We examined the external ability of the loan loss provision (LLP) model to detect extreme cases of earnings management (EM). According to the literature, the LLP model is the most useful in examining EM in banking institutions. We used it herein to explore the time-series behaviour of a fraudulent business group in the Portuguese banking sector between 1992 and 2013 − the Banco Espírito Santo Group (GBES). We conclude that GBES did not make discretionary use of LLP (i.e., DLLP) in the fraud period (2008 to 2013) when compared with the pre-fraud years (1992 to 2007). However, the level of LLP was significantly higher in the latter period; this was consistent with the procyclical nature of GBES's LLP. The results of a difference-in-difference approach did not reveal any significant differences between GBES's DLLP and non-fraudulent banks in the fraud period. Interestingly, the full bank sample (including GBES) provided evidence of the procyclical nature of LLP. Additional tests did not support the hypothesis of income smoothing via LLP, either amongst the bank sample as a whole or by GBES. The proven facts of the fraud indicated a significant undervaluation of loans and financial instruments and an underestimation of LLP. Thus, we expected to find negative DLLP in the fraud period and significantly different DLLP between the pre-fraud period and the fraud period itself. The DLLP of GBES should also have been significantly different from non-fraudulent banks in the fraud period. The LLP model proved ineffective in detecting GBES fraud and assessing the decisions of the bank's leader and his team, while the use of DLLP was effective. The evidence collected in our study will be of benefit to scholars and banking regulators.
Citation: Tânia Menezes Montenegro, Filomena Antunes Brás. Scandal in the Portuguese banking sector – how a banking specific earnings management model predicted the fall of a family business group[J]. Green Finance, 2022, 4(3): 364-386. doi: 10.3934/GF.2022018
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We examined the external ability of the loan loss provision (LLP) model to detect extreme cases of earnings management (EM). According to the literature, the LLP model is the most useful in examining EM in banking institutions. We used it herein to explore the time-series behaviour of a fraudulent business group in the Portuguese banking sector between 1992 and 2013 − the Banco Espírito Santo Group (GBES). We conclude that GBES did not make discretionary use of LLP (i.e., DLLP) in the fraud period (2008 to 2013) when compared with the pre-fraud years (1992 to 2007). However, the level of LLP was significantly higher in the latter period; this was consistent with the procyclical nature of GBES's LLP. The results of a difference-in-difference approach did not reveal any significant differences between GBES's DLLP and non-fraudulent banks in the fraud period. Interestingly, the full bank sample (including GBES) provided evidence of the procyclical nature of LLP. Additional tests did not support the hypothesis of income smoothing via LLP, either amongst the bank sample as a whole or by GBES. The proven facts of the fraud indicated a significant undervaluation of loans and financial instruments and an underestimation of LLP. Thus, we expected to find negative DLLP in the fraud period and significantly different DLLP between the pre-fraud period and the fraud period itself. The DLLP of GBES should also have been significantly different from non-fraudulent banks in the fraud period. The LLP model proved ineffective in detecting GBES fraud and assessing the decisions of the bank's leader and his team, while the use of DLLP was effective. The evidence collected in our study will be of benefit to scholars and banking regulators.
The nonlinear Ginzburg-Landau equation plays an important role in the studies of physics, which describes many interesting phenomena and has been studied extensively (see [1] for a more detailed description). The fractional Ginzburg-Landau equation [2,3,4] is employed to describe processes in media with fractional dispersion or long-range interaction. It becomes very popular because the fractional derivative and fractional integral have broad applications in different fields of science [5,6,7,8,9,10].
Our work focuses on the existence of invariant measures of the autonomous fractional stochastic delay Ginzburg-Landau equations on Rn:
du(t)+(1+iν)(−Δ)αu(t)dt+(1+iμ)|u(t)|2βu(t)dt+λu(t)dt=G(x,u(t−ρ))dt,+∞∑k=1(σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t), t>0, | (1.1) |
with initial condition
u(s)=φ(s),s∈[−ρ,0], | (1.2) |
where u(x,t) is a complex-valued function on Rn×[0,+∞). In (1.1), i is the imaginary unit, α,β,μ,ν and λ are real constants with β>0,λ>0 and ρ>0. (−Δ)α with 0<α<1 is the fractional Laplace operator, σ1,k(x)∈L2(Rn) and σ2,k(u):C→R are nonlinear functions, κ(x)∈L2(Rn)⋂L∞(Rn) and {Wk}∞k=1 is a sequence of independent standard real-valued Wiener process on a complete filtered probability space (Ω, F, {Ft}t∈R,P), where {Ft}t∈R is an increasing right continuous family of sub-σ-algebras of F that contains all P-null sets.
The Ginzburg-Landau equation with fractional derivative was first introduced in [2]. There is a large amount of literature which was used for investigating fractional deterministic Ginzburg-Landau equations such as [1] and stochastic equations such as [11,12,13,14,15,16,17]. These papers had respectively researched the long-time deterministic as well as random dynamical systems of fractional equations with autonomous forms and non-autonomous forms. However, in spite of quite a lot of contribution of the works, no result is provided for the existence of pathwise pullback random attractors and invariant measures for the delay stochastic Ginzburg-Landau equations.
The delay differential equations [18] was described the dynamical systems that rely on current and past historical states. For the past few years, researchers had made great progress in the study of linear and nonlinear delay differential equations, see [20,21]. Delay differential equations are widely used in many fields, so investigating the solutions of equations has profound significance. Therefore, it's necessary that we establish the dynamics of delay stochastic Ginzburg-Landau equations.
The goal of this paper is to prove the existence of invariant measures of the stochastic Eqs (1.1) and (1.2) in L2(Ω;C([−ρ,0],L2(Rn))) by applying Krylov-Bogolyubov's method. The main difficulty of this paper is that deducing the uniform estimates of solutions (because of the nonlinear term (1+iμ)|u(t)|2βu(t) and complex-valued solutions), proving the weak compactness of a set distribution laws of the segments of solutions in L2(Ω;C([−ρ,0],L2(Rn))) (because the standard Sobolev embeddings are not compact on unbounded domains Rn), and establishing the equicontinuity of solutions in L2(Ω;C([−ρ,0],L2(Rn))) (because the uniform estimates in L2(Ω;C([−ρ,0],L2(Rn))) are not sufficient, and the uniform estimates in L2(Ω;C([−ρ,0],H1(Rn))) are needed).
For the estimates of the nonlinear term (1+iμ)|u(t)|2βu(t), we apply integrating by parts and nonnegative definite quadratic form. There are Several methods to handle the noncompact on unbounded domain, including weighted spaces [22,23,24], weak Feller approach [25,26] and uniform tail-estimates [23,27]. We first obtain the uniform estimates of the tail of the solution as well as the technique of dyadic division, then establish the weak compactness of a set of probability distribution of solutions in C([−ρ,0],L2(Rn)) applying the Ascoli-Arzelˊa theorem.
Let S be the Schwartz space of rapidly decaying C∞ functions on Rn. The fractional Laplace operator (−Δ)α for 0<α<1 is defined by, for u∈S,
(−Δ)αu(x)=−12C(n,α)∫Rnu(x+y)+u(x−y)−2u(x)|y|n+2αdy, x∈Rn, |
where C(n,α) is a positive constant given by
C(n,α)=α4αΓ(n+2α2)πn2Γ(1−α). |
By [28], the inner product ((−Δ)α2u,(−Δ)α2v) in the complex field is defined by
((−Δ)α2u,(−Δ)α2v)=C(n,α)2∫Rn∫Rn(u(x)−u(y))(ˉv(x)−ˉv(y))|x−y|n+2αdxdy, |
for u∈Hα(Rn). The fractional Sobolev space Hα(Rn) is endowed with the norm
‖u‖2Hα(Rn)=‖u‖2L2(Rn)+2C(n,α)‖(−Δ)α2u‖2L2(Rn). |
About the fractional derivative of fractional Ginzburg-Landau equations, there is another statement in [29].
We organize the article as follows. In Section 2, we establish the well-posedness of (1.1) and (1.2) in L2(Ω;C([−ρ,0],H)). In Sections 3 and 4, we derive the uniform estimates of solutions in L2(Ω;C([−ρ,0],H)) and L2(Ω;C([−ρ,0],V)), respectively. In Section 5, the existence of invariant measures is obtained.
In this section, we show the nonlinear drift term and the diffusion term in (1.1) which are needed for the well-posedness of the stochastic delay Ginzburg-Landau Eqs (1.1) and (1.2) defined on Rn.
We assume that G:Rn×C→C is continuous and satisfies
|G(x,u)|≤|h(x)|+a|u|, ∀x∈Rn, u∈C | (2.1) |
and
|∇G(x,u)|≤|ˆh(x)|+ˆa|∇u|, ∀x∈Rn, u∈C, | (2.2) |
where a and ˆa>0 are constants and h(x),ˆh(x)∈L2(Rn). Moreover, G(x,u) is Lipschitz continuous in u∈C uniformly with respect to x∈Rn. More precisely, there exists a constant CG>0 such that
|G(x,u1)−G(x,u2)|≤CG|u1−u2|, ∀x∈Rn, u1,u2∈C. | (2.3) |
For the diffusion coefficients of noise, we suppose that for each k∈N+
∞∑k=1‖σ1,k‖2<∞, | (2.4) |
and that σ2,k(u):C→R is globally Lipschitz continuous; namely, for every k∈N+, there exists a positive number αk such that for all s1,s2∈C,
|σ2,k(s1)−σ2,k(s2)|≤αk|s1−s2|. | (2.5) |
We further assume that for each k∈N+, there exist positive numbers βk, ˆβk, γk and ˆγk such that
|σ2,k(s)|≤βk+γk|s|, ∀s∈C, | (2.6) |
and
|∇σ2,k(s)|≤ˆβk+ˆγk|∇s|, ∀s∈C, | (2.7) |
where ∞∑k=1(α2k+β2k+γ2k+ˆβ2k+ˆγ2k)<+∞. In this paper, we deal with the stochastic Eqs (1.1) and (1.2) in the space C([−ρ,0],L2(Rn)). In the following discussion, we denote by H=L2(Rn), V=H1(Rn).
A solution of problems (1.1) and (1.2) will be understood in the following sense.
Definition 2.1. We suppose that φ(s)∈L2(Ω,C([−ρ,0],H)) is F0-measurable. Then, a continuous H-valued Ft-adapted stochastic process u(x,t) is named a solution of problems (1.1) and (1.2), if
1) u is pathwise continuous on [0,+∞), and Ft-adapted for all t≥0,
u∈L2(Ω,C([0,T],H))⋂L2(Ω,L2([0,T],V)) |
for all T>0,
2) u(s)=φ(s) for −ρ≤s≤0,
3) For all t≥0 and ξ∈V,
(u(t),ξ)+(1+iν)∫t0((−Δ)α2u(s),(−Δ)α2ξ)ds+∫t0∫Rn(1+iμ)|u(s)|2βu(s)ξ(x)dxds+λ∫t0(u(s),ξ)ds=(φ(0),ξ)+∫t0(G(s,u(s−ρ)),ξ)ds+∞∑k=1∫t0(σ1,k(x)+κ(x)σ2,k(u(s)),ξ)dWk(s), | (2.8) |
for almost all ω∈Ω.
By the Galerkin method and the argument of Theorem 3.1 in [30], one can verify that if (2.1)–(2.7) hold true, then, for every F0-measurable function φ(s)∈L2(Ω,C([−ρ,0],H)), the problems (1.1) and (1.2) has a unique solution u(x,t) in the sense of Definition 2.1.
Now, we establish the Lipschitz continuity of the solutions of the problems (1.1) and (1.2) with respect to the initial data in L2(Ω,C([−ρ,0],H)).
Theorem 2.2. Suppose (2.1)–(2.6) hold, and F0-measurable function φ1,φ2∈L2(Ω,C([−ρ,0],H)). If u1=u(t,φ1) and u2=u(t,φ2) are the solutions of the problems (1.1) and (1.2) with initial data φ1 and φ2, respectively, then, for any t≥0,
E[sup−ρ≤s≤t‖u(s,φ1)−u(s,φ2)‖2]+E[∫t0‖u(s,φ1)−u(s,φ2)‖2Vds] |
≤C1e˜C1tE[sup−ρ≤s≤0‖φ1(s)−φ2(s)]‖2], |
where C1 and ˜C1 are positive constants independent of φ1 and φ2.
