Prostate cancer is ranked as the fourth most prevalent cancer commonly diagnosed among males over 40 years of age, according to the WHO Cancer Fact Sheet 2020, and it is additionally a leading cause of cancer mortality among males. The incidence of prostate cancer and mortality varied significantly across the globe. Diagnosis of prostate cancer hinders easier management of cases, and prostate-specific antigen (PSA) use for screening of prostate cancer has poor specificity and sensitivity, thereby yielding overdiagnosis and unnecessary biopsies. Radiologically guided (ultrasound/MRI) prostate biopsy, considered the gold standard, is invasive and can miss a significant number of metastatic cancers. Even though mild, other prostate biopsy complications occur on a large scale, and few severe ones are often recorded. Scientists intensify their search for biomarker(s) for non-invasive diagnosis of prostate cancer using proteomics, metabolomics, genomics, and bioinformatics—urinary biomarkers were uniquely on the lookout. Vasoactive intestinal peptide (VIP)/pituitary adenylate cyclase-activating peptide (PACAP) receptor 1 (VPAC1), which is overexpressed (a thousandfold) in prostate cancer at the onset of oncogenesis and is excreted in the urine on tumor cells, is a contender in the prostate cancer biomarker quest. VPAC1 is ubiquitous, expressed by normal and malignant cells, and interwoven in their cell membranes. Therefore, using urine samples limits the possibility of making the wrong diagnosis, since VPAC1 is not normally excreted in the urine. Nevertheless, studying transmembrane receptors is intricate. However, producing monoclonal antibodies against the N-terminal end of VPAC1 can provide a promising target for designing a non-invasive diagnostic assay for early detection of prostate cancer using a urine sample.
Citation: Mansur Aliyu, Ali Akbar Saboor-Yaraghi, Shima Nejati, Behrouz Robat-Jazi. Urinary VPAC1: A potential biomarker in prostate cancer[J]. AIMS Allergy and Immunology, 2022, 6(2): 42-63. doi: 10.3934/Allergy.2022006
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Prostate cancer is ranked as the fourth most prevalent cancer commonly diagnosed among males over 40 years of age, according to the WHO Cancer Fact Sheet 2020, and it is additionally a leading cause of cancer mortality among males. The incidence of prostate cancer and mortality varied significantly across the globe. Diagnosis of prostate cancer hinders easier management of cases, and prostate-specific antigen (PSA) use for screening of prostate cancer has poor specificity and sensitivity, thereby yielding overdiagnosis and unnecessary biopsies. Radiologically guided (ultrasound/MRI) prostate biopsy, considered the gold standard, is invasive and can miss a significant number of metastatic cancers. Even though mild, other prostate biopsy complications occur on a large scale, and few severe ones are often recorded. Scientists intensify their search for biomarker(s) for non-invasive diagnosis of prostate cancer using proteomics, metabolomics, genomics, and bioinformatics—urinary biomarkers were uniquely on the lookout. Vasoactive intestinal peptide (VIP)/pituitary adenylate cyclase-activating peptide (PACAP) receptor 1 (VPAC1), which is overexpressed (a thousandfold) in prostate cancer at the onset of oncogenesis and is excreted in the urine on tumor cells, is a contender in the prostate cancer biomarker quest. VPAC1 is ubiquitous, expressed by normal and malignant cells, and interwoven in their cell membranes. Therefore, using urine samples limits the possibility of making the wrong diagnosis, since VPAC1 is not normally excreted in the urine. Nevertheless, studying transmembrane receptors is intricate. However, producing monoclonal antibodies against the N-terminal end of VPAC1 can provide a promising target for designing a non-invasive diagnostic assay for early detection of prostate cancer using a urine sample.
