Alzheimer’s disease (AD), the leading cause of dementia, is a complex neurodegenerative disorder. The AD brain is characterized by the presence of Amyloid-β (Aβ) plaques, neurofibrillary tangles, and an increased inflammatory response. Microglia, the chief immune cells of the central nervous system, have been implicated in AD due to their strong association with Aβ plaques. The role of inflammation associated with microglia has been hotly contested in development of Alzheimer’s disease. A growing amount of genetic studies have implicated microglia in late-onset AD and their role in Aβ clearance. Although traditionally microglia have been considered to be either in resting or activated states, these cells are now known to exist in multiple heterogeneous populations and altered roles that appear to impact pathological states of the Alzheimer’s brain.
Citation: Craig T. Vollert, Jason L. Eriksen. Microglia in the Alzheimers brain: a help or a hindrance?[J]. AIMS Neuroscience, 2014, 1(3): 210-224. doi: 10.3934/Neuroscience.2014.3.210
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Alzheimer’s disease (AD), the leading cause of dementia, is a complex neurodegenerative disorder. The AD brain is characterized by the presence of Amyloid-β (Aβ) plaques, neurofibrillary tangles, and an increased inflammatory response. Microglia, the chief immune cells of the central nervous system, have been implicated in AD due to their strong association with Aβ plaques. The role of inflammation associated with microglia has been hotly contested in development of Alzheimer’s disease. A growing amount of genetic studies have implicated microglia in late-onset AD and their role in Aβ clearance. Although traditionally microglia have been considered to be either in resting or activated states, these cells are now known to exist in multiple heterogeneous populations and altered roles that appear to impact pathological states of the Alzheimer’s brain.
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 $ \mathbb{R}^n $:
$ 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 $ \mathbb{R}^{n}\times [0, +\infty) $. In (1.1), $ \text{i} $ is the imaginary unit, $ \alpha, \beta, \mu, \nu $ and $ \lambda $ are real constants with $ \beta > 0, \lambda > 0 $ and $ \rho > 0 $. $ {(-\Delta})^{\alpha} $ with $ 0 < \alpha < 1 $ is the fractional Laplace operator, $ \sigma_{1, k}(x)\in L^2(\mathbb{R}^n) $ and $ \sigma_{2, k}(u):\mathbb{C}\rightarrow \mathbb{R} $ are nonlinear functions, $ \kappa(x)\in L^2(\mathbb{R}^n)\bigcap L^\infty(\mathbb{R}^n) $ and $ \{W_k\}_{k = 1}^\infty $ is a sequence of independent standard real-valued Wiener process on a complete filtered probability space $ (\Omega $, $ \mathcal {F} $, $ \{{\mathcal {F}}_{t}\}_{t\in \mathbb{R}}, P) $, where $ \{\mathcal {F}_t\}_{t\in\mathbb{R}} $ is an increasing right continuous family of sub-$ \sigma $-algebras of $ \mathcal 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 $ L^2(\Omega; C([-\rho, 0], L^2(\mathbb{R}^n))) $ 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+\text{i}\mu)|u(t)|^{2\beta}u(t) $ and complex-valued solutions), proving the weak compactness of a set distribution laws of the segments of solutions in $ L^2(\Omega; C([-\rho, 0], L^2(\mathbb{R}^n))) $ (because the standard Sobolev embeddings are not compact on unbounded domains $ \mathbb{R}^n $), and establishing the equicontinuity of solutions in $ L^2(\Omega; C([-\rho, 0], L^2(\mathbb{R}^n))) $ (because the uniform estimates in $ L^2(\Omega; C([-\rho, 0], L^2(\mathbb{R}^n))) $ are not sufficient, and the uniform estimates in $ L^2(\Omega; C([-\rho, 0], H^1(\mathbb{R}^n))) $ are needed).
For the estimates of the nonlinear term $ (1+\text{i}\mu)|u(t)|^{2\beta}u(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([-\rho, 0], L^2(\mathbb{R}^n)) $ applying the Ascoli-Arzel$ \grave{a} $ theorem.
