Stability analysis of stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay

Stability analysis of stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay Shuang Yanga,b, Yangrong Lia, Qiangheng Zhanga, Tom´as Caraballob,∗ aSchool of Mathematics and Statistics, Southwest University, Chongqing 400715, China bDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, C/Tarfia s/n, 41012-Sevilla, Spain Abstract The asymptotic behaviour of stochastic three-dimensional Lagrangian-averaged Navier-Stokes equations with infinite delay and nonlinear hereditary noise is analysed. First, using Galerkin’s approximations and the monotonicity method, we prove the existence and uniqueness of solutions when the non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous with an additional assumption. Next, we show the existence and uniqueness of stationary solutions to the corresponding deterministic equation via the Lax-Milgram and the Schauder theorems. Later, we focus on the stability properties of stationary solutions. To begin with, we discuss the local stability of stationary solutions for general delay terms by using a direct method and then apply the abstract results to two kinds of infinite delays. Besides, the exponential stability of stationary solutions is also established in the case of unbounded distributed delay. Moreover, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we establish criteria on the polynomial asymptotic stability of stationary solutions for the special case of proportional delay. Keywords: Stochastic three-dimensional Lagrangian-averaged Navier-Stokes equations; Stationary solutions; Exponential convergence; Polynomial asymptotic stability; Infinite delay 2020 MSC: 35Q30, 35A01, 35B35 1. Introduction Navier-Stokes equations are known to model the motion of many important fluids which include water, air, oil, etc. Motivated by the first published paper of Leray [29], the authors in references [3, 6, 7, 9–18, 20, 23– 25, 27, 31–35, 37, 40] have extensively studied the well-posedness and long-time behavior of solutions to Navier- Stokes equations. Due to the importance of delay effects in many physical, biological and engineering models [1, 8, 42], delay differential equations have received much attention over the recent years. Besides, the future state of systems may not only depend on their current state, but also on their past history, which plays a nontrivial role in some cases. It is worth mentioning some typical examples, such as the investigation of high-viscosity liquids under the condition of low temperatures, the thermomechanical analysis with respect to polymers, population models, etc (see [22, 36] and the references therein). Thus, we need to take into account some hereditary characteristics such as aftereffect, time lag, memory and time delay in our models. Many researchers have focused on the relationship between the Navier-Stokes equations and the phenomenon of turbulence for a long time. It is worth stressing that the common assumption relates the onset of turbulence to the randomness of background movement. Moreover, the systems we study are affected by a variety of random factors in real life, so it is necessary to consider some kind of noise in our models. The concept of random dynamical systems was first introduced by Ulam and von Neumann [41] in 1945. Due to the fact that stochastic differential equations originated from random dynamical systems, a growing number of people have studied random dynamical systems since 1980s. Amongst the many notable results, it is remarkable that the importance of the work in [4], Bensoussan and Temam in this article discussed the stochastic Navier-Stokes equations driven by white noise and random forces, providing a more realistic model to solve the problem. ∗Corresponding author. Email addresses: [email protected] (Shuang Yang), [email protected] (Yangrong Li), [email protected] (Qiangheng Zhang), [email protected] (Tom´as Caraballo) Preprint submitted to JDDE December 20, 2022 Based on the previous discussion, we investigate in this paper the stochastic dynamics of the following nonautonomous stochastic three-dimensional Lagrangian-averaged Navier-Stokes (LANS) equation with infinite delay and nonlinear hereditary noise:                ∂t(u−α∆u) + ν(Au −α∆(Au)) + (u·∇)(u−α∆u) −α∇u∗·∆u+∇p=f(t) + g1(t, ut) + g2(t, ut)˙ W, in (τ, +∞)×O, ∇·u= 0,in (τ, +∞)×O, u= 0, Au = 0 on (τ, +∞)×∂O, u(τ+s, x) = φ(s, x), s ∈(−∞,0], x ∈ O, (1.1) where O ⊆ R3is a bounded open set with sufficiently regular boundary ∂O, τ ∈R,Ais the Stokes operator, the pair (ν, α) of positive coefficients denotes the kinematic viscosity of the fluid and the square of the spatial scale at which fluid motion is filtered respectively, the symbol ∗denotes the transpose of a matrix, u= (u1, u2, u3) is the averaged (or large-scale) velocity of the fluid, utdenotes the segment of solutions up to time t, i.e. ut(s) = u(t+s) for all s≤0, pis the pressure of the fluid, fis a non-delayed external force field, the terms g1, g2contain some hereditary characteristics, such as memory, unbounded variable or infinite distributed delay, etc, φis an initial velocity field defined in (−∞,0] ×O, and ˙ Wdenotes the generalized derivative (white noise) of a cylindrical Wiener process, which will be specified later. The LANS model was first used to deal with the turbulence closure problem by