prostate-specific antigen;
vasoactive intestinal peptide;
pituitary adenylate cyclase-activating peptide;
VIP/PACAP receptor 1;
age-standardized rate;
benign prostatic hypertrophy;
5α-dihydrotestosterone;
quantitative reverse transcription polymerase chain reaction;
magnetic resonance imaging;
multiparametric MRI;
dynamic contrast-enhanced MRI;
diffusion-weighted imaging MRI;
trans-rectal ultrasound scan;
enzyme-linked immunosorbent assay;
prostate cancer antigen 3;
transmembrane protease serine 2;
glutathione S-transferase P1;
digital rectal examination;
area under the curve;
hepatocyte growth factor;
insulin-like growth factor binding protein 3;
Plasma osteopontin;
lymph node carcinoma of the prostate;
G protein-coupled receptors;
receptor activity modifying proteins;
seven-transmembrane domain;
extracellular domain;
Chinese hamster ovary;
TATA-box binding protein;
cyclooxygenase-2;
metalloproteinase 9;
urokinase plasminogen activator;
uPA receptor;
4,6-diamidino-2-phenylindole;
adenylyl cyclase;
cAMP response element-binding protein;
phosphokinase A;
inducible nitric oxide synthase;
CREB binding protein;
nuclear factor-κβ;
extracellular signal-regulated kinase;
MAP/ERK kinase (MEK) kinase 1;
IFN regulatory factor-1;
inhibitory κβ kinase;
bombesin-like peptides;
gastrin-releasing peptide receptor;
phospholipase C;
diacylglycerol;
1,4,5-triphosphate;
protein kinase C;
mitogen-activated protein kinase;
mitogen-activated protein kinase kinase;
ETS like-1 protein;
serum response element
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
p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds]≤[4(p−1)ϵ2p2p−21∞∑k=1(‖σ1,k‖2+‖κ‖2β2k)+4p‖κ‖2L∞∞∑k=1γ2k]E[∫t∧τn0eμs‖u(s)‖2pds] +2μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k))eμt. | (3.52) |
For the fifth term on the right-hand side of (3.47), applying (3.52), we have
2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]≤2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2p−2‖σ1,k+κσ2,k(u(s))‖2ds]≤[8(p−1)2ϵ2p2p−21∞∑k=1(‖σ1,k‖2+‖κ‖2β2k)+8p(p−1)‖κ‖2L∞∞∑k=1γ2k]E[∫t∧τn0eμs‖u(s)‖2pads] +4(p−1)μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k))eμt. | (3.53) |
From (3.47), (3.48), (3.52) and (3.53), we obtain that for t≥0,
E[eμ(t∧τn)‖u(t∧τn)‖2p]+2pE[∫t∧τn0eμs‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds]≤(1+aρeμρ2p(4p−2)2p−12p)E[‖φ‖2pCH] +(μ−2pλ+aeμρ2p21−12p(2p−1)2p−12p+4(p−1)(2p−1)ϵ2p2p−21 ×∞∑k=1(‖σ1,k‖2+‖κ‖2β2k)+4p(2p−1)‖κ‖2L∞∞∑k=1γ2k)E[∫t∧τn0eμs‖u(s)‖2pds] +1μ(4p−2a2peμρ)2p−12p‖h‖2peμt+4(p−1)μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k))eμt. | (3.54) |
Then by (3.42) and (3.54), we obtain that for t≥0,
E[eμ(t∧τn)‖u(t∧τn)‖2p]+2pE[∫t∧τn0eμs‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds]≤(1+aρeμρ2p(4p−2)2p−12p)E[‖φ‖2pCH]+1μ(4p−2a2paeμρ)2p−12p‖h‖2peμt +4(p−1)μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k))eμt. | (3.55) |
Letting n→∞, by Fatou's Lemma, we deduce that for t≥0,
E[eμt‖u(t)‖2p]+2pE[∫t0eμs‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds]≤(1+aρeμρ2p(4θ−2)2θ−12p)E[‖φ‖2pCH]+1μ(4p−2a2peμρ)2p−12p‖h‖2peμt +4(p−1)μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k))eμt. |
Hence, we have for t≥0,
E[‖u(t)‖2p]+2pE[∫t0eμ(s−t)‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds]≤(1+aρeμρ2p(4p−2)2p−12p)E[‖φ‖2pCH]+1μ(4p−2a2peμρ)2p−12p‖h‖2p +4(p−1)μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k)). |
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([−ρ,0],V). To the end, we assume that for each k∈N, the function σ1,k∈V and
∞∑k=1‖σ1,k‖2V<∞. | (4.1) |
Furthermore, we assume that the function κ∈V and there exists a constant C>0 such that
|∇κ(x)|≤C. | (4.2) |
In the sequel, we further assume that the constant a, ˆγk in (2.7) are sufficiently small in the sense that there exists a constant p≥2 such that
ˆa21−12p(2p−1)2p−12p+2p(2p−1)‖κ‖2L∞∞∑k=1(β2k+ˆβ2k+γ2k+ˆγ2k)<pλ2. | (4.3) |
By (4.3), we can find
√2ˆa+2‖κ‖2L∞∞∑k=1ˆγ2k<λ2. | (4.4) |
Lemma 4.1. Suppose (2.1)–(2.7) and (4.4) hold. If φ(s)∈L2(Ω;C([−ρ,0],V)), then, for all t≥0, there exists a positive constant μ4 such that the solution u of (1.1) and (1.2) satisfies
sups≥−ρE[‖∇u(t)‖2]+sups≥0E[∫t0eμ4(s−t)‖(−Δ)α+12u(s)‖2ds]≤M4(E[‖φ‖2CV]+1), | (4.5) |
where M4 is a positive constant independent of φ.