Proof. Since both u1 and u2 are the solutions of the problems (1.1) and (1.2), we have, for all t≥0,
u1−u2+(1+iν)∫t0(−Δ)α(u1−u2)ds+(1+iμ)∫t0(|u1|2βu1−|u2|2βu2)ds+λ∫t0(u1−u2)ds=φ1(0)−φ2(0)+∫t0(G(x,u1(s−ρ))−G(x,u2(s−ρ)))ds+∞∑k=1∫t0κ(x)(σ2,k(u1)−σ2,k(u2))dWk. | (2.9) |
By (2.9), the integration by parts of Ito's formula and taking the real parts, we get, for all t≥0,
‖u1−u2‖2+2∫t0‖(−Δ)α2(u1−u2)‖2ds+2Re∫t0∫Rn(ˉu1−ˉu2)[|u1|2βu1−|u2|2βu2]dxds+2λ∫t0‖u1−u2‖2ds=‖φ1(0)−φ2(0)‖2+2Re∫t0(u1−u2,G(x,u1(s−ρ))−G(x,u2(s−ρ)))ds+∞∑k=1∫t0‖κ(x)(σ2,k(u1)−σ2,k(u2))‖2ds+2Re∫t0(u1−u2,∞∑k=1κ(x)(σ2,k(u1)−σ2,k(u2)))dWk(s). | (2.10) |
For the third term in the first row of (2.10), one has
2Re∫t0∫Rn(ˉu1−ˉu2)[|u1|2βu1−|u2|2βu2]dxds=∫t0∫Rn2|u1|2β+2+2|u2|2β+2−2Re(u1ˉu2)(|u1|2β+|u2|2β)dxds≥∫t0∫Rn2|u1|2β+2+2|u2|2β+2−2|u1||u2|(|u1|2β+|u2|2β)dxds≥∫t0∫Rn2|u1|2β+2+2|u2|2β+2−(|u1|2+|u2|2)(|u1|2β+|u2|2β)dxds=∫t0∫Rn|u1|2β+2+|u2|2β+2−|u1|2β|u2|2−|u2|2β|u1|2dxds=∫t0∫Rn(|u1|2β−|u2|2β)(|u1|2−|u2|2)dxds≥0. |
By (2.10), we deduce that for t≥0,
E[sup0≤r≤t‖u1(r)−u2(r)‖2]≤E[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]+2E[∫t0‖u1−u2‖⋅‖G(x,u1(s−ρ))−G(x,u2(s−ρ))‖ds] +∞∑k=1E[∫t0‖κ(σ2,k(u1)−σ2,k(u2))‖2ds] +2E[sup0≤r≤t|∞∑k=1∫r0(u1−u2,κ(x)(σ2,k(u1)−σ2,k(u2))dWk(s))|]. | (2.11) |
For the second term on the right-hand side of (2.11), by (2.3), one has
2E[∫t0‖u1−u2‖⋅‖G(x,u1(s−ρ))−G(x,u2(s−ρ))‖ds]≤E[∫t0‖u1−u2‖2ds]+E[∫t0‖G(x,u1(s−ρ))−G(x,u2(s−ρ))‖2ds]≤E[∫t0‖u1−u2‖2ds]+C2GE[∫t0‖u1(s−ρ)−u2(s−ρ)‖2ds]=E[∫t0‖u1−u2‖2ds]+C2GE[∫t−ρ−ρ‖u1−u2‖2ds]≤(1+C2G)E[∫t0‖u1−u2‖2ds]+C2GE[∫0−ρ‖φ1(s)−φ2(s)‖2ds]≤(1+C2G)∫t0E[sup0≤r≤s‖u1−u2‖2]ds+ρC2GE[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]. |
For the third term on the right-hand side of (2.11), by (2.5), we have
∞∑k=1E[∫t0‖κ(x)(σ2,k(u1)−σ2,k(u2))‖2ds]≤‖κ(x)‖2L∞∞∑k=1α2kE[∫t0‖u1−u2‖2ds]≤‖κ(x)‖2L∞∞∑k=1α2k∫t0E[sup0≤r≤s‖u1−u2‖2]ds. | (2.12) |
For the forth term on the right-hand side of (2.11), by Burkholder-Davis-Gundy's inequality, one has
2E[sup0≤r≤t|∞∑k=1∫r0(u1−u2,κ(x)(σ2,k(u1)−σ2,k(u2))dWk(s))|]≤B1E[(∫t0∞∑k=1|(u1−u2,κ(x)(σ2,k(u1)−σ2,k(u2)))|2ds)12]≤B1E[(∫t0∞∑k=1‖u1−u2‖2⋅‖κ‖2L∞⋅‖σ2,k(u1)−σ2,k(u2)‖2ds)12]≤B1E[sup0≤s≤t‖u1−u2‖⋅‖κ‖L∞⋅(∞∑k=1α2k)12(∫t0‖u1−u2‖2ds)12]≤12E[sup0≤s≤t‖u1−u2‖2]+12B21‖κ‖2L∞∞∑k=1α2kE[∫t0sup0≤r≤s‖u1−u2‖2ds], | (2.13) |
where B1 is a constant produced by Burkholder-Davis-Gundy's inequality.
It follows from (2.11)–(2.13) that for all t≥0,
E[sup0≤r≤t‖u1(r)−u2(r)‖2]≤2(1+ρC2G)E[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]+2[1+C2G+(1+12B21)‖κ‖2L∞∞∑k=1α2k]∫t0E[sup0≤r≤s‖u1(r)−u2(r)‖2]ds. | (2.14) |
Applying Gronwall inequality to (2.14), we obtain that for all t≥0,
E[sup0≤r≤t‖u1(r)−u2(r)‖2]≤2(1+ρC2G)ec1tE[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2], | (2.15) |
where c1=2[1+C2G+(1+12B21)‖κ‖2L∞∞∑k=1α2k]. By (2.10), there exists c2 such that for all t≥0,
E[∫t0‖u1−u2‖2Vds]≤˜c2ec2tE[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]. |
We assume that a, αk and γk are small enough in the sense, there exists a constant p≥2 such that
21−12p(2p−1)2p−12pa+2p(2p−1)‖κ‖2L∞∞∑k=1(α2k+γ2k)<pλ. | (3.1) |
By (3.1), one has
2‖κ‖2L∞∞∑k=1γ2k<λ, | (3.2) |
and
√2a+2‖κ‖2L∞∞∑k=1γ2k<λ. | (3.3) |
The inequalities (3.1)–(3.3) are used to establish the uniform tail-estimate of the solution of (1.1) and (1.2).
Lemma 3.1. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)∈L2(Ω;C([−ρ,0],H)), then, for all t≥0, there exists a positive constant μ1 such that the solution u of (1.1) and (1.2) satisfies
E[‖u(t)‖2]+∫t0eμ1(s−t)E(‖u(s)‖2V)ds+∫t0eμ1(s−t)E(‖u(s)‖2β+2L2β+2)ds |
≤M1E[sup−ρ≤s≤0‖φ(s)‖2]+~M1, | (3.4) |
and
∫t+ρ0E[‖u(s)‖2V]ds≤(M1(t+ρ)+1+√2aρC(n,α))E[sup−ρ≤s≤0‖φ(s)‖2]+√2(t+ρ)aC(n,α)‖h(x)‖2 |
+2(t+ρ)C(n,α)∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)+˜M1(t+ρ), |
where ˜M1 is a positive constant independent of φ.
Proof. By (1.1) and the integration by parts of Ito's formula, we have for all t≥0,
‖u(t)‖2+2∫t0‖(−Δ)α2u(s)‖2ds+2∫t0‖u(s)‖2β+2L2β+2ds+2λ∫t0‖u(s)‖2ds=2Re∫t0(u(s),G(x,u(s−ρ)))ds+‖φ(0)‖2+∞∑k=1∫t0‖σ1,k(x)+κ(x)σ2,k(u(s))‖2ds+2Re∫t0(u(s),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(s)))dWk(s). | (3.5) |
The system (3.5) can be rewritten as
d(‖u(t)‖2)+2‖(−Δ)α2u(t)‖2dt+2‖u(t)‖2β+2L2β+2dt+2λ‖u(t)‖2dt=2Re(u(t),G(x,u(t−ρ)))dt+∞∑k=1‖σ1,k(x)+κ(x)σ2,k(u(t))‖2dt+2Re(u(t),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t). | (3.6) |
Assume that μ1 is a positive constant, one has
eμ1t‖u(t)‖2+2∫t0eμ1s‖(−Δ)α2u(s)‖2ds+2∫t0eμ1s‖u(s)‖2β+2L2β+2ds=(μ1−2λ)∫t0eμ1s‖u(s)‖2ds+‖φ(0)‖2+2Re∫t0eμ1s(u(s),G(x,u(s−ρ)))ds+∞∑k=1∫t0eμ1s‖σ1,k+κσ2,k(u(s))‖2ds+2Re∫t0eμ1s(u(s),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(s)))dWk(s). |
Taking the expectation, we have for all t≥0,
eμ1tE(‖u(t)‖2)+2E[∫t0eμ1s‖(−Δ)α2u(s)‖2ds]+2E[∫t0eμ1s‖u(s)‖pLpds]=E(‖φ(0)‖2)+(μ1−2λ)E[∫t0eμ1s‖u(s)‖2ds]+2E[∫t0eμ1sRe(u(s),G(x,u(s−ρ)))ds]+∞∑k=1E[∫t0eμ1s‖σ1,k(x)+κ(x)σ2,k(u(s))‖2ds]. | (3.7) |
For the third term on the right-hand side (3.7), by (2.1), we have
2E[∫t0eμ1sRe(u(s),G(x,u(s−ρ)))ds]≤2∫t0eμ1sE[‖u(s)‖‖G(x,u(s−ρ))‖]ds≤√2a∫t0eμ1sE(‖u(s)‖2)ds+√22a∫t0eμ1sE[‖G(x,u(s−ρ))‖2]ds≤√2a∫t0eμ1sE(‖u(s)‖2)ds+√2a∫t0eμ1s‖h(x)‖2ds+√2a∫t0eμ1sE[‖u(s−ρ)‖2]ds≤√2a(1+eμ1ρ)∫t0eμ1sE[‖u(s)‖2]ds+√2a‖h(x)‖2∫t0eμ1sds+√2aeμ1ρ∫0−ρeμ1sE[‖φ(s)‖2]ds≤√2a(1+eμ1ρ)∫t0eμ1sE[‖u(s)‖2]ds+√2eμ1taμ1‖h(x)‖2+√2aρeμ1ρE[sup−ρ≤s≤0‖φ(s)‖2]. | (3.8) |
For the forth term on the right-hand side (3.7), by (2.6), we have
∞∑k=1E[∫t0eμ1s‖σ1,k+κσ2,k(u(s))‖2ds]≤∞∑k=1E[∫t0eμ1s(2‖σ1,k‖2+2‖κσ2,k(u(s))‖2)ds]≤2μ1∞∑k=1‖σ1,k‖2eμ1t+4∞∑k=1∫t0eμ1sE[β2k‖κ‖2+γ2k‖κ‖2L∞‖u(s)‖2]ds≤2μ1∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)eμ1t+4∞∑k=1γ2k‖κ(x)‖2L∞∫t0eμ1sE(‖u(s)‖2)ds. | (3.9) |
By (3.7)–(3.9), we obtain for all t≥0,
eμ1tE(‖u(t)‖2)+2E[∫t0eμ1s‖(−Δ)α2u(s)‖2ds]+2E[∫t0eμ1s‖u(s)‖2β+2L2β+2ds]≤(1+√2aρeμ1ρ)E[sup−ρ≤s≤0‖φ(s)‖2]+[μ1−2λ+√2a(1+eμ1ρ)+4∞∑k=1γ2k‖κ‖2L∞]∫t0eμ1sE[‖u‖2]ds+√2aμ1eμ1t‖h(x)‖2+2μ1∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)eμ1t. | (3.10) |
By (3.2), there exists a positive constant μ1 sufficiently small such that
2μ1+√2a+√2aeμ1ρ+4∞∑k=1γ2k‖κ(x)‖2L∞≤2λ. |
Then, we have, for all t≥0,
E(‖u(t)‖2)+2∫t0eμ1(s−t)E(‖(−Δ)α2u(s)‖2)ds +μ1∫t0eμ1(s−t)E(‖u(s)‖2)ds+2∫t0eμ1(s−t)E(‖u(s)‖2β+2L2β+2)ds≤(1+√2aρeμ1ρ)E(sup−ρ≤s≤0‖φ(s)‖2)+1μ1(√2a‖h(x)‖2+2∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)), |
which completes the proof of (3.4).
Integrating (3.6) on [0,t+ρ] and taking the expectation, one has
E[‖u(t+ρ)‖2]+2E[∫t+ρ0‖(−Δ)α2u(s)‖2ds]+2E[∫t+ρ0‖u(s)‖2β+2L2β+2ds]+2λE[∫t+ρ0‖u(s)‖2ds]=E[‖φ(0)‖2]+2E[∫t+ρ0Re(u(s),G(x,u(s−ρ)))ds]+∞∑k=1E[∫t+ρ0‖σ1,k+κ(x)σ2,k(u(s))ds]. | (3.11) |
For the second term on the right-hand side of (3.11), by (2.1), we have
2E[∫t+ρ0Re(u,G(x,u(s−ρ)))ds]≤2√2aE[∫t+ρ0‖u‖2ds]+√2aρE[sup−ρ≤s≤0‖φ(s)‖2]+√2(t+ρ)a‖h‖2. | (3.12) |
For the third term on the right-hand side of (3.11), one has
∞∑k=1E[∫t+ρ0‖σ1,k+κσ2,k(u(s))ds]≤2(t+ρ)∞∑k=1(‖σ1,k‖2+2β2k‖κ‖2)+4∞∑k=1γ2k‖κ(x)‖2L∞E[∫t+ρ0‖u‖2ds]. | (3.13) |
Then, by (3.2) and (3.11)–(3.13), for all t≥0, we obtain,
2E[∫t+ρ0‖(−Δ)α2u(s)‖2ds]≤(1+√2aρ)E[sup−ρ≤s≤0‖φ(s)‖2] |
+2(t+ρ)∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)+√2(t+ρ)a‖h(x)‖2. |
The result then follows from (3.4).
The next lemma is used to obtain the uniform estimates of the segments of solutions in C([−ρ,0],H).
