existence-uniqueness and exponential estimate of

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 37, Number 4, April 2017

doi:10.3934/dcds.2017093 pp. 2161–2180

EXISTENCE-UNIQUENESS AND EXPONENTIAL ESTIMATE OF PATHWISE SOLUTIONS OF RETARDED STOCHASTIC EVOLUTION SYSTEMS WITH TIME SMOOTH DIFFUSION COEFFICIENTS

Daoyi Xu∗ Yangtze Center of Mathematics, Sichuan University Chengdu 610064, China

Weisong Zhou College of Mathematics and Physics, Chongqing University of Posts and Telecommunications Chongqing 400065, China

(Communicated by Massimiliano Gubinelli) Abstract. In this paper, we study the existence-uniqueness and exponential estimate of the pathwise mild solution of retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. Firstly, the existence-uniqueness of the maximal local pathwise mild solution are given by the generalized local Lipschitz conditions, which extend a classical Pazy theorem on PDEs. We assume neither that the noise is given in additive form or that it is a very simple multiplicative noise, nor that the drift coefficient is global Lipschitz continuous. Secondly, the existence-uniqueness of the global pathwise mild solution are given by establishing an integral comparison principle, which extends the classical Wintner theorem on ODEs. Thirdly, an exponential estimate for the pathwise mild solution is obtained by constructing a delay integral inequality. Finally, the results obtained are applied to a retarded stochastic infinite system and a stochastic partial functional differential equation. Combining some known results, we can obtain a random attractor, whose condition overcomes the disadvantage in existing results that the exponential converging rate is restricted by the maximal admissible value for the time delay.

1. Introduction. In this paper, we will study the stochastic evolution systems described by the following stochastic functional differential equation (SFDE) du(t) = [A(t)u(t) + f (t, ut )]dt + g(t, ut )dW (t), t ∈ [0, a), (1) u(t) = ξ(t), t ∈ [−τ, 0] in a Banach space X, where a is a constant or a = ∞, A(t) : D ⊂ X → X is a family of densely defined closed linear operators, and D is independent of t and dense in X, f and g are appropriate nonlinear terms and called the drift terms and the diffusion 2010 Mathematics Subject Classification. Primary: 60H15, 35R60; Secondary: 34K30, 34K50. Key words and phrases. Stochastic PDEs, pathwise solutions, existence, uniqueness, exponential estimate. The first author is supported by National Natural Science Foundation of China grant No. 11271270 and the second author is supported by the Doctor Start-up Funding of Chongqing University of Posts and Telecommunications grant No. A2016-80. ∗ Corresponding author: Daoyi Xu.

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DAIYI XU AND WEISONG ZHOU

terms, respectively. W is a two-sided Wiener process with values in a separable Hilbert space U . The term ut is given by ut (s) = u(t + s) with s ∈ [−τ, 0], where τ > 0 and the initial data ξ(t) is a continuous function on [−τ, 0]. As we all known, stochastic differential equations (SDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic and social sciences. Therefore, the theory of SDEs has been developed very quickly [1, 2, 4, 5, 6, 8, 9, 12, 14, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 30, 31, 32, 33, 35, 37, 38]. In the general way of studying SDEs, the solutions satisfying the equations almost surely for the sample point ω in the sample space Ω are often considered. For the class of the solutions, there are a lot of results on the existence and uniqueness [9, 21, 24, 26, 31, 32, 33, 35], stability [5, 6, 17, 18, 19, 21, 27], periodic solutions [16, 30, 37], attracting set [19, 38], and so on. On the other hand, ones are interested in analyzing the long-time behavior of the mild solution to (1) by obtaining the random attractor associated to the random dynamical system generated by the mild solution [2, 4, 8, 20]. Nevertheless, it is well-known that a large class of partial differential equations (PDEs) with stationary random coefficients and Itˆ o stochastic ordinary differential equations (ODEs) can generate random dynamical systems, see the monograph by Arnold [1]. However, as pointed out in [4], the stochastic PDEs driven by Brownian motion may not be so simple since the stochastic integral is only defined almost surely where the exceptional set may depend on the initial state, which contradicts the definition of the cocycle property, and on the other, Kolmogorov’s theorem in an appropriate form is only true for finite dimensional random fields, see Kunita [14, Theorem 1.4.1]. Therefore, the pathwise solution (i.e. the solution satisfying the equations for every ω ∈ Ω, see e.g. [2, 4, 8, 20] and references therein) must be considered by transforming the stochastic equation into a random one, which is possible to deal with the latter by using deterministic techniques. This transformation is known as cohomology. A advantage of the cohomology is that the drift terms in the equation (1) may be dealt with as the same as the deterministic equations [2, 8, 20], for example, the local Lipschitz condition is valid for the existence of a solution [23, Theorem 6.1.4]. But the diffusion terms in the cohomology is restricted since it is applicable only when considering an additive noise or very particular cases of multiplicative noise. Recently, Bessaih, Garrido-Atienza and Schmalfuss [4] obtain the existenceuniqueness and a random attractor of pathwise solutions of the equation (1) with the autonomous drift and diffusion terms by considering diffusion terms with the smooth property. Their results relax the conditions on diffusion term under the global Lipschitz conditions. Motivated by the above discussions, we will give some new results on the existence-uniqueness and exponential estimate of pathwise solutions of the equation of (1) under the local Lipschitz conditions. The ideas of removing the Itˆo stochastic integral and establishing the existenceuniqueness of the maximal local mild solution are partially borrowed from Bessaih, Garrido-Atienza and Schmalfuss [4] and Xu, Wang and Yang [32]. Nevertheless, the main novelty in this article is to establish a method so that the deterministic techniques are effective for both of the drift terms and the diffusion terms with the smooth property and give an exponential estimate of the pathwise mild solution of the system (1) so that the condition on the attractiveness has not the restriction on the time delay with appropriate smallness. Our technique on the global existenceuniqueness is establishing an integral comparison principle and by means of the

