Existence and asymptotic behavior of solutions for neutral stochastic partial integrodifferential equations with infinite delays

Existence and asymptotic behavior of solutions for neutral stochastic partial integrodifferential equations with infinite delays Mamadou Abdoul Diop∗ Universit´e Gaston Berger de Saint-Louis, UFR SAT D´epartement de Math´ematiques, B.P234, Saint-Louis, S´en´egal Tom´as Caraballo† Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain Mahamat Mahamat Zene‡ Universit´e Gaston Berger de Saint-Louis, UFR SAT, D´epartement de Math´ematiques B.P234, Saint-Louis, S´en´egal Abstract In this work we study the existence, uniqueness and asymptotic behavior of mild solutions for neutral stochastic partial integrodifferential equations with infinite delays. To prove the results, we use the theory of resolvent operators as developed by R. Grimmer [12], as well as a version of the fixed point principle. We establish sufficient conditions ensuring that the mild solutions are exponentially stable in pth-moment. An example is provided to illustrate the abstract results. Keys words: Resolvent operators, C0-semigroup, neutral stochastic partial integrodifferential equations, Wiener process, infinite delay, mild solutions, exponential stability. 1 Introduction The study and importance of nonlinear stochastic delay partial differential equations delay are motivated by the fact that when one wants to model some evolution phenomena arising in Physics, Biology, Engineering, etc., some hereditary characteristics such as aftereffect, time lag, memory, and time delay can appear in the variables of the problem. Typical examples arise from the researches of materials with thermal memory, biochemical reactions, population models, etc. (see, for instance, Hale and Lunel [15], Murray [20], Ruess [24, 25], Wu [28], Caraballo et al. [4, 5, 6], Caraballo and Real [7], and references therein). The existence, uniqueness and asymptotic behavior of solutions of stochastic partial differential equations have been considered by many authors (see for example [1, 2, 3, 8, 9, 10, 11, 14, 17, 26, 27]). Caraballo and Liu ∗E-mail address: mamadou-ab[email protected]. Corresponding author †E-mail address: [email protected] ‡E-mail address: mht.mh[email protected] 1 [8], Liu and Mao [19] and Taniguchi [26] discussed the exponential stability of the strong and mild solutions, by imposing some kind of coercivity condition, using the Lyapunov method and by a direct estimate of solutions, respectively. In particular, the Lyapunov direct method has some difficulties with the theory and application to specific problems when discussing the asymptotic behavior of solutions in stochastic differential equations. More recently, Luo [18] has studied the asymptotic stability of mild solutions of stochastic partial differential equations with finite delays using a fixed point approach which shows that some of these difficulties can be