Lemma 3.2. Suppose (2.1)–(2.6) and (3.2) hold. Then, for any φ(s)∈L2(Ω,F0;C([−ρ,0],H)), the solution of (1.1) satisfies that, for all t≥ρ,
E(supt−ρ≤r≤t‖u(r)‖2)≤M2E[sup−ρ≤s≤0‖φ(s)‖2]+˜M2, |
where M2 and ˜M2 are positive constants independent of φ.
Proof. By (1.1) and integration by parts of Ito's formula and taking the real part, we get for all t≥ρ and t−ρ≤r≤t,
‖u(r)‖2+2∫rt−ρ‖(−Δ)α2u(s)‖2ds+2∫rt−ρ‖u(s)‖2β+2L2β+2ds+2λ∫rt−ρ‖u(s)‖2ds=‖u(t−ρ)‖2+2Re∫rt−ρ(u(s),G(x,u(s−ρ)))ds+∞∑k=1∫rt−ρ‖σ1,k(x)+κ(x)σ2,k(u(s)))‖2ds+2Re∞∑k=1∫rt−ρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s)). | (3.14) |
For the second term on the right-hand side of (3.14), by (2.1) we have, for all t≥ρ and t−ρ≤r≤t,
2Re∫rt−ρ(u(s),G(x,u(s−ρ)))ds≤2∫rt−ρ‖u(s)‖⋅‖G(x,u(s−ρ))‖ds≤∫rt−ρ‖u(s)‖2ds+∫rt−ρ‖G(x,u(s−ρ))‖2ds≤∫rt−ρ‖u(s)‖2ds+2∫rt−ρ‖h‖2ds+2a2∫rt−ρ‖u(s−ρ)‖2ds≤∫rt−ρ‖u(s)‖2ds+2ρ‖h‖2+2a2∫t−ρt−2ρ‖u(s)‖2ds. | (3.15) |
For the third term on the right-hand side of of (3.14), for all t≥ρ and t−ρ≤r≤t, by (2.6), we have
∞∑k=1∫rt−ρ‖σ1,k(x)+κ(x)σ2,k(u(s))‖2ds≤2ρ∞∑k=1‖σ1,k‖2+4ρ‖κ‖2∞∑k=1β2k+4‖κ‖2L∞∞∑k=1γ2k∫rt−ρ‖u(s)‖2ds. | (3.16) |
By (3.14)–(3.16), we obtain for all t≥ρ and t−ρ≤r≤t,
‖u(r)‖2≤c3+‖u(t−ρ)‖2+c4∫rt−2ρ‖u(s)‖2ds |
+2Re∞∑k=1∫rt−ρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s)), | (3.17) |
where c3=2ρ‖h‖2+2ρ∞∑k=1‖σ1,k‖2+4ρ‖κ‖2∞∑k=1β2k and c4=1+2a2+4‖κ‖2L∞∞∑k=1γ2k. By (3.17), we find that for all t≥ρ,
E[supt−ρ≤r≤t‖u(r)‖2]≤c3+E[‖u(t−ρ)‖2]+c4∫tt−2ρE[‖u(s)‖2]ds+2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s))|]. | (3.18) |
For the second term and the third term on the right-hand side of (3.18), by Lemma 3.1, we deduce for all t≥ρ,
E[‖u(t−ρ)‖2]≤sups≥0E[‖u(s)‖2]≤M1E[sup−ρ≤s≤0‖φ‖2]+˜M1 | (3.19) |
and
c4∫tt−2ρE[‖u(s)‖2]ds≤2ρc4sups≥−ρE[‖u(s)‖2]≤c5E[sup−ρ≤s≤0‖φ‖2]+c5. | (3.20) |
For the last term on the right-hand side of (3.18), by Burkholder-Davis-Gundy's inequality and Lemma 3.1, we obtain for all t≥ρ,
2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρ(u(s),σ1,k(x)+κ(x)σ2,k(u(s))dWk(s))|]≤2B2E[(∞∑k=1∫tt−ρ|(u(s),σ1,k+κσ2,k(u(s)))|2ds)12]≤12E[supt−ρ≤s≤t‖u(s)‖2]+2B22E[∞∑k=1∫tt−ρ‖σ1,k+κσ2,k(u(s))‖2ds]≤12E[supt−ρ≤s≤t‖u(s)‖2]+2B22(2ρ∞∑k=1‖σ1,k‖2+4ρ‖κ‖2∞∑k=1β2k)+8B22ρ‖κ‖2L∞∞∑k=1γ2ksups≥0E[‖u(s)‖2]. | (3.21) |
By Lemma 3.1 and (3.18)–(3.21), we deduce that for all t≥ρ,
E[supt−ρ≤r≤t‖u(r)‖2]≤M2E[sup−ρ≤s≤0‖φ(s)‖2]+˜M2. |
This completes the proof.
To establish the tightness of a family of distributions of solutions, we now derive uniform estimates on the tails of solutions to the problems (1.1) and (1.2).
Lemma 3.3. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)∈L2(Ω,C([−ρ,0],H)). Then, for all t≥0, the solution u of (1.1) and (1.2) satisfies
lim supm→∞supt≥−ρ∫|x|≥mE[|u(t,x)|2]dx=0. |
Proof. We suppose that θ(x):Rn→R is a smooth function with 0≤θ(x)≤1, for all x∈Rn defined by
θ(x)={0if |x|≤1,1if |x|≥2. |
For fixed m∈N, we denote that θm(x)=θ(xm). By (1.1), we have
d(θmu)+(1+iν)θm(−Δ)αudt+(1+iμ)θm|u|2βudt+λθmudt=θmG(x,u(t−ρ))dt |
+∞∑k=1θm(σ1,k+κσ2,k)dWk(t). | (3.22) |
By (3.2), We can find μ2 sufficiently small such that
μ2+2√2a+4‖κ‖2L∞∞∑k=1γ2k−2λ<0. | (3.23) |
By (3.22) and integration by parts of Ito's formula and taking the expectation, we obtain
E[‖θmu‖2]+2∫t0eμ2(s−t)E[∫Rnθ2m|u|2β+2dx]ds=e−μ2tE[‖θmφ(0)‖2]−2∫t0eμ2(s−t)E[Re(1+iν)((−Δ)α2u,(−Δ)α2(θ2mu))]ds+(μ2−2λ)∫t0eμ2(s−t)E[‖θmu‖2]ds+2∫t0eμ2(s−t)E[Re(θmu,θmG(x,u(s−ρ)))]ds+∞∑k=1∫t0eμ2(s−t)E[‖θm(σ1,k+κ(x)σ2,k(u(s)))‖2]ds. | (3.24) |
For the first term in the second row of (3.24), since φ(s)∈L2(Ω,C([−ρ,0],H)), we have for all s∈[−ρ,0], E[‖φ(0)‖2]<∞. It follows that for any ε>0, there exists a positive N1=N1(ε,φ)≥1, for all m≥N1, one has ∫|x|≥mE[φ2(0,x)]dx<ε. Consequently,
E[‖θmφ(0)‖2]=E[∫Rn|θ(xm)φ(0,x)|2dx]=E[∫|x|≥m|θ(xm)φ(0,x)|2dx]≤∫|x|≥mE[|φ(0,x)|2]dx<ε, ∀m≥N1. | (3.25) |
Now we consider the second term on the right-hand side of (3.24). We first have
−2E[Re(1+iν)((−Δ)α2u(s),(−Δ)α2(θ2mu(s)))]=−C(n,α)E[Re(1+iν)∫Rn∫Rn[u(x)−u(y)][θ2m(x)ˉu(x)−θ2m(y)ˉu(y)]|x−y|n+2α]dxdy=−C(n,α)E[Re(1+iν)∫Rn∫Rn[u(x)−u(y)][θ2m(x)(ˉu(x)−ˉu(y))+ˉu(y)(θ2m(x)−θ2m(y))]|x−y|n+2α]dxdy=−C(n,α)E[Re(1+iν)∫Rn∫Rnθ2m(x)|u(x)−u(y)|2|x−y|n+2αdxdy]−C(n,α)E[Re(1+iν)∫Rn∫Rn(u(x)−u(y))(θ2m(x)−θ2m(y))ˉu(y)|x−y|n+2αdxdy]≤−C(n,α)E[Re(1+iν)∫Rn∫Rn(u(x)−u(y))(θ2m(x)−θ2m(y))ˉu(y)|x−y|n+2αdxdy]≤C(n,α)√1+ν2E[|∫Rn∫Rn(u(x)−u(y))(θ2m(x)−θ2m(y))ˉu(y)|x−y|n+2αdxdy|]≤2C(n,α)√1+ν2E[∫Rn|ˉu(y)|(∫Rn|(u(x)−u(y))(θm(x)−θm(y))||x−y|n+2αdx)dy]≤2C(n,α)√1+ν2E[‖u(s)‖(∫Rn(∫Rn|(u(x)−u(y))(θm(x)−θm(y))||x−y|n+2αdx)2dy)12]≤2C(n,α)√1+ν2E[‖u(s)‖(∫Rn(∫Rn|u(x)−u(y)|2|x−y|n+2αdx∫Rn|(θm(x)−θm(y))|2|x−y|n+2αdx)dy)12]. | (3.26) |
We now prove the following inequality:
∫Rn|(θm(x)−θm(y))|2|x−y|n+2αdx≤c6m2α. | (3.27) |
Let x−y=h and hm=z, then, we obtain,
∫Rn|(θm(x)−θm(y))|2|x−y|n+2αdx=∫Rn|θ(y+hm)−θ(ym)|2|h|n+2αdh=∫Rn|θ(ym+z)−θ(ym)|2mn+2α|z|n+2αmndz=1m2α∫Rn|θ(ym+z)−θ(ym)|2|z|n+2αdz=1m2α∫|z|≤1|θ(ym+z)−θ(ym)|2|z|n+2αdz+1m2α∫|z|>1|θ(ym+z)−θ(ym)|2|z|n+2αdz≤c∗6m2α∫|z|≤1|z|2|z|n+2αdz+4m2α∫|z|>11|z|n+2αdz≤c∗6m2α∫|z|≤11|z|n+2α−2dz+4m2α∫|z|>11|z|n+2αdz≤c∗6ˉc6m2α+4˜c6m2α=c∗6ˉc6+4˜c6m2α. | (3.28) |
This proves (3.27) with c6:=c∗6ˉc6+4˜c6. By (3.26) and (3.27), we obtain,
−2E[Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))]≤2√c6(1+ν2)C(n,α)m−αE[‖u(s)‖√∫Rn∫Rn|u(x)−u(y)|2|x−y|n+2αdxdy]≤√c6(1+ν2)C(n,α)m−α(E(‖u(s)‖2)+E(∫Rn∫Rn|u(x)−u(y)|2|x−y|n+2αdxdy))≤√c6(1+ν2)C(n,α)m−αE(‖u(s)‖2)+2√c6(1+ν2)m−αE(‖(−Δ)α2u(s)‖2). | (3.29) |
By (3.29), for the second term on the right-hand side of (3.24), we get
−2∫t0eμ2sE[Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))]ds≤√c6(1+ν2)C(n,α)m−α∫t0eμ2sE[‖u(s)‖2]ds+2√c6(1+ν2)m−α∫t0eμ2sE[‖(−Δ)α2u(s)‖2]ds. | (3.30) |
By Lemma 3.1, we have
√c6(1+ν2)C(n,α)m−α∫t0eμ2(s−t)E[‖u(s)‖2]ds≤√c6(1+ν2)C(n,α)m−α[M1E[sup−ρ≤s≤0‖φ(s)‖2]+˜M1]∫t0eμ2(s−t)ds≤√c6(1+ν2)C(n,α)m−α1μ2[M1E[sup−ρ≤s≤0‖φ(s)‖2]+˜M1]. | (3.31) |
By (3.31), we deduce that there exists N2(ε,φ)≥N1, for all t≥0 and m≥N2,
√c6(1+ν2)C(n,α)m−α∫t0eμ2(s−t)E[‖u(s)‖2]ds<ε. |
By Lemma 3.1, there exists N3(ε,φ)≥N2 such that for all t≥0 and m≥N3,
2√c6(1+ν2)m−α∫t0eμ2(s−t)E[‖(−Δ)α2u‖2]ds≤2√c6(1+ν2)m−α[M1E[sup−ρ≤s≤0‖φ(s)‖2]+˜M1]<ε. |
For the forth term on the right-hand side of (3.24), we obtain that there exists N4(ε,φ)≥N3, for all t≥0 and m≥N4,
2∫t0eμ2(s−t)E[Re(θmu,θmG(x,u(s−ρ)))]ds≤√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds+1√2a∫t0eμ2(s−t)E[‖θmG(x,u(s−ρ))‖2]ds≤√2aμ2∫|x|≥mh2(x)dx+√2a∫0−ρeμ2(s−t)E[‖θmφ(s)‖2]ds+2√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds≤√2aμ2ε+√2a∫0−ρeμ2(s−t)E[‖θmφ(s)‖2]ds+2√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Since {φ(s)∈L2(Ω,H)|s∈[−ρ,0]} is compact, it has a open cover of balls with radius √ε2 which denoted by {B(φi,√ε2)}li=1. Since φi=φ(si)∈L2(Ω;C([−ρ,0],H)) for i=1,2,⋯,l, we obtain that for given ε>0,
{φ(s)∈L2(Ω;C([−ρ,0],H))}⊆∪li=1{X∈L2(Ω,H)|‖X−φi‖L2(Ω,H)<√ε2}. |
Since φi∈L2(Ω,H), there exists a positive constant N5=N5(ε,φ)≥N4, for m≥N5, we have
supi=1,2,⋯,l∫|x|≥mE[|φ(si,x)|2]dx<ε4. |
Then,
sups∈[−ρ,0]∫|x|≥mE[|φ(s,x)|2]dx<ε2,∀m≥N5. |
Consequently, one has
2∫t0eμ2(s−t)E[Re(θmu,θmG(x,u(s−ρ)))]ds≤√2aμ2ε+√2aρε2+2√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. | (3.32) |
For the fifth term on the right-hand side of (3.24), by (2.6), we obtain
∞∑k=1∫t0eμ2(s−t)E[‖θm(σ1,k+κ(x)σ2,k(u(s)))‖2]ds≤2∞∑k=1∫t0eμ2(s−t)‖θmσ1,k‖2ds+2∞∑k=1∫t0eμ2(s−t)E[‖θmκ(x)σ2,k(u(s))‖2]ds≤2μ2∞∑k=1∫|x|≥m|σ1,k(x)|2dx+4μ2∞∑k=1β2k∫|x|≥mκ2(x)dx+4‖κ(x)‖2L∞∞∑k=1γ2k∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Since ∞∑k=1‖σ1,k‖2<∞ and κ(x)∈L2(Rn)⋂L∞(Rn), there exists N6=N6(ε,φ)≥N5, for all t≥0 and m≥N6, we have
∞∑k=1∫|x|≥m|σ1,k(x)|2dx+∫|x|≥mκ2(x)dx<ε. |
Consequently, for the fifth term on the right-hand side of (3.24), we get for all t≥0 and m≥N6,
∞∑k=1∫t0eμ2(s−t)E[‖θm(σ1,k+κσ2,k)‖2]ds≤2μ2(1+2∞∑k=1β2k)ε |
+4‖κ‖2L∞∑∞k=1γ2k∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Therefore, for all t≥0 and m≥N6,
E[‖θmu(t)‖2]≤[2+e−μ2t+√2aμ2+√22aρ+2μ2(1+2∞∑k=1β2k)]ε |
+(μ2−2λ+2√2a+4‖κ‖2L∞∑∞k=1γ2k)∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Taking the limit in the above equation and by (3.23), we have
lim supm→∞supt≥−ρ∫|x|≥mE[|u(t,x)|2]dx=0, |
which completes the proof.