EXISTENCE-UNIQUENESS AND EXPONENTIAL ESTIMATE OF RETARDED SES

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existence-uniqueness of the solution of a delay integral equation, which extend the classical Wintner theorem [13]. The technique for obtaining the exponential estimate is not based on the integral inequality in [4], rather on establishing a new delay integral inequality, which avoids the delay parameter derived by the procedure of reducing the integral inequality in [4] and overcomes the restriction on the time delay with appropriate smallness. This paper is organized as follows. Firstly, we give the existence-uniqueness theorem of the maximal local pathwise mild solution for the SFDE under the generalized local Lipschitz conditions by establishing a existence-uniqueness lemma of the global solution under the generalized global Lipschitz conditions. This theorem is new even for the deterministic equation (see [23, Theorem 6.1.4]). Secondly, by establishing an integral comparison principle, we obtain an existence-uniqueness theorem of the global solution for the SFDE, which extend the classical Wintner theorem. Thirdly, we obtain an exponential estimate of the global mild solution by constructing a new delay inequality. Moreover, our conditions presented in theorems are much weaker than some existing related results in [4, 13, 15, 23, 28]. For example, we need neither that the noise is given in additive form or a very simple multiplicative noise, nor that the drift coefficient is global Lipschitz continuous. In the last section, two applications are provided to show the effectiveness of the theoretical results. 2. Preliminaries. In this section, we introduce some notations and recall some basic definitions. R+ = {x ∈ R : x ≥ 0}, N denotes the natural numbers. Let X and Y be Banach spaces, and C(X; Y) be the set of continuous functions on X with values in Y and L(X; Y) be the Banach space of all bounded linear operators from X to Y. We denote the norms of elements in X, Y and L(X; Y) by symbols | · |, | · |Y and | · |L(X;Y) , respectively. In particular, we use the notation L(X) when X = Y. Motivated by the previous works [4], we say that (Ω, F, P,N θ) is a metric dynamical system if (Ω, F, P) is a probability space and θ is a B(R) F, F measurable flow θ = (θt )t∈R , i.e. θt ◦ θτ = θt+τ , θ0 = idΩ ,

t, τ ∈ R

such that P is ergodic with respect to θ. In the following, we consider the Brownian motion metric dynamical system: let U be a separable Hilbert space that are zero at zero equipped with the compact open topology. We consider the Wiener measure P on B(C(R; U )) having a trace-class covariance operator Q on U . Then the canonical probability space (C(R; U ), B(C(R; U )), P) becomes an ergodic metric dynamical system if we add the Wiener shift θt ω(·) = ω(· + t) − ω(t),

ω ∈ C(R; U ).

(2)

Let us consider for some β ∈ (0, 1/2) the set of paths in C(R; U ) which have a β-H¨ older-seminorm on any interval [−k, k], k ∈ N. Denote by k · kC β ([a,b];U ) (and very often simply by k · kβ ) the seminorm of the space C β ([a, b]; U ). Lp (J; Rd ) = R ∆ {f : J → Rd | J |f (t)|p dt < ∞, p > 0, J ⊆ R}. Especially, denote C = C([−τ, 0]; X) with the norm kϕk = sup |ϕ(s)| and τ > 0 is given. −τ ≤s≤0

We shall assume that A(·) generates an evolution operator T(t, s) ∈ L(X) for 0 ≤ s ≤ t < a. That is, T(t, s) satisfies the following conditions (see for instance [23, p.129]):

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(i) T(s, s) = I (the identity mapping in X), T(t, r)T(r, s) = T(t, s), for 0 ≤ s ≤ r ≤ t < a; (ii) (t, s) → T(t, s) is strongly continuous for 0 ≤ s ≤ t < a; ∂ ∂ (iii) ∂t T(t, s)u = A(t)T(t, s)u, ∂s T(t, s)u = −T(t, s)A(s)u for all u ∈ D(A(t)) ⊂ X. We also assume that (S1 ) g is smoothing in the following sense: for any u ∈ C([−τ, a); X), the derivative of the mapping [0, a) 3 t 7−→ g(t, ut ) ∈ C 1 ([0, a) × C; L(U ; X)) is given by another operator K : [0, a) × C 7−→ L(U ; X) with a special structure, that is dg(t, ut ) = K(t, ut ). dt We shall assume that there is a Banach space (Y, | · |Y ) compactly embedded in X with |x|Y ≤ η|x| for a constant η > 0 such that (S2 ) For any t ∈ (0, a), the function s → A(s)T(t, s) defined from [0, t] into L(Y; X) is continuous and there is a function S ∈ L1 ([0, t]; R+ ) such that |A(s)T(t, s)|L(Y;X) ≤ S(t − s)/η,

∀ s ∈ [0, t].

Remark 1. Assume that Condition (S2 ) holds and u ∈ C([0, t]; X), then from the Bochner’s criterion for integrable functions and the estimate |A(s)T(t, s)u(s)| ≤ |A(s)T(t, s)|L(Y;X) |u(s)|Y ≤ S(t − s)|u(s)|,

∀ s ∈ [0, t],

(3)

we have that the function s → A(s)T(t, s)u(s) is integrable over [0, t]. 3. Existence-uniqueness of pathwise mild solution. In this section, the existence-uniqueness of the pathwise mild solution u(t) of (1) will be given. Since d[g(t, ut )ω(t)] = K(t, ut )ω(t)dt + g(t, ut )dω under Condition (S1 ), (1) becomes d[u(t) − g(t, ut )ω(t)] ={A(t)[u(t) − g(t, ut )ω(t)] (4) + A(t)g(t, ut )ω(t) − K(t, ut )ω(t) + f (t, ut )}dt. Then the pathwise mild solution of (1) is defined by the following integral equation Z t u(t) =T(t, 0)ξ(0) + T(t, s)[f (s, us ) − K(s, us )ω(s)]ds 0 (5) Z t + g(t, ut )ω(t) + A(s)T(t, s)g(s, us )ω(s)ds, ∀ ω ∈ Ω. 0