overcome with this fixed point theory. Moreover, systems with infinite delay deserve a study because they describe several interesting problems which are present in the real world. Therefore, it is interesting to study the stability problems for stochastic systems with infinite delays. However, to the best of our knowledge, no work has been reported on the existence of solutions and stability problems for stochastic integrodifferential equations with infinite delays. Motivated by the above considerations, in this paper we will establish sufficient conditions ensuring the existence and asymptotic stability of mild solutions to the following stochastic partial integrodifferential equations with infinite delays,                            d[x(t) + G(t, x(t−ρ(t)))] = A[x(t) + G(t, x(t−ρ(t)))] dt +Zt 0 B(t−s) [x(s) + G(s, x(s−ρ(s)))ds]dt +b(t, Z0 −∞ g(θ, x(t+θ))dθ)dt +h(t, Z0 −∞ σ(θ, x(t+θ))dθ)dw(t), t ≥0, x0=ϕ∈ B, (1.1) here, the state x(·) takes values in a separable real Hilbert spaces Hwith inner product h·, .·iHand norm k·kH, Ais the infinitesimal generator of a strongly continous semigroup of bounded linear operators S(t), t ≥0 on H, with domain D(A)⊂H, and B(t), t≥0 is a closed linear operator on H. The history xt: (−∞,0] → H, xt(θ) = x(t+θ), for t≥0, belongs to some abstract phase space Bwhich will be described axiomatically in Section 2. Let Kbe another separable Hilbert spaces with inner product h·, .·iKand norm k·kK. Suppose that {w(s):0≤s≤t}is a given K-valued Wiener process with covariance operator Q≥0 defined on a complete probability space (Ω,F,{Ft}t≥0,P) equipped with a normal filtration {Ft}t≥0which is generated by the Wiener process w(·). We are also using the same notation k·k for the norm L(K;H), where L(K;H) denotes the space of all bounded linear operator from Kinto H. We assume that G, b : [0,+∞)×H→H,h: [0,+∞)×H→ L0 2(K, H), g, σ : (−∞,0] ×H→Hare all Borel measurable, ρ(t) : [0,+∞)→[0, r] is continuous. Here L0 2=L2(K0;H) denotes the space of all Q-Hilbert-Schmidt operators (see [22]) from K0to Hwith the norm |ξ|2 L0 2:= tr(ξQξ∗)<∞, ξ ∈ L(K, H). The initial data ϕ={ϕ(t) : −∞ < t ≤0}is an F0-adapted, B-valued random variable independent of the Wiener process wwith finite second moment. Our main results concerning (1.1), rely essentially on techniques based on the use of a strongly continuous family of operators R(t), t ≥0 defined on the Hilbert space Hand called their resolvent (the precise definition will be given below). The paper is organized as follows: in Section 2 we recall some preliminaries which are used throughout this paper. In Section 3 we state the existence, uniqueness and asymptotic behavior of a mild solution. Finally, in Section 4, an example is given to illustrate our abstract results. 