Let $ \mathcal S $ be the Schwartz space of rapidly decaying $ C^\infty $ functions on $ \mathbb{R}^n $. The fractional Laplace operator $ (-\Delta)^{\alpha} $ for $ 0 < \alpha < 1 $ is defined by, for $ u\in\mathcal S $,
$ (-\Delta)^{\alpha}u(x) = -\frac12C(n,\alpha)\int_{\mathbb{R}^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2\alpha}}dy,\ \ \ \ x\in\mathbb{R}^n, $ |
where $ C(n, \alpha) $ is a positive constant given by
$ C(n,\alpha) = \frac{\alpha4^{\alpha}\Gamma(\frac{n+2\alpha}{2})}{\pi^{\frac{n}{2}}\Gamma(1-\alpha)}. $ |
By [28], the inner product $ \left((-\Delta)^{\frac\alpha2}u, (-\Delta)^{\frac\alpha2}v\right) $ in the complex field is defined by
$ \left((-\Delta)^{\frac\alpha2}u,(-\Delta)^{\frac\alpha2}v\right) = \frac{C(n,\alpha)}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(u(x)-u(y))(\bar{v}(x)-\bar{v}(y))}{|x-y|^{n+2\alpha}}dxdy, $ |
for $ u\in H^\alpha(\mathbb{R}^n) $. The fractional Sobolev space $ H^\alpha(\mathbb{R}^n) $ is endowed with the norm
$ \|u\|^2_{H^\alpha(\mathbb{R}^n)} = \|u\|^2_{L^2(\mathbb{R}^n)}+\frac{2}{C(n,\alpha)}\|(-\Delta)^{\frac\alpha2}u\|^2_{L^2{\left( \mathbb{R}^n\right) }}. $ |
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 $ L^2(\Omega; C([-\rho, 0], H)) $. In Sections 3 and 4, we derive the uniform estimates of solutions in $ L^2(\Omega; C([-\rho, 0], H)) $ and $ L^2(\Omega; C([-\rho, 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 $ \mathbb{R}^n $.
We assume that $ G:\mathbb{R}^n\times \mathbb{C}\rightarrow \mathbb{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 $ \hat{a} > 0 $ are constants and $ h(x), \hat{h}(x)\in L^2(\mathbb{R}^n) $. Moreover, $ G(x, u) $ is Lipschitz continuous in $ u\in\mathbb{C} $ uniformly with respect to $ x\in \mathbb{R}^n $. More precisely, there exists a constant $ C_G > 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\in\mathbb{N}^+ $
$ ∞∑k=1‖σ1,k‖2<∞, $ | (2.4) |
and that $ \sigma_{2, k}(u):\mathbb{C}\rightarrow \mathbb{R} $ is globally Lipschitz continuous; namely, for every $ k\in\mathbb{N}^+ $, there exists a positive number $ \alpha_k $ such that for all $ s_1, s_2\in\mathbb{C} $,
$ |σ2,k(s1)−σ2,k(s2)|≤αk|s1−s2|. $ | (2.5) |
We further assume that for each $ k\in\mathbb{N}^+ $, there exist positive numbers $ \beta_k $, $ \hat{\beta}_k $, $ \gamma_k $ and $ \hat{\gamma}_k $ such that
$ |σ2,k(s)|≤βk+γk|s|, ∀s∈C, $ | (2.6) |
and
$ |∇σ2,k(s)|≤ˆβk+ˆγk|∇s|, ∀s∈C, $ | (2.7) |
where $ \sum\limits_{k = 1}^\infty(\alpha^2_k+\beta^2_k+\gamma^2_k+\hat{\beta}^2_k+\hat{\gamma}^2_k) < +\infty $. In this paper, we deal with the stochastic Eqs (1.1) and (1.2) in the space $ C([-\rho, 0], L^2(\mathbb{R}^n)) $. In the following discussion, we denote by $ H = L^2(\mathbb{R}^n) $, $ V = H^1(\mathbb{R}^n) $.
A solution of problems (1.1) and (1.2) will be understood in the following sense.