Lemma 3.1. Assume that hypotheses (S1 ) and (S2 ) are fulfilled. Suppose that for any b ∈ [0, a), f (t, 0) ∈ L1 ([0, b]; X), g(t, 0) ∈ C([0, b]; L(U ; Y)), K(t, 0) ∈ L1 ([0, b]; L(U ; X)), (6) in addition, f, g and K satisfy the generalized global Lipschitz conditions, that is, |f (t, ϕ) − f (t, ψ)| ∨ |K(t, ϕ) − K(t, ψ)|L(U ;X) ≤ L1 (t)kϕ − ψk,

(7)

|g(t, ϕ) − g(t, ψ)|L(U ;Y) ≤ L2 (t)kϕ − ψk,

(8)

for all t ∈ [0, b] and ϕ, ψ ∈ C, where L1 (t) ∈ L1 ([0, b]; R+ ), L2 (t) ∈ C([0, b]; R+ ). Then, for any ξ ∈ C and any ω ∈ Ω, there exists an unique global mild solution u(t) ∈ C([−τ, a); X).


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Remark 2. (6), (7) and (8) become the usual global Lipschitz conditions and linear growth conditions of f, g and K in the corresponding spaces when f (t, 0), g(t, 0), K(t, 0), L1 (t), L2 (t) are constants. However, here f (t, 0), K(t, 0), L1 (t), L2 (t) may be unbounded (see [31]). Note that from the above conditions we can just conclude that g(t, 0) ∈ C([0, b]; L(U ; X)) and g is a generalized global Lipschitz mapping from C into L(U ; X). Moreover, the generalized global Lipschitz condition of g in L(U ; X) is still denoted by (8) with the norm | · |L(U ;X) in its right hand. Proof. We will prove Lemma 3.1 by using the classical contraction mapping principle. Let us write for a while T instead of T (ω). Firstly, we consider the complete metric subspace CTξ ([−τ, T ]; X) of functions u ∈ C([−τ, T ]; X) with u(s) = ξ(s) for s ∈ [−τ, 0] and kY kC ξ = T

sup |Y (t)| < ∞, f or Y ∈ CTξ ([−τ, T ]; X).

t∈[−τ,T ]

For such a T > 0 to be determined later, consider the operator Γ : CTξ ([−τ, T ]; X) → CTξ ([−τ, T ]; X) defined, for t ≥ 0, by Z t Γ(u)(t) =T(t, 0)ξ(0) + T(t, s)[f (s, us ) − K(s, us )ω(s)]ds 0 (9) Z t + g(t, ut )ω(t) + A(s)T(t, s)g(s, us )ω(s)ds. 0

By Condition (S2 ), u ∈ C([0, t]; X), Bochner’s criterion for integrable functions, we have that the function s → A(s)T(t, s)u(s) is integrable over [0, t] for all t ∈ [0, a). Then from the integrability of f, g, K on [0, t], we obtain the above integrals define continuous mappings from [0, T ] into X which are zero for the time parameter zero. Therefore, it follows easily that t → Γ(u)(t) is continuous. Furthermore, from (7) and (8), we get |Γ(u)(t)| ≤|T(t, 0)|L(X) |ξ(0)| + [|g(t, ut ) − g(t, 0)|L(U ;Y) + |g(t, 0)|L(U ;Y) ]|ω(t)|U Z t + |T(t, s)|L(X) |[|f (s, us ) − f (s, 0)| + |f (s, 0)|]ds 0 Z t + |T(t, s)|L(X) |[|K(s, us ) − K(s, 0)|L(U ;X) + |K(s, 0)|L(U ;X) ]|ω(s)|U ds 0 Z t + S(t − s)[|g(s, us ) − g(s, 0)|L(U ;Y) + |g(s, 0)|L(U ;Y) ]|ω(s)|U ds 0

≤N (b)kξk + [L2 (t)kut k + |g(t, 0)|L(U ;Y) ]kωkβ tβ Z t + N (b) (L1 (s)kus k + |f (s, 0)|)ds 0 Z t + N (b) (L1 (s)kus k + |K(s, 0)|L(U ;X) )kωkβ sβ ds 0 Z t + S(t − s)[L2 (s)kus k + |g(s, 0)|L(U ;Y) ]kωkβ sβ ds, 0

(10) where N (b) =

sup 0≤s≤t≤b

∆

|T(t, s)|L(X) and kωkβ = kωkC β ([0,t];U ) .

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It follows from (6) and the continuity of L2 that for any T ∈ [0, a) Z

t

t

Z |f (s, 0)|ds ∨

0

t

Z |g(s, 0)|L(U ;Y) ds ∨

0

t

Z |K(s, 0)|L(U ;X) ds ∨

0

S(t − s)L2 (s)ds < ∞. 0

(11)

Combing with the continuity of g and the integrability of L1 , we can get Γ(u)(t) ∈ CTξ ([−τ, T ]; X) if u(t) ∈ CTξ ([−τ, T ]; X). Thus, Γ maps CTξ ([−τ, T ]; X) into itself. Now we shall show that Γ has an unique fixed point. we take u1 , u2 ∈ CTξ ([−τ, T ]; X). From Conditions (7) and (8), for t ∈ [0, T ], we get |(Γ(u1 ) − Γ(u2 ))(t)| ≤|g(t, u1t ) − g(t, u2t )|L(U ;Y) |ω(t)|U Z t + |T(t, s)|L(X) |f (s, u1s ) − f (s, u2s )|ds 0 Z t + |T(t, s)|L(X) |K(s, u1s ) − K(s, u2s )|L(U ;X) |ω(s)|U ds 0 Z t + |A(s)T(t, s)|L(Y;X) |g(s, u1s ) − g(s, u2s )|L(U ;Y) |ω(s)|U ds 0 Z t 1 2 β ≤L2 (t)kut − ut kkωkβ t + N (b) L1 (s)ku1s − u2s kds 0 Z t + N (b) L1 (s)ku1s − u2s kkωkβ sβ ds 0 Z t S(t − s)L2 (s)ku1s − u2s kkωkβ sβ ds + 0 Z t ≤N (b) sup ku1s − u2s k L1 (s)ds + kωkβ {L2 (t)tβ 0≤s≤t

0

t

Z

[N (b)L1 (s) + L2 (s)S(t − s)]sβ ds} sup ku1s − u2s k.