2 2 Preliminary Notes 2.1 Wiener process Let (Ω,F,{Ft}t≥0,P) be a filtered complete probability space satisfying the usual conditions. We set Ft=F0for t < 0. We denote by W= (W)t≥0aK-valued Wiener process defined on the probability space (Ω,F,{Ft}t≥0, P) with covariance operator Q. That is, Ehw(t), xiKhw(t), yiK= (t∧s)hQx, yiK,∀x, y ∈K, where Qis a positive, self-adjoint trace class operator on K. For the construction of stochastic integral in Hilbert spaces, we refer to Da Prato and Zabczyk [22]. 2.2 Partial integrodifferential equations in Banach spaces In this section, we recall some fundamental facts needed to establish our results. Regarding the theory of resolvent operators we refer the reader to [12, 23]. Throughout the paper, Hwill denote a Banach space with norm k · kH,Aand B(t) are closed linear operators on H.Yrepresents the Banach space D(A), the domain of operator A, equipped with the graph norm kykY:= kAykH+kykHfor y∈Y. The notation C([0,+∞); Y) stands for the space of all continuous functions from [0,+∞) into Y. We then consider the following Cauchy problem    v0(t) = Av(t) + Zt 0 B(t−s)v(s)ds for t≥0, v(0) = v0∈H. (2.1) Definition 2.1. ([12]) A resolvent operator for equation (2.1) is a bounded linear operator valued function R(t)∈ L(H) for t≥0, satisfying the following properties : (i) R(0) = Iand kR(t)k ≤ Neβt for some constants Nand β. (ii) For each x∈H,R(t)xis strongly continuous for t≥0. (iii) For x∈Y, R(·)x∈C1([0,+∞); H)∩C([0,+∞); Y) and R0(t)x=AR(t)x+Zt 0 B(t−s)R(s)xds =R(t)Ax +Zt 0 R(t−s)B(s)xds for t≥0. For additional details on resolvent operators, we refer the reader to [12, 23]. The resolvent operator plays an important role to study the existence of solutions and to establish a variation of constants formula for non–linear systems. For this reason, we need to know when the linear system (2.1) possesses a resolvent operator. Theorem 2.2 below provides a satisfactory answer to this problem. In what follows we suppose the following assumptions: 3 (H1) A is the infinitesimal generator of a C0-semigroup (T(t))t≥0which is compact for t > 0 on H. (H2) For all t≥0, B(t) is a continuous linear operator from (Y, k·kY) into (H, k·kH). Moreover, there exists an integrable function c: [0,+∞)→R+such that for any y∈Y,t7→ B(t)ybelongs to W1,1([0,+∞), H) and     d dtB(t)y   H ≤c(t)kykYfor y∈Yand t≥0. Theorem 2.2. ([12]) Assume that hypotheses (H1) and (H2) hold. Then equation (2.1) admits a resolvent operator (R(t))t≥0. Theorem 2.3. ([16]) Assume that hypotheses (H1) and (H2) hold. Then, the corresponding resolvent operator R(t)of the equation (2.1) is continuous for t > 0in the operator norm, namely, for all t0>0, it holds that lim h→0kR(t0+h)−R(t0)k= 0. In the sequel, we recall some results from [12] concerning the existence of solutions for the following integrodifferential equation    v0(t) = Av(t) + Zt 0 B(t−s)v(s)ds +q(t) for t≥0, v(0) = v0∈H, (2.2) where q: [0,+∞[→His a continuous function. Definition 2.4. ([12]) A continuous function v: [0,+∞)→His said to be a strict solution of equation (2.2) if (i) v∈C1([0,+∞); H)∩C([0,+∞); Y), (ii) vsatisfies Eq. (2.2) for t≥0. Remark 2.5.From this definition we deduce that v(t)∈D(A), and the function B(t−s)v(s) is integrable, for all t > 0 and s∈[0,+∞). Theorem 2.6. ([12]) Assume that (H1)-(H2) hold. If vis a strict solution of the (2.2), then the following variation of constants formula holds v(t) = R(t)v0+Zt 0 R(t−s)q(s)ds for t≥0.