using the method of Lagrangian averaging. The study of such models, including the well-posedness of solutions and the existence, uniqueness and asymptotic stability of stationary solutions, has been carried out in some papers, e.g., [12–14, 16] and the references therein. In the special case that α= 0, the system (1.1) is reduced to the usual three-dimensional Navier-Stokes equation whose dynamical behavior had been widely investigated in [9, 15, 17, 18, 25, 27, 37]. The deterministic and non-delay version of (1.1), i.e. g2= 0 and g1is independent of u, has been considered in [20, 35]. In the stochastic case without delay, Caraballo et al. in [16] studied the stochastic dynamics of such a system for the first time. In addition, the delay version of Navier-Stokes equations has also received much attention over the last years. The analysis of Navier-Stokes equations with some hereditary features was first studied by Caraballo and Real in [15] and developed in [9, 12, 14, 23–25, 30–34]. On the one hand, for bounded delay, the authors have discussed several issues including the existence, uniqueness, asymptotic behavior and regularity of solutions, the existence, uniqueness and stability of stationary solutions, the existence, uniqueness of global or pullback (random) attractors. On the other hand, the case of unbounded or infinite delays has been analyzed in [24, 26, 30–34]. We may choose several phase spaces for dealing with the infinite delay as given in above references. The first one is the Banach space Cγ(H) = {ϕ∈C((−∞,0]; H) : lim s→−∞eγsϕ(s) exists in H},where γ > 0,(1.2) where His the 3D Lebesgue-type Hilbert space. The second one is C−∞(H) = {ϕ∈C((−∞,0]; H) : lim s→−∞ϕ(s) exists in H},(1.3) see [30, 31]. We also use Cγ(V) and C−∞(V), where Vis the Sobolev-type subspace instead of Hin (1.2) and (1.3). Our first goal in this paper is to prove the existence and uniqueness of solutions to the stochastic three dimensional LANS Eq. (1.1) in the Banach space Cγ(V). As done by Liu and Caraballo [31] for the usual two-dimensional system, we need to assume that the non-delayed external force fis locally integrable (see Hypothesis F) and the delay forcing terms gi(t, ut)(i= 1,2) are globally Lipschitz continuous (see Hypothesis G). An example is given in the last part of Section 2. The calculation shows that the example (corresponding to infinite distributed delay) satisfies all conditions of Hypothesis G in the space Cγ(V) for γ > 0. Besides, we also need an extra assumption on the nonlinear diffusion term g2of noise (see Hypothesis I). Under the above assumptions, we use the Galerkin method to construct an approximating sequence. We then give a priori estimates for the approximating sequence ensuring the solutions exist for the whole time interval [τ, τ +T] for all T > 0. Next, by the monotonicity method established in [39], we obtain the well-posedness of solutions. Another interesting and challenging topic is to consider the asymptotic behaviour of solutions for Eq. (1.1) towards to the stationary solution. This issue will provide some useful information on future evolution of the 2 system. Thanks to the Lax-Milgram and the Schauder theorems, we first prove the existence and uniqueness of stationary solutions to the corresponding deterministic equation. We then show the local stability of the stationary solution for the general delay term by using a direct method, where the general delay contains the unbounded variable delay and the infinite distributed delay in C−∞(V). Next, we prove the global stability of the stationary solution. However, to obtain stability results in Cγ(V) with γ > 0, the exponential stability in the case of unbounded variable delay fails to be proved in general (see [31] and [34] for more details). Fortunately, in the case of infinite distributed delay, we are able to prove not only stability of stationary solutions in Cγ(V), but also exponential asymptotic stability. Since we can not analyze the exponential stability in the unbounded variable delay case in Cγ(V), we will explore, at least, the asymptotic stability in C−∞(V), by using the Lyapunov functionals construction proposed by Kolmanovskii and Shaikhet [28]. Furthermore, we eventually discuss the polynomial asymptotic stability in the particular case of proportional delay (also known as pantograph delay). The article is organized as follows. In Section 2, we introduce the cylindrical Wiener process, some notations and linear operators, describe some suitable assumptions about the non-delayed external force fand delay terms gi(i= 1,2). In Section 3, we prove the existence and uniqueness of solutions to system (1.1). In Section 4, stationary solutions and their stability results are established. More precisely, on the one hand, we provide some sufficient conditions ensuring the existence and uniqueness of stationary solutions. On the other hand, we further study the convergence of stationary solutions including local stability, exponential stability, asymptotic stability via Lyapunov method and polynomial asymptotic stability. 2. Preliminaries 2.1. The cylindrical Wiener process Let (Ω,F,{Ft}t∈R, P) be a complete filtered probability space such that {Ft}t∈Ris an increasing right continuous family of sub σ-algebras of F, which contains all P-null sets, and further Ft=F0for all t≤0. Let {βj t, t ≥0, j = 1,2,3, . . .