E ≤ c p,n (cid:0) δ p − ǫ /ǫ p + 1 /ǫ (cid:1) Z tkδ ǫ E k ˆ v ǫ,n ( r ) − v ǫ,n ( r ) k p E dr. (6.13)If we denote Λ ǫ,n ( t ) := ˆ v ǫ,n ( t ) − v ǫ,n ( t ) and ϑ ǫ,n ( t ) := Λ ǫ,n ( t ) − ρ ǫ,n ( t ), we have dϑ ǫ,n ( t ) = 1 ǫ [( A ( t ) − α ) ϑ ǫ,n ( t ) + B ,n ( t, u ǫ,n ( kδ ǫ ) , ˆ v ǫ,n ( t )) − B ,n ( t, u ǫ,n ( t ) , v ǫ,n ( t ))] dt = 1 ǫ [( A ( t ) − α ) ϑ ǫ,n ( t ) + B ,n ( t, u ǫ,n ( kδ ǫ ) , ˆ v ǫ,n ( t )) − B ,n ( t, u ǫ,n ( t ) , ˆ v ǫ,n ( t )) − τ ( t, u ǫ,n ( t ) , ˆ v ǫ,n ( t ) , v ǫ,n ( t )) ( ϑ ǫ,n ( t ) + ρ ǫ,n ( t ))] dt (6.14)By proceeding as [13, Lemma 6.3], we can get k ϑ ǫ,n ( t ) k E ≤ c n ǫ Z tkδ ǫ e − αǫ ( t − r ) k u ǫ,n ( kδ ǫ ) − u ǫ,n ( r ) k E dr + 1 ǫ sup r ∈ [ kδ ǫ ,t ] k ρ ǫ,n ( r ) k E Z tkδ ǫ exp (cid:16) − ǫ Z tr τ ǫ,n ( σ ) dσ (cid:17) τ ǫ,n ( r ) dr ≤ c p,n (cid:0) k x k m pC θ ( ¯ O ) + k y k p E (cid:1)(cid:0) δ β ( θ ) pǫ + δ ǫ (cid:1) + c p,n (cid:0) δ p − ǫ /ǫ p + 1 /ǫ (cid:1) Z tkδ ǫ E k ˆ v ǫ,n ( r ) − v ǫ,n ( r ) k p E dr. (6.15)where τ ǫ,n ( r ) := τ ( ξ ǫ,n ( t ) , u ǫ,n ( t, ξ ǫ,n ( t )) , ˆ v ǫ,n ( t, ξ ǫ,n ( t )) , v ǫ,n ( t, ξ ǫ,n ( t ))) , and ξ ǫ,n ( t ) ∈ ¯ O , satisfy | ϑ ǫ,n ( t, ξ ǫ,n ( t )) | = k ϑ ǫ,n ( t ) k E . Due to (6.13) and (6.15), for any p ≥
1, we have E k ˆ v ǫ ( t ) − v ǫ ( t ) k p E ≤ c p,n (cid:0) k x k m pC θ ( ¯ O ) + k y k m p E (cid:1)(cid:0) δ β ( θ ) pǫ + δ ǫ (cid:1) c p,n (cid:0) δ p − ǫ /ǫ p + 1 /ǫ (cid:1) Z tkδ ǫ E k ˆ v ǫ,n ( r ) − v ǫ,n ( r ) k p E dr. From the Gronwall lemma, this means E k ˆ v ǫ,n ( t ) − v ǫ,n ( t ) k p E ≤ c p,n (cid:0) k x k m pC θ ( ¯ O ) + k y k p E (cid:1)(cid:0) δ β ( θ ) pǫ + δ ǫ (cid:1) e c p,n ( δ p − ǫ /ǫ p +1 /ǫ ) δ ǫ . For t ∈ [0 , T ], selecting δ ǫ = ǫ ln ǫ − κ , then if we take κ < β ( θ ) pβ ( θ ) p +2 c p,n ∧ c p,n , we have (6.10). (cid:3) Finally, under the same assumptions as in Theorem 6.1, by proceeding as [19, Lemma 8.2] and[20, Lemma 6.4], for any
T >
0, we can getlim ǫ → E sup t ∈ [0 ,T ] k R ǫ ( t ) k E = 0 . Through the above proof, Theorem 6.1 is established. (cid:3)
7. Conclusions
In this paper, we study the averaging principle for a class of non-autonomous slow-fast systemwith polynomial growth. First, using the Sobolev embedding theorem, fxed point theorem andstopping technique, the existence and uniqueness of the mild solution is proved. Next, by means ofthe comparison theorem and the properties of transition operator, the existence of time-dependentevolution family of measures associated with the fast equation is studied, and the averaged coefcientis obtained. Finally, through the truncation technique, the averaging principle for a class of non-autonomous slow-fast systems with polynomial growth is presented.