Lemma 3.4. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)∈L2(Ω,C([−ρ,0],H)), then the solution u of (1.1) and (1.2) satisfies
lim supm→∞supt≥0E[supr∈[t−ρ,t]∫|x|≥m|u(r,x)|2dx]=0. |
Proof. By (3.22) and integration by parts of Ito's formula and taking the real part, for all t≥ρ and r∈[t−ρ,t], we have
eμ2r‖θmu(r)‖2+2∫rt−ρeμ2s∫Rnθ2m|u|2β+2dxds=eμ2(t−ρ)‖θmu(t−ρ)‖2−2∫rt−ρeμ2sRe(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))ds+(μ2−2λ)∫rt−ρeμ2s‖θmu(s)‖2ds+2Re∫rt−ρeμ2s(θmu(s),θmG(x,u(s−ρ)))ds+∞∑k=1∫rt−ρeμ2s‖θm(σ1,k+κ(x)σ2,k(u(s)))‖2ds+2Re∞∑k=1∫rt−ρeμ2s(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s). | (3.33) |
By (3.33), we deduce,
E[supt−ρ≤r≤t‖θmu(r)‖2]≤E[‖θmu(t−ρ)‖2]−2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)Re(1+iν)((−Δ)α2u,(−Δ)α2θ2mu)ds]+|μ2−2λ|E[supt−ρ≤r≤t∫rt−ρ‖θmu‖2eμ2(s−r)ds]+2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmu‖⋅‖θmG(x,u(s−ρ))‖ds]+∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θm(σ1,k+κσ2,k(u(s)))‖2ds]+2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρeμ2(s−r)(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s)|]. | (3.34) |
For the first term on the right-hand side of (3.34), by Lemma 3.3, one has for any ε>0, there exists ˜N1(ε,φ)≥1 such that for all m≥˜N1 and t≥ρ,
E[‖θmu(t−ρ)‖2]≤∫|x|≥mE[|u(t−ρ,x)|2]dx<ε. | (3.35) |
For the second term on the right-hand side of (3.34), by (3.29), we have
−2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))ds]≤2√c6(1+ν2)C(n,α)m−αE[supt−ρ≤r≤t(∫rt−ρeμ2(s−r)‖u(s)‖‖(−Δ)α2u(s)‖ds)]≤2√c6(1+ν2)C(n,α)m−αeμ2ρE[(∫tt−ρeμ2(s−t)‖u(s)‖‖(−Δ)α2u(s)‖ds)]≤√c6(1+ν2)C(n,α)m−αeμ2ρ{∫tt−ρeμ2(s−t)E[‖u‖2]ds+E[∫tt−ρeμ2(s−t)‖(−Δ)α2u‖2ds]}≤√c6(1+ν2)C(n,α)m−αeμ2ρ{ρsups∈[t−ρ,t]E[‖u(s)‖2]+E[∫tt−ρeμ2(s−t)‖(−Δ)α2u‖2ds]}. | (3.36) |
By Lemma 3.1 and (3.36), we deduce that there exists ˜N2(ε,φ)≥˜N1 such that for all m≥˜N2 and t≥ρ,
−2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))ds]<ε. | (3.37) |
For the third term on the right-hand side of (3.34), by Lemma 3.3, we obtain that for all m≥˜N2 and t≥ρ,
|μ2−2λ|E[supt−ρ≤r≤t∫rt−ρ‖θmu(s)‖2eμ2(s−r)ds]≤|μ2−2λ|E[∫tt−ρ‖θmu(s)‖2ds] |
≤|μ2−2λ|ρsupt−ρ≤s≤tE[‖θmu(s)‖2]<|μ2−2λ|ρε. | (3.38) |
For the forth term on the right-hand side of (3.34), by (2.1), we obtain
2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmu(s)‖⋅‖θmG(x,u(s−ρ))‖ds]≤∫tt−ρE[‖θmu(s)‖2]ds+2ρ‖θmh‖2+2a2∫t−ρt−2ρE[‖θmu(s)‖2]ds≤ρsupt−ρ≤s≤tE[‖θmu(s)‖2]+2ρ‖θmh‖2+2a2ρsupt−2ρ≤s≤t−ρE[‖θmu(s)‖2], |
which along with Lemma 3.3, we deduce that there exists ˜N3(ε,φ)≥˜N2 such that for all m≥˜N3 and t≥ρ,
2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmu(s)‖⋅‖θmG(x,u(s−ρ))‖ds]<(3+2a2)ρε. | (3.39) |
For the fifth term on the right-hand side of (3.34), by (2.6), we have
∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θm(σ1,k+κσ2,k(u(s)))‖2ds]≤2ρ∞∑k=1‖θmσ1,k‖2+2ρ∞∑k=1supt−ρ≤s≤tE[‖θmκ(x)σ2,k(u(s))‖2]≤2ρ∞∑k=1∫|x|≥m|σ1,k(x)|2dx+4ρ∞∑k=1β2k∫|x|≥m|κ(x)|2dx+4ρ‖κ(x)‖2L∞∞∑k=1γ2ksupt−ρ≤s≤tE[‖θmu(s)‖2]. |
By the condition κ(x)∈L2(Rn)⋂L∞(Rn), (2.4) and Lemma 3.3, we deduce that there exists ˜N4(ε,φ)≥˜N3 such that for all m≥˜N4 and t≥ρ,
∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θm(σ1,k+κσ2,k(u(s)))‖2ds]<2ρ(1+λ+2∞∑k=1β2k)ε. | (3.40) |
For the sixth term on the right-hand side of (3.34), by (2.6), (3.40) and Burkholder-Davis-Gundy's inequality, we have,
2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρeμ2(s−r)(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s)|]≤2e−μ2(t−ρ)E[supt−ρ≤r≤t|∞∑k=1∫rt−ρeμ2s(θmu(s),θmσ1,k+θmκ(x)σ2,k(u(s)))dWk(s)|]≤2˜B2e−μ2(t−ρ)E[(∫tt−ρe2μ2s∞∑k=1|(θmu(s),θmσ1,k+θmκ(x)σ2,k(u(s)))|2ds)12]≤2˜B2e−μ2(t−ρ)E[supt−ρ≤s≤t‖θmu(s)‖(∫tt−ρe2μ2s∞∑k=1‖θmσ1,k+θmκσ2,k(u(s))‖2ds)12]≤12E[supt−ρ≤s≤t‖θmu(s)‖2]+2˜B22E[e2μ2ρ∫tt−ρe2μ2(s−t)∞∑k=1‖θmσ1,k+θmκ(x)σ2,k(u(s))‖2ds]≤12E[supt−ρ≤s≤t‖θmu(s)‖2]+2˜B22e2μ2ρ∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmσ1,k+θmκ(x)σ2,k(u(s))‖2ds]≤12E[supt−ρ≤s≤t‖θmu(s)‖2]+4ρ(1+λ+2∞∑k=1β2k)˜B22e2μ2ρε. |
Above all, for all m≥˜N4 and t≥ρ, we obtain,
E[supt−ρ≤r≤t‖θmu(r)‖2]≤[4+2|μ2−2λ|ρ+(6+4a2)ρ+4ρ(1+2˜B22e2μ2ρ)(1+λ+2∞∑k=1β2k)]ε. |
Therefore, we conclude
lim supm→∞supt≥0E[supt−ρ≤r≤t∫|x|≥m|u(r,x)|2dx]=0. |
Lemma 3.5. Suppose (2.1)–(2.6) and (3.1) hold. If φ(s)∈L2(Ω,C([−ρ,0],H)), then there exists a positive constant μ3 such that the solution u of (1.1) and (1.2) satisfies
supt≥−ρE[‖u(t)‖2p]+supt≥0E[∫t0eμ3(s−t)‖u(s)‖2p−2‖(−Δ)α2u(s)‖2ds] ≤(1+aρeμρ2p(4p−2)2p−12p)E[‖φ‖2pCH]+M3, | (3.41) |
where M3 is a positive constant independent of φ.
Proof. By (3.1), there exist positive constants μ and ϵ1 such that
μ+aeμρ2p21−12p(2p−1)2p−12p+4(p−1)(2p−1)ϵ2p2p−21∞∑k=1(‖σ1,k‖2+‖κ‖2β2k) +4θ(2p−1)‖κ‖2L∞∞∑k=1γ2k≤2pλ. | (3.42) |
Given n∈N, let τn be a stopping time as defined by
τn=inf{t≥0:‖u(t)‖>n}, |
and as usual, we set τn=+∞ if {t≥0:‖u(t)‖>n}=∅. By the continuity of solutions, we have
limn→∞τn=+∞. |
Applying Ito's formula, we obtain
d(‖u(t)‖2p)=d((‖u(t)‖2)p)=p‖u(t)‖2(p−1)d(‖u(t)‖2)+2p(p−1)‖u(t)‖2(p−2) ×∞∑k=1|(u(t),σ1,k+κσ2,k(u(t)))|2dt. | (3.43) |
Substituting (3.6) into (3.43), we infer
d(‖u(t)‖2p)=−2p‖u(t)‖2(p−1)‖(−Δ)α2u(t)‖2dt−2p‖u(t)‖2(p−1)‖u(t)‖2β+2L2β+2dt−2pλ‖u(t)‖2pdt +2p‖u(t)‖2(p−1)Re(u(t),G(x,u(t−ρ)))dt +p‖u(t)‖2(p−1)∞∑k=1‖σ1,k(x)+κ(x)σ2,k(u(t))‖2dt +2p‖u(t)‖2(p−1)Re(u(t),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t) +2p(p−1)‖u(t)‖2(p−2)∞∑k=1|(u(t),σ1,k+κσ2,k(u(t)))|2dt. | (3.44) |
We also get the formula
d(eμt‖u(t)‖2p)=μeμt‖u(t)‖2pdt+eμtd(‖u(t)‖2p). | (3.45) |
Substituting (3.44) into (3.45) and integrating on (0,t∧τn) with t≥0, we deduce
eμ(t∧τn)‖u(t∧τn)‖2p+2p∫t∧τn0eμs‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds=−2p∫t∧τn0eμs‖u(s)‖2(p−1)‖u(s)‖2β+2L2β+2ds+‖φ(0)‖2p+(μ−2pλ)∫t∧τn0eμs‖u(s)‖2pds +2p∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds +p∞∑k=1∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds +2p∞∑k=1∫t∧τn0eμs‖u(t)‖2(p−1)Re(u(s),σ1,k+κσ2,k(u(s)))dWk(s) +2p(p−1)∞∑k=1∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds. | (3.46) |
Taking the expectation, we obtain for t≥0,
E[eμ(t∧τn)‖u(t∧τn)‖2p]+2pE[∫t∧τn0eμs‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds]=−2pE[∫t∧τn0eμs‖u(s)‖2(p−1)‖u(s)‖2β+2L2β+2ds]+E[‖φ(0)‖2p]+(μ−2pλ)E[∫t∧τn0eμs‖u(s)‖2pds] +2pE[∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds] +p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds] +2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]≤E[‖φ(0)‖2p]+(μ−2pλ)E[∫t∧τn0eμs‖u(s)‖2pds] +2pE[∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds] +p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds] +2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]. | (3.47) |
Next, we estimate the terms on the right-hand side of (3.47).