}be a sequence of mutually independent standard real valued Ft-Wiener processes and Ka separable Hilbert space with an orthonormal basis {ej;j= 1,2,3, . . .}. Suppose that {W(t); t≥0}be a K-valued cylindrical Wiener process (with the covariance operator Q:K→K) given by W(t) = ∞ X j=1 βj(t)ej, t ≥0.(2.1) Given a separable Hilbert space H0, we denote by L2(K, H0) the space of Hilbert-Schmidt operators from Kinto H0with following norm kSk2 L2(K,H0)= tr(SQS∗),∀S∈ L2(K, H0),(2.2) where tr denotes the trace of an operator and S∗is the adjoint operator of S. For any separable Banach space X, interval (a, b)⊂Rand p≥1, we denote by Ip(a, b;X) the Banach space of all processes ϕ∈Lp(Ω×(a, b),F⊗B((a, b)), dP ⊗dt;X) such that ϕ(t) is Ft-progressively measurable for a.e. t∈(a, b), where B(·) denotes the Borel σ-algebra. We also denote by Lp(Ω,F, dP;C(a, b;X)) with p≥1 the space of all continuous and Ft-progressively measurable X-valued processes ϕsuch that E(supa≤t≤bkϕ(t)kp X)< ∞, where C(a, b;X) is the Banach space of all continuous functions from [a, b] into X. For convenience, we write Lp(Ω,F, dP;C(a, b;X)) as Lp(Ω; C(a, b;X)). For any positive constant T > 0, process Φ ∈I2(τ, τ +T;L2(K, H0)) and t∈[τ, τ +T], the stochastic integral Rt τΦ(s)dW(s) is defined by the unique continuous H0-valued Ft-martingale such that Zt τ Φ(s)dW(s), wH0 =∞ X j=1 Zt τΦ(s)ej, wH0 dβj(s),∀w∈H0,(2.3) where the integral with respect to βj(s) is the Ito integral. By [21], if Φ ∈I2(τ, τ +T;L2(K, H0)) and φ∈L2(Ω, L∞(τ, τ +T;H0)) is Ft-progressively measurable, then ∞ X j=1 Zt τΦ(s)ej, φ(s)H0 dβj(s) =: Zt τΦ(s)dW(s), φ(s), τ ≤t≤τ+T, converges in L1(Ω, C(τ, τ +T)). Recall that given a function φ: (−∞, τ +T]→X, for each t∈(τ, τ +T), the segment φtof φis defined by φt(s) = φ(t+s),∀s∈(−∞,0].(2.4) 3 2.2. Notations and hypotheses In this subsection, we introduce some notations and linear operators, recall some properties with respect to the nonlinear term (u·∇)(u−α∆u)−α∇u∗·∆uin the problem (1.1), and impose some suitable assumptions. Denote by L2(O) := (L2(O))3,H1 0(O) := (H1 0(O))3,C∞ 0(O)=(C∞ 0(O))3and V={u∈C∞ 0(O) : ∇·u= 0 in O}.(2.5) Let Hbe the closure of Vin L2(O). Then His a Hilbert space with the inner product and norm (u, v) = 3 X j=1 ZO uj(x)vj(x)dx, |u|2= (u, u),∀u, v ∈H. (2.6) Let Vbe the closure of Vin H1 0(O). Then Vis a Hilbert space with the inner product ((u, v)) = (u, v) + α(∇u, ∇v)=(u, v) + α 3 X i,j=1 ZO ∂uj ∂xi ∂vj ∂xi dx, ∀u, v ∈V, (2.7) and the norm kuk2= ((u, u)). We have V⊂H⊂V∗, where V∗is the dual space of V, the injections are dense, continuous and compact. Denote by Pthe Leray projector from L2(O) onto Hand define the Stokes operator Aby Aw =−P(∆w),∀w∈D(A) = H2(O)∩V, (2.8) where H2(O)=(H2(O))3. We deduce (Au, v)=(∇u, ∇v),kukH2(O)≤C1|Au|,∀u∈D(A), v ∈V, (2.9) where C1is a positive constant. In particular, D(A) is a Hilbert space. Denote by h·,·i the duality product between (D(A))∗and D(A), and define a continuous linear operator e A∈ L(D(A),(D(A))∗) by he Au, vi=ν(Au, v) + να(Au, Av),∀u, v ∈D(A) =: D(e A).(2.10) It is well-known that the Stokes operator Ahas a sequence {λk:k∈N}of eigenvalues satisfying 0<inf v∈V\{0}kvk2 |v|2=λ1≤λ2≤ ··· , λk→ ∞ (2.11) and a sequence {ξk∈D(A) : k∈N}of eigenvectors which is orthonormal in H. From (2.10) we have he Aξk, vi=νλk((ξk, v)) (2.12) and the eigenvalues of the operator e Aare given by e λk:= νλk. By (2.10)-(2.12), the operator e A∈ L(D(A),(D(A))∗) satisfies the following conditions: (A1) e Ais self-adjoint; (A2) For all u∈D(A), 2he Au, ui ≥ eα(Au, Au), where eα= 2να; (A3) e Aξk=e λkξkwith e λk=νλk. As in [14], we associate another inner product on D(A) = D(e A), defined by (u, v)D(A):= he Au, vi,and so e λ1kuk2≤ kuk2 D(A),∀u, v ∈D(A).(2.13) By (2.9), the above is equivalent to the original inner product ((u, v)) + (Au, Av) for u, v ∈D(A). For u∈D(A) and v∈L2(O), we regard (u·∇)vas the element of (H−1(O))3=: H−1(O) given by h(u·∇)v, wi−1= 3 X i,j=1h∂ivj, uiwji−1,∀w∈H1 0(O),(2.14) 4 where h·,·i−1denotes the duality product between H−1(O) and H1 0(O) or between H−1(O) and H1 0(O), and uiwj∈H1 0(O) due to the continuous injections of H2(O)⊂L∞(O) and H1 0(O)⊂L6(O). Hence, there exists a positive constant C2:= C2(O) such that |h(u·∇)v, wi−1| ≤ C2|Au||v|kwk,∀(u, v, w)∈D(A)×L2(O)×H1 0(O).(2.15) If u∈D(A), then ∇u∗∈(H1(O))3⊂(L6(O))3, where H1(O) = (H1(O))3and L6(O) = (L6(O))3. For all v∈L2(O), we have ∇u∗·v∈(L3/2(O))3⊂H−1(O) satisfying h∇u∗·v, wi−1= 3 X i,j=1 ZO (∂jui)viwjdx, ∀w∈H1 0(O),(2.16) which implies that there exists a positive constant C3:= C3(O) such that |h∇u∗·v, wi−1| ≤ C3|Au||v|kwk,∀(u, v, w)∈D(A)×L2(O)×H1 0(O).(2.17) Now, we introduce the trilinear operator as follows: b#(u, v, w) = h(u·∇)v, wi−1+h∇u∗·v, wi−1,∀(u, v, w)∈D(A)×L2(O)×H1 0(O).(2.18) By [16, Proposition 2.2], we obtain b#(u, v, w) = −b#(w, v, u),∀(u, v, w)∈D(A)×L2(O)×D(A),(2.19) which implies that b#(u, v, u) = 0,∀(u, v)∈D(A)×L2(O). Moreover, there exists a positive constant c#:= c#(O) such that |b#(u, v, w)| ≤ c#|Au||v|kwk,∀(u, v, w)∈D(A)×L2(O)×H1 0(O),(2.20) and |b#(u, v, w)| ≤ c#kuk|v||Aw|,∀(u, v, w)∈D(A)×L2(O)×D(A).