Asymptotic behavior of a stochastic differential system with infinite delay and α-stable process

Asymptotic behavior of a stochastic differential system with infinite delay and α-stable process Hai Huang1and Tom ´ as Caraballo2,3,∗ 1School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, P. R. China 2Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico Facultad de Matem´ aticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain 3Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, P. R. China Abstract This work is devoted to the asymptotic behavior of solutions for a stochastic differential system with infinite delay and α-stable process. By applying the theory of semigroup, fixed point theorem and results on stochastic convolutions in [13] (see Lemma 2.1), we study respectively the existence, uniqueness, global attracting sets and stability in distribution of mild solutions for the considered equation. An example is provided in the end to illustrate the applications of the obtained results. Key words: Stochastic differential equations, infinite delay, α-stable process, global attracting sets, stability in distribution. AMS Subject Classification (2020): 35R10, 39B72, 39B82, 47D06, 60H15. 1 Introduction Stochastic partial differential equations (SPDEs) involve the deterministic states and the stochastic noise which can be found in many applications of various sciences such as physics, chemistry, engineering and economics. Recently, some quantitative and qualitative properties of solutions of stochastic evolution equations have also received much attention (see [1,2,4,6,9,14,17,20,23,24,27,30,32] and the references therein). Note that the Wiener process is not suitable to represent a noise process when, in applications, the typical heavy tail property is modeled. This restriction rules out the interesting stochastic differential equations driven by α-stable processes. In the recent years, there has been a number of very outstanding results on SPDEs driven by α-stable processes, see [5,13,26,31]. In particular, the study of stability in distribution for these kinds of systems is an important area of research. By the Banach fixed-point theorem, Zang and Li [31] obtained the existence, uniqueness and stability in distribution (see Definition 5.1) of mild solutions for the following neutral stochastic differential equation with finite delay:        dX(t)+g(X(t−τ))=AX(t)+f(X(t−τ))dt +dZ(t),0≤t≤T, X0(·)=ξ(·)∈D([−τ, 0]; H), where Z(·) is a cylindrical α-stable process. A few years later, Liu and Li [13] extended the results in [31] to the following stochastic integro-differential equation with finite delay:        dX(t)+g(t,X(t−τ))=AX(t)+f(t,X(t−τ))dt +σ(t)dZ(t),t≥0, X0(·)=ξ(·)∈D([−τ, 0]; H),(1) where σ(·)∈[0,+∞)→[0,+∞) is a locally integrable function. They also discussed the existence of global attracting sets of exponential decay in the p-th moment sense of mild solutions for system (1). The main tools are an integral inequality and a result on stochastic convolutions (see Lemma 2.1) newly established. Lately, sufficient ∗The corresponding author, email: [email protected] 1 conditions were established to guarantee the existence and uniqueness of the stationary distribution for a stochastic system with finite memory driven by an α-stable process [28]. It is well known that global attracting sets are important