Proof. By (4.4), there exists a positive constant μ1 such that
μ1−2λ+8‖κ‖2L∞∞∑k=1ˆγ2k<0. | (4.6) |
By (1.1) and applying Ito's formula to eμ1t‖∇u(t)‖2, we have for t≥0,
eμ1t‖∇u(t)‖2+2∫t0eμ1s‖(−Δ)α+12u(s)‖2ds+2∫t0eμ1sRe((1+iμ)|u(s)|2βu(s),−Δu(s))ds=(μ1−2λ)∫t0eμ1s‖∇u(s)‖2ds+‖∇φ(0)‖2+2Re∫t0eμ1s(G(x,u(s−ρ)),−Δu(s))ds +∞∑k=1∫t0eμ1s‖∇(σ1,k+κσ2,k(u(s)))‖2ds +2∞∑k=1Re∫t0eμ1s(σ1,k(x)+κ(x)σ2,k(u(s)),−Δu(s))dWk(s). |
Taking the expectation, we have for all t≥0,
eμ1tE[‖∇u(t)‖2]+2E[∫t0eμ1s‖(−Δ)α+12u(s)‖2ds]+2E[∫t0eμ1sRe((1+iμ)|u(s)|2βu(s),−Δu(s))ds]=(μ1−2λ)E[∫t0eμ1s‖∇u(s)‖2ds]+E[‖∇φ(0)‖2]+2E[Re∫t0eμ1s(G(x,u(s−ρ)),−Δu(s))ds] +∞∑k=1E[∫t0eμ1s‖∇(σ1,k+κσ2,k(u(s)))‖2ds]. | (4.7) |
First, we estimate the third term on the left-hand side of (4.7). Applying integrating by parts, we have
Re((1+iμ)|u|2βu,Δu)=−Re(1+iμ)∫Rn((β+1)|u|2β|∇u|2+β|u|2(β−1)(u∇¯u)2)dx=∫Rn|u|2(β−1)(−(β+1)|u|2|∇u|2+β(1+iμ)2(u∇¯u)2+β(1−iμ)2(¯u∇u)2)dx=∫Rn|u|2(β−1)trace(YMYH), | (4.8) |
where
Y=(¯u∇uu∇¯u)H,M=(−β+12β(1+iμ)2β(1−iμ)2−β+12), |
and YH is the conjugate transpose of the matrix Y. We observe that the condition β≤1√1+μ2−1 implies that the matrix M is nonpositive definite. Hence, we obtain
2E[∫t0eμ1sRe((1+iμ)|u(s)|2βu(s),Δu(s))ds]≤0. | (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
2E[Re∫t0eμ1s(G(x,u(s−ρ)),−Δu(s))ds]≤2E[∫t0eμ1s‖∇u(s)‖‖∇G(x,u(s−ρ))‖ds]≤E[∫t0eμ1s‖∇u(s)‖2ds]+E[∫t0eμ1s‖∇G(x,u(s−ρ))‖2ds]≤E[∫t0eμ1s‖∇u(s)‖2ds]+2E[∫t0eμ1s‖ˆh(x)‖2ds]+2ˆa2E[∫t0eμ1s‖∇u(s−ρ)‖2ds]≤E[∫t0eμ1s‖(−Δ)α+12u(s)‖2ds]+2μ1‖ˆh(x)‖2eμ1t +cμ1sups≥0E[‖u(s)‖2]eμ1t+2ˆa2μ1sup−ρ≤s≤0E[‖∇φ(s)‖2]eμ1t, | (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
∞∑k=1E[∫t0eμ1s‖∇(σ1,k+κσ2,k(u(s)))‖2ds]≤2∞∑k=1E[∫t0eμ1s(‖∇σ1,k‖2+‖∇(κσ2,k(u(s)))‖2)ds]≤2μ1∞∑k=1‖∇σ1,k‖2eμ1t+8∞∑k=1E[∫t0eμ1s(β2k‖∇κ‖2+ˆβ2k‖κ‖2+γ2kC2‖u(s)‖2+ˆγ2k‖κ‖2L∞‖∇u(s)‖2)ds]≤2μ1∞∑k=1(‖∇σ1,k‖2+4β2k‖∇κ‖2+4ˆβ2k‖κ‖2+4C2γ2ksups≥0E[‖u(s)‖2])eμ1t +8∞∑k=1ˆγ2k‖κ(x)‖2L∞E[∫t0eμ1s‖∇u(s)‖2ds]. | (4.11) |
By (4.7), (4.10) and (4.11), we obtain
E[‖∇u(t)‖2]+E[∫t0eμ1(s−t)‖(−Δ)α+12u(s)‖2ds] ≤E[‖∇φ(0)‖2]e−μ1t+2μ1‖ˆh(x)‖2+(μ1−2λ+8‖κ‖2L∞∞∑k=1ˆγ2k)E[∫t0eμ1s‖∇u(s)‖2ds] +2μ1(c2+4(C2∞∑k=1γ2k+c‖κ‖2L∞∞∑k=1ˆγ2k))sups≥−ρE[‖u(s)‖2] +2μ1∞∑k=1(‖∇σ1,k‖2+4(β2k+ˆβ2k)‖κ‖2V)+2ˆa2μ1sup−ρ≤s≤0E[‖∇φ(s)‖2]. | (4.12) |