(2.21) We then define a bilinear mapping e B:D(A)×D(A)→(D(A))∗, denoted by he B(u, v), wi=b#(u, v −α∆v, w),∀(u, v, w)∈D(A)×D(A)×D(A),(2.22) and e B(u) := e B(u, u) for all u∈D(A). By the definition and properties of b#, we find that there exists a positive constant ec:= ec(O) such that (B1) he B(u, v), ui= 0 and he B(u), vi=−he B(v, u), ui,∀(u, v)∈D(A)×D(A); (B2) ke B(u, v)k(D(A))∗≤eckukkvkD(A),∀(u, v)∈D(A)×D(A); (B3) |he B(u, v), wi| ≤ eckukD(A)kvkD(A)kwk,∀(u, v, w)∈D(A)×D(A)×D(A). Recall the phase space Cγ(V) = {ϕ∈C((−∞,0]; V) : lim s→−∞eγsϕ(s) exists in V},where γ > 0,(2.23) which is a Banach space with the sup norm kϕkCγ(V)= sup s∈(−∞,0] eγskϕ(s)k.(2.24) We now establish some assumptions on the non-delayed external force and delay terms respectively. Hypothesis F.f∈I2(τ, τ +T;H−1(O)) for any τ∈Rand T > 0. Hypothesis G. Let g1:R×Cγ(V)→H−1(O) and g2:R×Cγ(V)→ L2(K, L2(O)) satisfy the following conditions. (G1) For any η∈Cγ(V), gi(·, η) are measurable, i= 1,2. (G2) gi(·,0) = 0, i = 1,2. (G3) There exists Lgi>0 (i= 1,2) such that for all t∈Rand η, ζ ∈Cγ(V), kg1(t, η)−g1(t, ζ)kH−1(O)≤Lg1kη−ζkCγ(V), 5 kg2(t, η)−g2(t, ζ)kL2(K,L2(O)) ≤Lg2kη−ζkCγ(V); (G4) There exists Cgi>0 (i= 1,2) such that for all t∈Rand η, ζ ∈Cγ(V), Zt τkg1(s, us)−g1(s, vs)k2 H−1(O)ds ≤C2 g1Zt −∞ ku(s)−v(s)k2ds, Zt τkg2(s, us)−g2(s, vs)k2 L2(K,L2(O))ds ≤C2 g2Zt −∞ ku(s)−v(s)k2ds; (G5) There exists e Cgi>0 (i= 1,2) such that for all τ∈R, t ≥τ, all decreasing function $∈C0([τ, t]) and u, v ∈C0((−∞, t]; V) Zt τ $(s)kg1(s, us)−g1(s, vs)k2 H−1(O)ds ≤e Cg1Zt τ $(s)ku(s)−v(s)k2ds, Zt τ $(s)kg2(s, us)−g2(s, vs)k2 L2(K,L2(O))ds ≤e Cg2Zt τ $(s)ku(s)−v(s)k2ds. We infer from (G2)-(G3) that, for all η∈Cγ(V), kg1(t, η)kH−1(O)≤Lg1kηkCγ(V),kg2(t, η)kL2(K,L2(O)) ≤Lg2kηkCγ(V). Next, let us define e f(t) as (( e f(t), w)) = hf(t), wi−1,∀(t, w)∈R×V. By the hypothesis F,e f∈I2(τ, τ +T; (D(A))∗) for any τ∈Rand T > 0. In addition, we define eg1:R×Cγ(V)→Vsuch that ((eg1(t, η), w)) = hg1(t, η), wi−1,∀(t, η, w)∈R×Cγ(V)×V. Finally, we define eg2:R×Cγ(V)→ L2(K, V ) such that eg2(t, η) = (I+αA)−1◦P ◦g2(t, η),∀(t, η)∈R×Cγ(V), where Iis the identity operator in Hand I+αA :D(A)→His bijective, moreover, (((I+αA)−1u, w)) = (u, w),∀u∈H, w ∈V. Hence, for the orthonormal basis {ej}of K, we have (g2(t, η)ej, w) = ((I+αA)eg2(t, η)ej, w) = ((eg2(t, η)ej, w)), for all j≥1 and (t, η, w)∈R×Cγ(V)×D(A), by (2.3), we further obtain that Zt τ g2(s, η)dW (s), w=∞ X j=1 Zt τ (g2(s, η)ej, w)dβj(s) =∞ X j=1 Zt τeg2(s, η)ej, wdβj(s) =Zt τeg2(s, η)dW(s), w.(2.25) By the same method as in [13], one can prove that eg1:R×Cγ(V)→Vand eg2:R×Cγ(V)→ L2(K, V ) satisfy the following conditions: (H1) For any η∈Cγ(V), egi(·, η) are measurable, i= 1,2; (H2) egi(·,0) = 0, i = 1,2; (H3) Taking Leg1=Lg1, Leg2=Lg2/√1 + αλ1, we deduce, for all t∈Rand η, ζ ∈Cγ(V), keg1(t, η)−eg1(t, ζ)k ≤ Leg1kη−ζkCγ(V), 6 keg2(t, η)−eg2(t, ζ)kL2(K,V )≤Leg2kη−ζkCγ(V); (H4) Setting Ceg1=Cg1, Ceg2=Cg2/√1 + αλ1, we obtain, for all t∈Rand η, ζ ∈Cγ(V), Zt τkeg1(s, us)−eg1(s, vs)k2ds ≤C2 eg1Zt −∞ ku(s)−v(s)k2ds, Zt τkeg2(s, us)−eg2(s, vs)k2 L2(K,V )ds ≤C2 eg2Zt −∞ ku(s)−v(s)k2ds; (H5) Letting e Ceg1=e Cg1,e Ceg2=e Cg2/√1 + αλ1such that for all τ∈R, t ≥τ, and all decreasing function $∈C0([τ, t]) and u, v ∈C0((−∞, t]; V) Zt τ $(s)keg1(s, us)−eg1(s, vs)k2ds ≤e Ceg1Zt τ $(s)ku(s)−v(s)k2ds, Zt τ $(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds ≤e Ceg2Zt τ $(s)ku(s)−v(s)k2ds. It follows from (H2)-(H3) that for all t∈Rand η∈Cγ(V), keg1(t, η)k ≤ Leg1kηkCγ(V),keg2(t, η)kL2(K,V )≤Leg2kηkCγ(V).(2.26) An example of the delayed terms with (H1)-(H5) is given as follows. Example 1: For all t∈Rand ξ∈Cγ(V), let egi(t, ξ) = Z0 −∞ e Gi(t, s, ξ(s))ds, i = 1,2,(2.27) where e G1:R×(−∞,0]×V→Vwith e G1(t, s, 0) = 0, and e G2:R×(−∞,0]×V→ L2(K, V ) with e G2(t, s, 0) = 0, and both are measurable. Assume that there exist Le Gi∈L2(−∞,0) (i= 1,2) with Le Gi(·)e−(γ+θ)·∈L2(−∞,0) for certain θ > 0 such that for all t∈R,s∈(−∞,0] and η, ζ ∈V, ke G1(t, s, η)−e G1(t, s, ζ)k ≤ Le G1(s)kη−ζk, ke G2(t, s, η)−e G2(t, s, ζ)kL2(K,V )≤Le G2(s)kη−ζk. Thus, we can rewrite the delay terms egi(i= 1,2) in our problem as egi(t, ut) = R0 −∞ e Gi(t, s, u(t+s))ds (i= 1,2). It follows that the example is within our framework, and egi(i= 1,2) fulfill conditions (H1)-(H5) (e.g., see [34] for more details). 3. Well-posedness of stochastic 3D LANS equations with infinite delay In this section, we prove the well-posedness of the stochastic Eq. (1.1), which can be transferred into the following abstract equation:    du dt +e Au(t) + e B(u(t)) = e f(t) + eg1(t, ut) + eg2(t, ut)dW dt ,∀t > τ, u(τ+s) = φ(s), s ∈(−∞,0]. (3.1) Definition 3.1. Suppose that φ∈L2(Ω, Cγ(V)) (which is a F0-progressively measurable V-valued processes) and τ∈R. A stochastic process udefined on Ris called a solution to system (3.1) if u∈I2(τ, τ +T;D(A)) ∩L2(Ω, L∞(τ, τ +T;V)),∀T > 0, uτ=φand P-almost surely ((u(t), w)) + Zt τhe Au(s), wids +Zt τhe B(u(s)), wids = ((φ(0), w)) + Zt τe f(s) + eg1(s, us), wds +w, Zt τeg2(s, us)dW(s) (3.2) for all t≥τand w∈D(A). 7 Lemma 3.2. For all u, v ∈D(A), we have h−e Aw −2( e B(u)−e B(v)), wi ≤ σkwk2kvk2 D(A),(3.3) where w=u−vand σ=ec2. Proof. Note that h−e Aw, wi=−kwk2 D(A).(3.4) By the property (B1) of the operator e B, we have he B(u), wi=−he B(w, u), ui=−he B(w, u), vi,(3.5) and similarly he B(v), wi=−he B(w, v), vi.(3.6) Subtracting (3.6) from (3.5), he B(u)−e B(v), wi=−he B(w), vi,(3.7) which, together with (B2), implies that |he B(u)−e B(v), wi| =|he B(w), vi| ≤ ke B(w)k(D(A))∗kvkD(A) ≤eckwkkwkD(A)kvkD(A) ≤1 2kwk2 D(A)+ec2 2kwk2kvk2 D(A).