+

0≤s≤t

0

(12) Let H(t, kωkβ ) = N (b) s)]sβ ds}, thus,

Rt 0

L1 (s)ds + kωkβ {L2 (t)tβ +

Rt 0

[N (b)L1 (s) + L2 (s)S(t −

k(Γ(u1 ) − Γ(u2 ))(t)kC ξ ≤ H(t, kωkβ )k(Γ(u1 ) − Γ(u2 ))(t)kC ξ . T

(13)

T

It is trivial if H(t, kωkβ ) ≤ 21 for all t ∈ (0, a). Otherwise, from the continuity of H(t, kωkβ ) and H(0, kωkβ ) = 0, there must be T := T (ω) such that H(t, kωkβ ) ≤

1 f or t ∈ [0, T ], 2

(14)

which shows that Γ is contractive. Consequently, Γ has an unique fixed point u in CTξ ([−τ, T ]; X), which is the unique mild solution of (1). Up to now we have been able to prove the existence of an unique local mild solution u ∈ C([−τ, T ]; X) to (1). To obtain the existence-uniqueness for the global mild solution u(t), we shall prove that for any b ∈ (0, a), the solution u(t) exists on the interval [−τ, b]. Denote T1 (ω) = T (ω) (which in the next computations will be denoted by T1 for short) and it is trivial if T1 ≥ b. For another case, let us build the solution in the next time interval, say [T1 , T2 ], i.e., we need to find T2 such that


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we also have the existence-uniqueness in the last interval. This problem has been come down by Theorem 3 in [4] to solving the following equation  R ˆ + s T(s, r)[f (r, yr ) − K(r, yr )θT ω(s)]dr  T(s, 0)ξ(0) 1 0 Rs +g(s, ys )θT1 ω(s) + 0 A(r)T(s, r)g(r, yr )θT1 ω(r)dr, s ∈ [0, b], (15) y(s) =  ˆ ξ(s) = u1 (s + T1 ), −τ ≤ s ≤ 0, . where u1 ∈ C([−τ, T1 ]; X) is the solution to (1) and s above is given by s = t − T1 . This means that we have to solve the same problem than in the previous step, but with initial condition ξˆ and noise θT1 ω. Therefore, following the same steps than before, under the inequality (13) we obtain a new piece defined now in the interval [T1 − τ, T1 + T1 (θT1 ω)], and thus we define T2 (ω) as T2 (ω) = T1 (ω) + T1 (θT1 ω). If T2 (ω) ≥ b, we are done. Otherwise, b > T2 (ω) = T1 (ω) + T1 (θT1 (ω) ω), i.e., T1 (θT1 (ω) ω) < b − T1 (ω). Moreover, for this case, we let k = kωkC β ([0,b];U ) and t∗ = inf{t ∈ (0, a)|H(t, k) ≥

1 }. 2

Noting that the above H is monotone in k and for any b ∈ (T1 , a) kθT1 (ω) ωkC β ([0,b−T1 (ω)];U ) ≤ kωkC β ([0,b];U ) = k, we can get t∗ < T1 (θT1 (ω) ω) and therefore T2 (ω) ≥ 2t∗ . Repeating this method it turns out that there exists i ∈ N such that Ti (ω) > it∗ > b. This shows that the solution u(t) exists on [0, b]. Thus, the arbitrariness of b ∈ [0, a) shows that u(t) exists on [0, a) and the proof is completed. Theorem 3.2. Assume that hypotheses (S1 ) and (S2 ) are fulfilled and the functions f, g and K satisfy Condition (6) and the generalized local Lipschitz conditions, that is, for any n > 0, b ∈ (0, a) there exist L1n (t) ∈ L1 ([0, b]; R+ ), L2n (t) ∈ C([0, b]; R+ ) such that |f (t, ϕ) − f (t, ψ)| ∨ |K(t, ϕ) − K(t, ψ)|L(U ;X) ≤ L1n (t)kϕ − ψk,

(16)

|g(t, ϕ) − g(t, ψ)|L(U ;Y) ≤ L2n (t)kϕ − ψk,

(17)

for all t ∈ [0, b] and ϕ, ψ ∈ C with kϕk ∨ kψk ≤ n. Then there must be a stopping time β = β(ω) ∈ (0, a) such that the equation (1) has an unique maximal local mild solution u(t) for t ∈ [−τ, β). That is, u(t) is a mild solution of (1) and the sample path u(t) = u(t, ω) explodes (i.e., limt→β − |u(t, ω)| = ∞) at β(ω) if β(ω) < a for some ω ∈ Ω, and β(ω) is called the explosion time of the sample path x(t, ω). Proof. For the above n, define functions fn and gn as follows n ∧ kut k n ∧ kut k ut ), gn (t, ut ) = g(t, ut ), kut k kut k n ∧ kut k Kn (t, ut ) =K(t, ut ), kut k fn (t, ut ) =f (t,

(18)

kut k where we set ku = 1 when ut ≡ 0. Then it is obvious that fn , gn and Kn satisfy tk the generalized Lipschitz conditions (7) and (8) on [0, b] × C. By Lemma 3.1, the

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following equation for t ∈ [0, b] ⊂ [0, a), Z t T(t, s)[fn (s, (un )s ) − Kn (s, (un )s )ω(s)]ds un (t) =T(t, 0)un (0) + 0 Z t A(s)T(t, s)gn (s, (un )s )dω(s), + gn (t, (un )t )ω(t) +

(19)

0

(un )0 = un (s) =

ξ, 0,

if kξk ≤ n, if kξk > n,

s ∈ [−τ, 0]

(20)

has an unique mild solution un (t) ∈ C([−τ, b]; X). Define a sequence of stopping time δn by δn = b ∧ inf{t ∈ (0, b] : |un (t)| ≥ n}. From (18) and (20), for s ∈ [0, δn ], we have known that   fn+1 (s, (un )s ) = fn (s, (un )s ) = f (s, (un )s ), gn+1 (s, (un )s ) = gn (s, (un )s ) = g(s, (un )s ),  Kn+1 (s, (un )s ) = Kn (s, (un )s ) = K(s, (un )s ).