characteristics for the understanding of the asymptotic behavior of SPDEs and have been extensively studied in the past years. Many results on this topic for various stochastic (impulsive) functional differential equations with finite delay have been reported until now, see for instance [3,8,11–13,21]. Clearly, it is generally more challenging and difficult to deal with stochastic evolution systems with infinite delay than the equations with finite delay, since the phase spaces for infinite delay lose some nice properties of those for finite delay (such as the compactness property). Very recently, Li and Wang [10] considered the fractional stochastic evolution equations with infinite delay and Wiener process:            CDα ty(t)=Ay(t)+f(t,yt)+g(t,yt)dW(t) dt ,t≥0,0<α<1 2, y(t)=φ(t),t∈(−∞,0], (2) and            CDα ty(t)=Ay(t)+I1−α tf(t,yt)+hI1−α tg(t,yt)idW(t) dt ,t≥0,0<α<1, y(t)=φ(t),t∈(−∞,0], (3) where CDα tis the Caputo fractional derivative, I1−α tis the (1 −α)-order fractional integral operator. They first proved the existence, uniqueness and continuous dependence of mild solutions of system (2). Then the existence and some properties of a global forward attracting set of system (3) in the mean-square topology were established. Later, Xu et al. [29] improved the results of [10] to a stochastic evolution system with impulses and fractional noise under some suitable assumptions. We also refer to [16,19,22] for more details and further references related to stochastic evolution differential equations with infinite delay. Inspired by the above mentioned work, in this article, we investigate the existence, uniqueness and global attracting sets of the mild solutions, as well as their stability in distribution, for the following stochastic differential equation with infinite delay and α-stable process:        dx(t)=[Ax(t)+f(t,xt)]dt +σ(t)dZ(t),t≥0, x0=ϕ∈Cγ F,(4) where x(t)∈Xtakes values in a separable Hilbert space X. The (unbounded) linear operator Agenerates a compact C0-semigroup on X. The history of the solution, xt, is defined in the usual way by xt(θ)=x(t+θ) for θ∈(−∞,0]. The properties of function f(·,·) will be specified later and the definition of σ(·) is the same as in system (1). Additionally, the α-stable process and the space Cγ Fwill be described below. Eq. (4) is the abstract form of some stochastic partial functional differential equations which have a wide practical background such as some heat flow in materials with fading memory and an α-stable process, and this is exactly the motivation of this work. We will first show the global existence of mild solutions to problem (4) by applying the Schauder fixed point principle and the techniques of stochastic analysis. After establishing the ultimate boundedness of solutions, we will be able to further obtain the existence of a minimal global attracting set via limit arguments in the p−square topology with p∈(1, α). Furthermore, in the case of uniqueness of solutions, it is possible to provide more information about the geometrical structure of such global attracting set. In particular, it is proved that the minimal compact globally attracting set for the solutions of the problem becomes a singleton. In addition, we shall derive some sufficient conditions for stability in distribution of mild solutions for the considered equation. The main technique in all our