(2.3) Accordingly, we can establish the following definition. Definition 2.7. ([12]) A function v: [0,+∞)→His called a mild solution of equation (2.2), for v0∈H, if vsatisfies the variation of constants formula (2.3). The next theorem provides sufficient conditions ensuring the regularity of solutions of the equation (2.2). Theorem 2.8. ([12]) Let q∈C1([0,+∞); H)and vbe defined by (2.3). If v0∈D(A), then vis a strict solution of equation (2.2). In the sequel, we suppose that the phase space is axiomatically defined, and we use the approach proposed by Hale and Kato in [13]. To establish the axioms of the phase space Bwe follow the terminology used in Hino et al. [29]. The axioms of the phase space Bare established for F0-measurable functions from (−∞,0] into H, endowed with a seminorm and are the following: 4 (A1) If x: (−∞, T]→H, T > 0 is such that x0∈ B, then, for every t∈[0, T ], the following conditions hold: (i) xt∈ B, (ii) kx(t)k ≤ LkxtkB, (iii) kxtkB≤u(t) sup 0≤s≤t kx(s)kH+v(t)kx0kB, where L > 0 is a constant; u(·), v(·) : [0,+∞)→[1,+∞), u(.) is continuous, v(·) is locally bounded, and L, u(·), v(·) are independent of x(·) (A2) The space Bis complete. (A3) For the functions x(·) in (A1), t→xtis a B-valued continuous function for t∈[0, T ]. (A4) If (ϕn)n∈Nis a sequence of continuous functions with compact support defined from (−∞,0] into H, which converges to ϕuniformly on compact subsets of (−∞,0], then ϕ∈ B and kϕn−ϕkB→0 as n→ ∞. Before starting and proving the main results, we present the definition of the mild solution to (1.1). Definition 2.9. An H-valued stochastic process {x(t), t ∈(−∞, T]}, 0 ≤T≤ ∞, is called a mild solution of equation (1.1) if (i) x(t) is an Ft-adapted, continuous process with ZT 0 kx(t)kp Hdt < ∞almost surely; (ii) for t≥0, x(t) satisfies the following integral equation: x(t) = R(t)[ϕ(0) + G(0, ϕ(−ρ(0)))] −G(t, x(t−ρ(t))) +Zt 0 R(t−s)b(s, Z0 −∞ g(θ, x(s+θ))dθ)ds +Zt 0 R(t−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)dW(s), and x0=ϕ∈ B, i.e. x(t) = ϕ(t) for t≤0. Definition 2.10. Let p≥2 be an integer. The mild solution x(t) of (1.1) with an initial value ϕ∈ B is said to decay exponentially to zero in pth-moment if there exist some constants M≥1, η > 0 such that Ekx(t)kp H< ME sup θ≤0 kϕ(θ)kp He−ηt, t ≥0. 