(3.8) Combining (3.4) and (3.8), we obtain (3.3) as desired. In the following, we present the well-posedness of problem (3.1). For this end, we further assume Hypothesis I. For all u, v ∈L2(−∞, τ +T;D(A)) and t∈[τ, τ +T], Eq. (3.1) satisfies Zt τkeg2(s, us)−eg2(s, vs)k2 L2(K,V )ds + 2 Zt τ ((eg1(s, us)−eg1(s, vs), u(s)−v(s)))ds ≤σZt τkv(s)k2 D(A)ku(s)−v(s)k2ds + 2 Zt τhe A(u(s)−v(s)) + e B(u(s)) −e B(v(s)), u(s)−v(s)ids, (3.9) where σis given by (3.3) in Lemma 3.2. Remark 3.3. Let C2 eg2+2 e λ1 C2 eg1≤e λ1 2and u(s+τ) = v(s+τ) = φ(s), s ≤0.(3.10) Then (3.9) in hypothesis I is satisfied. Indeed, by (2.13) and Lemma 3.2, we only need to prove that the following inequality holds: Zt τkeg2(s, us)−eg2(s, vs)k2 L2(K,V )ds + 2 Zt τ ((eg1(s, us)−eg1(s, vs), u(s)−v(s)))ds ≤e λ1Zt τku(s)−v(s)k2ds. (3.11) The Young inequality, (H4) and (3.10) imply Zt τkeg2(s, us)−eg2(s, vs)k2 L2(K,V )ds + 2 Zt τ ((eg1(s, us)−eg1(s, vs), u(s)−v(s)))ds 8 ≤C2 eg2Zt −∞ ku(s)−v(s)k2ds +2 e λ1Zt τkeg1(s, us)−eg1(s, vs)k2ds +e λ1 2Zt τku(s)−v(s)k2ds ≤C2 eg2+2 e λ1 C2 eg1Zt −∞ ku(s)−v(s)k2ds +e λ1 2Zt τku(s)−v(s)k2ds =C2 eg2+2 e λ1 C2 eg1+e λ1 2Zt τku(s)−v(s)k2ds ≤e λ1Zt τku(s)−v(s)k2ds, (3.12) which implies (3.11) as desired. Theorem 3.4. Suppose that hypotheses F, G, I hold, moreover, φ∈L4(Ω, Cγ(V)) and e f∈I4(τ, τ +T;V), then there exists a unique solution uto (3.1), which satisfies in addition, u∈I4(τ, τ +T;V)∩L4(Ω, L∞(τ, τ +T;V)).(3.13) In fact, there exists a positive constant Rdepending on T,Ekφk4 Cγ(V)and ERτ+T τke f(s)k4dssuch that Esup τ≤r≤τ+Tkurk4 Cγ(V)+EZτ+T τku(s)k4ds≤R. (3.14) Proof. We split the proof into several steps as follows. Step 1:We use the Galerkin method to construct an approximating sequence. Consider the Hilbert basis {wj;j∈N} ⊂ D(A) of Vsuch that e Awj=e λjwj,∀j≥1, denote by Vmthe linear space spanned by {w1, w2, . . . , wm}for m∈N, and then put um(t) = m X j=1 am,j(t)wj,(3.15) where am,j(t) (j= 1,··· , m) will be obtained as the solution of the following finite dimensional system:                        ((um(t), wj)) + Zt τhe Aum(s), wjids +Zt τhe B(um(s)), wjids = ((um(τ), wj)) + Zt τe f(s) + eg1(s, um s), wjds +wj,Zt τeg2(s, um s)dW(s),∀t∈[τ, τ +T], j ∈[1, m], P-a.s. um(τ+s) = Pmφ(s),∀s∈(−∞,0], (3.16) where Pm:V→Vmis the projector. By the similar argument in [5], for each m∈N, the stochastic ODE (3.16) possesses a (local) solution {am,j(·)}m j=1 in [τ, tm) with τ < tm(by the initial condition, the value of am,j(·) in (−∞, τ] is well defined). From this, um(·) is well-defined in [τ, tm) (and thus in (−∞, tm)). Next, we will give a priori estimate to ensure that the solutions umis global, i.e. tm= +∞. Step 2:We give a-priori estimates for the approximating sequence. We first claim that, for any T > 0, the following inequality holds: Esup τ≤r≤τ+Tkum rk2 Cγ(V)+EZτ+T τkum(s)k2 D(A)ds≤R1,(3.17) where R1is a positive constant depending on T,Ekφk2 Cγ(V)and ERτ+T τke f(s)k2ds. Indeed, multiplying (3.16) by am,j, summing those relations for j= 1,··· , m and applying Ito’s formula to kum(t)k2, we obtain that kum(t)k2+ 2 Zt τkum(s)k2 D(A)ds =kum(τ)k2+ 2 Zt τe f(s) + eg1(s, um s), um(s)ds (3.18) 9 ≥ −EZt τ σe−σ%(s)ky(s)k2 D(A)ku(s)−y(s)k2ds+ 2EZt τ e−σ%(s)hι, u(s)−y(s)ids −2EZt τ e−σ%(s)h−e Ay(s)−e B(y(s)), u(s)−y(s)ids + 2EZt τ e−σ%(s)κ1−eg1(s, ys), u(s)−y(s)ds+EZt τ e−σ%(s)kκ2−eg2(s, ys)k2 L2(K,V )ds.(3.49) Taking y(t) = u(t) in (3.49), since e−σ%(t)is bounded with respect to t∈[τ, τ +T], we find that κ2=eg2(t, ut), t ∈ [τ, τ +T]. By (3.48), we derive Ym=Ee−σ%(s)kum(t)k2−Ekum(τ)k2−2EZt τ e−σ%(s)e f(s), um(s)ds,(3.50) which, together with (3.47), implies lim inf m→∞ Ym≥Ee−σ%(s)ku(s)k2−Ekφ(0)k2−2EZt τ e−σ%(s)e f(s), u(s)ds =−EZt τ σe−σ%(s)ky(s)k2 D(A)ku(s)k2ds+ 2EZt τ e−σ%(s)hι, u(s)ids + 2EZt τ e−σ%(s)κ1, u(s)ds+EZt τ e−σ%(s)keg2(s, us)k2 L2(K,V )ds.(3.51) Besides, lim inf m→∞ Zm≥ −EZt τ σe−σ%(s)ky(s)k2 D(A)ky(s)k2−2(u(s), y(s))ds+ 2EZt τ e−σ%(s)hι, −y(s)ids −2EZt τ e−σ%(s)h−e Ay(s)−e B(y(s)), u(s)−y(s)ids −2EZt τ e−σ%(s)eg1(s, ys), u(s)−y(s)ds+ 2EZt τ e−σ%(s)κ1,−y(s)ds +EZt τ e−σ%(s)eg2(s, ys)−2eg2(s, us),eg2(s, ys)L2(K,V )ds.(3.52) Therefore, by (3.49), (3.51) and (3.52), we have 0≥lim inf m→∞ Xm= lim inf m→∞ Ym+ lim inf m→∞ Zm ≥ −EZt τ σe−σ%(s)ky(s)k2 D(A)ku(s)−y(s)k2ds+ 2EZt τ e−σ%(s)hι, u(s)−y(s)ids + 2EZt τ e−σ%(s)κ1−eg1(s, ys), u(s)−y(s))ds−2EZt τ e−σ%(s)h−e Ay(s)−e B(y(s)), u(s)−y(s)ids +EZt τ e−σ%(s)keg2(s, us)−eg2(s, ys)k2 L2(K,V )ds.(3.53) We further obtain 0≤EZt τ e−σ%(s)keg2(s, us)−eg2(s, ys)k2 L2(K,V )ds ≤2EZt τ e−σ%(s)h−e Ay(s)−e B(y(s)), u(s)−y(s)ids + 2EZt τ e−σ%(s)eg1(s, ys)−κ1, u(s)−y(s)ds −2EZt τ e−σ%(s)hι, u(s)−y(s)ids+σEZt τ e−σ%(s)ky(s)k2 D(A)ku(s)−y(s)k2ds.(3.54) 16 Let y(t) = u(t)−ϑz(t) with for any z∈I2(τ, τ +T;D(A)) ∩I4(τ, τ +T;V) and ϑ∈[0,1], then 0≤2EZt τ e−σ%(s)h−e A(u−ϑz)−e B(u−ϑz), ϑzids + 2EZt τ e−σ%(s)eg1(s, us−ϑz)−κ1, ϑzds(3.55) −2EZt τ e−σ%(s)hι, ϑzids+σϑ2EZt τ e−σ%(s)ky(s)k2 D(A)kzk2ds. Dividing by ϑon both sides of (3.55), and then letting ϑ→0, we have EZt τ e−σ%(s)hι+e Au(s) + e B(u(s)), zids+EZt τ e−σ%(s)κ1−eg1(s, us), zds≤0.(3.56) Since I2(τ, τ +T;D(A)) ∩I4(τ, τ +T;V) is dense in I2(τ, τ +T;V), we find e−σ%(s)ι+e Au(s) + e B(u(s)) + κ1−eg1(s, us)= 0, a.e. t ∈[τ, τ +T], ω ∈Ω.(3.57) Note that κ2=eg2(t, ut), t ∈[τ, τ +T], we can rewrite (3.46) as u(t) + Zt τe Au(s)ds +Zt τe B(u(s))ds =φ(0) + Zt τ (e f(s) + eg1(s, us))ds +Zt τeg2(s, us)dW(s), a.e. t ∈[τ, τ +T], ω ∈Ω.(3.58) Therefore, the existence of a weak solution has been proved. Step 4:We derive the estimate (3.14). For each n∈Nand T > 0, we can define a stopping time τm nas follows. τm n= inf t≤τ+T:kum(t)k2+Zt τkum(s)k2 D(A)ds ≥n.