Then by (4.6) and (4.12), we obtain that for all t≥0,
E[‖∇u(t)‖2]+E[∫t0eμ1(s−t)‖(−Δ)α+12u(s)‖2ds] ≤E[‖∇φ(0)‖2]e−μ1t+2μ1‖ˆh(x)‖2+2μ1(c2+4(C2∞∑k=1γ2k+c‖κ‖2L∞∞∑k=1ˆγ2k))sups≥−ρE[‖u(s)‖2] +2μ1∞∑k=1(‖∇σ1,k‖2+4(β2k+ˆβ2k)‖κ‖2V)+2ˆa2μ1sup−ρ≤s≤0E[‖∇φ(s)‖2]. | (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 φ(s)∈L2(Ω;C([−ρ,0],V)), then the solution u of (1.1) and (1.2) satisfies
supt≥ρ{E[supt−ρ≤r≤t‖∇u(r)‖2]}≤M5(E[‖φ‖2CV]+1), | (4.14) |
where M5 is a positive constant independent of φ.
Proof. By (1.1) and Ito's formula, we get for all t≥ρ and t−ρ≤r≤t,
‖∇u(r)‖2+2∫rt−ρ‖(−Δ)α+12u(s)‖2ds +2∫rt−ρRe((1+iμ)|u(s)|2βu(s),−Δu(s))ds+2λ∫rt−ρ‖∇u(s)‖2ds=‖∇u(t−ρ)‖2+2Re∫rt−ρ(G(x,u(s−ρ)),−Δu(s))ds +∞∑k=1∫rt−ρ‖∇(σ1,k+κσ2,k(u(s)))‖2ds +2∞∑k=1Re∫rt−ρ(σ1,k(x)+κ(x)σ2,k(u(s)),−Δu(s))dWk(s). | (4.15) |
For the third term on the left-hand side of (4.15), applying (4.8), we have
−2∫rt−ρRe((1+iμ)|u(s)|2βu(s),−Δu(s))ds≤0. | (4.16) |
For the second term on the right-hand side of (4.15), applying (2.2) and Gagliardo-Nirenberg inequality, we have
2Re∫rt−ρ(G(x,u(s−ρ)),−Δu(s))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‖ˆh(x)‖2ds+2ˆa2∫rt−ρ‖∇u(s−ρ)‖2ds≤∫rt−ρ‖(−Δ)α+12u(s)‖2ds+2ρ‖ˆh(x)‖2+2ˆa2∫r−ρt−2ρ‖∇u(s)‖2ds+c∫rt−ρ‖u(s)‖2ds. | (4.17) |
For the third term on the right-hand side of (4.15), applying (2.6) and (2.7), we have
∞∑k=1∫rt−ρ‖∇(σ1,k+κσ2,k(u(s)))‖2ds≤2∞∑k=1∫rt−ρ(‖∇σ1,k‖2+‖∇(κσ2,k(u(s)))‖2)ds≤2ρ∞∑k=1‖∇σ1,k‖2+8ρ(‖∇κ‖2∞∑k=1β2k+‖κ‖2∞∑k=1ˆβ2k) +8C2∞∑k=1γ2k∫rt−ρ‖u(s)‖2ds+8‖κ‖2L∞∞∑k=1ˆγ2k∫rt−ρ‖∇u(s)‖2ds. | (4.18) |
By (4.4) and (4.15)–(4.18), we infer that for all t≥ρ and t−ρ≤r≤t,
‖∇u(r)‖2≤c1+‖∇u(t−ρ)‖2+c2∫rt−2ρ‖u(s)‖2ds+2ˆa2∫rt−2ρ‖∇u(s)‖2ds +2∞∑k=1Re∫rt−ρ(σ1,k(x)+κ(x)σ2,k(u(s)),−Δu(s))dWk(s), | (4.19) |
where c1 and c2 are positive constants. By (4.19), we deduce that for all t≥ρ,
E[supt−ρ≤r≤t‖∇u(r)‖2]≤c1+E[‖∇u(t−ρ)‖2]+c2∫rt−2ρE[‖u(s)‖2+‖∇u(s)‖2]ds +2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρ(σ1,k(x)+κ(x)σ2,k(u(s)),−Δu(s))dWk(s)|]. | (4.20) |
For the second term on the right-hand side of (4.20), by Lemma 4.1 we infer that for all t≥ρ,