Definition 2.1. We suppose that $ \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) $ is $ \mathcal{F}_0 $-measurable. Then, a continuous $ H $-valued $ \mathcal{F}_t $-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, +\infty) $, and $ \mathcal{F}_t $-adapted for all $ t\geq0 $,
$ u\in L^2(\Omega,C([0,T],H))\bigcap L^2(\Omega,L^2([0,T],V)) $ |
for all $ T > 0 $,
2) $ u(s) = \varphi(s) $ for $ -\rho\leq s\leq0 $,
3) For all $ t\geq0 $ and $ \xi\in 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 $ \omega\in \Omega $.
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 $ \mathcal {F}_0 $-measurable function $ \varphi(s)\in L^2(\Omega, C([-\rho, 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 $ L^2(\Omega, C([-\rho, 0], H)) $.
Theorem 2.2. Suppose (2.1)–(2.6) hold, and $ \mathcal {F}_0 $-measurable function $ \varphi_1, \varphi_2\in L^2(\Omega, C([-\rho, 0], H)) $. If $ u_1 = u(t, \varphi_1) $ and $ u_2 = u(t, \varphi_2) $ are the solutions of the problems (1.1) and (1.2) with initial data $ \varphi_1 $ and $ \varphi_2 $, respectively, then, for any $ t\geq0 $,
$ \mathbb{E}\left[\sup\limits_{-\rho\leq s\leq t}\|u(s,\varphi_1)-u(s,\varphi_2)\|^2\right]+\mathbb{E}\left[\int^t_0\|u(s,\varphi_1)-u(s,\varphi_2)\|^2_Vds\right] $ |
$ \leq C_1e^{\tilde{C}_1t}\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq0}\|\varphi_1(s)-\varphi_2(s)]\|^2\right], $ |
where $ C_1 $ and $ \tilde{C}_1 $ are positive constants independent of $ \varphi_1 $ and $ \varphi_2 $.
Proof. Since both $ u_1 $ and $ u_2 $ are the solutions of the problems (1.1) and (1.2), we have, for all $ t\geq 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\geq0 $,
$ ‖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\geq0 $,
$ 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 $ B_1 $ is a constant produced by Burkholder-Davis-Gundy's inequality.
It follows from (2.11)–(2.13) that for all $ t\geq 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\geq0 $,
$ 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 $ c_1 = 2\left[1+C_G^2+\left(1+\frac12B_1^2\right)\|\kappa\|_{L^\infty}^2\sum\limits_{k = 1}^\infty\alpha_k^2\right] $. By (2.10), there exists $ c_2 $ such that for all $ t\geq0 $,
$ \mathbb{E}\left[\int^t_0\|u_1-u_2\|^2_Vds\right]\leq \tilde{c}_2 e^{c_2t}\mathbb{E}[\sup\limits_{-\rho\leq s\leq0}\|\varphi_1(s)-\varphi_2(s)\|^2]. $ |
We assume that $ a $, $ \alpha_k $ and $ \gamma_k $ are small enough in the sense, there exists a constant $ p\geq2 $ 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 $ \varphi(s)\in L^2(\Omega; C([-\rho, 0], H)) $, then, for all $ t\geq0 $, there exists a positive constant $ \mu_1 $ such that the solution $ u $ of (1.1) and (1.2) satisfies
$ \mathbb{E}[\|u(t)\|^2]+\int^t_0e^{\mu_1(s-t)}\mathbb{E}(\|u(s)\|^2_{V})ds+\int^t_0e^{\mu_1(s-t)}\mathbb{E}(\|u(s)\|^{2\beta+2}_{L^{2\beta+2}})ds $ |
$ ≤M1E[sup−ρ≤s≤0‖φ(s)‖2]+~M1, $ | (3.4) |
and
$ \int^{t+\rho}_0\mathbb{E}[\|u(s)\|^2_V]ds\leq \left(M_1(t+\rho)+\frac{1+\sqrt2a\rho}{C(n,\alpha)}\right)\mathbb{E}[\sup\limits_{-\rho\leq s\leq0}\|\varphi(s)\|^2]+\frac{\sqrt2(t+\rho)}{aC(n,\alpha)}\|h(x)\|^2 $ |
$ +\frac{2(t+\rho)}{C(n,\alpha)}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\beta^2_k\|\kappa(x)\|^2)+\tilde{M}_1(t+\rho), $ |
where $ \tilde{M}_1 $ is a positive constant independent of $ \varphi $.