(3.59) For fixed m, the sequence {τm n;n≥1}is increasing to τ+T. By (3.42) and (3.59), we obtain Esup r∈[τ,t∧τm n]kum rk4 Cγ(V)+EZt∧τm n τkum(s)k4ds≤R, ∀t∈[τ, τ +T].(3.60) Thanks to (3.60) and Fatou’s lemma, we deduce that (3.14) holds for every T > 0. Step 5:We prove the uniqueness of solutions to Eq. (3.1). Let u, v be two solutions of Eq. (3.1) with the same initial condition u(s) = v(s) = φ(s−τ), s ≤τ, and let w:= u−v. For every n∈Nand T > 0, we can define a stopping time Tnby Tn= inf nt≤τ+T:Zt τkv(s)k2 D(A)ds ≥no.(3.61) In addition, let ς(t) := e−σRt τkv(s)k2 D(A)ds, where σis given by (3.3) in Lemma 3.2 and E(Rt τkv(s)k2 D(A)ds) is finite due to the steps 2-3. Applying Ito’s formula to the process ς(t)kw(t)k2, we infer from (3.3) in Lemma 3.2 that ς(t∧Tn)kw(t∧Tn)k2=−σZt∧Tn τ ς(s)kv(s)k2 D(A)kw(s)k2ds + 2 Zt∧Tn τ ς(s)h−e Aw(s)−e B(u) + e B(v), w(s)ids + 2 Zt∧Tn τ ς(s)eg1(s, us)−eg1(s, vs),w(s)ds + 2 Zt∧Tn τ ς(s)w(s),eg2(s, us)−eg2(s, vs)dW(s) 17 +Zt∧Tn τ ς(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds ≤ −Zt∧Tn τ ς(s)kw(s)k2 D(A)ds + 2 Zt∧Tn τ ς(s)eg1(s, us)−eg1(s, vs), w(s)ds + 2 Zt∧Tn τ ς(s)w(s),eg2(s, us)−eg2(s, vs)dW(s) +Zt∧Tn τ ς(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds. (3.62) Taking supremum and expectation of (3.62), we find Esup τ≤r≤t ς(r∧Tn)kw(r∧Tn)k2+EZt τ ς(s∧Tn)kw(s∧Tn)k2 D(A)ds ≤2Esup τ≤r≤t∧TnZr τ ς(s)eg1(s, us)−eg1(s, vs), w(s)ds + 2Esup τ≤r≤t∧TnZr τ ς(s)w(s),eg2(s, us)−eg2(s, vs)dW(s) +Esup τ≤r≤t∧TnZr τ ς(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds.(3.63) The Young inequality and (H5) imply 2Esup τ≤r≤t∧TnZr τ ς(s)eg1(s, us)−eg1(s, vs), w(s)ds ≤EZt∧Tn τ ς(s)keg1(s, us)−eg1(s, vs)k2ds+EZt∧Tn τ ς(s)kw(s)k2ds ≤(e Ceg1+ 1)EZt τ ς(s∧Tn)kw(s∧Tn)k2ds ≤(e Ceg1+ 1)EZt τ sup τ≤θ≤s ς(θ∧Tn)kw(θ∧Tn)k2ds.(3.64) By the Burkholder-Davis-Gundy inequality and (H5), we have 2Esup τ≤r≤t∧TnZr τ ς(s)w(s),eg2(s, us)−eg2(s, vs)dW(s) ≤2c10E Zt∧Tn τς2(s)kw(s)k2keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds1 2! ≤2c10E sup τ≤s≤t ς1/2(s∧Tn)kw(s∧Tn)kZt∧Tn τ ς(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds1 2! ≤1 2E sup τ≤s≤t ς(s∧Tn)kw(s∧Tn)k2!+c11EZt∧Tn τ ς(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds ≤1 2E sup τ≤s≤t ς(s∧Tn)kw(s∧Tn)k2!+c11 e Ceg2Zt τ E sup τ≤θ≤s ς(θ∧Tn)kw(θ∧Tn)k2!ds, (3.65) where c11 = 2c2 10. By (H5), the last line of (3.63) is bounded by Esup τ≤r≤t∧TnZr τ ς(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds ≤E Zt∧Tn τ ς(s)keg2(s, us)−eg2(s, vs)k2 L2(K,V )ds! 18 ≤e Ceg2Zt τ E sup τ≤θ≤s ς(θ∧Tn)kw(θ∧Tn)k2!ds. (3.66) It follows from (3.63)-(3.66) that Esup τ≤r≤t ς(r∧Tn)kw(r∧Tn)k2≤c12 Zt τ Esup τ≤θ≤s ς(θ∧Tn)kw(θ∧Tn)k2ds, ∀t∈[τ, τ +T], where c12 = 2e Ceg1+1+c11 e Ceg2+e Ceg2.The Gronwall Lemma, together with 0 < ς ≤1, implies Esup τ≤r≤tkw(r∧Tn)k2= 0,∀t∈[τ, τ +T], and thus, u(r∧Tn) = v(r∧Tn),a.e., ω∈Ω. Furthermore, by Markov’s inequality, P(Tn< τ +T) = PZt τkv(s)k2 D(A)ds ≥n≤ ERt τkv(s)k2 D(A)ds n. We infer from ERt τkv(s)k2 D(A)ds<∞that Tn→τ+Tas n→ ∞. Therefore, u(r) = v(r), a.e., ω∈Ω for all r≤τ+T. The proof is concluded. 4. Stationary solutions and their stability results In this section, we are concerned with existence, uniqueness and stability properties of the stationary solutions to (1.1). For this end, we need to assume that e f(t)≡e f∈(D(A))∗(i.e. f(t)≡f∈H−1(O)), which is independent of the time. 4.1. Existence and uniqueness of stationary solutions We now consider the abstract equation associated to Eq. (1.1):    du dt +e Au(t) + e B(u(t)) = e f+eg1(t, ut) + eg2(t, ut)dW dt ,∀t > 0, u(t) = φ(t), t ∈(−∞,0]. (4.1) We denote by u(t) := u(t;φ) the solution of (1.1) with τ= 0, where φ=u0. By a stationary solution to (4.1), we mean a constant solution (in other words, an equilibrium point) of (4.1). Therefore, u∞∈D(A) will be a stationary solution if formally e Au∞+e B(u∞) = e f+eg1(t, u∞) + eg2(t, u∞)dW dt ,∀t≥0.(4.2) However, this equation depends on tand a noisy term. Therefore, we would need to assume that eg1and eg2 would not depend on t, moreover, to get rid of the noise, we must assume that eg2(t, u∞) = 0. Consequently, we will focus on the existence of stationary solutions for the deterministic equation (i.e. eg2= 0 in (4.1)) which will be any u∞∈D(A) such that e Au∞+e B(u∞) = e f+eg1(t, u∞),∀t≥0,(4.3) and then analyze the behavior of the solutions to (4.1) around these stationary solutions of (4.3). Now, in order to study the existence of solutions to (4.3), we have to restrict ourselves to assume that for constant elements ξ∈Cγ(V), egi(t, ξ) (i= 1,2) can be rewritten as egi(t, ξ) = e Gi(ξ∗) if ξ(s) = ξ∗,∀s≤0,(4.4) 19 where e G1:V→Vwith e G1(0) = 0, e G2:V→ L2(K, V ) with e G2(0) = 0, they are Lipschitz continuous, that is, there exist Le Gi>0 (i= 1,2), for all η, ζ ∈V, ke G1(η)−e G1(ζ)k ≤ Le G1kη−ζk,(4.5) ke G2(η)−e G2(ζ)kL2(K,V )≤Le G2kη−ζk.(4.6) For example, if egi(i= 1,2) are driven by unbounded variable delay, defined by egi(t, ξ) = e Gi(ξ(−h(t))), i = 1,2,(4.7) with e Gisatisfying conditions (4.4)-(4.6), where h∈C1([0,+∞)), h(t)≥0 and h∗= supt≥0h0(t)<1. In this case, the delay terms egi(i= 1,2) in our problem become egi(t, ut) = e Gi(u(t−h(t))). Another example is the case of infinite distributed delay, that is, the delay terms egi(i= 1,2) are defined by egi(t, ξ) = Z0 −∞ e Hi(s, ξ(s))ds, (4.8) where e H1: (−∞,0] ×V→Vwith e H1(s, 0) = 0, and e H2: (−∞,0] ×V→ L2(K, V ) with e H2(s, 0) = 0 are measurable, and they are Lipschitz continuous with respect to their second variable, that is, there exist Le Hi(s)∈ L2(−∞,0) (i= 1,2) with Le Hi(·)e−(γ+θ)·∈L2(−∞,0), for certain θ > 0, such that for all s∈(−∞,0], η, ζ ∈V, ke H1(s, η)−e H1(s, ζ)k ≤ Le H1(s)kη−ζk,(4.9) ke H2(s, η)−e H2(s, ζ)kL2(K,V )≤Le H2(s)kη−ζk.