For the third term on the right-hand side of (3.47), by Young's inequality and (2.1), we infer
2θE[∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds]≤2θE[∫t∧τn0eμs‖u(s)‖2p−1‖G(x,u(s−ρ))‖2ds]≤aeμρ2p21−12p(2p−1)2p−12pE[∫t∧τn0eμs‖u(s)‖2pds] +(2p−122p−1a2peμρ)2p−12pE[∫t∧τn0eμs‖G(x,u(s−ρ))‖2ds]≤aeμρ2p21−12p(2p−1)2p−12pE[∫t∧τn0eμs‖u(s)‖2pds] +22p−1(2p−122p−1a2peμρ)2p−12pE[∫t∧τn0eμs(‖h‖2p+a2p‖u(s−ρ)‖2p)ds]≤aeμρ2p21−12p(2p−1)2p−12pE[∫t∧τn0eμs‖u(s)‖2pds] +1μ(4p−2a2peμρ)2p−12p‖h‖2peμt+aρeμρ2p(4p−2)2p−12pE[‖φ‖2pCH]. | (3.48) |
For the forth term on the right-hand side of (3.47), we infer
p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds]≤2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k‖2ds] +2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖κσ2,k(u(s))‖2ds]. | (3.49) |
For the first term on the right-hand side of (3.49), we have
2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k‖2ds]≤2(p−1)ϵ2p2p−21∞∑k=1‖σ1,k‖2E[∫t∧τn0eμs‖u(s)‖2pds]+2μϵp1∞∑k=1‖σ1,k‖2eμt. | (3.50) |
For the second term on the right-hand side of (3.49), we have
2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖κσ2,k(u(s))‖2ds]≤4p‖κ‖2∞∑k=1β2kE[∫t∧τn0eμs‖u(s)‖2(p−1)ds]+4p‖κ‖2L∞∞∑k=1γ2kE[∫t∧τn0eμs‖u(s)‖2pds]≤4(p−1)ϵ2p2p−21‖κ‖2∞∑k=1β2kE[∫t∧τn0eμs‖u(s)‖2pds] +4μϵp1‖κ‖2∞∑k=1β2keμt+4p‖κ‖2L∞∞∑k=1γ2kE[∫t∧τn0eμs‖u(s)‖2pds]. | (3.51) |
By (3.49)–(3.51), we obtain
\begin{align} \begin{split} &\ \ \ \ p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2(p-1)}\|\sigma_{1,k}+\kappa\sigma_{2,k}(u(s))\|^2ds\right]\\& \leq \left[4(p-1)\epsilon_1^{\frac{2p}{2p-2}}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +\|\kappa\|^2\beta^2_k)+4p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\gamma^2_k\right]\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2p}ds\right]\\&\ \ \ \ +\frac{2}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{split} \end{align} | (3.52) |
For the fifth term on the right-hand side of (3.47), applying (3.52), we have
\begin{align} \begin{split} &\ \ \ \ 2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2(p-2)}\left|(u(s),\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\right|^2ds\right]\\&\leq 2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2p-2}\|\sigma_{1,k}+\kappa\sigma_{2,k}(u(s))\|^2ds\right]\\&\leq \left[8(p-1)^2\epsilon_1^{\frac{2p}{2p-2}}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +\|\kappa\|^2\beta^2_k)+8p(p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\gamma^2_k\right]\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2pa}ds\right]\\&\ \ \ \ +\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{split} \end{align} | (3.53) |
From (3.47), (3.48), (3.52) and (3.53), we obtain that for t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[e^{\mu(t\wedge\tau_n)}\|u(t\wedge\tau_n)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4p-2)^{\frac{2p-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]\\&\ \ \ +\left(\mu-2p\lambda+ae^{\frac{\mu\rho}{2p}}2^{1-\frac{1}{2p}}(2p-1)^{\frac{2p-1}{2p}} +4(p-1)(2p-1)\epsilon_1^{\frac{2p}{2p-2}}\right.\\&\ \ \ \ \left.\times\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +\|\kappa\|^2\beta^2_k)+4p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\gamma^2_k\right)\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2p}ds\right]\\&\ \ \ \ \ \ + \frac 1\mu\left(\frac{4p-2}{a^{2p}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}e^{\mu t}+\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{split} \end{align} | (3.54) |
Then by (3.42) and (3.54), we obtain that for t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[e^{\mu(t\wedge\tau_n)}\|u(t\wedge\tau_n)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4p-2)^{\frac{2p-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]+ \frac 1\mu\left(\frac{4p-2}{a^{2pa}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}e^{\mu t}\\&\ \ \ +\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{split} \end{align} | (3.55) |
Letting n\rightarrow \infty , by Fatou's Lemma, we deduce that for t\geq 0 ,
\begin{align*} &\ \ \ \ \mathbb{E}\left[e^{\mu t}\|u(t)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t}e^{\mu s}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4\theta-2)^{\frac{2\theta-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]+ \frac 1\mu\left(\frac{4p-2}{a^{2p}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}e^{\mu t}\\&\ \ \ +\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{align*} |
Hence, we have for t\geq 0 ,
\begin{align*} &\ \ \ \ \mathbb{E}\left[\|u(t)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t}e^{\mu (s-t)}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4p-2)^{\frac{2p-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]+ \frac 1\mu\left(\frac{4p-2}{a^{2p}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}\\&\ \ \ +\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k)). \end{align*} |
This implies the desired estimate.
In this section, we establish the uniform estimates of solutions of problems (1.1) and (1.2) with initial data in C([-\rho, 0], V) . To the end, we assume that for each k\in\mathbb{N} , the function \sigma_{1, k}\in V and
\begin{align} \sum\limits^\infty_{k = 1}\|\sigma_{1,k}\|^2_V < \infty. \end{align} | (4.1) |
Furthermore, we assume that the function \kappa\in V and there exists a constant C > 0 such that
\begin{align} |\nabla\kappa(x)|\leq C. \end{align} | (4.2) |
In the sequel, we further assume that the constant a , \hat{\gamma}_k in (2.7) are sufficiently small in the sense that there exists a constant p\geq 2 such that
\begin{align} \hat{a}2^{1-\frac{1}{2p}}(2p-1)^{\frac{2p-1}{2p}}+2p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}(\beta^2_k+\hat{\beta}^2_k+\gamma_k^2+\hat{\gamma}^2_k) < p\frac\lambda 2. \end{align} | (4.3) |
By (4.3), we can find
\begin{align} \sqrt{2}\hat{a}+2\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k < \frac \lambda 2. \end{align} | (4.4) |
Lemma 4.1. Suppose (2.1)–(2.7) and (4.4) hold. If \varphi(s)\in L^2(\Omega; C([-\rho, 0], V)) , then, for all t\geq0 , there exists a positive constant \mu_4 such that the solution u of (1.1) and (1.2) satisfies
\begin{align} \begin{split} \sup\limits_{s\geq -\rho}\mathbb{E}[\|\nabla u(t)\|^2]+\sup\limits_{s\geq 0}\mathbb{E}\left[\int^t_0e^{\mu_4(s-t)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right] \leq M_4\left(\mathbb{E}[\|\varphi\|^2_{C_V}]+1\right), \end{split} \end{align} | (4.5) |
where M_4 is a positive constant independent of \varphi .
Proof. By (4.4), there exists a positive constant \mu_1 such that
\begin{align} \mu_1-2\lambda+8\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k < 0. \end{align} | (4.6) |
By (1.1) and applying Ito's formula to e^{\mu_1 t}\|\nabla u(t)\|^2 , we have for t\geq 0 ,
\begin{align*} &\ \ \ \ e^{\mu_1t}\|\nabla u(t)\|^2+2\int^t_0e^{\mu_1s}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds+2\int^t_0e^{\mu_1s}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\\& = (\mu_1-2\lambda)\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds+\|\nabla\varphi(0)\|^2+2\mbox{Re}\int^t_0e^{\mu_1s}(G(x,u(s-\rho)),-\Delta u(s))ds\\& \ \ \ \ +\sum\limits^\infty_{k = 1}\int^t_0e^{\mu_1s}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\\& \ \ \ \ \ +2\sum\limits^\infty_{k = 1}\mbox{Re}\int^t_0e^{\mu_1s}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s). \end{align*} |
Taking the expectation, we have for all t\geq0 ,
\begin{align} \begin{split} &\ \ \ \ e^{\mu_1t}\mathbb{E}[\|\nabla u(t)\|^2]\!\!+\!\!2\mathbb{E}\left[\int^t_0e^{\mu_1s}\|(\!\!-\!\!\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right] \!\!+\!\!2\mathbb{E}\left[\int^t_0e^{\mu_1s}\mbox{Re}\left((1\!\!+\!\!\text{i}\mu)|u(s)|^{2\beta}u(s),\!\!-\!\!\Delta u(s)\right)ds\right]\\& = (\mu_1-2\lambda)\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]\!\!+\!\!\mathbb{E}\left[\|\nabla\varphi(0)\|^2\right]\!\!+\!\!2\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\\& \ \ \ \ \!\!+\!\!\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla(\sigma_{1,k}\!\!+\!\!\kappa\sigma_{2,k}(u(s)))\|^2ds\right]. \end{split} \end{align} | (4.7) |
First, we estimate the third term on the left-hand side of (4.7). Applying integrating by parts, we have
\begin{align} \begin{split} &\ \ \ \ \mbox{Re}\left((1+\text{i}\mu)|u|^{2\beta}u,\Delta u\right)\\& = -\mbox{Re}(1+\text{i}\mu)\int_{\mathbb{R}^n}\left((\beta+1)|u|^{2\beta}|\nabla u|^2+\beta|u|^{2(\beta-1)}(u\nabla \overline{u})^2\right)dx\\& = \int_{\mathbb{R}^n}|u|^{2(\beta-1)}\left(-(\beta+1)|u|^{2}|\nabla u|^2+\frac{\beta(1+\text{i}\mu)}{2}(u\nabla \overline{u})^2+\frac{\beta(1-\text{i}\mu)}{2}(\overline{u}\nabla u)^2\right)dx\\& = \int_{\mathbb{R}^n}|u|^{2(\beta-1)}trace(YMY^H), \end{split} \end{align} | (4.8) |
where
Y = \left( \begin{array}{c} \overline{u}\nabla u \\ u\nabla \overline{u} \\ \end{array} \right)^H, M = \left( \begin{array}{cc} -\frac{\beta+1}{2} & \frac{\beta(1+\text{i}\mu)}{2} \\ \frac{\beta(1-\text{i}\mu)}{2} & -\frac{\beta+1}{2} \\ \end{array} \right), |
and Y^H is the conjugate transpose of the matrix Y . We observe that the condition \beta\leq \frac{1}{\sqrt{1+\mu^2}-1} implies that the matrix M is nonpositive definite. Hence, we obtain
\begin{align} \begin{split} 2\mathbb{E}\left[\int^t_0e^{\mu_1s}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),\Delta u(s)\right)ds\right]\leq 0. \end{split} \end{align} | (4.9) |
Next, we estimate the terms on the right-hand side of (4.7). For the third term on the right-hand side of (4.7), applying (2.2) and Gagliardo-Nirenberg inequality, we have
\begin{align} \begin{split} &\ \ \ \ 2\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\leq2\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|\|\nabla G(x,u(s-\rho))\|ds\right]\\& \leq\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]+\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla G(x,u(s-\rho))\|^2ds\right]\\& \leq\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]+2\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\hat{h}(x)\|^2ds\right]+2\hat{a}^2\mathbb{E}\left[\int^{t}_0e^{\mu_1s}\|\nabla u(s-\rho)\|^2ds\right]\\& \leq\mathbb{E}\left[\int^t_0e^{\mu_1s}\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2ds\right]\!\!+\!\!\frac{2}{\mu_1}\|\hat{h}(x)\|^2e^{\mu_1t}\\&\ \ \ \ \!\!+\!\!\frac{c}{\mu_1}\sup\limits_{s\geq 0}\mathbb{E}[\| u(s)\|^2]e^{\mu_1t}\!\!+\!\!\frac{2\hat{a}^2}{\mu_1}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}[\|\nabla \varphi(s)\|^2]e^{\mu_1t}, \end{split} \end{align} | (4.10) |
where c is a positive constant from Gagliardo-Nirenberg inequality. For the forth term on the right-hand side of (4.7), applying (2.6) and (2.7), we have
\begin{align} \begin{split} &\ \ \ \ \sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right] \leq2\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}(\|\nabla\sigma_{1,k}\|^2\!\!+\!\!\|\nabla(\kappa\sigma_{2,k}(u(s)))\|^2)ds\right]\\& \leq\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\|\nabla \sigma_{1,k}\|^2e^{\mu_1t} \!\!+\!\!8\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\left(\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!\hat{\beta}^2_k\|\kappa\|^2\!\!+\!\!{\gamma}^2_kC^2\|u(s)\|^2\!\!+\!\!\hat{\gamma}^2_k\|\kappa\|^2_{L^\infty}\|\nabla u(s)\|^2\right)ds\right]\\& \leq\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla\kappa\|^2+4\hat{\beta}^2_k\|\kappa\|^2+4C^2{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}[\|u(s)\|^2]\right)e^{\mu_1t}\\&\ \ \ \ \ +8\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\|\kappa(x)\|^2_{L^\infty}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]. \end{split} \end{align} | (4.11) |
By (4.7), (4.10) and (4.11), we obtain
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}[\|\nabla u(t)\|^2]\!\!+\!\!\mathbb{E}\left[\int^t_0e^{\mu_1(s-t)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \leq \mathbb{E}\left[\|\nabla\varphi(0)\|^2\right]e^{-\mu_1t}+\frac{2}{\mu_1}\|\hat{h}(x)\|^2 +\left(\mu_1-2\lambda+8\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right)\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]\\&\ \ \ \ +\frac{2}{\mu_1}\left(\frac c2+4\left(C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k+c\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right)\right)\sup\limits_{s\geq -\rho}\mathbb{E}[\|u(s)\|^2]\\& \ \ \ \ +\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4(\beta^2_k+\hat{\beta}^2_k)\|\kappa\|^2_V\right)+\frac{2\hat{a}^2}{\mu_1}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}[\|\nabla \varphi(s)\|^2]. \end{split} \end{align} | (4.12) |
Then by (4.6) and (4.12), we obtain that for all t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}[\|\nabla u(t)\|^2]+\mathbb{E}\left[\int^t_0e^{\mu_1(s-t)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \leq \mathbb{E}\left[\|\nabla\varphi(0)\|^2\right]e^{-\mu_1t}\!+\!\frac{2}{\mu_1}\|\hat{h}(x)\|^2\!+\!\frac{2}{\mu_1}\left(\frac c2\!+\!4(C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k\!+\!c\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k)\right)\sup\limits_{s\geq -\rho}\mathbb{E}[\|u(s)\|^2]\\& \ \ \ \ +\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4(\beta^2_k+\hat{\beta}^2_k)\|\kappa\|^2_V\right)+\frac{2\hat{a}^2}{\mu_1}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}[\|\nabla \varphi(s)\|^2]. \end{split} \end{align} | (4.13) |
Then by (4.13) and Lemma 3.1, we obtain the estimates (4.5).