(21)

That is, (19) and the following equations for t ∈ [0, b], Z t un+1 (t) =T(t, 0)un+1 (0) + T(t, s)[fn+1 (s, (un+1 )s ) − Kn+1 (s, (un+1 )s )ω(s)]ds 0 Z t + gn+1 (t, (un+1 )t )ω(t) + A(s)T(t, s)gn+1 (s, (un+1 )s )dω(s) 0

(22) have the same coefficients for t ∈ [0, δn ] and the initial data overlap in D = {ut ∈ C : kut k ≤ n}. Thus, we can get un+1 (t) = un (t), t ∈ [−τ, δn ]. Furthermore, this implies that δn is increasing in n. So we can define δ(w) = lim δn . Then u(t) = lim un (t) is continuous and satisfies the integral equation n→+∞

n→+∞

(5) for t ∈ [t0 , δ). Moreover, the stopping time δ depends on b, that is, δ = δ(b, ω). Taking β(ω) = lim δ(b, ω), the proof is completed. b→a

Remark 3. Theorem 3.2 is a generalization of classical Pazy existence-uniqueness theorem of the maximal local mild solution (see [23, Theorem 6.1.4]), which can be obtained by taking g(·) ≡ τ = 0 and L1n (t) = L(n, b) = constant in Theorem 3.2. In order to obtain a global mild solution, we firstly give an integral comparison principle as the differential and discrete comparison principle in [3, 10, 22, 29]. m m Lemma 3.3. Let J = [−τ, a), h1 ∈ C(J × C; R+ ), h2 : J × J × C → R+ , and h1 (t, ut ) be monotone nondecreasing in ut for each fixed t and h1 (0, ·) = 0. Moreover, let h2 (t, s, ut ) be measurable in (t, s) ∈ J × J for each fixed ut ∈ C, continuous and monotone nondecreasing in ut for each fixed (t, s). If x(t) satisfies Z t x(t) ≤ x0 (t) + h1 (t, xt ) + h2 (t, s, xs )ds, t ∈ [0, a), (23) 0

where x(t), x0 (t) ∈ C(J; Rm ). Assume that y(t) ∈ C(J; Rm ) is a solution of Z t y(t) = x0 (t) + h1 (t, yt ) + h2 (t, s, ys )ds, t ∈ [0, a). 0

(24)


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Then for all t ∈ [0, a), x(t) ≤ y(t) as x(t) ≡ y(t) = x0 (t),

t ∈ [−τ, 0].

(25)

Proof. For any positive constant ε > 0 and E = [1, ..., 1]T ε, denote by y ε (t) the solution of the delay integral equation Z t ε ε y (t) = x0 (t) + h1 (t, yt ) + h2 (t, s, ysε )ds + E, t ≥ 0, 0

with the initial condition y ε (t) = x0 (t) + E, ∀ t ∈ [−τ, 0]. At first, We shall prove that x(t) < y ε (t), t ≥ 0. (26) If the above inequality (26) is not true, from the continuity of x and y, there must be a t∗ > 0, and some integer k such that xk (t∗ ) = ykε (t∗ ), xi (t) < yiε (t),

t ∈ [−τ, t∗ ), i = 1, . . . , m.

Thus, the inequality (23) implies that Z x(t∗ ) ≤x0 (t∗ ) + h1 (t∗ , xt∗ ) +

(27)

t∗

h2 (t∗ , s, xs )ds

0 ∗

∗

Z

t∗

0, the solution of the scalar differential equation Rt γ[1 + tβ F1 (t, Wt ) + 0 F2 (t, s, Ws )ds], f or t ≥ 0, (31) W(t) = γ, f or t ∈ [−τ, 0] exists on [0, a). Then, for any ξ ∈ C and any ω ∈ Ω, the initial problem (1) has an unique global mild solution u(t) for t ∈ [−τ, a). Moreover, |u(t)| ≤ W(t), ∀ t ∈ [0, a) provided that one chooses a suitable initial condition of (31).

(32)

2170


Proof. Theorem 3.4 guarantees that there is a maximal local mild solution u(t) of . (1), that is, there exists β = β(ω) > 0 such that u(t) satisfies  Rt  T(t, 0)ξ(0) + Rg(t, ut )ω(t) + 0 T(t, s)[f (s, us ) − K(s, us )ω(s)]ds t u(t) = (33) + 0 A(s)T(t, s)g(s, us )ω(s)ds, t ∈ [0, β);  ξ(t), t ∈ [−τ, 0]. In what follows, we will derive β = a. From (29), (30) and (33), for any b ∈ (0, a) and t ∈ [0, b], we can get Z t T(t, s)[f (s, us ) − K(s, us )ω(s)]ds| |u(t)| ≤ |T(t, 0)ξ(0) + g(t, ut )ω(t)| + | 0 Z t +| A(s)T(t, s)g(s, us )ω(s)ds| 0

≤ N (b)kξk + kωkβ tβ F1 (t, |u|t ) Z t + N (b) [|f (s, us )| + |K(s, us )|L(U ;X) |ω(s)|U ]ds 0 Z t + S(t − s)|g(s, us )|L(U ;Y) kωkβ sβ ds 0 Z t ≤ γ[1 + tβ F1 (t, |u|t ) + F2 (t, s, |u|s )ds],

(34)

0

where γ = max{kωkβ , N (b), N (b)kξk, N (b)kωkβ bβ , kωkβ bβ }. Moreover, we assume Z t ∆ v(t) = γ[1 + tβ F1 (t, |u|t ) + F2 (t, s, |u|s )ds], t ∈ [0, b].