E[‖∇u(t−ρ)‖2]≤sups≥−ρE[‖∇u(s)‖2]≤c3E[‖φ‖2CV]+c3. | (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≥ρ,
c2∫rt−2ρE[‖u(s)‖2+‖∇u(s)‖2]ds≤2ρc2sups≥−ρE[‖u(s)‖2+‖∇u(s)‖2]≤c4E[‖φ‖2CV]+c4. | (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≥ρ,
2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρ(σ1,k(x)+κ(x)σ2,k(u(s)),−Δu(s))dWk(s)|]≤2c5E[(∞∑k=1∫tt−ρ|(σ1,k(x)+κ(x)σ2,k(u(s)),−Δu(s))|2ds)12]≤2c5E[(∞∑k=1∫tt−ρ‖∇u(s)‖2‖∇(σ1,k(x)+κ(x)σ2,k(u(s)))‖2ds)12]≤2c5E[supt−ρ≤s≤t‖∇u(s)‖(∞∑k=1∫tt−ρ‖∇(σ1,k(x)+κ(x)σ2,k(u(s)))‖2ds)12]≤12E[supt−ρ≤s≤t‖∇u(s)‖2]+2c25E[∞∑k=1∫tt−ρ‖∇(σ1,k(x)+κ(x)σ2,k(u(s)))‖2ds]≤12E[supt−ρ≤s≤t‖∇u(s)‖2]+c6+c6∫tt−ρE[‖u(s)‖2+‖∇u(s)‖2]ds≤12E[supt−ρ≤s≤t‖∇u(s)‖2]+c6+ρc6(sups≥0E‖u(s)‖2+sups≥0‖∇u(s)‖2)≤12E[supt−ρ≤s≤t‖∇u(s)‖2]+c7E[‖φ‖2CV]+c7. | (4.23) |
By (4.20)–(4.23), we obtain that for all t≥ρ,
E[supt−ρ≤r≤t‖∇u(r)‖2]≤c8E[‖φ‖2CV]+c9, |
which completes the proof.
Lemma 4.3. Suppose (2.1)–(2.7) and (3.1) hold. If φ(s)∈L2p(Ω,C([−ρ,0],V)), then there exists a positive constant μ5 such that the solution u of (1.1) and (1.2) satisfies
(4.24) |
where is a positive constant independent of .
Proof. By (3.1), there exist positive constants and such that
(4.25) |
By (1.1) and applying Ito's formula to , we get for ,
Taking the expectation, we have for ,
(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
(4.27) |
For the forth term on the right-hand side of (4.26), applying (2.7), we infer
(4.28) |
Then applying Young's inequality, (4.28) can be estimated by
(4.29) |
For the fifth term on the right-hand side of (4.26), applying integrating by parts and (4.29), we get
(4.30) |
By (4.26), (4.27), (4.29) and (4.30), we obtain
(4.31) |
Then by (4.25) and (4.31), we deduce that for all ,
(4.32) |
Therefore, by (4.32) and Lemma 3.5, there exists a constant independent of such that
(4.33) |
For convenience, we write . Then, similar to Theorem 6.5 in [31], the solution of (1.1) and (1.2) can be expressed as
(4.34) |
The next lemma is concerned with the Hlder 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 , then the solution of (1.1) and (1.2) satisfies, for any ,
(4.35) |
where is a positive constant depending on , but independent of and .
