The existences and asymptotic behavior of solutions to stochastic semilinear generalized Rayleigh

Abstract

We consider a functional stochastic delay semilinear Rayleigh–Stokes equation involving Riemann–Liouville derivative. Our aim is using the resolvent theory, fixed point argument to prove the global solvability and gives some sufficient conditions to ensure the asymptotic stability of mild solutions in the mean square moment.

Keywords

Rayleigh–Stokes problem stability stochastic equation fractional Brownian motion nonlocal PDE

1. Introduction

Fractional differential equations are naturally used to model anomalous diffusion processes [8]. This is due to both the rapid development of fractional calculus theory and its applications in various fields, such as physics, fluid mechanics, viscoelasticity, heat conduction in materials with memory, chemistry, and engineering [4,11,12,16,23]. As a result, fractional differential equations have gained significant attention in recent years.

Stochastic differential equations are commonly used in many different fields, including physics, chemistry, economics, social sciences, finance, and engineering. The existence of solutions for fractional stochastic differential equations has been studied in the literature [7,22,24]. Fractional Brownian motion (fBm) is a type of Gaussian process with continuous sample paths. It is indexed by the Hurst parameter $H \in (0, 1)$ . fBm is a self-similar process with stationary increments, and it has long-range memory when $H > \frac{1}{2}$ . These properties make fBm a naturally occurring noise component in many physical phenomena, as well as in mathematical finance, communication networks, hydrology, and medicine. For more information on fBm, please see the following articles [1,5–7,13,14,21,27].

The Rayleigh–Stokes equation is a mathematical equation that describes the flow of non-Newtonian fluids in cylinders. It is a nonlinear equation, which makes it difficult to solve analytically. However, there are a number of numerical methods that have been developed to solve the equation, and there are also some analytic solutions available for special cases. Let $O \subset R^{d}$ be a bounded domain with smooth boundary $\partial O$ and $(Ω, F, P)$ be a probability space. We consider following problem

\begin{aligned} \partial_{t} u - (1 + θ \partial_{t}^{α}) Δ u = f (t, u_{h}) + g (t) d W^{H} (t) in O, t > 0, \end{aligned}

(1.1)

\begin{aligned} u = 0 on \partial O, t ⩾ 0, \end{aligned}

(1.2)

\begin{aligned} u (x, s) = ϕ (x, s), x \in O, s \in [- ζ, 0], \end{aligned}

(1.3)

where

θ > 0

α \in (0, 1)

\partial_{t} = \frac{\partial}{\partial t}

\partial_{t}^{α}

stands for the Riemann–Liouville derivative of order α defined by

\begin{aligned} \partial_{t}^{α} v (t) = \frac{d}{d t} \int_{0}^{t} G_{1 - α} (t - s) v (s) d s \end{aligned}

where

G_{ρ} (t) = \frac{t^{ρ - 1}}{Γ (ρ)}

for

ρ > 0

t > 0

. In this model,

u_{h}

is defined by

u_{h} (x, t) = u (x, t - h (t))

with h being a continuous function on

R^{+}

such that

- ζ ⩽ t - h (t) ⩽ t

and

lim_{t \to \infty} (t - h (t)) = \infty

. For convenience, throughout this paper we denote

X = L^{2} (O)

then

f : R^{+} \times L^{2} (Ω, X) \to L^{2} (Ω, X)

is a nonlinear map,

g : [0, \infty) \to L_{2}^{0} (Y, X)

which will be specified in Section 2,

W^{H}

is a fractional Brownian motion with

H \in (\frac{1}{2}; 1)

on a real line and a separable Hilbert space

Y

ϕ \in C ([- ζ, 0]; L^{2} (Ω, X))

is given.

Equation (1.1) comes from a generalized Rayleigh–Stokes problem, which was described in [15,26]. The fractional derivative term in the equation is important for describing the viscoelasticity of the fluids. In the literature, various numerical methods have been developed for solving the Rayleigh–Stokes problem in the linear case, such as those described in [2,3,9,10,25,30]. Analytic solutions to this problem have also been obtained in some special cases [15,17,26,29,31]. Recently, some inverse problems involving equation (1.1) have been addressed in [19,20,28], where the state function is identified from its terminal value.