3 Main Results In this section we discuss the existence, uniqueness and asymptotic behavior of the mild solution to equation (1.1). In order to obtain our main result, we shall impose the following assumptions: (H3) The resolvent operator given by Theorem 2.2 satisfies the following condition: kR(t)k ≤ e−γt for some constant γ > 0. (H4) For p≥2, there exists a constant KG>0 such that for any x, y ∈H, and t≥0, kG(t, x)−G(t, y)kp H≤KGkx−ykp H. 5 (H5) The mappings b: [0,+∞)×H→H, h : [0,+∞)×H→ L0 2(K, H) satisfy Lipschitz conditions, i.e., there exist positive constants Kb, Khsuch that, for any x, y ∈H, and t∈R, kb(t, x)−b(t, y)kH≤Kbkx−ykH,kh(t, x)−h(t, y)kL0 2≤Khkx−ykH. Moreover, we assume that kb(t, 0)kH=kh(t, 0)kL0 2= 0. (H6) There exist some positive constants Lg, Lσ, ξ, with 0 < ξ < γ, and such that for all t∈R, x, y ∈H, kg(t, x)−g(t, y)kH≤Lge−ξ|t|kx−ykH; kσ(t, x)−σ(t, y)kH≤Lσe−ξ|t|kx−ykH, we further assume that kG(t, 0)kH=kσ(t, 0)kH=kg(t, 0)kH= 0. Our main goal is to state and prove the following theorem. Theorem 3.1. Let p≥2be an integer and assume that (H1)-(H6) are satisfied. Suppose also that 3p−1hKG+Kp bLp g(ξγ)−p+ (2γ) −p 2Kp hLp σξ−pCpi<1,(3.1) where Cp=p(p−1) 2p 2 . If an initial value ϕ∈ B satisfies Ekϕ(t)kp H≤M0Ekϕ(0)kp He−µt, t ≤0, for some M0≥1and 0< µ < ξ, then there exists a unique mild solution to (1.1) associated to ϕ(t)and decays exponentially to zero in pth moment. In order to prove the theorem, we first recall a useful lemma. Lemma 3.2. (Burkholder-Davis-Gundy inequality) ([22], p. 182) Let l≥1. Then for an arbitrary L0 2-valued predictable process Φ(t), sup 0≤s≤t E   Zs 0 φ(u)dw(u)    2l H ≤(l(2l−1))lZt 0EkΦ(s)k2l L0 21 ldsl .(3.2) Proof of Theorem 3.1. Without loss of generality, we assume that 0 < η < ξ. For the given initial datum ϕ∈ B, we denote by Sthe subset of the Banach space of all Ft-adapted continuous processes x(·) : R→H endowed with the norm kxkS:= sup t∈R Ekx(t)kp H, such that x(t) = ϕ(t) for t≤0, and there exist some constants M∗≥1, η > 0 and η < ξ depending on x(·), satisfying Ekx(t)kp H< M∗Esup θ≤0 kϕ(θ)kp He−ηt, t ≥0, which is a complete metric space for the distance induced by this norm. Define a mapping π:S→Sby π(x)(t) = ϕ(t) for t≤0 and π(x)(t) = R(t)[ϕ(0) + G(0, ϕ(−ρ(0)))] −G(t, x(t−ρ(t))) +Zt 0 R(t−s)b(s, Z0 −∞ g(θ, x(s+θ))dθ)ds +Zt 0 R(t−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)dW(s) for t≥0 =I1+I2+I3+I4.(3.3) 6 We need to prove that π(S)⊂Sand that is contractive. Let us first prove the continuity of (πx)(t) on t≥0. To this end, let x∈S, t1≥0 and |r|>0 be sufficiently small. Notice that Ek(πx)(t1+r)−(πx)(t1)kp H≤4p−1 4 X i=1 EkIi(t1+r)−Ii(t1)kp H. Applying Lemma 3.2 together with assumption (H3), it follows that EkI4(t1+r)−I4(t1)kp H =E   Zt1+r 0 R(t1+r−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)dW(s) −Zt1 0 R(t1−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)dW(s)    p H ≤2p−1Cp     Zt1 0 E    (R(t1+r−s)−R(t1−s))h(s, Z0 −∞ σ(θ, x(s+θ))dθ)    p L0 2!2 p ds  p 2 + Zt1+r t1 E    R(t1+r−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)    p L0 2!2 p ds  p 2     ≤2p−1Cp     Zt1 0 kR(t1+r−s)−R(t1−s)kpE    h(s, Z0 −∞ σ(θ, x(s+θ))dθ)    p L0 2!2 p ds  p 2 + Zt1+r t1 kR(t1+r−s)kpE    h(s, Z0 −∞ σ(θ, x(s+θ))dθ)    p L0 2!2 p ds  p 2     . Noting that for any s∈[0, T ],0≤T < ∞, we