discussion are the theory of semigroup, the Gronwall inequality and results on stochastic convolutions in (see Lemma 2.1). In comparison with the above literature, the contribution of our work is two fold. One is to extend the asymptotic properties results for some semi-linear differential equations driven by Wiener process and fractional Brownian motion introduced in [10,29] to stochastic differential equations with α-stable process. The second one is to extend the results of stability in distribution [13,31] to the case in which the considered equations are affected by infinite delay. It is also worth stressing here that, thanks to the p−moment exponential stability of solutions for system (4) (see Lemma 4.2) we are able to prove the existence of attracting sets and provide interesting information. More importantly, compared to [11–13,28,31], this lemma makes the proof procedure of stability in distribution simpler. Additionally, to the best of our knowledge, up to now there is no other work dealing with global attracting sets and 2 stability in distribution for this kind of semi-linear stochastic differential equations at some times. Needless to say that the obtained results in this work extend and improve many existing works such as [10–13,28,29,31]. The organization of this article is as follows. In Section 2 we recall some definitions, hypotheses and the concept of α-stable process to be used throughout the paper. In Section 3 we discuss, by using Schauder fixed point theorem, the global existence of mild solutions. Following that, in Section 4 we carry on the existence of a compact global attracting set for system (4). Moreover, if we impose a Lipschitz condition on the nonlinear functions f(·,·), ensuring uniqueness of solutions, we can prove that the compact global attracting set becomes a singleton in the p−square topology. Section 5 is devoted to proving the stability in distribution of the system (4). Finally, in Section 6, we exhibit an example to illustrate the applications of the obtained results. 2 Preliminaries In this section, we recall some definitions, notations, the phase spaces for infinite delay and α-stable process to be used in the article. Let Xand Kbe two separable Hilbert spaces and let L(X;K) denote the space of all bounded linear operators from Xinto K; we abbreviate it to L(X) whenever K=X. For a closed linear operator (A,D(A)) on Xin Eq. (4), Y is the Banach space (D(A),k·k) with the graph norm k·kYgiven by kxkY=kAxk+kxk, for x∈D(A). Throughout this work, we assume that (H1) The operator (A,D(A)) is a self-adjoint operator on Xadmitting a discrete spectrum, −∞ ←− −λn≤ −λn−1≤ · · · ≤ −λ2≤ −λ1<0,n−→ +∞, with corresponding eigenvector {en}n≥0which form a basis of X, and generating a compact C0−semigroup (S(t))t≥0 on X(for the theory of operator semigroup we refer to the books [7] and [18]) such that kS(t)k ≤ Me−λ1t,for M≥1 and t≥0. Now, we turn to state some preliminary results about α-stable process. Let Z(t) be a cylindrical α-stable process for α∈(0,2), defined by Z(t)= ∞ X n=1 βnZn(t)ξn,t≥0,(5) where βn≥0,(n=1,2,· · · ),are nonnegative real numbers which denote the intensity of the noise so that the series (5) is well defined in a proper sense, Zn(t),(n=1,2,· · · ),are independent, real-valued, normalized, symmetric α-stable L´ evy processes defined on a stochastic basis (Ω,F,{Ft}t≥0,P) and