In this work, we study a nonlinear stochastic model with a delayed term in the nonlinearity f. This model describes a situation in which the external force depends on the history state of the system. It is important to note that the presence of a delayed term can reduce the performance and affect the stability of the system. A typical example of a delayed term is $h (t) = (1 - q) t + ζ$ , $u_{h} (x, t) = u (x, q t - ζ)$ , where $q \in (0, 1]$ . This is called a proportional delay. For this model, the long-time behavior of solutions has not been addressed in the literature, and we aim to fill this gap. We first prove that the problem is globally solvable. Then, we analyze some sufficient conditions for the asymptotic stability of our system.

Our paper is structured as follows. In the next section, we review the key properties of fractional Brownian motion, relaxation functions, and resolvent operators. Section 3 is devoted to proving the solvability of our problem. In the final section, we present the mean square moment asymptotic stability of the solutions.

2. Preliminaries

2.1. Fractional Brownian motion

We first recall the definition of Wiener integrals with respect to an infinite dimensional fractional Brownian motion with Hurst index $H > \frac{1}{2}$ (see [21]).

Let $(Ω, F, P)$ be a complete probability space with a normal filtration ${F_{t}}_{t \in [0, T]}$ and $T > 0$ be an arbitrary fixed horizon. An one-dimensional fractional Brownian motion (fBm) with Hurst parameter $H \in (0; 1)$ is a centered Gaussian process $β^{H} = {β^{H} (t), 0 ⩽ t ⩽ T}$ with the covariance function

\begin{aligned} R (t, s) & = E [β^{H} (t) β^{H} (s)] \\ = \frac{1}{2} (| t |^{2 H} + | s |^{2 H} - | t - s |^{2 H}) \end{aligned}

It is known that

β^{H} (t)

with

H > \frac{1}{2}

admits the following Volterra representation

\begin{aligned} β^{H} (t) = \int_{0}^{t} K (t, s) d β (s) \end{aligned}

where β is a standard Brownian motion and the Volterra kernel

K (t, s)

is given by

\begin{aligned} K (t, s) = C_{H} \int_{s}^{t} (u - s)^{H - \frac{3}{2}} {(\frac{u}{s})}^{H - \frac{1}{2}} d u; t ⩾ s . \end{aligned}

For the deterministic function $φ \in L^{2} ([0; T])$ , the fractional Wiener integral of φ with respect to $β^{H}$ is defined by

\begin{aligned} \int_{0}^{T} φ (s) d β^{H} (s) = \int_{0}^{T} K_{H}^{*} φ (s) d β (s) \end{aligned}

where

K_{H}^{*} φ (s) = \int_{s}^{T} φ (r) \frac{\partial K}{\partial r} (r, s) d r

Let $X$ and $Y$ be two real, separable Hilbert spaces and $L (Y, X)$ be the space of bounded linear operators from $Y$ to $X$ . For the sake of convenience, we shall use the same notation to denote the norms in $X$ , $Y$ and $L (Y, X)$ . Let ${ξ_{n}; n = 1, 2, \dots}$ be a complete orthonormal basis in $Y$ and $Q \in L (Y, X)$ be an operator defined by $Q ξ_{n} = λ_{n} ξ_{n}$ with finite trace $t r Q = \sum_{n = 1}^{\infty} λ_{n}^{*} < \infty$ , where $λ_{n}^{*}; n = 1, 2, \dots$ are non-negative real numbers. We define the infinite dimensional fBm on Y with covariance Q as $W^{H} (t) = \sum_{n = 1}^{\infty} \sqrt{λ_{n}^{*}} ξ_{n} β_{n}^{H} (t)$ where $β_{n}^{H} (t)$ are real, independent fBm’s. This process is a $Y$ -valued Gaussian, it starts from 0, has zero mean and covariance:

\begin{aligned} E ⟨ W^{H} (t), x ⟩ ⟨ W^{H} (s), y ⟩ = R (t, s) ⟨ Q (x), y ⟩ for all x, y \in Y ~and~ t, s \in [0; T] . \end{aligned}

In order to define Wiener integrals with respect to the Q-fBm $W^{H} (t)$ , we introduce the space $L_{2}^{0} := L_{2}^{0} (Y, X)$ of all Q-Hilbert–Schmidt operators $ψ : Y \to X$ . We recall that $ψ \in L (Y, X)$ is called a Q-Hilbert–Schmidt operator if

\begin{aligned} {‖ ψ ‖}_{L_{2}^{0}} = \sum_{n = 1}^{\infty} {‖ \sqrt{λ_{n}^{*}} ψ ξ_{n} ‖}^{2} < \infty \end{aligned}

and that the space

L_{2}^{0}

equipped with the inner product

⟨ φ, ψ ⟩_{L_{2}^{0}} = \sum_{n = 1}^{\infty} ⟨ φ ξ_{n}, ψ ξ_{n} ⟩

is a separable Hilbert space.