(35)

0

From (34), |u(t)| ≤ v(t), and combining the monotonicity of F1 and F2 , we have Z t β v(t) ≤ γ[1 + t F1 (t, vt ) + F2 (t, s, |v|s )ds]. (36) 0 β

Let h1 (t, xt ) = γt F1 (t, vt ) and h2 (t, s, xt ) = γF2 (t, s, |v|s ) so that all conditions of Lemma 6 are satisfied, then we obtain that |u(t)| ≤ v(t) ≤ W(t), ∀ t ∈ [0, b].

(37)

This shows that the solution u(t) may not be exploded in [0, b], that is β > b. Thus, the arbitrariness of b ∈ [0, a) shows that u(t) exists on [0, a) and the proof is completed. Remark 5. Theorem 3.4 is a generalization of Theorem 3 in [4]. In fact, the global Lipschitz conditions in [4] and Condition (S1 ) imply the right hand of (29) and (30) are linear functions with integrable coefficients, which guarantee the existence of the global mild solution of (31). Corollary 1. Let all conditions of Theorem 3.2 and (29) hold, S(·) ∈ C((0, b]; R+ ) for any b ∈ (0, a) and the upper right-sided Dini derivative D+ F1 (t, |u|t ) with respect to t ∈ [0, b] exists. Assume that there are functions Ψ ∈ C(R+ ; R+ ) and κ ∈ C(R+ ; R+ ) such that D+ F1 (t, |u|t ) ∨ F1 (t, |u|t ) ∨ |f (t, ut )| ∨ |K(t, ut )|L(U ;X) ≤ κ(t)Ψ(kut k)

(38)


2171

R ∞ ds where Ψ(·) > 0 is monotone nondecreasing and satisfies Ψ(s) = ∞. Then, for any ξ ∈ C and any ω ∈ Ω, the initial problem (1) has an unique global mild solution u(t) for t ∈ [−τ, a). Proof. From (38), we have |f (s, us )| + S(t − s)|g(s, us )|L(U ;Y) + |K(s, us )|L(U ;X) ≤ [2 + S(t − s)]κ(s)Ψ(kus k). (39) In order to apply Theorem 3.4, we shall prove that, for arbitrary constant γ > 0, the solution of the integral equation Rt γ{1 + tβ F1 (t, Wt ) + 0 [2 + S(t − s)]κ(s)Ψ(W(s))]ds}, ∀t ∈ [0, a), W(t) = (40) γ, f or t ∈ [−τ, 0] exists globally, where W(t) = sup−τ ≤s≤t W(s). If not, there must be a δ > 0 such that lim W(t) = ∞. Without loss of generality, we let δ be the first explosion time t→δ

of W(t), that is, W(t) < ∞ f or t ∈ [0, δ), and lim W(t) = ∞. t→δ

(41)

Thus, we have v(t) → ∞ as t → δ, where v(t) is a solution of the following integral equation Rt γ{1 + tβ F1 (t, vt ) + 0 [2 + S(δ − s)]κ(s)Ψ(¯ v (s))ds}, ∀ t ∈ [0, δ), v(t) = (42) γ, ∀ t ∈ [−τ, 0]. By the continuity of Ψ, S and κ and Condition (38), we have D+ v(t) =γ{βtβ−1 F1 (t, vt ) + tβ D+ F1 (t, vt ) + [2 + S(δ − t)]κ(t)Ψ(¯ v (t))} ≤γ[βtβ−1 + tβ + 2 + S(δ − t)]κ(t)Ψ(¯ v (t)), ∀ t ∈ [0, δ).

(43)

By Lemma 4 in [35], we get D+ v¯(t) ≤ γ[βtβ−1 + tβ + 2 + S(δ − t)]κ(t)Ψ(¯ v (t)), ∀ t ∈ [0, δ). Since Ψ > 0 for any t ∈ (0, δ), then for any t ∈ (0, δ), Z v¯(t) Z t d¯ v ≤γ [βsβ−1 + sβ + 2 + S(δ − s)]κ(s)ds. Ψ(¯ v (s)) γ 0

(44)

(45)

From (42), we have v(t) → ∞ as t → δ. This leads to a contradiction for the left side of (45) tends to ∞ but the right side of (45): Z t γ [βsβ−1 + sβ + 2 + S(δ − s)]κ(s)ds < ∞ as t → δ. 0

So, the solution of the integral equation (40) exists globally. Then, by Theorem 3.4, for any ξ ∈ C and any ω ∈ Ω, the mild solution u(t) of (1) exists on [0, ∞). Remark 6. Indeed, it is clear that (38) is only required for large |u|. Admissible choices of Ψ(u), for example, Ψ(u) = Cu, Cu ln u, . . . for large u and a constant C. Then the classical Wintner theorem [13, p.29] has been extended to the global existence of pathwise solutions of stochastic PDE, which does not require the global Lipschitz condition.