have E    h(s, Z0 −∞ σ(θ, x(s+θ))dθ)    p L0 2 ≤Kp hEZ0 −∞ kσ(θ, x(s+θ))kHdθ)p ≤Kp hLp σZs −∞ eξ(τ−s)dτp−1Zs −∞ eξ(τ−s)Ekx(τ)kp Hdτ ≤Kp hLp σξ1−pZ0 −∞ eξ(τ−s)M0Ekϕ(0)kp He−µτ dτ +Zs 0 eξ(τ−s)M∗Esup θ≤0 kϕ(θ)kp He−ητ dτ ≤Kp hLp σξ1−pM∗Esupθ≤0kϕ(θ)kp H ξ−ηe−ηs +M0Esupθ≤0kϕ(θ)kp H ξ−µe−ξs ≤L, 7 where Lis a positive constant. Using the norm continuity of R(t) for t > 0 and applying Lebesgue’s dominated convergence theorem, it follows that EkI4(t1+r)−I4(t1)kp H→0 as r→0. Similarly, it is not difficult to check that EkIi(t1+r)−Ii(t1)kp H→0 as r→0, i = 1,2,3. Next, we show that π(S)⊂S. Let x∈S. From the definition of π, we have E|π(x)(t)|p H≤4p−1E|R(t)[ϕ(0) + G(0, ϕ)]|p H+ 4p−1E|G(t, x(t−ρ(t)))|p H(3.4) +4p−1E   Zt 0 R(t−s)b(s, Z0 −∞ g(θ, x(s+θ))dθ)ds    p H +4p−1E   Zt 0 R(t−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)dW(s)    p H := 4p−1(I1+I2+I3+I4). By assumption (H4), we obtain EkG(t, x(t−ρ(t)))kp H≤EkG(t, x(t−ρ(t))) −G(t, 0)kp H(3.5) ≤KGE|x(t−ρ(t))|p H ≤KGM∗eηrEsup θ≤0 kϕ(θ)kp He−ηt +M0eµrEkϕ(0)kp Heµt. For the term I3, by an application of H¨older inequality and assumption (H4), it follows that I3≤EZt 0    R(t−s)b(s, Z0 −∞ g(θ, x(s+θ))dθ)   H dsp ≤Kp bEZt 0 e−γ(t−s)   Z0 −∞ g(θ, x(s+θ))dθ   H dsp ≤Kp bZt 0 e−γ(t−s)dsp−1 EZt 0    e −γ(t−s) pZ0 −∞ g(θ, x(s+θ))dθ    p H ds ≤Kp bγ1−pEZt 0Zs −∞ Lge −γ(t−s) peξ(τ−s)kx(τ)kHdτp ds ≤Kp bγ1−pLp gZt 0"Zs −∞ eξ(τ−s)dτp−1Zs −∞ e−γ(t−s)eξ(τ−s)Ekx(τ)kp Hdτ#ds ≤Kp bγ1−pLp gξ1−pZt 0Z0 −∞ e−γ(t−s)eξ(τ−s)M0Ekϕ(0)kp He−µτ dτds +Kp bγ1−pLp gξ1−pZt 0Zs 0 e−γ(t−s)eξ(τ−s)M∗Esup θ≤0 kϕ(θ)kp He−ητ dτds ≤Kp bγ1−pLp gξ1−pM∗Esupθ≤0kϕ(θ)kp H (γ−η)(ξ−η)e−ηt +M0Ekϕ(0)kp H (γ−ξ)(ξ−µ)e−ξt.(3.6) Taking into account Lemma 3.2 and assumption (H5), we obtain that 8 I4=E   Zt 0 R(t−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)dW(s)    p H ≤Cp  Zt 0"E    R(t−s)h(s, Z0 −∞ σ(θ, x(s+θ))dθ)    p L0 2#2 p ds   p 2 ≤Kp hCp  Zt 0 e−2γ(t−s)"E   Z0 −∞ σ(θ, x(s+θ))dθ    p H#2 p ds   p 2 , where Cp=p(p−1 2p 2 . By assumption (H5) and H¨older inequality, we deduce I4≤Kp hCpLp σ(Zt 0 e−2γ(t−s)EZs −∞ eξ(τ−s)kx(τ)kHdτp2 p ds) p 2 ≤Kp hCpLp σ  Zt 0 e−2γ(t−s)"Zs −∞ eξ(τ−s)dτp−1Zs −∞ eξ(τ−s)Ekx(τ)kp Hdτ#2 p ds   p 2 ≤Kp hCpLp σξ1−p(Zt 0 e−2γ(t−s)Zs −∞ eξ(τ−s)Ekx(τ)kp Hdτ2 p ds) p 2 . Noting that p≥2, we then have I4≤Kp hCpLp σξ1−p(Zt 0 e−2γ(t−s)Zs 0 eξ(τ−s)M∗Esup θ≤0 kϕ(θ)kp He−ητ dτ +Z0 −∞ eξ(τ−s)M0Ekϕ(0)kp He−µτ dτ 2 p ds) p 2 ≤Kp hCpLp σξ1−p(Zt 0 e−2γ(t−s)"M∗Esupθ≤0kϕ(θ)kp H ξ−ηe−ηs 2 p +M0Ekϕ(0)kp H ξ−µe−ξs2 p#ds) p 2 (3.7) ≤2p−2 2Kp hCpLp σξ1−p"M∗Esupθ≤0kϕ(θ)kp H ξ−ηp 2pγ −2ηp 2 e−ηt +M0Ekϕ(0)kp H ξ−µp 2pγ −2ξp 2 e−ξt#. Recalling (3.3), from (3.4) to (3.7), we can deduce that there exists M1≥1 such that Ek(πx)(t)kp H≤M1Esup θ≤0 kϕ(θ)kp He−ηt. Since each term of (πx)(t) is Ft-measurable then the Ft-measurability of (πx)(t) is easily verified. It follows that πis well defined. Thus, we conclude that π(S)⊂S. It remains to show that πhas a unique fixed point. For any x, y ∈S, we have 9