Proof. By (1.1) and the integration by parts of Ito's formula, we have for all $ t\geq0 $,
$ ‖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 $ \mu_1 $ 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\geq0 $,
$ 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\geq0 $,
$ 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 $ \mu_1 $ sufficiently small such that
$ 2\mu_1+\sqrt2a+\sqrt2ae^{\mu_1\rho}+4\sum\limits^\infty_{k = 1}\gamma^2_k\|\kappa(x)\|^2_{L^\infty}\leq2\lambda. $ |
Then, we have, for all $ t\geq0 $,
$ 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+\rho] $ 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\geq0 $, we obtain,
$ 2\mathbb{E}\left[\int^{t+\rho}_0\|(-\Delta)^{\frac\alpha2}u(s)\|^2ds\right]\leq(1+\sqrt2a\rho)\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq0}\|\varphi(s)\|^2\right] $ |
$ +2(t+\rho)\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\beta^2_k\|\kappa(x)\|^2)+\frac{\sqrt2(t+\rho)}{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([-\rho, 0], H) $.
Lemma 3.2. Suppose (2.1)–(2.6) and (3.2) hold. Then, for any $ \varphi(s)\in L^2(\Omega, \mathcal {F}_0;C([-\rho, 0], H)) $, the solution of (1.1) satisfies that, for all $ t\geq\rho $,
$ \mathbb{E}\left(\sup\limits_{t-\rho\leq r\leq t}\|u(r)\|^2\right)\leq M_2\mathbb{E}[\sup\limits_{-\rho\leq s\leq0}\|\varphi(s)\|^2]+\tilde{M}_2, $ |
where $ M_2 $ and $ \tilde{M}_2 $ are positive constants independent of $ \varphi $.
Proof. By (1.1) and integration by parts of Ito's formula and taking the real part, we get for all $ t\geq\rho $ and $ t-\rho\leq r\leq 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\geq\rho $ and $ t-\rho\leq r\leq 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\geq\rho $ and $ t-\rho\leq r\leq 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\geq\rho $ and $ t-\rho\leq r\leq t $,
$ \|u(r)\|^2\leq c_3+\|u(t-\rho)\|^2+c_4\int^r_{t-2\rho}\|u(s)\|^2ds $ |
$ +2Re∞∑k=1∫rt−ρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s)), $ | (3.17) |
where $ c_3 = 2\rho\|h\|^2+2\rho\sum\limits^\infty_{k = 1}\|\sigma_{1, k}\|^2+4\rho\|\kappa\|^2\sum\limits^\infty_{k = 1}\beta^2_k $ and $ c_4 = 1+2a^2+4\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\gamma^2_k $. By (3.17), we find that for all $ t\geq\rho $,
$ 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\geq\rho $,
$ 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\geq\rho $,
$ 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\geq\rho $,
$ \mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|u(r)\|^2\right]\leq M_2\mathbb{E}[\sup\limits_{-\rho\leq s\leq0}\|\varphi(s)\|^2]+\tilde{M}_2. $ |
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 $ \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) $. Then, for all $ t\geq0 $, the solution $ u $ of (1.1) and (1.2) satisfies
$ \limsup\limits_{m\rightarrow \infty}\sup\limits_{t\geq-\rho}\int_{|x|\geq m}\mathbb{E}[|u(t,x)|^2]dx = 0. $ |
Proof. We suppose that $ \theta(x):\mathbb{R}^n\rightarrow \mathbb{R} $ is a smooth function with $ 0\leq\theta(x)\leq1 $, for all $ x\in\mathbb{R}^n $ defined by