ξn(n=1,2,· · · ) is a complete orthonormal basis in X. We recall that the stochastic process {l(t) : t≥0}is said to be α-stable L´ evy process if it satisfies (i)l(0) =0 a.s.; (ii)l(t) has independent increments; (iii)l(t)−l(s)∼ηfor any 0 ≤s<t<∞, where ηstands for an α-stable random variable, which is uniquely determined by its characteristic function involved with four parameters: α∈(0,2), the index of stability; β∈[−1,1], the skewness parameter; δ∈(0,∞), the scale parameter and µ∈(−∞,∞), the shift, which has the characteristic function φη(u)=Eexp(iuη)=exp −δα|u|α(1 −iβsgn(u)Φ(u)) +iµu,u∈R, where the expectation Eis defined by Ev=RΩv(ω)dP,Φ(u)=tan(απ 2) for α,1 and Φ(u)=−2 πlog |u|for α=1. We call ηstrictly α-stable whenever µ=0 and if β=0, ηis said to be symmetric α-stable. For a real-valued normalized (standard) symmetric α-stable L´ evy process l(t), α∈(0,2), it has the characteristic function φl(t)(u)=Eexp(iul(t)) =exp {−t|u|α},u∈R, 3 and the associated L´ evy measure λα(dx)=cα |x|1+α,x∈R− {0}, where cαis a constant. For more details on the theory of α-stable process, we can refer the reader to [2] and [25]. We recall that fis said to be Ft-adapted if f(t,·) : Ω→Xis Ft-measurable, a.e.t∈[0,T], 0 ≤T<∞. Suppose that y(t) : Ω→X,t≥0, is a continuous, Ft-adapted, X-valued stochastic process. Then, we can associate it with another process (history process) yt:Ω→X,t≥0, by setting yt(s)(ω)=y(t+s)(ω), s∈(−∞,0]. Thus, we say that the process ytis induced by the process y(t). In what follows, Lp(Ω;X) denotes the Banach space of all strongly measurable, p-integrable X-valued random variables equipped with the norm kϕ(·)kLp=(Ekϕ(·)kp)1 p<∞for 0 <p< α. And C([a,b]; Lp(Ω;X)) represents the Banach space of all continuous mappings from [a,b] into Lp(Ω;X) with the norm sup t∈[a,b] Ekx(t)kp!1 p <∞. The abstract phase space Cγ Fis defined by Cγ F=(ϕ∈C((−∞,0]; Lp(Ω;X)) : sup θ∈(−∞,0] eγθEkϕ(θ)kp<∞), for some fixed parameter γ > 0, its norm is defined by kϕkCγ F= sup θ∈(−∞,0] eγθEkϕ(θ)kp!1 p <∞, ϕ ∈Cγ F. Obviously, Cγ F,k·kCγ Fis a Banach space. Let x∈C([0,T]; Lp(Ω;X)) with x(0) =ϕ(0) and ϕ∈Cγ F. Then for s∈[0,T], we denote the mapping x∨sϕfrom (−∞,0] to Lp(Ω;X) defined by x∨sϕ(θ)=       x(s+θ), θ ∈(−s,0], ϕ(s+θ), θ ≤ −s. Finally, we end this section by stating an important lemma to be used in our later discussion. Lemma 2.1. ([13], Lemma 3.3) Let Z be a cylindrical α-stable process, α∈(0,2). Assume that condition (H1) holds. Then, for any t ≥0and p ∈(0, α), EZt 0 S(t−s)σ(s)dZ(s) p ≤Cp,α  ∞ X n=1 βα nZt 0 e−αλn(t−s)σα(s)ds p α , where {βn}n≥1is the sequence given in (5) and constant Cp,α >0depends only on p and α. 3 Existence of mild solutions To begin, we first study in this section the existence of mild solutions to system (4). To this end, we always assume p∈(1, α) and impose the assumptions on the function f(·,·) as below. (H2) Function f: [0,+∞)×Cγ F→Xis continuous in the second variable. And there exist two nonnegative continuous functions k1,k2∈L1([0,+∞)), such that fsatisfies Ekf(t, ϕ)kp≤k1(t)+k2(t)kϕkp Cγ F , for t∈[0,+∞) and every ϕ∈Cγ F. Remark 3.1. Due to the fact that the continuous functions ki∈L1([0,+∞))appearing in conditions (H2)are nonnegative, it follows Z+∞ 0 ki(s)ds :=Ki<∞,i=1,2, where Kiare positive constants. Definition 3.1. A stochastic process x(·), defined on (−∞,T], is said to be a mild solution of system (4) if the following conditions are satisfied: 4 (1) x(t, ω)is measurable as a function from [0,T]×Ωto X and x(t)is Ft−adapted; (2) x0(·)=ϕ∈C((−∞,0]; Lp(Ω;X)) on (−∞,0]; (3) the process x(·)satisfies the integral equation x(t)=S(t)ϕ(0) +Zt 0 S(t−s)f(s,xs)ds +Zt 0 S(t−s)σ(s)dZ(s),t∈[0,T]. First we establish the following existence result for system (4). Theorem 3.1. Assume that assumptions (H1) and (H2) are fulfilled. Also, define the following positive constant: v1=sup t≥0 ∞ X n=1 βα nZt 0 e−αλn(t−s)σα(s)ds p α .(6) Then, for any initial value ϕ∈Cγ F, system (4) has at least one global mild solution x(·;ϕ). Proof. Based on the Schauder fixed point theorem, we shall show the existence of local solutions for (4) on an interval (−∞,T1] for some T1>0. Once this is proved, we will extend the solution to (−∞,+∞). First, we have from Remark 3.1 that the function g(t)=Rt 0k2(s)ds is nondecreasing for t∈[0,+∞). Thus, we can take some 0 <T1<Tsuch that k0B3p−1 p−1 λ1p!p−1ZT1 0 k2(s)ds <1.(7) Due to (H1), (H2), (6) and (7), for any fixed T>0, let ϕ∈Cγ Fbe given, and define the following two positive constants: ρ0B 3p−1MpEkϕ(0)kp+Mpp−1 λ1pp−1K1+K2kϕkp Cγ F+Cp,αv1 1−k0 ,(8) a0Bsup s∈[0,T1]Ekf(s,x∨sϕ)kpkx∨sϕkp Cγ F ≤ρ0+kϕkp Cγ F.(9) Define the set E(T1, ρ0) by E(T1, ρ0)B(x(·)∈C([0,T1]; Lp(Ω;X)) x(0) =ϕ(0) and sup t∈[0,T1] Ekx(t)kp≤ρ0). Obviously, E(T1, ρ0) is closed, bounded and convex. On the set E(T1, ρ0) we define an operator Qas: (Qx)(t)=S(t)ϕ(0) +Zt 0 S(t−s)f(s,x∨sϕ)ds +Zt 0 S(t−s)σ(s)dZ(s),t∈[0,T1],(10) for any x∈E(T1, ρ0). We show now Qmaps E(T1, ρ0) into itself, i.e. Q(E(T1, ρ0)) ⊂E(T1, ρ0). Let x∈E(T1, ρ0), combining Lemma 2.1, (H1), (H2), (6) and using the H¨ older inequality, we deduce Ek(Qx)(t)kp≤3p−1EkS(t)ϕ(0)kp+3p−1EZt 0 S(t−s)f(s,x∨sϕ)ds p +3p−1EZt 0 S(t−s)σ(s)dZ(s) p ≤3p−1MpEkϕ(0)kp+3p−1Mp Zt 0 e−λ1p p−1(t−s)ds!p−1Zt 0 Ekf(s,x∨sϕ)kpds +3p−1Cp,α sup t≥0 ∞ X n=1 βα nZt 0 e−αλn(t−s)σα(s)ds p α ≤3p−1MpEkϕ(0)kp+3p−1Mp p−1 λ1p!p−1 K1+K2kϕkp Cγ F +ρ0ZT1 0 k2(s)ds!+3p−1Cp,αv1 ≤ρ0,(by (8)). 5 This implies that Qmaps E(T1, ρ0) into itself since clearly (Qx)(0) =ϕ(0) and, it is easy to see that (Qx)(·) is continuous on the interval [0,T1]. In order to apply the Schauder fixed point theorem, we need to prove that Qis a compact operator. For this purpose, we first prove that Qis continuous on E(T1, ρ0). Let {xn} ⊂ E(T1, ρ0) with xn→x(n→+∞) for some x∈E(T1, ρ0). Then, for all s∈[0,T1], kxn∨sϕ−x∨sϕkp Cγ F =sup θ∈(−∞,0] eγθEkxn∨sϕ(θ)−x∨sϕ(θ)kp, ≤sup s∈[0,T1] Ekxn(s)−x(s)kp−→ 0 as n−→ +∞,(11) and kx∨sϕkp Cγ F =sup θ∈(−∞,0] eγθEkx∨sϕ(θ)kp≤ kϕkp Cγ F +sup s∈[0,T1] Ekx(s)kp.(12) From (11) and (12), we obtain that, for all nsufficiently large, kxn∨sϕkp Cγ F ≤2p−1kxn∨sϕ−x∨sϕkp Cγ F +2p−1kx∨sϕkp Cγ F ≤1+2p−1kϕkp Cγ F +2p−1sup s∈[0,T1] Ekx(s)kp.(13) By (H2) and (13), we find that there exists a positive constant b0such that, for all nsufficiently large and t∈[0,T1], Zt 0Ekf(s,xn∨sϕ)kp+Ekf(s,x∨sϕ)kpds ≤b0.