(4.10) In this case, we can rewrite the delay terms egi(i= 1,2) in our problem as egi(t, ut) = R0 −∞ e Hi(s, u(t+s))ds (i= 1,2). The above both situations are within our framework, the conditions (H1)-(H5) are fulfilled for the infinite distributed delay in Cγ(V) for γ > 0, but not necessarily for the unbounded variable delay. However, conditions (H1)-(H5) are satisfied for both delays in C−∞(V). Now, we are interested in studying the existence and uniqueness of a stationary solution to Eq. (4.3). Theorem 4.1. Assume that the above assumptions and notations hold. If e λ1> Leg1, then: (a) For all e f∈(D(A))∗, then there exists at least one stationary solution to (4.3), which belongs to D(A)if e f∈V; (b) If (1 −e λ−1 1Leg1)2>ece λ−1 1ke fk, then the stationary solution to (4.3) is unique. Proof. We can prove this result by using the same method as in [14, Theorem 10] (or [13, Lemma 3.2]), which is based on the Lax-Milgram, the Schauder theorems. Therefore, we omit the details. 4.2. Local stability of stationary solutions In this subsection, we will prove the local stability of stationary solutions to (4.3) for general delay terms by using a direct method and then apply the abstract results to two specific situations. Theorem 4.2. Suppose that the same hypotheses and notations in Theorem 3.4 and Theorem 4.1 hold. In addition, let 2e λ1≥2ecke fk 1−e λ−1 1Leg1 + 2Ceg1+C2 eg2.(4.11) If u(·)is any solution of Eq. (4.1),u∞is the unique stationary solution of Eq. (4.3) and w(t) = u(t)−u∞, then Ekw(t)k2≤Ekw(0)k2+ (Ceg1+C2 eg2)Z0 −∞ Ekφ(s)−u∞k2ds. (4.12) 20 Proof. Applying Ito’s formula to kw(t)k2, we obtain kw(t)k2=kw(0)k2+ 2 Zt 0h−e Aw(s)−e B(u(s)) + e B(u∞), w(s)ids + 2 Zt 0eg1(s, us)−eg1(s, u∞), w(s)ds + 2 Zt 0eg2(s, us)−eg2(s, u∞), w(s)dW (s) +Zt 0keg2(s, us)−eg2(s, u∞)k2 L2(K,V )ds. (4.13) Taking expectation of (4.13), thanks to Fubini’s theorem, Ekw(t)k2+ 2 Zt 0 Ekw(s)k2 D(A)ds =Ekw(0)k2−2Zt 0 Ehe B(u(s)) −e B(u∞), w(s)ids + 2EZt 0eg1(s, us)−eg1(s, u∞), w(s)ds +Zt 0 Ekeg2(s, us)−eg2(s, u∞)k2 L2(K,V )ds. (4.14) By (2.13), (B2) and (3.7), we deduce −2Zt 0 Ehe B(u(s)) −e B(u∞), w(s)ids =−2Zt 0 Ehe B(u∞, w(s)), w(s)ids ≤2ecZt 0 E(ku∞kkw(s)k2 D(A))ds ≤2ece λ−1 2 1Zt 0 E(ku∞kD(A)kw(s)k2 D(A))ds. (4.15) By (2.13), (2.26) and (4.3), we find ku∞k2 D(A)=he Au∞, u∞i = (( e f, u∞)) + ((eg1(t, u∞), u∞)) ≤e λ−1 2 1ke fkku∞kD(A)+e λ−1 1Leg1ku∞k2 D(A),(4.16) which, together with e λ1> Leg1, implies that ku∞kD(A)≤e λ−1 2 1ke fk 1−e λ−1 1Leg1 .(4.17) Thanks to (4.16)-(4.17), we can rewrite (4.15) as −2Zt 0 Ehe B(u(s)) −e B(u∞), w(s)ids ≤2ece λ−1 1ke fk 1−e λ−1 1Leg1Zt 0 E(kw(s)k2 D(A))ds. (4.18) We now estimate the last two terms of (4.14) respectively. On the one hand, by (H4), (2.13) and the Young inequality, with 0>0 to be specified later on, we deduce 2EZt 0eg1(s, us)−eg1(s, u∞), w(s)ds ≤2e λ−1 2 1Zt 0 Ekeg1(s, us)−eg1(s, u∞)kkw(s)kD(A)ds ≤1 0Zt 0 Ekw(s)k2 D(A)ds +0e λ−1 1C2 eg1Zt −∞ Ekw(s)k2ds ≤1 0Zt 0 Ekw(s)k2 D(A)ds +0e λ−1 1C2 eg1Z0 −∞ Ekφ(s)−u∞k2ds 21 +e λ−1 1Zt 0 Ekw(s)k2 D(A)ds.(4.19) On the other hand, by (H4) and (2.13), we find Zt 0 Ekeg2(s, us)−eg2(s, u∞)k2 L2(K,V )ds ≤C2 eg2Z0 −∞ Ekφ(s)−u∞k2ds +e λ−1 1Zt 0 Ekw(s)k2 D(A)ds.(4.20) It follows from the above inequalities that Ekw(t)k2≤Ekw(0)k2+2ece λ−1 1ke fk 1−e λ−1 1Leg1 +1 0 +0e λ−2 1C2 eg1+e λ−1 1C2 eg2−2× Zt 0 Ekw(s)k2 D(A)ds+ (0e λ−1 1C2 eg1+C2 eg2)Z0 −∞ Ekφ(s)−u∞k2ds ≤Ekw(0)k2+e λ−1 12ecke fk 1−e λ−1 1Leg1 +e λ1 0 +0e λ−1 1C2 eg1+C2 eg2−2e λ1× Zt 0 Ekw(s)k2 D(A)ds+ (0e λ−1 1C2 eg1+C2 eg2)Z0 −∞ Ekφ(s)−u∞k2ds. (4.21) In order to minimize the right-hand side of (4.21), we choose 0=e λ1C−1 eg1such that e λ1 0+0e λ−1 1C2 eg1achieves its minimum value 2Ceg1. Then, by (4.11), we have (4.12) as desired. In what follows, we will discuss the local stability of stationary solutions to (4.3) when the delay terms have particular forms in C−∞(V), and establish some sufficient conditions in the next corollaries. In this way, it is much easier for us to check the conditions than (4.11) in practical application. Corollary 4.3. Under the same hypotheses and notations in Theorem 3.4 and Theorem 4.1, let the delay terms egi(t, ut) = e Gi(u(t−h(t)))(i= 1,2) satisfy (4.5)-(4.7), moreover, 2e λ1≥2ecke fk 1−e λ−1 1Leg1 +2(1 −h∗)1 2Le G1+L2 e G2 1−h∗(4.22) is satisfied. If u(·)is any solution of Eq. (4.1),u∞is the unique stationary solution of Eq. (4.3) and w(t) = u(t)−u∞, then Ekw(t)k2≤Ekw(0)k2+(1 −h∗)1 2Le G1+L2 e G2 1−h∗Z0 −∞ Ekφ(s)−u∞k2ds. (4.23) Proof. Taking eh=s−h(s), we obtain ds = 1/(1 −h0(s))deh≤1/(1 −h∗)deh. Then, by (4.5), it follows Zt 0keg1(s, us)−eg1(s, vs)k2ds =Zt 0ke G1(u(s−h(s))) −e G1(v(s−h(s)))k2ds ≤L2 e G1Zt 0ku(s−h(s)) −v(s−h(s))k2ds ≤L2 e G1 1−h∗Zt −∞ ku(s)−v(s)k2ds, (4.24) Let Ceg1:= Le G1 √1−h∗, we deduce that, there exists Ceg1>0 such that the first inequality in (H4) holds. Similarly, there exists Ceg2=Le G2 √1−h∗>0, such that the last one in (H4) holds. Thanks to (4.22), we find 2e λ1≥2ecke fk 1−e λ−1 1Leg1 +2(1 −h∗)1 2Le G1+L2 e G2 1−h∗ 22 ≥2ecke fk 1−e λ−1 1Leg1 + 2Ceg1+C2 eg2,(4.25) which implies (4.11). Therefore, by