Lemma 4.2. Suppose (2.1)–(2.7) and (4.4) hold. If \varphi(s)\in L^2(\Omega; C([-\rho, 0], V)) , then the solution u of (1.1) and (1.2) satisfies
\begin{align} \begin{split} \sup\limits_{t\geq \rho}\left\{\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|\nabla u(r)\|^2\right]\right\} \leq M_5\left(\mathbb{E}[\|\varphi\|^2_{C_V}]+1\right), \end{split} \end{align} | (4.14) |
where M_5 is a positive constant independent of \varphi .
Proof. By (1.1) and Ito's formula, we get for all t\geq \rho and t-\rho\leq r\leq t ,
\begin{align} \begin{split} &\ \ \ \ \|\nabla u(r)\|^2+2\int^r_{t-\rho}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\\&\ \ \ \ +2\int^r_{t-\rho}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds+2\lambda\int^r_{t-\rho}\|\nabla u(s)\|^2ds\\& = \|\nabla u(t-\rho)\|^2+2\mbox{Re}\int^r_{t-\rho}(G(x,u(s-\rho)),-\Delta u(s))ds\\& \ \ \ \ +\sum\limits^\infty_{k = 1}\int^r_{t-\rho}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\\& \ \ \ \ \ +2\sum\limits^\infty_{k = 1}\mbox{Re}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s). \end{split} \end{align} | (4.15) |
For the third term on the left-hand side of (4.15), applying (4.8), we have
\begin{align} \begin{split} -2\int^r_{t-\rho}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\leq 0. \end{split} \end{align} | (4.16) |
For the second term on the right-hand side of (4.15), applying (2.2) and Gagliardo-Nirenberg inequality, we have
\begin{align} \begin{split} &\ \ \ \ 2\mbox{Re}\int^r_{t-\rho}(G(x,u(s-\rho)),-\Delta u(s))ds\leq2\int^r_{t-\rho}\|\nabla u(s)\|\|\nabla G(x,u(s-\rho))\|ds\\& \leq\int^r_{t-\rho}\|\nabla u(s)\|^2ds+\int^r_{t-\rho}\|\nabla G(x,u(s-\rho))\|^2ds\\& \leq\int^r_{t-\rho}\|\nabla u(s)\|^2ds+2\int^r_{t-\rho}\|\hat{h}(x)\|^2ds+2\hat{a}^2\int^r_{t-\rho}\|\nabla u(s-\rho)\|^2ds\\& \leq\int^r_{t-\rho}\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2ds\!+\!2\rho\|\hat{h}(x)\|^2+2\hat{a}^2\int^{r-\rho}_{t-2\rho}\|\nabla u(s)\|^2ds\!+c\int^r_{t-\rho}\|u(s)\|^2ds. \end{split} \end{align} | (4.17) |
For the third term on the right-hand side of (4.15), applying (2.6) and (2.7), we have
\begin{align} \begin{split} &\ \ \ \ \sum\limits^\infty_{k = 1}\int^r_{t-\rho}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds \leq2\sum\limits^\infty_{k = 1}\int^r_{t-\rho}(\|\nabla\sigma_{1,k}\|^2+\|\nabla(\kappa\sigma_{2,k}(u(s)))\|^2)ds\\& \leq 2\rho\sum\limits^\infty_{k = 1}\|\nabla \sigma_{1,k}\|^2+8\rho\left(\|\nabla \kappa\|^2\sum\limits^\infty_{k = 1}\beta^2_k+\|\kappa\|^2\sum\limits^\infty_{k = 1}\hat{\beta}^2_k\right)\\&\ \ \ \ \ +8C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k\int^r_{t-\rho}\|u(s)\|^2ds +8\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\int^r_{t-\rho}\|\nabla u(s)\|^2ds. \end{split} \end{align} | (4.18) |
By (4.4) and (4.15)–(4.18), we infer that for all t\geq \rho and t-\rho\leq r\leq t ,
\begin{align} \begin{split} &\ \ \ \ \|\nabla u(r)\|^2\leq c_1+\|\nabla u(t-\rho)\|^2+c_2\int^r_{t-2\rho}\|u(s)\|^2ds+2\hat{a}^2\int^{r}_{t-2\rho}\|\nabla u(s)\|^2ds\\& \ \ \ \ \ +2\sum\limits^\infty_{k = 1}\mbox{Re}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s), \end{split} \end{align} | (4.19) |
where c_1 and c_2 are positive constants. By (4.19), we deduce that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|\nabla u(r)\|^2\right]\leq c_1+\mathbb{E}\left[\|\nabla u(t-\rho)\|^2\right]+c_2\int^r_{t-2\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]ds\\& \ \ \ \ \ +2\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\left|\sum\limits^\infty_{k = 1}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s)\right|\right]. \end{split} \end{align} | (4.20) |
For the second term on the right-hand side of (4.20), by Lemma 4.1 we infer that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\|\nabla u(t-\rho)\|^2\right]\leq \sup\limits_{s\geq-\rho}\mathbb{E}\left[\|\nabla u(s)\|^2\right]\leq c_3\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_3. \end{split} \end{align} | (4.21) |
For the third term on the right-hand side of (4.20), by Lemmas 3.1 and 4.1 we infer that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ c_2\int^r_{t-2\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]ds\leq 2\rho c_2 \sup\limits_{s\geq-\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]\leq c_4\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_4. \end{split} \end{align} | (4.22) |
For the last term on the right-hand side of (4.20), by BDG inequality, (4.18), Lemmas 3.1 and 4.1, we deduce that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ 2\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\left|\sum\limits^\infty_{k = 1}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s)\right|\right]\\& \leq2c_5\mathbb{E}\left[\left(\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\left|\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)\right|^2ds\right)^{\frac 12}\right]\\& \leq2c_5\mathbb{E}\left[\left(\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\|\nabla u(s)\|^2\|\nabla\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s))\right)\|^2ds\right)^{\frac 12}\right]\\&\leq2c_5\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|\left(\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\|\nabla\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s))\right)\|^2ds\right)^{\frac 12}\right]\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+2c_5^2\mathbb{E}\left[\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\|\nabla\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s))\right)\|^2ds\right]\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+c_6+c_6\int^t_{t-\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]ds\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+c_6+\rho c_6\left(\sup\limits_{s\geq 0}\mathbb{E}\|u(s)\|^2+\sup\limits_{s\geq 0}\|\nabla u(s)\|^2\right)\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+c_7\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_7. \end{split} \end{align} | (4.23) |
By (4.20)–(4.23), we obtain that for all t\geq \rho ,
\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|\nabla u(r)\|^2\right]\leq c_8\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_9, |
which completes the proof.
Lemma 4.3. Suppose (2.1)–(2.7) and (3.1) hold. If \varphi(s)\in L^{2p}(\Omega, C([-\rho, 0], V)) , then there exists a positive constant \mu_5 such that the solution u of (1.1) and (1.2) satisfies
\begin{align} \begin{split} &\ \sup\limits_{t\geq -\rho}\mathbb{E}[\|\nabla u(t)\|^{2p}]+\sup\limits_{t\geq 0}\mathbb{E}\left[\int^t_0e^{\mu_5(s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \ \ \ \ \ \ \ \leq M_5\left(\mathbb{E}\left[\|\varphi\|^{2p}_{C_V}\right]+1\right), \end{split} \end{align} | (4.24) |
where M_5 is a positive constant independent of \varphi .
Proof. By (3.1), there exist positive constants \mu and \epsilon_1 such that
\begin{align} \begin{split} &\ \mu+4(p-1)\epsilon_1^{\frac {p}{p-1}}+\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p} +8C^2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k +8p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\\&\ \ \ \ \ \ \ \ \ +2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\leq 2p\lambda. \end{split} \end{align} | (4.25) |
By (1.1) and applying Ito's formula to e^{\mu t}\|\nabla u(t)\|^{2p} , we get for t\geq 0 ,
\begin{align*} &\ \ \ \ e^{\mu t}\|\nabla u(t)\|^{2p}+2p\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\\&\ \ \ \ \ \ +2p\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\\& = \|\nabla\varphi(0)\|^{2p}+(\mu-2p\lambda)\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\\&\ \ \ \ \ +2p\mbox{Re}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(G(x,u(s-\rho)),-\Delta u(s))ds\\& \ \ \ \ \ +p\sum\limits^\infty_{k = 1}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\\& \ \ \ \ \ +2p\sum\limits^\infty_{k = 1}\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-1)}\left(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s)\\& \ \ \ \ \ +2p(p-1)\sum\limits^\infty_{k = 1}\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-2)}\left|(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s))\right|^2ds. \end{align*} |
Taking the expectation, we have for t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ e^{\mu t}\mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \ \ \ \ \ +2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\right]\\& = \mathbb{E}\left[\|\nabla\varphi(0)\|^{2p}\right]+(\mu-2p\lambda)\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ \ +2p\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\\& \ \ \ \ \ +p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\& \ \ \ \ \ +2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-2)}\left|(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s))\right|^2ds\right]. \end{split} \end{align} | (4.26) |
By (4.8), we get the third term on the left-hand side of (4.26) is nonnegative. Next, we estimate each term on the right-hand side of (4.26). For the third term on the right-hand side of (4.26), applying (2.2), Gagliardo-Nirenberg inequality and Young's inequality, we deduce
\begin{align} \begin{split} &\ \ \ \ 2p\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\\&\leq2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s)\|\|\nabla G(x,u(s-\rho))\|ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s)\|^2ds\right]+p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla G(x,u(s-\rho))\|^2ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s)\|^2ds\right]\\&\ \ \ \ +2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\hat{h}\|^2ds\right]+2p\hat{a}^2\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s-\rho)\|^2ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2+c\|u(s)\|^2)ds\right]+\frac 2{\epsilon_1^p}\|\hat{h}(x)\|^{2p}\int^t_0e^{\mu s}ds\\&\ \ \ \ +4(p-1)\epsilon_1^{\frac {p}{p-1}}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right] +\frac{2p\hat{a}^{2}}{\epsilon_1^p}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2ds\right]+\left(4(p-1)\epsilon_1^{\frac {p}{p-1}}+\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\right)\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ +\frac 1{\mu\epsilon_1^p}\|\hat{h}(x)\|^{2p}e^{\mu t} +\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}\left[\|\nabla \varphi(s)\|^{2p}\right]e^{\mu t}+c\sup\limits_{s\geq 0}\mathbb{E}\left[\| u(s)\|^{2p}\right]e^{\mu t}. \end{split} \end{align} | (4.27) |
For the forth term on the right-hand side of (4.26), applying (2.7), we infer
\begin{align} \begin{split} &\ \ \ \ p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\& \leq2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p\!-\!1)}\|\nabla\sigma_{1,k}\|^2ds\right] \!+\!2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p\!-\!1)}\|\nabla(\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\&\leq 2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla\sigma_{1,k}\|^2ds\right]\\&\ \ \ \ +8p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\left(\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!\hat{\beta}^2_k\|\kappa\|^2\!\!+\!\!{\gamma}^2_kC^2\|u(s)\|^2\!\!+\!\!\hat{\gamma}^2_k\|\kappa\|^2_{L^\infty}\|\nabla u(s)\|^2\right)ds\right]\\&\leq2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)ds\right]\\&\ \ \ \ +8C^2p\sum\limits^\infty_{k = 1}{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|u(s)\|^2ds\right] +8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]. \end{split} \end{align} | (4.28) |