$ \theta(x) = {0if |x|≤1,1if |x|≥2. $ |
For fixed $ m\in\mathbb{N} $, we denote that $ \theta_m(x) = \theta(\frac xm) $. By (1.1), we have
$ d(\theta_m u)+(1+\text{i}\nu)\theta_m(-\Delta)^{\alpha} udt+(1+\text{i}\mu)\theta_m|u|^{2\beta}u dt+\lambda\theta_mudt = \theta_mG(x,u(t-\rho))dt $ |
$ +∞∑k=1θm(σ1,k+κσ2,k)dWk(t). $ | (3.22) |
By (3.2), We can find $ \mu_2 $ 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 $ \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) $, we have for all $ s\in[-\rho, 0] $, $ \mathbb{E}[\|\varphi(0)\|^2] < \infty $. It follows that for any $ \varepsilon > 0 $, there exists a positive $ N_1 = N_1(\varepsilon, \varphi)\geq1 $, for all $ m\geq N_1 $, one has $ \int_{|x|\geq m}\mathbb{E}[\varphi^2(0, x)]dx < \varepsilon. $ 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 $ \frac 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 $ c_6: = c_6^{*}\bar c_6+4\tilde{c}_6 $. 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 $ N_2(\varepsilon, \varphi)\geq N_1 $, for all $ t\geq0 $ and $ m\geq N_2 $,
$ \sqrt{c_6(1+\nu^2)}C(n,\alpha)m^{-\alpha}\int^t_0e^{\mu_2(s-t)}\mathbb{E}[\|u(s)\|^2]ds < \varepsilon. $ |
By Lemma 3.1, there exists $ N_3(\varepsilon, \varphi)\geq N_2 $ such that for all $ t\geq0 $ and $ m\geq N_3 $,
$ 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 $ N_4(\varepsilon, \varphi)\geq N_3 $, for all $ t\geq0 $ and $ m\geq N_4 $,
$ 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 $ \{\varphi(s)\in L^2(\Omega, H)|s\in[-\rho, 0]\} $ is compact, it has a open cover of balls with radius $ \frac{\sqrt\varepsilon}{2} $ which denoted by $ \{B(\varphi^i, \frac{\sqrt\varepsilon}{2})\}^l_{i = 1} $. Since $ \varphi^i = \varphi(s_i)\in L^2(\Omega; C([-\rho, 0], H)) $ for $ i = 1, 2, \cdots, l $, we obtain that for given $ \varepsilon > 0 $,
$ \{\varphi(s)\in L^2(\Omega;C([-\rho,0],H))\}\subseteq\cup^{l}_{i = 1}\left\{X\in L^2(\Omega,H)| \|X-\varphi^i\|_{L^2(\Omega,H)} < \frac{\sqrt\varepsilon}{2}\right\}. $ |
Since $ \varphi^i\in L^2(\Omega, H) $, there exists a positive constant $ N_5 = N_5(\varepsilon, \varphi)\geq N_4 $, for $ m\geq N_5 $, we have
$ \sup\limits_{i = 1,2,\cdots,l}\int_{|x|\geq m}\mathbb{E}[|\varphi(s_i,x)|^2]dx < \frac{\varepsilon}{4}. $ |
Then,
$ \sup\limits_{s\in[-\rho,0]}\int_{|x|\geq m}\mathbb{E}[|\varphi(s,x)|^2]dx < \frac{\varepsilon}{2}, \forall m\geq N_5. $ |
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 $ \sum\limits^\infty_{k = 1}\|\sigma_{1, k}\|^2 < \infty $ and $ \kappa(x)\in L^2(\mathbb{R}^n)\bigcap L^\infty(\mathbb{R}^n) $, there exists $ N_6 = N_6(\varepsilon, \varphi)\geq N_5 $, for all $ t\geq0 $ and $ m\geq N_6 $, we have
$ \sum\limits^\infty_{k = 1}\int_{|x|\geq m}|\sigma_{1,k}(x)|^2dx+\int_{|x|\geq m}\kappa^2(x)dx < \varepsilon. $ |
Consequently, for the fifth term on the right-hand side of (3.24), we get for all $ t\geq0 $ and $ m\geq N_6 $,
$ \sum\limits^{\infty}_{k = 1}\int^t_0e^{\mu_2(s-t)}\mathbb{E}\left[\|\theta_m\left(\sigma_{1,k}+\kappa\sigma_{2,k}\right)\|^2\right]ds\leq \frac2{\mu_2}(1+2\sum\limits^\infty_{k = 1}\beta^2_k)\varepsilon $ |
