On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem

1.   Introduction

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 $\mathbb{R}^n$:

with initial condition

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 $u\in\mathcal S$,

where $C(n, \alpha)$ is a positive constant given by

By ^[28], the inner product $\left((-\Delta)^{\frac\alpha2}u, (-\Delta)^{\frac\alpha2}v\right)$ in the complex field is defined by

for $u\in H^\alpha(\mathbb{R}^n)$. The fractional Sobolev space $H^\alpha(\mathbb{R}^n)$ is endowed with the norm

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.

2.   Preliminaries

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

and

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

For the diffusion coefficients of noise, we suppose that for each $k\in\mathbb{N}^+$

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 $s_1, s_2\in\mathbb{C}$,

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

and

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 $t\geq0$,

for all $T > 0$,

2) $u(s) = \varphi(s)$ for $-\rho\leq s\leq0$,

3) For all $t\geq0$ and $\xi\in V$,

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 $t\geq0$,

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 $t\geq 0$,

By (2.9), the integration by parts of Ito's formula and taking the real parts, we get, for all $t\geq0$,

For the third term in the first row of (2.10), one has

By (2.10), we deduce that for $t\geq0$,

For the second term on the right-hand side of (2.11), by (2.3), one has

For the third term on the right-hand side of (2.11), by (2.5), we have

For the forth term on the right-hand side of (2.11), by Burkholder-Davis-Gundy's inequality, one has

where $B_1$ is a constant produced by Burkholder-Davis-Gundy's inequality.

It follows from (2.11)–(2.13) that for all $t\geq 0$,

Applying Gronwall inequality to (2.14), we obtain that for all $t\geq0$,

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 $t\geq0$,

3.   Uniform estimates of solutions with initial data in $C([-\rho, 0], H)$

We assume that $a$, $\alpha_k$ and $\gamma_k$ are small enough in the sense, there exists a constant $p\geq2$ such that

By (3.1), one has

and

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

and

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 $t\geq0$,

The system (3.5) can be rewritten as

Assume that $\mu_1$ is a positive constant, one has

Taking the expectation, we have for all $t\geq0$,

For the third term on the right-hand side (3.7), by (2.1), we have

For the forth term on the right-hand side (3.7), by (2.6), we have

By (3.7)–(3.9), we obtain for all $t\geq0$,

By (3.2), there exists a positive constant $\mu_1$ sufficiently small such that

Then, we have, for all $t\geq0$,

which completes the proof of (3.4).

Integrating (3.6) on $[0, t+\rho]$ and taking the expectation, one has

For the second term on the right-hand side of (3.11), by (2.1), we have

For the third term on the right-hand side of (3.11), one has

Then, by (3.2) and (3.11)–(3.13), for all $t\geq0$, we obtain,

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 $t\geq\rho$,

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 $t-\rho\leq r\leq t$,

For the second term on the right-hand side of (3.14), by (2.1) we have, for all $t\geq\rho$ and $t-\rho\leq r\leq t$,

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

By (3.14)–(3.16), we obtain for all $t\geq\rho$ and $t-\rho\leq r\leq t$,

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 $t\geq\rho$,

For the second term and the third term on the right-hand side of (3.18), by Lemma 3.1, we deduce for all $t\geq\rho$,

and

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 $t\geq\rho$,

By Lemma 3.1 and (3.18)–(3.21), we deduce that for all $t\geq\rho$,

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

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

For fixed $m\in\mathbb{N}$, we denote that $\theta_m(x) = \theta(\frac xm)$. By (1.1), we have

By (3.2), We can find $\mu_2$ sufficiently small such that

By (3.22) and integration by parts of Ito's formula and taking the expectation, we obtain

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,

Now we consider the second term on the right-hand side of (3.24). We first have

We now prove the following inequality:

Let $x-y = h$ and $\frac hm = z$, then, we obtain,

This proves (3.27) with $c_6: = c_6^{*}\bar c_6+4\tilde{c}_6$. By (3.26) and (3.27), we obtain,

By (3.29), for the second term on the right-hand side of (3.24), we get

By Lemma 3.1, we have

By (3.31), we deduce that there exists $N_2(\varepsilon, \varphi)\geq N_1$, for all $t\geq0$ and $m\geq N_2$,

By Lemma 3.1, there exists $N_3(\varepsilon, \varphi)\geq N_2$ such that for all $t\geq0$ and $m\geq N_3$,

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 $m\geq N_4$,

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 $\varepsilon > 0$,

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

Then,

Consequently, one has

For the fifth term on the right-hand side of (3.24), by (2.6), we obtain

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

Consequently, for the fifth term on the right-hand side of (3.24), we get for all $t\geq0$ and $m\geq N_6$,

Therefore, for all $t\geq0$ and $m\geq N_6$,

Taking the limit in the above equation and by (3.23), we have

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

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

By (3.33), we deduce,

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 $t\geq\rho$,

For the second term on the right-hand side of (3.34), by (3.29), we have

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 $t\geq\rho$,

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 $t\geq\rho$,

For the forth term on the right-hand side of (3.34), by (2.1), we obtain

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 $t\geq\rho$,

For the fifth term on the right-hand side of (3.34), by (2.6), we have

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 $t\geq \rho$,

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,

Above all, for all $m\geq\tilde{N}_4$ and $t\geq\rho$, we obtain,

Therefore, we conclude

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

where $M_3$ is a positive constant independent of $\varphi$.

Proof. By (3.1), there exist positive constants $\mu$ and $\epsilon_1$ such that

Given $n\in \mathbb{N}$, let $\tau_n$ be a stopping time as defined by

and as usual, we set $\tau_n = +\infty$ if $\{t\geq 0:\|u(t)\| > n\} = \emptyset.$ By the continuity of solutions, we have

Applying Ito's formula, we obtain

Substituting (3.6) into (3.43), we infer

We also get the formula

Substituting (3.44) into (3.45) and integrating on $(0, t\wedge\tau_n)$ with $t\geq 0$, we deduce

Taking the expectation, we obtain for $t\geq 0$,

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

For the forth term on the right-hand side of (3.47), we infer

For the first term on the right-hand side of (3.49), we have

For the second term on the right-hand side of (3.49), we have

By (3.49)–(3.51), we obtain

For the fifth term on the right-hand side of (3.47), applying (3.52), we have

From (3.47), (3.48), (3.52) and (3.53), we obtain that for $t\geq 0$,

Then by (3.42) and (3.54), we obtain that for $t\geq 0$,

Letting $n\rightarrow \infty$, by Fatou's Lemma, we deduce that for $t\geq 0$,

Hence, we have for $t\geq 0$,

This implies the desired estimate.

4.   Uniform estimates of solutions with initial data in $C([-\rho, 0], V)$

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

Furthermore, we assume that the function $\kappa\in V$ and there exists a constant $C > 0$ such that

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

By (4.3), we can find

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

where $M_4$ is a positive constant independent of $\varphi$.

Proof. By (4.4), there exists a positive constant $\mu_1$ such that

By (1.1) and applying Ito's formula to $e^{\mu_1 t}\|\nabla u(t)\|^2$, we have for $t\geq 0$,

Taking the expectation, we have for all $t\geq0$,

First, we estimate the third term on the left-hand side of (4.7). Applying integrating by parts, we have

where

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

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

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

By (4.7), (4.10) and (4.11), we obtain

Then by (4.6) and (4.12), we obtain that for all $t\geq 0$,

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

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 $t-\rho\leq r\leq t$,

For the third term on the left-hand side of (4.15), applying (4.8), we have

For the second term on the right-hand side of (4.15), applying (2.2) and Gagliardo-Nirenberg inequality, we have