Then applying Young's inequality, (4.28) can be estimated by
\begin{align} \begin{split} &\ \ \ \ p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\& \leq\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\times\mathbb{E}\left[\int^t_0e^{\mu s}\left((2p-2)\epsilon_1^{\frac p{p-1}}\|\nabla u(s)\|^{2p}+\frac2{\epsilon_1^p}\right)ds\right]\\&\ \ \ \ +2C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\left((4p-4)\epsilon_1^{\frac p{p-1}}\|\nabla u(s)\|^{2p}+\frac4{\epsilon_1^p}\|u(s)\|^{2p}\right)ds\right]\\&\ \ \ \ +8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\& = \left[(2p-2)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\right.\\&\ \ \ \ \left.+2C^2(4p-4)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k+8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right] \mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ \!+\!\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2\!+\!4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac2{\epsilon_1^p}e^{\mu t} \!+\!\frac{8C^2}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]e^{\mu t}. \end{split} \end{align} | (4.29) |
For the fifth term on the right-hand side of (4.26), applying integrating by parts and (4.29), we get
\begin{align} \begin{split} &\ \ \ \ 2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-2)}\left|(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s))\right|^2ds\right]\\&\leq2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2p-2}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\&\leq2(p-1)\left[(2p-2)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\right.\\&\ \ \ \ \left.+2C^2(4p-4)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k+8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right] \mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ \ \!+\!\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2\!+\!4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac{4(p\!-\!1)}{\epsilon_1^p}e^{\mu t} \!+\!\frac{16C^2(p\!-\!1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]e^{\mu t}. \end{split} \end{align} | (4.30) |
By (4.26), (4.27), (4.29) and (4.30), we obtain
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+p\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\& \leq\mathbb{E}\left[\|\nabla\varphi(0)\|^{2p}\right]e^{-\mu t}+\left[\mu-2p\lambda+4(\varrho-1)\epsilon_1^{\frac {p}{p-1}}+\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p} +8C^2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k\right.\\&\ \ \ \ \left.+8p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k+2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\right]\\&\ \ \ \ \times\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2p}ds\right]+\frac 1{\mu\epsilon_1^p}\|\hat{h}(x)\|^{2p} +\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}\left[\|\nabla \varphi(s)\|^{2p}\right]\\&\ \ \ \ +c\sup\limits_{s\geq 0}\mathbb{E}\left[\| u(s)\|^{2p}\right] +\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac{4p-2}{\epsilon_1^p}\\&\ \ \ \ +\frac{8C^2(2p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]. \end{split} \end{align} | (4.31) |
Then by (4.25) and (4.31), we deduce that for all t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+p\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\& \leq\mathbb{E}\left[\|\nabla\varphi(0)\|^{2p}\right]e^{-\mu t}+\frac 1{\mu\epsilon_1^p}\|\hat{h}(x)\|^{2p} +\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}\left[\|\nabla \varphi(s)\|^{2p}\right]\\&\ \ \ \ +c\sup\limits_{s\geq 0}\mathbb{E}\left[\| u(s)\|^{2p}\right] +\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac{4p-2}{\epsilon_1^p}\\&\ \ \ \ +\frac{8C^2(2p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]. \end{split} \end{align} | (4.32) |
Therefore, by (4.32) and Lemma 3.5, there exists a constant M_5 independent of \varphi such that
\begin{align} \begin{split} &\ \ \ \ \sup\limits_{t\geq -\rho}\mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+\sup\limits_{t\geq 0}\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\& \leq M_5(\mathbb{E}\left[\|\varphi\|^{2p}_{C_V}\right]+1). \end{split} \end{align} | (4.33) |
For convenience, we write A = (1+\text{i}\nu)(-\triangle)^\alpha+\lambda I . Then, similar to Theorem 6.5 in [31], the solution of (1.1) and (1.2) can be expressed as
\begin{align} \begin{split} &\ u(t) = e^{-At}u(0)-\int_0^te^{-A(t-s)}(1+\text{i}\mu)|u(s)|^{2\beta}u(s)ds\\&\ \ \ +\int_0^te^{-A(t-s)}G(\cdot,u(s-\rho))ds+\sum\limits^\infty_{k = 1}\int_0^te^{-A(t-s)}(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))dW_k(s). \end{split} \end{align} | (4.34) |
The next lemma is concerned with the H \ddot{o} lder continuity ofsolutions in time which is needed to prove the tightness of distributions of solutions.
Lemma 4.4. Suppose (2.1)–(2.7) and (3.1) hold. If \varphi(s)\in L^{2p}(\Omega, C([-\rho, 0], V)) , then the solution u of (1.1) and (1.2) satisfies, for any t > r\geq 0 ,
\begin{align} \begin{split} \mathbb{E}[\|u(t)-u(r)\|^{2p}]\leq M_6(|t-r|^{p}+|t-r|^{2p}), \end{split} \end{align} | (4.35) |
where M_6 is a positive constant depending on \varphi , but independent of t and r .
Proof. By (4.34), we get for t > r\geq 0 ,
\begin{align} \begin{split} &\ u(t) = e^{-A(t-r)}u(r)-\int_r^te^{-A(t-s)}(1+\text{i}\mu)|u(s)|^{2\beta}u(s)ds\\&\ \ \ +\int_r^te^{-A(t-s)}G(\cdot,u(s-\rho))ds+\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))dW_k(s). \end{split} \end{align} | (4.36) |
Then we infer
\begin{align} \begin{split} &\ \|u(t)-u(r)\|^{2p}\leq \frac{5^{2p}}{4}\left[\|(e^{-A(t-r)}-I)u(r)\|^{2p}+\|\int_r^te^{-A(t-s)}(1+\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}\right.\\&\ \ \ \left.+\|\int_r^te^{-A(t-s)}G(\cdot,u(s-\rho))ds\|^{2p}+\|\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))dW_k(s)\|^{2p}\right]. \end{split} \end{align} | (4.37) |
Taking the expectation of (4.36), we have for all t > r\geq 0 ,
\begin{align} \begin{split} &\ \mathbb{E}[\|u(t)-u(r)\|^{2p}]\leq \frac{5^{2p}}{4}\mathbb{E}[\|(e^{-A(t-r)}\!-\!I)u(r)\|^{2p}]\!+\!\frac{5^{2p}}{4}\mathbb{E}\left[\|\int_r^te^{-A(t-s)}(1\!+\!\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}\right]\\&\ \ \ \!+\!\frac{5^{2p}}{4}\mathbb{E}\left[\|\int_r^te^{-A(t-s)}G(\cdot,u(s\!-\!\rho))ds\|^{2p}\right] \!+\!\frac{5^{2p}}{4}\mathbb{E}\left[\|\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}\!+\!\kappa\sigma_{2,k}(u(s)))dW_k(s)\|^{2p}\right]. \end{split} \end{align} | (4.38) |
For the first term on the right-hand side of (4.38), by Theorem 1.4.3 in [32], we find that there exists a positive number C_0 depending on \varrho such that for all t > r\geq 0 ,
\begin{align*} \frac{5^{2p}}{4}\mathbb{E}[\|(e^{-A(t-r)}\!-\!I)u(r)\|^{2p}]\leq C_0(t-r)^p\mathbb{E}[\|u(r)\|^{2p}_{C_V}]. \end{align*} |
Applying Lemmas 3.5 and 4.3, we obtain for all t > r\geq 0 ,
\begin{align} \begin{split} \frac{5^{2p}}{4}\mathbb{E}[\|(e^{-A(t-r)}\!-\!I)u(r)\|^{2p}]\leq C_1(t-r)^p. \end{split} \end{align} | (4.39) |
For the second term on the right-hand side of (4.38), by the contraction property of e^{-At} , we infer that for all t > r\geq 0 ,
\begin{align*} &\ \ \ \mathbb{E}[\|\int_r^te^{-A(t-s)}(1\!+\!\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}] \leq(1\!+\!\mu^2)^p\mathbb{E}\left[\left(\int_r^t\||u(s)|^{2\beta+1}\|ds\right)^{2p}\right]\\& \leq (1\!+\!\mu^2)^p\mathbb{E}\left[\left(\int_r^t\||u(s)|^{2\beta+1}\|^{2p}ds\right)\right](t-r)^{2p-1}\\& \leq(1\!+\!\mu^2)^p\sup\limits_{s\geq 0}\mathbb{E}\left[\left(\|u(s)\|^{2(2\beta+1)}_{L^{2(2\beta+1)}}\right)^{p}\right](t-r)^{2p}. \end{align*} |
We deduce the estimate \sup\limits_{s\geq 0}\mathbb{E}\left[\left(\|u(s)\|^{2(2\beta+1)}_{L^{2(2\beta+1)}}\right)^{p}\right]\leq M'\left(\mathbb{E}[\|\varphi\|^2_{C_V}]+1\right) similarly to Lemma 3.5 together with Lemma 3.3 in [1]. Hence, the second term on the right-hand side of (4.38) can be estimated by
\begin{align} \begin{split} \mathbb{E}\left[\|\int_r^te^{-A(t-s)}(1\!+\!\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}\right] \leq C_2(t-r)^{2p}. \end{split} \end{align} | (4.40) |
For the third term on the right-hand side of (4.38), by the contraction property of e^{-At} and (2.1) and Lemma 3.5, we deduce that for all t > r\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \frac{5^{2p}}{4}\mathbb{E}\left[\|\int_r^te^{-A(t-s)}G(\cdot,u(s\!-\!\rho))ds\|^{2p}\right] \leq \frac{5^{2p}}{4}\mathbb{E}\left[\left(\int_r^t\|G(\cdot,u(s\!-\!\rho))\|ds\right)^{2p}\right]\\& \leq\frac{5^{2p}}{4}\mathbb{E}\left[\left(\int_r^t(\|h\|+a\|u(s\!-\!\rho)\|)ds\right)^{2p}\right]\\& \leq\frac{5^{2p}}{4}\mathbb{E}\left[\left(\int_r^t(\|h\|+a\|u(s\!-\!\rho)\|)^{2p}ds\right)\right](t-r)^{2p-1}\\& \leq\frac{10^{2p}}{8}(t-r)^{2p-1}\int_r^t\left(\|h\|^{2p}+a^{2p}\mathbb{E}\left[\|u(s\!-\!\rho)\|^{2p}\right]\right)ds\\& \leq\frac{10^{2p}}{8}\left(\|h\|^{2p}+a^{2p}\sup\limits_{t\geq -\rho}\mathbb{E}\left[\|u(s)\|^{2p}\right]\right)(t-r)^{2p}\leq C_3(t-r)^{2p}. \end{split} \end{align} | (4.41) |
For the forth term the right-hand side of (4.38), from the BDG inequality, the contraction property of e^{-At} , (2.6) H \ddot{o} lder's inequality and Lemma 3.5, we deduce
\begin{align*} &\frac{5^{2p}}{4}\mathbb{E}\left[\|\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}\!+\!\kappa\sigma_{2,k}(u(s)))dW_k(s)\|^{2p}\right]\\ &\leq\frac{5^{2p}}{4}C_4\mathbb{E}\left[\left(\int_r^t\sum\limits^\infty_{k = 1}\|e^{-A(t-s)}(\sigma_{1,k}\!+\!\kappa\sigma_{2,k}(u(s)))\|^2ds\right)^{p}\right]\\ &\leq\frac{5^{2p}}{4}C_4\mathbb{E}\left[\left(\int_r^t\sum\limits^\infty_{k = 1}2(\|\sigma_{1,k}\|^2+\|\kappa\sigma_{2,k}(u(s))\|^2)ds\right)^{p}\right]\\ &\leq\frac{5^{2p}}{4}C_4\mathbb{E}\left[\left(\int_r^t\sum\limits^\infty_{k = 1}2 (\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k+2\|\kappa\|^2_{L^\infty}\gamma^2_k\|(u(s))\|^2)ds\right)^{p}\right] \\ &\leq\frac{5^{2p}}{2}C_4\mathbb{E}\left[\left(2\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k)(t-r) +4\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\int_r^t\|u(s)\|^2ds\right)^{p}\right]\\& \leq\frac{10^{2p}}{8}C_4\left(\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k\right)^p(t-r)^p +\frac{10^{2p}}{8}C_4\left(2\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\right)^p\mathbb{E}\left[\left(\int_r^t\|u(s)\|^2ds\right)^{p}\right]\\& \leq\frac{10^{2p}}{8}C_4\left(\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k\right)^p(t-r)^p\\&\ \ \ \ +\frac{10^{2p}}{8}C_4\left(2\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\right)^p(t-r)^{p-1} \int_r^t\mathbb{E}\left[\|u(s)\|^{2p}\right]ds\\& \leq\frac{10^{2p}}{8}C_4\left(\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k\right)^p(t-r)^p\\&\ \ \ \ +\frac{10^{2p}}{8}C_4\left(2\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\right)^p(t-r)^{p} \sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]\\&\leq C_5(t-r)^p. \end{align*} | (4.42) |
Therefore, from (4.38)–(4.42), we obtain there exists C_6 > 0 independent of t and r , such that for all t > r\geq 0 ,
\mathbb{E}[\|u(t)-u(r)\|^{2p}]\leq C_6(|t-r|^{p}+|t-r|^{2p}). |
The proof is complete.
In this section, we first recall the definition of invariant measure and transition operator. Then we construct a compact subset of C([-\rho, 0];H) in order to prove the tightness of the sequence of invariant measure m_k on C([-\rho, 0];H) .