(14) By virtue of the definition of Qit yields that Ek(Qxn)(t)−(Qx)(t)kp≤EZt 0 S(t−s)(f(s,xn∨sϕ)−f(s,x∨sϕ))ds p ≤Mp Zt 0 e−λ1p p−1(t−s)ds!p−1Zt 0 Ekf(s,xn∨sϕ)−f(s,x∨sϕ)kpds ≤Mp p−1 λ1p!p−1Zt 0 Ekf(s,xn∨sϕ)−f(s,x∨sϕ)kpds. Thus, by carrying on the estimation as above and (14) we can apply the Lebesgue dominated convergence theorem to obtain sup 0≤t≤T1 k(Qxn)(t)−(Qx)(t)k−→ 0, as n−→ +∞, i.e., Qis continuous. Next, we prove that the family V(·)={(Qx)(·) : x∈E(T1, ρ0)}is an equicontinuous family of functions. To this end, let 0 <t1<t2≤T1and  > 0 be small enough such that 0 <  < t1, then Ek(Qx)(t2)−(Qx)(t1)kp≤7p−1Ek(S(t2)−S(t1))ϕ(0)kp+7p−1EZt1− 0 (S(t2−s)−S(t1−s))f(s,x∨sϕ)ds p +7p−1EZt1− 0 (S(t2−s)−S(t1−s))σ(s)dZ(s) p +7p−1EZt1 t1− (S(t2−s)−S(t1−s))f(s,x∨sϕ)ds p +7p−1EZt1 t1− (S(t2−s)−S(t1−s))σ(s)dZ(s) p +7p−1EZt2 t1 S(t2−s)f(s,x∨sϕ)ds p +7p−1EZt2 t1 S(t2−s)σ(s)dZ(s) p :=7p−1Ek(S(t2)−S(t1))ϕ(0)kp+ 6 X i=1 Ii. 6 In view of (6), for any t≥0, we obtain from the absolute continuity of the integral that there exists δ > 0, small enough, such that  ∞ X n=1 βα nZt+δ t e−αλn(t+δ−s)σα(s)ds p α →0,as δ→0+.(15) Using (H2), (14) and the H¨ older inequality again, we derive I1≤7p−1b0 Zt1− 0 kS(t2−s)−S(t1−s)k p p−1ds!p−1 . From Lemma 2.1, (H1)−(H2) and the semigroup property, we have I2=7p−1E(S(t2−t1+)−S()) Zt1− 0 S(t1−−s)σ(s)dZ(s) p ≤7p−1Cp,α kS(t2−t1+)−S()kp ∞ X n=1 βα nZt1− 0 e−αλn(t1−−s)σα(s)ds p α . Also, there holds I3≤7p−1b0 Zt1 t1− kS(t2−s)−S(t1−s)k p p−1ds!p−1 ≤21 p−1·7p−1b0 Zt1 t1− kS(t2−s)k p p−1+kS(t1−s)k p p−1ds!p−1 ≤21 p−1·7p−1Mpb0 Zt1 t1− 2ds!p−1 =21 p−1·14p−1Mpb0p−1. On the other hand, by (15), we find that I4=7p−1E(S(t2−t1)−I)Zt1 t1− S(t1−s)σ(s)dZ(s) p ≤7p−1Cp,α kS(t2−t1)−Ikp ∞ X n=1 βα nZt1 t1− e−αλn(t1−s)σα(s)ds p α ≤14p−1Cp,α(Mp+1)  ∞ X n=1 βα nZt1 t1− e−αλn(t1−s)σα(s)ds p α . For I5and I6, in a similar way as above, it follows immediately that I5≤7p−1b0 Zt2 t1 kS(t2−s)k p p−1ds!p−1 ≤7p−1b0Mp(t2−t1)p−1, and I6≤7p−1Cp,α  ∞ X n=1 βα nZt2 t1 e−αλn(t2−s)σα(s)ds p α , which, together with the previous estimates, implies that k(Qx)(t2)−(Qx)(t1)ktends to zero independently of x∈ E(T1, ρ0) as t2−t1→0 and →0+, since S(t) is uniformly continuous for t∈(0,T1]. Hence, V(·) is equicontinuous on (0,T1]. Meanwhile, its equicontinuity at t=0 is trivial. Now, it remains to show that, for each t∈[0,T1], the set V(t)={(Qx)(t) : x∈E(T1, ρ0)}is relatively compact in the space Lp(Ω;X). Clearly, we only need to prove it for t∈(0,T1]. Let t∈(0,T1] be fixed and 0 <  < t, for x∈E(T1, ρ0), we denote (Qx)(t)=S(t)ϕ(0) +S()Zt− 0 S(t−−s)f(s,x∨sϕ)ds +S()Zt− 0 S(t−−s)σ(s)dZ(s). 7 Note that EZt− 0 S(t−−s)f(s,x∨sϕ)ds p +EZt− 0 S(t−−s)σ(s)dZ(s) p ≤b0MpTp−1 1+Cp,αv1<+∞, which implies that the set (Zt− 0 S(t−−s)f(s,x∨sϕ)ds +Zt− 0 S(t−−s)σ(s)dZ(s) : x∈E(T1, ρ0)) is bounded in Lp(Ω;X). Thus, we deduce that the set V(t)={(Qx)(t) : x∈E(T1, ρ0)}is relatively compact in Lp(Ω;X) since S(t) is compact for every t>0. Observe that Ek(Qx)(t)−(Qx)(t)kp≤2p−1EZt t− S(t−s)f(s,x∨sϕ)ds p +2p−1EZt t− S(t−s)σ(s)dZ(s) p ≤2p−1b0Mpp−1+2p−1Cp,α  ∞ X n=1 βα nZt t− e−αλn(t−s)σα(s)ds p α →0, uniformly for x∈E(T1, ρ0) as →0+. Therefore, V(t) is relatively compact in Lp(Ω;X) as well for every t∈(0,T1]. The above arguments enable us to infer from Arzel` a-Ascoli theorem that Q:E(T1, ρ0)→E(T1, ρ0) is compact and, therefore, by the Schauder fixed point theorem we conclude Qpossesses a fixed point on E(T1, ρ0) whose extension by ϕon (−∞,0] is by Definition 3.1 a mild solution for (4). Now, consider Eq. (4) on the interval [T1,2T1] with initial data xT1. Similarly, there is a mild solution of Eq. (4) with x0=ϕand x(·)∈C([0,2T1]; Lp(Ω;X)). Next, turn our attention to the existence of solution of (4) on [0,+∞). The subsequent proofs are very similar to those in Theorem 4.6 of [10] and we omit them here. This completes the proof of this theorem.  