Theorem 4.2, we obtain (4.12) as desired, and thus, Ekw(t)k2≤Ekw(0)k2+ (Ceg1+C2 eg2)Z0 −∞ Ekφ(s)−u∞k2ds ≤Ekw(0)k2+(1 −h∗)1 2Le G1+L2 e G2 1−h∗Z0 −∞ Ekφ(s)−u∞k2ds. (4.26) The proof is concluded. Corollary 4.4. Assume that the same hypotheses and notations in Theorem 3.4 and Theorem 4.1 hold. Let the delay terms egi(t, ut) = R0 −∞ e Hi(s, u(t+s))ds (i= 1,2) satisfy (4.8)-(4.10), moreover, 2e λ1≥2ecke fk 1−e λ−1 1Leg1 + 2kLe H1kL2(−∞,0) +kLe H2k2 L2(−∞,0) (4.27) holds. If u(·)is any solution of Eq. (4.1),u∞is the unique stationary solution of Eq. (4.3) and w(t) = u(t)−u∞, then Ekw(t)k2≤Ekw(0)k2+kLe H1kL2(−∞,0) +kLe H2k2 L2(−∞,0)Z0 −∞ Ekφ(s)−u∞k2ds. (4.28) Proof. The proof is similar to the one of Corollary 4.3. It follows from (4.9) and (4.10) that there exist Cegi=kLe HikL2(−∞,0) >0 (i= 1,2) such that (H4) hold, and then by (4.27), we obtain (4.11). By Theorem 4.2, we deduce (4.28) as desired. Remark 4.5. In the case of infinite distributed delay, we can prove not only stability of stationary solutions in C−∞(V) (see Corollary 4.4) even in Cγ(V), but also their exponential asymptotic stability will be established as follows. 4.3. Exponential convergence of stationary solutions Under suitable assumptions, we prove that the solution u(t) to problem (4.1) with infinite distributed delay converges exponentially to the unique stationary solution u∞of Eq. (4.3) in Cγ(V) for γ > 0. Theorem 4.6. Assume that the same hypotheses and notations in Theorem 3.4 and Theorem 4.1 hold. Let the delay terms egi(t, ut) = R0 −∞ e Hi(s, u(t+s))ds (i= 1,2) satisfy (4.9)-(4.10), and moreover, there exists a constant 0<ρ<2γsuch that for all t≥0, 2e λ1≥2ecke fk 1−e λ−1 1Leg1 + 2(2ρ)−1 2kLe H1(·)e−(γ+ρ)·kL2(−∞,0) +1 2ρkLe H2(·)e−(γ+ρ)·k2 L2(−∞,0) +ρ(4.29) is satisfied. If u(·)is any solution of Eq. (4.1),u∞is the unique stationary solution of Eq. (4.3) and w(t) = u(t)−u∞, then Ekw(t)k2≤e−ρt1 + 1 2ρ(2γ−ρ)(2ρ)1 2kLe H1(·)e−(γ+ρ)·kL2(−∞,0) +kLe H2(·)e−(γ+ρ)·k2 L2(−∞,0)Ekφ−u∞k2 Cγ(V),(4.30) and Ekwtk2 Cγ(V)≤e−ρt2 + 1 2ρ(2γ−ρ)(2ρ)1 2kLe H1(·)e−(γ+ρ)·kL2(−∞,0) +kLe H2(·)e−(γ+ρ)·k2 L2(−∞,0)Ekφ−u∞k2 Cγ(V).(4.31) 23 Proof. Applying Ito’s formula to eρtkw(t)k2with 0 <ρ<2γ, we find, for all t≥0, eρtkw(t)k2=kw(0)k2+ρZt 0 eρskw(s)k2ds + 2 Zt 0 eρsh−e A(w(s)) −e B(u(s)) + e B(u∞), w(s)ids + 2 Zt 0 eρsZ0 −∞ e H1(r, u(s+r)) −e H1(r, u∞)dr, w(s)ds +Zt 0 eρsZ0 −∞ e H2(r, u(s+r)) −e H2(r, u∞)dr 2 L2(K,V )ds + 2 Zt 0 eρsw(s),Z0 −∞ e H2(r, u(s+r)) −e H2(r, u∞)drdW.(4.32) Taking expectation of (4.32), then using (2.13), we obtain Eeρtkw(t)k2+ 2 Zt 0 Eeρskw(s)k2 D(A)ds ≤Ekφ−u∞k2 Cγ(V)+ρe λ−1 1Zt 0 Eeρskw(s)k2 D(A)ds −2Zt 0 Eeρshe B(u(s)) −e B(u∞), w(s)ids + 2EZt 0 eρsZ0 −∞ e H1(r, u(s+r)) −e H1(r, u∞)dr, w(s)ds +EZt 0 eρsZ0 −∞ e H2(r, u(s+r)) −e H2(r, u∞)dr 2 L2(K,V )ds.(4.33) Thanks to (4.18), we deduce −2Zt 0 Eeρshe B(u(s)) −e B(u∞), w(s)ids ≤2ece λ−1 1ke fk 1−e λ−1 1Leg1Zt 0 Eeρskw(s)k2 D(A)ds. (4.34) By (2.13), (4.9) and the Young inequality with ˆ > 0 to be specified later on, the fourth line of (4.33) is bounded by 2EZt 0 eρsZ0 −∞ e H1(r, u(s+r)) −e H1(r, u∞)dr, w(s)ds ≤2e λ−1 2 1EZt 0 eρsZ0 −∞ Le H1(r)kw(s+r)kdr·kw(s)kD(A)ds ≤ˆe λ−1 1EZt 0 eρsZ0 −∞ Le H1(r)kw(s+r)kdr2ds+1 ˆEZt 0 eρskw(s)k2 D(A)ds =: ˆe λ−1 1I+1 ˆEZt 0 eρskw(s)k2 D(A)ds,(4.35) where Iis estimated as follows. By the H¨older inequality, I=EZt 0 eρsZ0 −∞ Le H1(r)kw(s+r)kdr2ds ≤EZt 0 eρsZ0 −∞ Le H1(r)e−γrkwskCγ(V)dr2ds =EZt 0 eρskwsk2 Cγ(V)Z0 −∞ Le H1(r)e−(γ+ρ)reρrdr2ds ≤EZt 0 eρskwsk2 Cγ(V)Z0 −∞ L2 e H1(r)e−2(γ+ρ)rdr Z0 −∞ e2ρrdrds 24 ≤1 2ρkLe H1(·)e−(γ+ρ)·k2 L2(−∞,0)EZt 0 eρskwsk2 Cγ(V)ds ≤1 2ρkLe H1(·)e−(γ+ρ)·k2 L2(−∞,0)× E Zt 0 eρs max sup θ≤−s e2γθkw(s+θ)k2,sup θ∈[−s,0] e2γθkw(s+θ)k2ds! ≤1 2ρkLe H1(·)e−(γ+ρ)·k2 L2(−∞,0)E Zt 0e−(2γ−ρ)skφ−u∞k2 Cγ(V) +e λ−1 1sup θ∈[−s,0] e(2γ−ρ)θeρ(s+θ)kw(s+θ)k2 D(A)ds!.(4.36) Thanks to (4.10), we obtain the following result by using the same method in (4.36): EZt 0 eρsZ0 −∞ e H2(r, u(s+r)) −e H2(r, u∞)dr 2 L2(K,V )ds ≤EZt 0 eρsZ0 −∞ Le H2(r)kw(s+r)kdr2ds ≤EZt 0 eρsZ0 −∞ Le H2(r)e−γrkwskCγ(V)dr2ds ≤1 2ρkLe H2(·)e−(γ+ρ)·k2 L2(−∞,0)E Zt 0e−(2γ−ρ)skφ−u∞k2 Cγ(V) +e λ−1 1sup θ∈[−s,0] e(2γ−ρ)θeρ(s+θ)kw(s+θ)k2 D(A)ds!.(4.37) Substituting (4.34)-(4.37) into (4.33), then by 0 <ρ<2γ, we have Eeρtkw(t)k2≤Ekφ−u∞k2 Cγ(V)+e λ−1 12ecke fk 1−e λ−1 1Leg1 +ˆ 2ρe λ1kLe H1(·)e−(γ+ρ)·k2 L2(−∞,0) +e λ1 ˆ +1 2ρkLe H2(·)e−(γ+ρ)·k2 L2(−∞,0) +ρ−2e λ1Zt 0 Emax r∈[0,s]{eρrkw(r)k2 D(A)}ds +1 2ρˆe λ−1 1kLe H1(·)e−(γ+ρ)·k2 L2(−∞,0) +kLe H2(·)e−(γ+ρ)·k2 L2(−∞,0)× Ekφ−u∞k2 Cγ(V)Zt 0 e−(2γ−ρ)sds. (4.38) Notice that min ˆ>0nˆ 2ρe λ1kLe H1(·)e−(γ+ρ)·k2 L2(−∞,0) +e λ1 ˆo= 2(2ρ)−1 2kLe H1(·)e−(γ+ρ)·kL2(−∞,0),(4.39) which is achieved by ˆ= (2ρ)1 2e λ1kLe H1(·)e−(γ+ρ)·k−1 L2(−∞,0). Then, we infer from (4.38), (4.39) and (4.29) that (4.30) holds. By (4.30), and by 0 <ρ<2γ, and thus e(2γ−ρ)θ≤1 when θ≤0, we find, for all t≥0, Ekwtk2 Cγ(V)=Esup θ≤0 e2γθkw(t+θ)k2 =Emax nsup θ∈(−∞,−t] e2γθkφ(t+θ)−u∞k2,sup θ∈[−t,0] e2γθkw(t+θ)k2o =Emax ne−2γtkφ−u∞k2 Cγ(V),sup θ∈[−t,0] e2γθkw(t+θ)k2o ≤max e−ρtEkφ−u∞k2 Cγ(V), e−ρt1 + 1 2ρ(2γ−ρ)(2ρ)1 2kLe H1(·)e−(γ+ρ)·kL2(−∞,0) 25 [14] T. Caraballo, A. M. M´arquez-Dur´an, J. Real, Asymptotic behaviour of the three-dimensional α-Navier- Stokes model with delays, J. Math. Anal. Appl. 340 (2008), 410–423. [15] T. Caraballo, J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), 2441–2453. [16] T. Caraballo, J. Real, T. 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