Recall that for any initial time t_0 and every \mathcal {F}_{t_0} -measurable function \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) , problems (1.1) and (1.2) has a unique solution u(t; t_0, \varphi) for t\in[t_0-\rho, \infty) . For convenience, given t\geq t_0 and \mathcal {F}_{t_0} -measurable function \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) , the segment of u(t; t_0, \varphi) on [t-\rho, t] is written as
u_t(t_0,\varphi)(s) = u(t+s;t_0,\varphi)\ for\ every\ s\in[-\rho,0]. |
Then u_t(t_0, \varphi)\in L^2(\Omega, C([-\rho, 0], H)) for all t\geq t_0 . We introduce the transition operator for (1.1). If \phi(s):C([-\rho, 0], H)\rightarrow \mathbb{R} is a bounded Borel function, then for initial time r with 0\leq r\leq t and \varphi(s)\in C([-\rho, 0], H) , we write
(p_{r,t}\phi)(\varphi) = \mathbb{E}[\phi(u_t(r,\varphi))]. |
Particularly, for \Gamma\in \mathcal {B}(C([-\rho, 0], H)) , 0\leq r\leq t and \varphi\in C([-\rho, 0], H) , we have
p(r,\varphi;t,\Gamma) = (p_{r,t}1_{\Gamma})(\varphi) = P\{\omega\in\Omega|u_t(r,\varphi)\in\Gamma\}, |
where 1_{\Gamma} is the characteristic function of \Gamma . Then p(r, \varphi; t, \cdot) is the distribution of u_t(0, \varphi) in C([-\rho, 0], H) . In the following context, we will write p_{0, t} as p_t .
Recall that a probability measure \mathscr{M} on C([-\rho, 0], H) is called an invariant measure, if for all t\geq0 and every bounded and continuous function \phi:C([-\rho, 0];H)\rightarrow \mathbb{R},
\int_{C([-\rho,0];H)}(p_t\phi)(\varphi)d\mathscr{M}(\varphi) = \int_{C([-\rho,0];H)}\phi(\varphi)d\mathscr{M}(\varphi),\ \ for\ all\ t\geq0. |
According to [33], we infer that the transition operator \{p_{r, t}\}_{0\leq r\leq t} has the following properties.
Lemma 5.1. Suppose (2.1)–(2.7) and (4.1)–(4.3) hold. One has
(a) The family \{p_{r, t}\}_{0\leq r\leq t} is Feller; that is, if \phi:C([-\rho, 0], H)\rightarrow \mathbb{R} is bounded and continuous, then for any 0\leq r\leq t , the function p_{r, t}\phi:C([-\rho, 0], H)\rightarrow \mathbb{R} is also bounded and continuous.
(b) The family \{p_{r, t}\}_{0\leq r\leq t} is homogeneous (in time); that is, for any 0\leq r\leq t ,
p(r,\varphi;t,\cdot) = p(0,\varphi;t-r,\cdot), \forall\varphi\in C([-\rho,0],H). |
(c) Given r\geq0 and \varphi\in C([-\rho, 0], H) , the process \{u_t(r, \varphi)\}_{t\geq r} is a C([-\rho, 0], H) -valued Markov process. Consequently, if \phi:C([-\rho, 0], H)\rightarrow \mathbb{R} is a bounded Borel function, then for any 0\leq s\leq r\leq t , P -almost surely,
(p_{s,t}\phi)(\varphi) = (p_{s,r}(p_{r,t}\phi))(\varphi), \forall\varphi\in C([-\rho,0],H), |
and the Chapman-Kolmogorov equation is valid:
p(s,\varphi;t,\Gamma) = \int_{C([-\rho,0],H)}p(s,\varphi;r,dy)p(r,y;t,\Gamma), |
for any \varphi\in C([-\rho, 0], H) and \Gamma\in\mathcal {B}(C([-\rho, 0], H)).
Now, we establish the existence of invariant measures of problems (1.1) and (1.2).
Theorem 5.2. Suppose (2.1)–(2.7) and (4.1)–(4.3) hold. Then (1.1) and (1.2) processes an invariant measure on C([-\rho, 0], H) .
Proof. We employ Krylov-Bogolyubov's method to the solution u(t, 0, 0) of problems (1.1) and (1.2), where the initial condition \varphi\equiv0 at the initial time 0. Because of this particular \varphi\in C([-\rho, 0], V)\subseteq C([-\rho, 0], H) , we know that all results obtained in the previous Sections 3 and 4 are valid. For simplicity, the solution u(t, 0, 0) is written as u(t) and the segment u_t(0, 0) as u_t . For k\in\mathbb{N}^+ , we set
\begin{align} \mathscr{M}_k = \frac1k\int^{k+\rho}_{\rho} p(0,0;t,\cdot)dt. \end{align} | (5.1) |
Step 1. We prove the tightness of \{\mathscr{M}_k\}_{k = 1}^\infty in C([-\rho, 0], H) . Applying Lemmas 3.2 and 4.2, we get that there exists C_1 > 0 such that for all t\geq \rho ,
\begin{align} \mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|^{2}_{V}\right]\leq C_1. \end{align} | (5.2) |
By (5.2) and Chebyshev's inequality, we have that for all t\geq \rho ,
\begin{align*} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|_{V}\geq R\right\}\right)\leq\frac 1{R^2}\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|^2_{V}\right]\leq\frac{C_1}{R^2}\rightarrow 0\ \ as\ \ R\rightarrow \infty, \end{align*} |
and hence for every \varepsilon > 0 , there exists R_1 = R_1(\varepsilon) > 0 such that for all t\geq \rho ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|_{V}\geq R_1\right\}\right)\leq \frac 13\varepsilon. \end{align} | (5.3) |
By Lemma 4.4, we get that there exists C_2 > 0 such that for all t\geq \rho and r, s\in[-\rho, 0] ,
\mathbb{E}[\|u_t(r)-u_t(s)\|^{2p}]\leq C_2(1+|r-s|^{p})|r-s|^{p}, |
and hence for all t\geq \rho and r, s\in[-\rho, 0] ,
\begin{align} \mathbb{E}[\|u_t(r)-u_t(s)\|^{2p}]\leq C_2(1+\rho^{p})|r-s|^{p}. \end{align} | (5.4) |
Since p\geq 2 , applying (5.4) and the usual technique of dyadic division, we obtain that there exists R_2 = R_2(\varepsilon) > 0 such that for all t\geq \rho ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq r\leq 0}\frac{\|u_t(r)-u_t(s)\|}{|r-s|^{\frac{p-1}{4p}}} \leq R_2\right\}\right)\geq1-\frac 13\varepsilon. \end{align} | (5.5) |
By Lemma 3.4, we get that for given \varepsilon > 0 and m\in\mathbb{N} , there exists an integer n_m = n_m(\varepsilon, m)\geq 1 such that for all t\geq \rho ,
\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx\right]\leq \frac{\varepsilon}{2^{2m+2}}, |
which implies that for all t\geq \rho and m\in\mathbb{N} ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx \geq \frac 1{2^m}\right\}\right)\leq 2^m\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx\right]\leq \frac{\varepsilon}{2^{m+2}}. \end{align} | (5.6) |
By (5.6), we infer that for all t\geq \rho ,
P\left(\bigcup\limits_{m = 1}^{\infty}\left\{\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx \geq \frac 1{2^m}\right\}\right)\leq \sum\limits^\infty_{k = 1}\frac{\varepsilon}{2^{m+2}}\leq \frac14\varepsilon, |
and hence for all t\geq \rho ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx \leq \frac 1{2^m}\ for \ all\ m\in\mathbb{N}\right\}\right)\geq 1-\frac14\varepsilon. \end{align} | (5.7) |
Let
\begin{align} \mathcal {M}_{1,\varepsilon} = \left\{\zeta:[-\rho,0]\rightarrow V,\sup\limits_{-\rho\leq s\leq 0}\|\zeta(s)\|_{V}\leq R_1(\varepsilon)\right\}, \end{align} | (5.8) |
\begin{align} \mathcal {M}_{2,\varepsilon} = \left\{\zeta\in C([-\rho,0],H):\sup\limits_{-\rho\leq s\leq r\leq 0}\frac{\|\zeta(r)-\zeta(s)\|}{|r-s|^{\frac{\varrho-1}{4\varrho}}} \leq R_2(\varepsilon)\right\}, \end{align} | (5.9) |
\begin{align} \mathcal {M}_{3,\varepsilon} = \left\{\zeta\in C([-\rho,0],H):\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|\zeta(s,x)|^{2}dx \leq \frac 1{2^m}\ for \ all\ m\in\mathbb{N}\right\}, \end{align} | (5.10) |
and
\begin{align} \mathcal {M}_{\varepsilon} = \mathcal {M}_{1,\varepsilon}\bigcap\mathcal {M}_{2,\varepsilon}\bigcap\mathcal {M}_{3,\varepsilon}. \end{align} | (5.11) |
From (5.3), (5.5) and (5.7)–(5.11), we obtain that for all t\geq \rho ,
\begin{align} P\left(u_t\in\mathcal {M}_{\varepsilon}\right) > 1-\varepsilon. \end{align} | (5.12) |
By (5.1) and (5.12), we deduce that for all k\in\mathbb{N} ,
\begin{align} \mathscr{M}_k\left(\mathcal {M}_{\varepsilon}\right) > 1-\varepsilon. \end{align} | (5.13) |
Next, we prove the set \mathcal {M}_{\varepsilon} is precompact in C([-\rho, 0], H) . First, we prove for every s\in[-\rho, 0] the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} is a precompact subset of H . By (5.8) and (5.11), we obtain that for every s\in[-\rho, 0] , the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} is bounded in V . Let \mathcal {Q}_{m_0} = \left\{x\in\mathbb{R}^n:|x| < n_{m_0}\right\} . Then we get that the set \{\zeta(s)|_{\mathcal {Q}_{m_0}}:\zeta\in\mathcal {M}_{\varepsilon}\} is bounded in H^1(\mathcal {Q}_{m_0}) and hence precompact in L^2(\mathcal {Q}_{m_0}) due to compactness of the embedding H^1(\mathcal {Q}_{m_0})\hookrightarrow L^2(\mathcal {Q}_{m_0}) . This implies that the set \{\zeta(s)|_{\mathcal {Q}_{m_0}}:\zeta\in\mathcal {M}_{\varepsilon}\} has a finite open cover of balls with radius \frac 12\delta in L^2(\mathcal {Q}_{m_0}) . Note that for every \delta > 0 , there exists m_0 = m_0(\delta)\in\mathbb{N} such that for all \zeta\in\mathcal {M}_{\varepsilon} ,
\begin{align} \int_{|x|\geq n_{m_0}}|\zeta(s,x)|^{2}dx \leq \frac 1{2^{m_0}} < \frac{\delta^2}8. \end{align} | (5.14) |
Hence, by (5.14), the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} has a finite open cover of balls with radius \frac 12\delta in L^2(\mathbb{R}^n) . Since \delta > 0 is arbitrary, we obtain that the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} is percompact in H . Then from (5.9) and (5.11), we obtain that \mathcal{M}_{\varepsilon} is equicontinuous in C([-\rho, 0], H) . Therefore, by the Ascoli-Arzel \grave{a} theorem we deduce that \mathcal {M}_{\varepsilon} is precompact in C([-\rho, 0], H) , which along with (5.13) shows that \{m_k\}_{k = 1}^\infty is tight on C([-\rho, 0], H) .
Step 2. We prove the existence of invariant measures of problems (1.1) and (1.2). Since the sequence \{\mathscr{M}_k\}_{k = 1}^\infty is tight on C([-\rho, 0];H) , there exists a probability measure m on C([-\rho, 0];H) , we take a subsequence of \{\mathscr{M}_k\} (not rebel) such that \mathscr{M}_k\rightarrow m, \ \ as\ \ k\rightarrow \infty. In the following, we prove \mathscr{M} is an invariant measure of (1.1) and (1.2). Applying (5.1) and the Chapman-Kolmogorov equation, we obtain that for every t\geq0 and every \phi:C([-\rho, 0];H)\rightarrow \mathbb{R} ,
\begin{align*} &\ \ \ \ \int_{C([-\rho,0];H)}\phi(v)d\mathscr{M}(v) = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\phi(v)p(0,0;s,dv)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho-t}_{\rho-t}\left(\int_{C([-\rho,0];H)}\phi(v)p(0,0;s+t,dv)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\phi(v)p(0,0;s+t,dv)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\left(\int_{C([-\rho,0];H)}\phi(v)p(s,\varphi;s+t,dv)\right)p(0,0;s,d\varphi)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\left(\int_{C([-\rho,0];H)}\phi(v)p(0,\varphi;t,dv)\right)p(0,0;s,d\varphi)\right)ds\\& = \int_{C([-\rho,0];H)}\left(\int_{C([-\rho,0];H)}\phi(v)p(0,\varphi;t,dv)\right)d\mathscr{M}(\varphi)\\& = \int_{C([-\rho,0];H)}(p_{0,t}\phi)(\varphi)d\mathscr{M}(\varphi), \end{align*} |
which completes the proof.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors are grateful to the anonymous referees whose suggestions have in our opinion, greatly improved the paper. This work is partially supported by the NSF of Shandong Province (No. ZR 2021MA055) and USA Simons Foundation (No. 628308).
The authors declare there is no conflict of interest.
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