$ +4\|\kappa\|^2_{L^\infty}\sum^\infty_{k = 1}\gamma^2_k\int^t_0e^{\mu_2(s-t)}\mathbb{E}[\|\theta_mu(s)\|^2]ds. $ |
Therefore, for all $ t\geq0 $ and $ m\geq N_6 $,
$ \mathbb{E}[\|\theta_mu(t)\|^2]\leq \left[2+e^{-\mu_2t}+\frac{\sqrt2}{a\mu_2}+\frac{\sqrt2}{2}a\rho+\frac{2}{\mu_2}(1+2\sum\limits^{\infty}_{k = 1}\beta^2_k)\right]\varepsilon $ |
$ +\left(\mu_2-2\lambda+2\sqrt2a+4\|\kappa\|^2_{L^\infty}\sum^\infty_{k = 1}\gamma^2_k\right)\int^t_0e^{\mu_2(s-t)}\mathbb{E}[\|\theta_mu(s)\|^2]ds. $ |
Taking the limit in the above equation and by (3.23), we have
$ \limsup\limits_{m\rightarrow \infty}\sup\limits_{t\geq-\rho}\int_{|x|\geq m}\mathbb{E}[|u(t,x)|^2]dx = 0, $ |
which completes the proof.
Lemma 3.4. Suppose (2.1)–(2.6) and (3.2) hold. If $ \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) $, then the solution $ u $ of (1.1) and (1.2) satisfies
$ \limsup\limits_{m\rightarrow \infty}\sup\limits_{t\geq 0}\mathbb{E}\left[\sup\limits_{r\in[t-\rho,t]}\int_{|x|\geq m}|u(r,x)|^2dx\right] = 0. $ |
Proof. By (3.22) and integration by parts of Ito's formula and taking the real part, for all $ t\geq\rho $ and $ r\in[t-\rho, 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 $ \varepsilon > 0 $, there exists $ \tilde{N}_1(\varepsilon, \varphi)\geq1 $ such that for all $ m\geq\tilde{N}_1 $ and $ t\geq\rho $,
$ 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 $ \tilde{N}_2(\varepsilon, \varphi)\geq \tilde{N}_1 $ such that for all $ m\geq\tilde{N}_2 $ and $ t\geq\rho $,
$ −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\geq\tilde{N}_2 $ and $ t\geq\rho $,
$ |\mu_2-2\lambda|\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\int^r_{t-\rho}\|\theta_mu(s)\|^2e^{\mu_2(s-r)}ds\right]\leq|\mu_2-2\lambda|\mathbb{E}\left[\int^t_{t-\rho}\|\theta_mu(s)\|^2ds\right] $ |
$ ≤|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 $ \tilde{N}_3(\varepsilon, \varphi)\geq\tilde{N}_2 $ such that for all $ m\geq\tilde{N}_3 $ and $ t\geq\rho $,
$ 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 $ \kappa(x)\in L^2(\mathbb{R}^n)\bigcap L^\infty(\mathbb{R}^n) $, (2.4) and Lemma 3.3, we deduce that there exists $ \tilde{N}_4(\varepsilon, \varphi)\geq \tilde{N}_3 $ such that for all $ m\geq\tilde{N}_4 $ and $ t\geq \rho $,
$ ∞∑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\geq\tilde{N}_4 $ and $ t\geq\rho $, we obtain,
$ \mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|\theta_m u(r)\|^2\right]\leq\left[4+2|\mu_2-2\lambda|\rho+(6+4a^2)\rho+4\rho(1+2\tilde{B}^2_2e^{2\mu_2\rho})(1+\lambda+2\sum\limits^\infty_{k = 1}\beta^2_k)\right]\varepsilon. $ |
Therefore, we conclude
$ \limsup\limits_{m\rightarrow \infty}\sup\limits_{t\geq 0}\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\int_{|x|\geq m}|u(r,x)|^2dx\right] = 0. $ |
Lemma 3.5. Suppose (2.1)–(2.6) and (3.1) hold. If $ \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) $, then there exists a positive constant $ \mu_3 $ 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 $ M_3 $ is a positive constant independent of $ \varphi $.