2172


4. Exponential estimate of pathwise solution. In order to study the asymptotic properties of the mild solution of Eq. (1) in detail, we firstly establish a delay differential inequality with variable coefficients. Lemma 4.1. Let I ≥ 0 and d ≥ 0 be constants. Assume that the measurable function y : J → R+ is a solution of the delay integral inequality Z t Rt (46) ψ(t, s)ys ds + de− 0 r(s)ds + I, t ∈ [0, a), y(t) ≤ c(t)¯ y (t) + 0

where y¯(t) = sup−τ ≤s≤t y(s), ψ(t, s) is finite measurable in (t, s) ∈ J × J, c(·) ∈ Rt C(R+ ; R+ ) and r(·) : J → R+ is a finite measurable function and sup t−τ r(s)ds 0≤t 0 such that Rt

I y(t) ≤ N e−λ 0 r(s)ds + 1−π , f or t ∈ [0, t¯], where [0, t¯] = [0, ∞) if t¯ = a = ∞,

(48)

provided that the initial condition satisfies y(t) ≡ ξ(t) ≤ N e−λ

Rt 0

r(s)ds

+

I , f or t ∈ [−τ, 0]. 1−π

(49)

Proof. For given d > 0 and π < 1, there must be positive constants λ ∈ (0, σ] small enough and N > 0 big enough such that d < 1. N ¯ > N , we shall prove that Firstly, for any N πeλr +

(50)

I ≡ z(t), f or t ∈ [0, t¯]. (51) 1−π If the inequality (51) is not true, then there must be a t∗ ∈ (0, t¯] such that ¯ e−λ y(t) < N

Rt 0

r(s)ds

+

y(t∗ ) ≥ z(t∗ ), y(t) < z(t), t ∈ [−τ, t∗ ). ∗

∗

(52)

Thus, for y(t ) < y¯(t ), from λ < 1, the inequalities (46), (47), (50), (52) and the definition of z(t), we have that Z t∗ R t∗ ∗ ∗ ∗ y(t ) ≤c(t )¯ y (t ) + ψ(t∗ , s)ys ds + de− 0 r(s)ds + I 0

Z t∗ I 0 e + ]+ ψ(t∗ , s) 1−π 0 R ∗ R ¯ e−λ 0s r(u)du eλr + I ]ds + de−λ 0t r(s)ds + I · [N 1−π Z t∗ R t∗ R t∗ ∗ −λ 0 r(s)ds λr ¯ ≤N e {e [c(t ) + ψ(t∗ , s)eλ s r(u)du ds] ¯ e−λ ≤c(t )[N ∗

R t∗

r(s)ds λr

0

d + } + [c(t∗ ) + N

Z 0

t∗

ψ(t∗ , s)ds]

I +I 1−π


¯ e−λ ≤N ¯ e−λ 0 such that |T(t)|L(X) ≤ e−λt ,

|(−A)α T(t)|L(Y;X) ≤ Cα t−α e−λt ,

t > 0,

(69)

α

where Y = D((−A) ) with the norm |(−A)α x|Y ≤ |(−A)α |L(Y) |x|Y ≤ η|x|Y α

(70) α

since (−A) is a bounded operator for α ∈ (0, 1), i.e. |(−A) |L(Y) ≤ η for some constant[23, p.71]. Theorem 4 in [4] (also see the generalization in Lemma 3.1) gives the global existence of a pathwise mild solution of (68) under the global Lipschitz conditions. Now we shall give the global existence of a pathwise mild solution of (68) under the local Lipschitz conditions Proposition 1. Assume that hypothesis (S1 ) with g(t, ut ) ≡ g(ut ) and K(t, ut ) ≡ K(ut ) is fulfilled and the functions f, g and K satisfy the local Lipschitz conditions, that is, for any n > 0 there exist constants L1n , L2n such that |f (ϕ) − f (ψ)| ≤ L1n kϕ − ψk,

(71)

|g(ϕ) − g(ψ)|L(U ;Y) ∨ |K(ϕ) − K(ψ)|L(U ;X) ≤ L2n kϕ − ψk,

(72)

for all ϕ, ψ ∈ C with kϕk ∨ kψk ≤ n. If there are functions F1 ∈ C(C; R+ ) and G ∈ C 1 (R+ ; R+ ) such that |g(ut )|L(U ;Y) ≤ F1 (|u|t ), +

(73)

D F1 (|u|t ) ∨ F1 (|u|t ) ∨ |f (ut )| ∨ |K(ut )|L(U ;X) ≤ G(kut k), (74) R ∞ ds where G(·) > 0 is monotone nondecreasing and satisfies G(s) = ∞. Then, for any ξ ∈ C and any ω ∈ Ω, the initial problem (68) has an unique global mild solution u(t) for t ∈ [−τ, ∞).

2176


Proof. For the autonomous system (68), Condition (6) holds obviously since f (0) and g(0) are constants. Moreover, Conditions (71) and (72) are special cases of (16) and (17) with L1n and L2n are constants. And (69) implies that |(−A)T(s)u(s)| =|(−A)1−α T(s)(−A)α u(s)| ∆

≤C1−α s1−α e−λs η|u(s)| = S(s)|u(s)|,

(75)

where S(s) = S1 (s)e−λs ∈ C((0, b]; R+ ) and S1 (s) = C1−α ηs1−α ∈ L1 ([0, b]; R+ ) for any b ∈ (0, a). This leads to Condition (S2 ) holds when taken A(s) = −A, T(t, s) = T(t − s). Applying Corollary 1, the proof is completed. Proposition 2. Suppose that the conditions (S1 ) and (72) in Proposition 1 are satisfied and the local Lipschtiz condition (71) is replaced by the global Lipschtiz condition, that is, for all ϕ, ψ ∈ C, |f (ϕ) − f (ψ)| ≤ Cf kϕ − ψk,

(76)

where Cf is a positive constant. If the functions g and K are bounded by M > 0, that is, |g(ut )|L(U ;Y) ∨ |K(ut )|L(U ;X) ≤ M (in general, let M ≥ |f (0)|)

(77)

and λ > Cf ,

(78)

then there exist an absorbing set B(ω) for (68). Proof. Firstly, Condition (55) can be implied by (69) and (56) is obvious since from the first inequality of (69), we have sup e−ζλ(t−0) t < ∞ f or any constant ζ ∈ (0, 1).

(79)

t∈[0,a)

Moreover, according to (76) and (77), we get |f (ut )| ≤ Cf kut k + |f (0)| ≤ Cf kut k + M.