4 Global attracting sets In this section, we will investigate the existence of global attracting sets of system (4) which is the main part of this paper. Let x(t, ϕ) be a mild solution of equation (4) with initial datum x0=ϕ∈Cγ F(xt(ϕ) being its history function). Meanwhile, the symbol Cwill denote a generic constant whose value may change from one line to another and even in the same line. 4.1 Existence of global attracting sets: General case In what follows, we shall obtain some estimates of solutions which will imply that the solutions are uniformly bounded with respect to bounded sets of initial conditions and positive values of time. This also implies the existence of an absorbing set for the solutions which is also a property on the ultimate boundedness of solutions. Theorem 4.1. Assume that all the hypotheses of Theorem 3.1 hold, and γ > pλ1.(16) Then, every mild solution x(·)of system (4) with x0=ϕ∈Cγ F, defined globally in time, satisfies kxtkp Cγ F ≤Ckϕkp Cγ F e−λ1t+C,for any t ≥0. 8 Proof. For any t≥0, it follows that Ekx(t)kp≤3p−1EkS(t)ϕ(0)kp+3p−1 Zt 0 kS(t−s)kds!p−1Zt 0 kS(t−s)kEkf(s,xs)kpds +3p−1Cp,α sup t≥0 ∞ X n=1 βα nZt 0 e−αλn(t−s)σα(s)ds p α ≤3p−1Mpe−pλ1tEkϕ(0)kp+3p−1Mp Zt 0 e−λ1(t−s)ds!p−1 ·Zt 0 e−λ1(t−s)k1(s)+k2(s)kxskp Cγ Fds +3p−1Cp,αv1 ≤3p−1Mpe−pλ1tkϕkp Cγ F +3p−1MpK1 λp−1 1 +3p−1Cp,αv1+3p−1Mp λp−1 1Zt 0 k2(s)e−λ1(t−s)kxskp Cγ F ds. (17) By (16), we have λ1<pλ1< γ, then e−(λ1−γ)θ<e−(pλ1−γ)θ≤1 holds immediately for any θ≤0. Multiplying (17) by eγθ and replacing tby t+θ, it follows sup θ∈(−t,0] eγθEkx(t+θ)kp≤3p−1Mpe−pλ1(t+θ)eγθkϕkp Cγ F +3p−1MpK1 λp−1 1 eγθ +3p−1Cp,αv1eγθ +3p−1Mp λp−1 1Zt+θ 0 k2(s)e−λ1(t+θ−s)eγθ kxskp Cγ F ds ≤3p−1Mpe−pλ1tkϕkp Cγ F +3p−1MpK1 λp−1 1 +3p−1Cp,αv1 +3p−1Mp λp−1 1Zt 0 k2(s)e−λ1(t−s)kxskp Cγ F ds. (18) Note that eγθEkx(t+θ)kp=e−γteγ(t+θ)Ekx(t+θ)kp≤e−γtkϕkp Cγ F ≤e−pλ1tkϕkp Cγ F ,for any θ∈(−∞,−t].(19) Therefore, eλ1tkxtkp Cγ F ≤3p−1Mpkϕkp Cγ F +3p−1MpK1 λp−1 1 eλ1t+3p−1Cp,αv1eλ1t+3p−1Mp λp−1 1Zt 0 k2(s)eλ1skxskp Cγ F ds. Applying the Gronwall inequality and the increasing property of function eλ1twith respect to t, we have for t≥0 that kxtkp Cγ F ≤3p−1MpM2kϕkp Cγ F e−λ1t+M1M2, where we used the notations M1=3p−1MpK1 λp−1 1 +3p−1Cp,αv1,M2=exp 3p−1MpK2 λp−1 1. The proof is finished.  A general result concerning the existence of a minimal compact set in Cγ Fwhich is globally attracting for the solutions of our problem will be proved in this subsection. To that end we first need the following compactness conclusion. Lemma 4.1. Assume the conditions of Theorem 4.1 hold. Then, for any bounded subset Φof Cγ F, any sequence {τn} with τn→+∞(n→+∞),{ϕn}with ϕn∈Φ, and any sequence of solutions {xn}of system (4) with xn 0=ϕn∈Φ, the sequence nxn τnois relatively compact in Cγ F. 9 In order to rewrite (38) into the abstract form (4), we take X=L2([0, π]) and define z(t)(·)Bz(t,·) and ϕ(t)(·)B ϕ(t,·). Let A:D(A)→Xbe the operator given by Aξ=ξ00, with the domain D(A)=H2([0, π]) ∩H1 0([0, π]) =nξ(·)∈X:ξ0, ξ00 ∈X, ξ(0) =ξ(π)=0o. Then Agenerates a strongly continuous semigroup (S(t))t≥0which is compact. Furthermore, Ahas a discrete spectrum, the eigenvalues are −n2,n∈N+, with the corresponding normalized eigenvectors en(x)=q2 πsin(nx), n=1,2,· · · . Then the following properties hold: (i) If ξ∈D(A), then Aξ=− ∞ X n=1 n2hξ, enien. (ii) For every ξ∈X, S(t)ξ= ∞ X n=1 e−n2thξ, enien. In