Proof. By (3.1), there exist positive constants $ \mu $ and $ \epsilon_1 $ 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\in \mathbb{N} $, let $ \tau_n $ be a stopping time as defined by
$ \tau_n = \inf\{t\geq 0:\|u(t)\| > n\}, $ |
and as usual, we set $ \tau_n = +\infty $ if $ \{t\geq 0:\|u(t)\| > n\} = \emptyset. $ By the continuity of solutions, we have
$ \lim\limits_{n\rightarrow \infty}\tau_n = +\infty. $ |
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\wedge\tau_n) $ with $ t\geq 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\geq 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\geq 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\geq 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\rightarrow \infty $, by Fatou's Lemma, we deduce that for $ t\geq 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\geq 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([-\rho, 0], V) $. To the end, we assume that for each $ k\in\mathbb{N} $, the function $ \sigma_{1, k}\in V $ and
$ ∞∑k=1‖σ1,k‖2V<∞. $ | (4.1) |
Furthermore, we assume that the function $ \kappa\in 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 $, $ \hat{\gamma}_k $ in (2.7) are sufficiently small in the sense that there exists a constant $ p\geq 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 $ \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
$ sups≥−ρE[‖∇u(t)‖2]+sups≥0E[∫t0eμ4(s−t)‖(−Δ)α+12u(s)‖2ds]≤M4(E[‖φ‖2CV]+1), $ | (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
$ μ1−2λ+8‖κ‖2L∞∞∑k=1ˆγ2k<0. $ | (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 $,
$ 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\geq0 $,
$ 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 = \left( ¯u∇uu∇¯u \right)^H, M = \left( −β+12β(1+iμ)2β(1−iμ)2−β+12 \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
$ 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\geq 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 $ \varphi(s)\in L^2(\Omega; C([-\rho, 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 $ 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 $,
$ ‖∇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\geq \rho $ and $ t-\rho\leq r\leq 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 $ c_1 $ and $ c_2 $ are positive constants. By (4.19), we deduce that for all $ t\geq \rho $,
$ 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\geq \rho $,
$ 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\geq \rho $,
$ 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\geq \rho $,