For the third term on the right-hand side of (4.15), applying (2.6) and (2.7), we have

By (4.4) and (4.15)–(4.18), we infer that for all $t\geq \rho$ and $t-\rho\leq r\leq t$,

where $c_1$ and $c_2$ are positive constants. By (4.19), we deduce that for all $t\geq \rho$,

For the second term on the right-hand side of (4.20), by Lemma 4.1 we infer that for all $t\geq \rho$,

For the third term on the right-hand side of (4.20), by Lemmas 3.1 and 4.1 we infer that for all $t\geq \rho$,

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 $t\geq \rho$,

By (4.20)–(4.23), we obtain that for all $t\geq \rho$,

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

where $M_5$ is a positive constant independent of $\varphi$.

Proof. By (3.1), there exist positive constants $\mu$ and $\epsilon_1$ such that

By (1.1) and applying Ito's formula to $e^{\mu t}\|\nabla u(t)\|^{2p}$, we get for $t\geq 0$,

Taking the expectation, we have for $t\geq 0$,

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

For the forth term on the right-hand side of (4.26), applying (2.7), we infer

Then applying Young's inequality, (4.28) can be estimated by

For the fifth term on the right-hand side of (4.26), applying integrating by parts and (4.29), we get

By (4.26), (4.27), (4.29) and (4.30), we obtain

Then by (4.25) and (4.31), we deduce that for all $t\geq 0$,

Therefore, by (4.32) and Lemma 3.5, there exists a constant $M_5$ independent of $\varphi$ such that

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

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 $t > r\geq 0$,

where $M_6$ is a positive constant depending on $\varphi$, but independent of $t$ and $r$.

Proof. By (4.34), we get for $t > r\geq 0$,

Then we infer

Taking the expectation of (4.36), we have for all $t > r\geq 0$,

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 $t > r\geq 0$,

Applying Lemmas 3.5 and 4.3, we obtain for all $t > r\geq 0$,

For the second term on the right-hand side of (4.38), by the contraction property of $e^{-At}$, we infer that for all $t > r\geq 0$,

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

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 $t > r\geq 0$,

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

Therefore, from (4.38)–(4.42), we obtain there exists $C_6 > 0$ independent of $t$ and $r$, such that for all $t > r\geq 0$,

The proof is complete.

5.   Existence of invariant measures

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

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

Particularly, for $\Gamma\in \mathcal {B}(C([-\rho, 0], H))$, $0\leq r\leq t$ and $\varphi\in C([-\rho, 0], H)$, we have

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 $\phi:C([-\rho, 0];H)\rightarrow \mathbb{R},$

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 $0\leq r\leq t$,

(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,

and the Chapman-Kolmogorov equation is valid:

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

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 $t\geq \rho$,

By (5.2) and Chebyshev's inequality, we have that for all $t\geq \rho$,

and hence for every $\varepsilon > 0$, there exists $R_1 = R_1(\varepsilon) > 0$ such that for all $t\geq \rho$,

By Lemma 4.4, we get that there exists $C_2 > 0$ such that for all $t\geq \rho$ and $r, s\in[-\rho, 0]$,

and hence for all $t\geq \rho$ and $r, s\in[-\rho, 0]$,

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 $t\geq \rho$,

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 $t\geq \rho$,

which implies that for all $t\geq \rho$ and $m\in\mathbb{N}$,

By (5.6), we infer that for all $t\geq \rho$,

and hence for all $t\geq \rho$,

Let

and

From (5.3), (5.5) and (5.7)–(5.11), we obtain that for all $t\geq \rho$,

By (5.1) and (5.12), we deduce that for all $k\in\mathbb{N}$,

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 $\zeta\in\mathcal {M}_{\varepsilon}$,

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 $\phi:C([-\rho, 0];H)\rightarrow \mathbb{R}$,

which completes the proof.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

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).

Conflict of interest

The authors declare there is no conflict of interest.