(80)

Then, combining (77) and (80), we have that (73) is satisfied with F1 (|u|t ) = M and ∆ D+ F1 ∨ F1 ∨ |f | ∨ |K|L(U ;X) ≤ Cf kut k + M = G(kut k), which that (74) is true since Cf s+M is monotone nondecreasing and satisfies R ∞ implies ds = ∞. Thus, by using Proposition 1, (68) has an unique global pathwise 0 Cf s+M mild solution u(t) for t ∈ [−τ, ∞). On the other hand, (77) and (80) indicate that (57) and (58) are satisfied with α1 = α2 = α3 = M . From (78), there must be a positive constant σ ∈ (0, 1) such that λ(1 − σ) > Cf , (81) then from (80), (81) and the first inequality of (69), we have m1 (t) ≡ Cf and Z b sup |T(b, s)|L(X) m1 (t)eλσ(b−s) ds 0≤b Cf ). Furthermore, the condition (78) overcomes the disadvantage that the exponential converging rate λ is restricted by the maximal admissible value for the time lag µ. 5.2. A stochastic partial functional differential equation. Let X = L2 ([0, π]; R) be the space of the functions which are square integrable. Now, we consider the operator A : D(A) ⊂ X → X defined by Ax = x00 , where D(A) = {x ∈ X : x00 ∈ X, x(0) = x(π) = 0}. It is well known that A is the infinitesimal generator of an analytic and compact semigroup (S(t))t≥0 on X. Moreover, A has a discrete spectrum, the eigenvalues are −n2 , n ∈ N, with corresponding normalized eigenfunctions zn (ξ) = (2/π)1/2 sin(nξ). In particular, we have that S(·) is an uniformly stable semigroup with |S(t)| ≤ e−t . Consider the non-autonomous stochastic partial differential system with diffusion  x) + f (t, u(t + θ, x))]dt  du(t, x) − ∆u(t, x)dt = [−a(t)u(t, Rt + t−τ ϑ(s, u(s, x))dsdω(t), t ≥ 0, θ ∈ [−τ, 0],  u(t, 0) = u(t, π) = 0, (83) with initial condition u(s, x) = ψ(s, x) ∈ C([−τ, 0] × [0, π]; X), ω(t) is a onedimensional standard Brownian motion, a(·) ∈ C(R; R+ ), f (·, ·) ∈ C(R+ × C; X), ϑ(·, ·) ∈ C(R+ × X; X). The system (83) may be rewritten as follows du = [A(t)u + f (t, ut )]dt + g(t, ut )dω(t), t ≥ 0, (84) u(s) = ψ(s), s ∈ [−τ, 0], Rt where A(t)u = Au(t, x) − a(t)u(t, x), and g(t, ut ) = t−τ ϑ(s, u(s, x))ds. Proposition 3. Assume that the functions f and ϑ satisfy the local Lipschitz conditions, that is, for any b ∈ (0, ∞) and n > 0 there exist constants L1n , L2n such that |f (t, ϕ) − f (t, ψ)| ≤ L1n kϕ − ψk,

(85)

|ϑ(t, u) − ϑ(t, v)|L(U ;Y) ≤ L2n |u − v|,

(86)

for all t ∈ [0, b], ϕ, ψ ∈ C with kϕk ∨ kψk ≤ n and u, v ∈ X with |u| ∨ |v| ≤ n. Moreover, there are functions m1 ∈ C(R+ ; R+ ), G ∈ C 1 (R+ ; R+ ) such that |f (t, ut )| ∨ |ϑ(t, u(t))| ≤ m1 (t)G(|u(t)|), (87) R ∞ ds where G(·) > 0 is monotone nondecreasing and satisfies G(s) = ∞. Then, for any ξ ∈ C and any ω ∈ Ω, the mild solution u(t) of (83) exists on [0, +∞).

2178


Proof. Since A(t) generates an evolution operator {T(t, s)}t≥s≥0 ∈ L(X), which can R − st a(v)dv be explicitly given by T(t, s) = S(t − s)e (see [7]). Moreover, using the properties of semigroup (S(t))t≥0 , we have t

1

|(−A) 2 S(t)| ≤

e− 2 t− 2 √ , 2

Rt

[1+a(v)]dv

1

t > 0,

(88)

t ≥ s ≥ 0.

(89)

and |T(t, s)|L(X) = e−

s

,

Then |(−A(s))T(t, s)u(s)| ≤|[−A − a(s)]T(t, s)u(s)| ≤|(−A)T(t, s)u(s)| + |a(s)T(t, s)u(s)| 1

1

1

Rt

≤|(−A) 2 T(t, s)(−A) 2 u(s)| + |T(t, s)|L(X) |a(s)u(s)| ≤|(−A) 2 S(t − s)e−

s

a(v)dv

1

|L(Y;X) |(−A) 2 u(s)|

+ |T(t, s)|L(X) |a(s)||u(s)| 1

≤

e− 2

Rt s

[1+a(v)]dv

≤[a(s)e−

√ Rt s

1

(t − s)− 2

2

[1+a(v)]dv

1

|(−A) 2 ||u| + a(s)e−

Rt s

[1+a(v)]dv

|u|

√ +

Rt 1 1 1 2 |(−A) 2 |e− 2 s [1+a(v)]dv (t − s)− 2 ]|u|. 2 (90)

Thus, Condition (S2 ) is satisfied with √ Rt Rt 1 1 1 2 |(−A) 2 |e− 2 s [1+a(v)]dv (t − s)− 2 ∈ C((0, b]; R+ ) S(t − s) = a(s)e− s [1+a(v)]dv + 2 for any b ∈ (0, a). Furthermore, let m(t) = sup m1 (θ), from (87), we have t−τ ≤θ≤t

Z

t

|g(t, ut )| ≤

∆

m1 (s)G(|u(s)|)ds = F1 (t, ut ) ≤ τ m(t)G(kut k), t−τ +

(91)

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Received August 2015; revised November 2016. E-mail address: [email protected] E-mail address: [email protected]