$ 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\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
$ supt≥−ρE[‖∇u(t)‖2p]+supt≥0E[∫t0eμ5(s−t)‖∇u(s)‖2(p−1)‖(−Δ)α+12u(s)‖2ds] ≤M5(E[‖φ‖2pCV]+1), $ | (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
$ μ+4(p−1)ϵpp−11+2pˆa2μϵp1+8C2(p−1)(2p−1)ϵpp−11∞∑k=1γ2k+8p(2p−1)‖κ‖2L∞∞∑k=1ˆγ2k +2(p−1)(2p−1)ϵpp−11∞∑k=1(‖∇σ1,k‖2+4β2k‖∇κ‖2+4ˆβ2k‖κ‖2)≤2pλ. $ | (4.25) |
By (1.1) and applying Ito's formula to $ e^{\mu t}\|\nabla u(t)\|^{2p} $, we get for $ t\geq 0 $,
$ eμt‖∇u(t)‖2p+2p∫t0eμs‖∇u(s)‖2(p−1)‖(−Δ)α+12u(s)‖2ds +2p∫t0eμs‖∇u(s)‖2(p−1)Re((1+iμ)|u(s)|2βu(s),−Δu(s))ds=‖∇φ(0)‖2p+(μ−2pλ)∫t0eμs‖∇u(s)‖2pds +2pRe∫t0eμs‖∇u(s)‖2(p−1)(G(x,u(s−ρ)),−Δu(s))ds +p∞∑k=1∫t0eμs‖∇u(s)‖2(p−1)‖∇(σ1,k+κσ2,k(u(s)))‖2ds +2p∞∑k=1Re∫t0eμ1s‖∇u(s)‖2(p−1)(σ1,k+κσ2,k(u(s)),−Δu(s))dWk(s) +2p(p−1)∞∑k=1Re∫t0eμ1s‖∇u(s)‖2(p−2)|(σ1,k+κσ2,k(u(s)),−Δu(s))|2ds. $ |
Taking the expectation, we have for $ t\geq 0 $,
$ eμtE[‖∇u(t)‖2p]+2pE[∫t0eμs‖∇u(s)‖2(p−1)‖(−Δ)α+12u(s)‖2ds] +2pE[∫t0eμs‖∇u(s)‖2(p−1)Re((1+iμ)|u(s)|2βu(s),−Δu(s))ds]=E[‖∇φ(0)‖2p]+(μ−2pλ)E[∫t0eμs‖∇u(s)‖2pds] +2pE[Re∫t0eμs‖∇u(s)‖2(p−1)(G(x,u(s−ρ)),−Δu(s))ds] +p∞∑k=1E[∫t0eμs‖∇u(s)‖2(p−1)‖∇(σ1,k+κσ2,k(u(s)))‖2ds] +2p(p−1)∞∑k=1E[Re∫t0eμ1s‖∇u(s)‖2(p−2)|(σ1,k+κσ2,k(u(s)),−Δu(s))|2ds]. $ | (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
$ 2pE[Re∫t0eμs‖∇u(s)‖2(p−1)(G(x,u(s−ρ)),−Δu(s))ds]≤2pE[∫t0eμs‖∇u(s)‖2(p−1)‖∇u(s)‖‖∇G(x,u(s−ρ))‖ds]≤pE[∫t0eμs‖∇u(s)‖2(p−1)‖∇u(s)‖2ds]+pE[∫t0eμs‖∇u(s)‖2(p−1)‖∇G(x,u(s−ρ))‖2ds]≤pE[∫t0eμs‖∇u(s)‖2(p−1)‖∇u(s)‖2ds] +2pE[∫t0eμs‖∇u(s)‖2(p−1)‖ˆh‖2ds]+2pˆa2E[∫t0eμs‖∇u(s)‖2(p−1)‖∇u(s−ρ)‖2ds]≤pE[∫t0eμs‖∇u(s)‖2(p−1)(‖(−Δ)α+12u(s)‖2+c‖u(s)‖2)ds]+2ϵp1‖ˆh(x)‖2p∫t0eμsds +4(p−1)ϵpp−11E[∫t0eμs‖∇u(s)‖2pds]+2pˆa2ϵp1E[∫t0eμs‖∇u(s)‖2pds]≤pE[∫t0eμs‖∇u(s)‖2(p−1)‖(−Δ)α+12u(s)‖2ds]+(4(p−1)ϵpp−11+2pˆa2μϵp1)E[∫t0eμs‖∇u(s)‖2pds] +1μϵp1‖ˆh(x)‖2peμt+2pˆa2μϵp1sup−ρ≤s≤0E[‖∇φ(s)‖2p]eμt+csups≥0E[‖u(s)‖2p]eμt. $ | (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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1. | Fabio Di Pietrantonio, Domenico Cannatà, Massimiliano Benetti, 2019, 9780128144015, 181, 10.1016/B978-0-12-814401-5.00008-6 | |
2. | Christina G. Siontorou, 2020, Chapter 25-2, 978-3-319-47405-2, 1, 10.1007/978-3-319-47405-2_25-2 | |
3. | Christina G. Siontorou, Georgia-Paraskevi Nikoleli, Marianna-Thalia Nikolelis, Dimitrios P. Nikolelis, 2019, 9780128157435, 375, 10.1016/B978-0-12-815743-5.00015-9 | |
4. | Christina G. Siontorou, 2019, Chapter 25-1, 978-3-319-47405-2, 1, 10.1007/978-3-319-47405-2_25-1 | |
5. | Christina G. Siontorou, 2022, Chapter 25, 978-3-030-23216-0, 707, 10.1007/978-3-030-23217-7_25 |