Acknowledgments
The research was supported in part by the NSF of China (11572247, 11802216) and the SeedFoundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical Uni-versity (ZZ2018027). B. Pei was an International Research Fellow of Japan Society for the Promo-tion of Science (Postdoctoral Fellowships for Research in Japan (Standard)). Y. Xu would like tothank the Alexander von Humboldt Foundation for the support.
ReferencesReferences [1] S. Cerrai, M. Freidlin, Averaging principle for a class of stochastic reaction–diffusion equations,Probability theory and related fields 144 (1-2) (2009) 137–177.[2] E. Harvey, V. Kirk, M. Wechselberger, J. Sneyd, Multiple timescales, mixed mode oscillationsand canards in models of intracellular calcium dynamics, J. Nonlinear Sci. 21 (5) (2011) 639–683.[3] F. K. Wu, T. Tian, J. B. Rawlings, G. Yin, Approximate method for stochastic chemicalkinetics with two-time scales by chemical Langevin equations, J. Chem. Phys. 144 (17) (2016)1195–1203.
4] N. N. Bogolyubov, Y. A. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Os-cillations, Gordon and Breach Science Publishers, New York, 1961.[5] R. Khasminskii, On the averaging principle for stochastic differential itˆo equations, Kyber-netika. 4 (1968) 260–279.[6] D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differentialsystems, Multiscale. Model. Sim. 6 (2007) 577–594.[7] M. Freidlin, A. Wentzell, Random Perturbations of Dynamical Systems, Springer Science andBusiness Media, Berlin Heidelberg, 2012.[8] J. Q. Duan, W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Else-vier, 2014.[9] Y. Xu, J. Q. Duan, W. Xu, An averaging principle for stochastic dynamical systems with L´evynoise, Physica D. 240 (2011) 1395–1401.[10] Y. Xu, B. Pei, Y. G. Li, Approximation properties for solutions to non-Lipschitz stochasticdifferential equations with L´evy noise, Math. Method Appl. Sci. 38 (2015) 2120–2131.[11] Y. Xu, B. Pei, J. L. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stoch. Dynam. 17 (2017) 1750013.[12] S. Cerrai, A khasminskii type averaging principle for stochastic reaction-diffusion equations,Ann. Appl. Probab. 19 (2009) 899–948.[13] S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomialnonlinearities perturbed by multiplicative noise, SIAM. J. Math. Anal. 43 (2011) 2482–2518.[14] W. Wang, A. Roberts, Average and deviation for slow-fast stochastic partial differential equa-tions, J. Differ. Equations. 253 (2012) 1265–1286.[15] B. Pei, Y. Xu, J. L. Wu, Two-time-scales hyperbolic-parabolic equations driven by Poissonrandom measures: Existence, uniqueness and averaging principles, J. Math. Anal. Appl. 447(2017) 243–268.[16] B. Pei, Y. Xu, G. Yin, Stochastic averaging for a class of two-time-scale systems of stochasticpartial differential equations, Nonlinear Anal. 160 (2017) 159–176.[17] B. Pei, Y. Xu, G. Yin, Averaging principles for SPDEs driven by fractional Brownian motionswith random delays modulated by two-time-scale Markov switching processes, Stoch. Dynam.18 (2018) 1850023.[18] J. Xu, Y. Miao, J. C. Liu, Strong averaging principle for slow-fast SPDEs with Poisson randommeasures, Discrete Cont. Dyn.-B. 20 (2015) 2233–2256.[19] S. Cerrai, A. Lunardi, Averaging principle for non-autonomous slow-fast systems of stochasticreaction-diffusion equations: The almost periodic case, SIAM. J. Math. Anal. 49 (2017) 2843–2884.[20] Y. Xu, R. F. Wang, B. Pei, Y. Z. Bai, J. Kurths, Averaging principles for non-autonomousslow-fast systems of stochastic reaction-diffusion equations with jumps, arXiv:1807.08068.
21] G. D. Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge UniversityPress, Cambridge, 2014.[22] D. Applebaum, L´evy Processes and Stochastic Calculus, Cambridge University Press, Cam-bridge, 2009.[23] S. Cerrai, Second Order PDEs in Finite and Infinite Dimension: A Probabilistic Approach,Springer-Verlag, Berlin, 2001.[24] S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitzreaction term, Probab. Theory Relat. Fields. 125 (2003) 271–304.21] G. D. Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge UniversityPress, Cambridge, 2014.[22] D. Applebaum, L´evy Processes and Stochastic Calculus, Cambridge University Press, Cam-bridge, 2009.[23] S. Cerrai, Second Order PDEs in Finite and Infinite Dimension: A Probabilistic Approach,Springer-Verlag, Berlin, 2001.[24] S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitzreaction term, Probab. Theory Relat. Fields. 125 (2003) 271–304.