Proof. By (4.34), we get for ,
(4.36) |
Then we infer
(4.37) |
Taking the expectation of (4.36), we have for all ,
(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 depending on such that for all ,
Applying Lemmas 3.5 and 4.3, we obtain for all ,
(4.39) |
For the second term on the right-hand side of (4.38), by the contraction property of , we infer that for all ,
We deduce the estimate 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
(4.40) |
For the third term on the right-hand side of (4.38), by the contraction property of and (2.1) and Lemma 3.5, we deduce that for all ,
(4.41) |
For the forth term the right-hand side of (4.38), from the BDG inequality, the contraction property of , (2.6) Hlder's inequality and Lemma 3.5, we deduce
(4.42) |
Therefore, from (4.38)–(4.42), we obtain there exists independent of and , such that for all ,
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 in order to prove the tightness of the sequence of invariant measure on .
Recall that for any initial time and every -measurable function , problems (1.1) and (1.2) has a unique solution for . For convenience, given and -measurable function , the segment of on is written as
Then for all . We introduce the transition operator for (1.1). If is a bounded Borel function, then for initial time with and , we write
Particularly, for , and , we have
where is the characteristic function of . Then is the distribution of in . In the following context, we will write as .
Recall that a probability measure on is called an invariant measure, if for all and every bounded and continuous function
According to [33], we infer that the transition operator has the following properties.
Lemma 5.1. Suppose (2.1)–(2.7) and (4.1)–(4.3) hold. One has
(a) The family is Feller; that is, if is bounded and continuous, then for any , the function is also bounded and continuous.
(b) The family is homogeneous (in time); that is, for any ,
(c) Given and , the process is a -valued Markov process. Consequently, if is a bounded Borel function, then for any , -almost surely,
and the Chapman-Kolmogorov equation is valid:
for any and
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 .
Proof. We employ Krylov-Bogolyubov's method to the solution of problems (1.1) and (1.2), where the initial condition at the initial time 0. Because of this particular , we know that all results obtained in the previous Sections 3 and 4 are valid. For simplicity, the solution is written as and the segment as . For , we set
(5.1) |
Step 1. We prove the tightness of in . Applying Lemmas 3.2 and 4.2, we get that there exists such that for all ,
(5.2) |
By (5.2) and Chebyshev's inequality, we have that for all ,
and hence for every , there exists such that for all ,
(5.3) |
By Lemma 4.4, we get that there exists such that for all and ,
and hence for all and ,
(5.4) |
Since , applying (5.4) and the usual technique of dyadic division, we obtain that there exists such that for all ,
(5.5) |
By Lemma 3.4, we get that for given and , there exists an integer such that for all ,
which implies that for all and ,
(5.6) |
By (5.6), we infer that for all ,
and hence for all ,
(5.7) |
Let
(5.8) |
(5.9) |
(5.10) |
and
(5.11) |
From (5.3), (5.5) and (5.7)–(5.11), we obtain that for all ,
(5.12) |
By (5.1) and (5.12), we deduce that for all ,
(5.13) |
Next, we prove the set is precompact in . First, we prove for every the set is a precompact subset of . By (5.8) and (5.11), we obtain that for every , the set is bounded in . Let . Then we get that the set is bounded in and hence precompact in due to compactness of the embedding . This implies that the set has a finite open cover of balls with radius in . Note that for every , there exists such that for all ,
(5.14) |
Hence, by (5.14), the set has a finite open cover of balls with radius in . Since is arbitrary, we obtain that the set is percompact in . Then from (5.9) and (5.11), we obtain that is equicontinuous in . Therefore, by the Ascoli-Arzel theorem we deduce that is precompact in , which along with (5.13) shows that is tight on .
Step 2. We prove the existence of invariant measures of problems (1.1) and (1.2). Since the sequence is tight on , there exists a probability measure on , we take a subsequence of (not rebel) such that In the following, we prove is an invariant measure of (1.1) and (1.2). Applying (5.1) and the Chapman-Kolmogorov equation, we obtain that for every and every ,
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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