particular, kS(t)k ≤ e−t. We assume that the following conditions hold for the system (38): (i) The functions f1: [0,+∞)×X→Ris continuous. Moreover, there exist continuous functions li: [0,+∞)× X→[0,+∞), i=1,2,3, are such that f1(t,x)≤l1(t)kxk+l2(t),for all (t,x)∈[0,+∞)×X. f1(t,x)−f1(t,y)≤l3(t)kx−yk,for all (t,x,y)∈[0,+∞)×X×X. (ii) The functions lisatisfy that Z0 −∞ `q 1(t+θ)e−qγθdθ!1 q ≤e−t, Z+∞ 0 Z0 −∞ `q 2(t+θ)e−qγθdθ!p q dt <∞, and |l3(t+θ)|<h(θ) with Z0 −∞ hq(θ)e−qγθdθ!p q <+∞, where 1 p+1 q=1. (iii) There exist some constants C1,C2>0 such that for n∈N, C1n−2β≤βn≤C2n−2β,for some β > 0. Now define the mapping f(·,·) : [0,+∞)×Cγ F→Xas f(t, ϕ)(x)=f(t, ϕ(·,x)) =Z0 −∞ f1(t+θ, ϕ(θ)(x))dθ, for any ϕ(·,x)=ϕ(·)(x)∈Cγ F. Then, with the above notations, system (38) is rewritten well into the form of (4). To obtain all the results of this paper, it is easy to see that, we only need to verify the conditions of Theorem 3.1 16 and Theorem 4.4. We now examine that the conditions in Theorem 3.1 are all fulfilled for this system. It is easy to check that (H1) is satisfied with M=λ1=1. For any ϕ∈Cγ F, by assumption (i), we have Ekf(t, ϕ)kp=E Zπ 0 |f(t, ϕ)(x)|2dx!p 2 ≤C Z0 −∞ (`1(t+θ)+`2(t+θ)Ekϕ(θ)k)e−γθeγθdθ!p ≤C Z0 −∞ `q 1(t+θ)e−qγθdθ!p q Z0 −∞ epγθdθ!+ Z0 −∞ `q 1(t+θ)e−qγθdθ!p q Z0 −∞ e(p−1)γθeγθEkϕ(θ)kpdθ! ≤C Z0 −∞ `q 1(t+θ)e−qγθdθ!p q +C Z0 −∞ `q 2(t+θ)e−qγθdθ!p q kϕkp Cγ F . Let k1(t)=CR0 −∞ `q 1(t+θ)e−qγθdθp qand k2(t)=CR0 −∞ `q 2(t+θ)e−qγθdθp qin (H2). Then the assumption (ii) ensures that (H2) hold true. Moreover, it is straightforward to see from assumption (iii) that sup t≥0       ∞ X n=1 βα nZt 0 e−αλn(t−s)σα(s)ds       p α ≤sup t≥0       ∞ X n=1 βα nZt 0 e−αn2(t−s)ds       p α ≤ ∞ X n=1 Cα αn2+2αβ  p α :=v1<∞. In other words, (6) in this case holds true. Thus, all conditions in Theorem 3.1 are satisfied. On the other hand, the function fsatisfies hypothesis (H3). From assumption (ii) we have, for any t≥0 and ϕ1, ϕ2∈Cγ F, |f(t, ϕ1)(x)−f(t, ϕ2)(x)|2≤ Z0 −∞ |f1(t+θ, ϕ1(θ)(x))−f1(t+θ, ϕ2(θ)(x))|dθ!2 ≤ Z0 −∞ l3(t+θ)kϕ1(θ)−ϕ2(θ)kdθ!2 ≤ Z0 −∞ h(θ)kϕ1(θ)−ϕ2(θ)kdθ!2 . Then, we deduce the following estimate: Ekf(t, ϕ1)−f(t, ϕ2)kp=E Zπ 0 |f(t, ϕ1)(x)−f(t, ϕ2)(x)|2dx!p 2 ≤πp 2 Z0 −∞ h(θ)e−γθeγθEkϕ1(θ)−ϕ2(θ)kdθ!p ≤πp 2 Z0 −∞ h(θ)e−qγθdθ!p q Z0 −∞ e(p−1)γθeγθEkϕ1(θ)−ϕ2(θ)kpdθ! ≤πp 2 Z0 −∞ h(θ)e−qγθdθ!p q kϕ1−ϕ2kp Cγ F . In addition, we suppose that γ > p>pπp 2R0 −∞ hq(θ)e−qγθdθp q, which implies that condition (30) is fulfilled. Last, it can still be obtained from assumptions (ii) and (iii), Z+∞ 0 esEkf(s,0)kpds ≤Z+∞ 0 esk1(s)ds =Z+∞ 0 es Z0 −∞ `q 1(s+θ)e−qγθdθ!p q ds ≤Z+∞ 0 e−(p−1)sds =1 p−1 :=v2<∞, 17 and sup t≥0       ∞ X n=1 βα nZt 0 e αλ1 pte−αλn(t−s)σα(s)ds       p α =sup t≥0       ∞ X n=1 βα nZt 0 eα pte−αn2(t−s)e−αsds       p α ≤sup t≥0       ∞ X n=1 βα nZt 0 e−1−1 pαn2(t−s)ds       p α ≤ ∞ X n=1 pCα (p−1)αn2+2αβ  p α :=v3<∞. Thanks to the above notations and conditions, we can thus prove Theorem 4.4. Acknowledgements This work is supported by Natural Science Foundation of Guangxi Province (2020JJG110003), Zhejiang Provincial National Natural Science Foundation of China (No. LQ24A010021), Spanish Ministerio de Ciencia e Innovaci´ on (MCI), Agencia Estatal de Investigaci´ on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PID2021-122991NB-C21. References [1] N. U. 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