The fractional Wiener integral of the function $ψ : [0; T] \to L_{2}^{0} (Y, X)$ with respect to Q-fBm is defined by

\begin{aligned} \int_{0}^{t} ψ (s) d W^{H} (s) & = \sum_{n = 1}^{\infty} \int_{0}^{t} \sqrt{λ_{n}^{*}} ψ (s) ξ_{n} d β_{n}^{H} (s) \\ = \sum_{n = 1}^{\infty} \int_{0}^{t} \sqrt{λ_{n}^{*}} K_{H}^{*} (ψ ξ_{n}) (s) d β_{n} (s), \end{aligned}

(2.1)

where

β_{n}

is the standard Brownian motion used to present

β_{n}^{H}

. We have the following fundamental inequality which was proved in [6]. Lemma 2.1.

If $ψ : [0; T] \to L_{2}^{0} (Y, X)$ satisfies $\int_{0}^{T} {‖ ψ (s) ‖}_{L_{2}^{0} (Y, X)}^{2} d s < \infty$ then the sum in ( 2.1 ) is well defined as an $X$ -valued random variable and we have

\begin{aligned} E {‖ \int_{0}^{t} ψ (s) d W^{H} (s) ‖}^{2} ⩽ 2 H t^{2 H - 1} \int_{0}^{t} {‖ ψ (s) ‖}_{L_{2}^{0} (Y, X)}^{2} d s . \end{aligned}

2.2. Resolvent operators and representation of solutions

In this section give a representation of solutions to (1.1)–(1.3) by using a resolvent operator and recall some basic properties of this operator.

Consider the relaxation problem

\begin{aligned} ω^{'} (t) + β (1 + θ \partial_{t}^{α}) ω (t) = 0, t > 0, \end{aligned}

(2.2)

\begin{aligned} ω (0) = 1, \end{aligned}

(2.3)

where the unknown ω is a scalar function, β and θ are positive parameters. We collect some properties of ω in Proposition 2.2

Proposition 2.2 [18]

Let ω be the solution of ( 2.2 )–( 2.3 ). Then

(1)
$0 < ω (t) ⩽ 1$ for all $t ⩾ 0$ .
(2)
The function ω is completely monotone for $t ⩾ 0$ , i.e. $(- 1)^{n} ω^{(n)} (t) ⩾ 0$ for $t ⩾ 0$ and $n \in N$ . Consequently, ω is a nonincreasing function.
(3)
$β ω (t) ⩽ (t + G_{2 - α} (t))^{- 1} ⩽ min {t^{- 1}, t^{α - 1}}$ , for all $t > 0$ .
(4)
$\int_{0}^{t} ω (s) d s ⩽ β^{- 1} (1 - ω (t))$ , for any $t > 0$ .
(5)
For fixed $t ⩾ 0$ and $θ > 0$ , the function $β \mapsto ω (t, β)$ is nonincreasing on $[0, \infty)$ .

Denote by $ω (\cdot, β)$ the solution of (2.2)–(2.3), respecting to parameter β. In what follows, we use the notation $u * v$ to express the Laplace convolution of u and v, i.e.
$\begin{aligned} (u * v) (t) = \int_{0}^{t} u (t - s) v (s) d s, u, v \in L_{l o c}^{1} (R^{+}) . \end{aligned}$
We now concern with the inhomogeneous problem
$\begin{aligned} v^{'} (t) + β (1 + θ \partial_{t}^{α}) v (t) = m (t), t > 0, \end{aligned}$
(2.4)

$\begin{aligned} v (0) = v_{0}, \end{aligned}$
(2.5)
where $β > 0$ , $θ > 0$ and g is a continuous function. The representation of v is given in Proposition 2.2.
Proposition 2.3 [18]

The solution of (2.4)–(2.5) is given by

\begin{aligned} v (t) = ω (t, β) v_{0} + ω (\cdot, β) * m (t), t ⩾ 0, \end{aligned}

where ω is defined by (2.2)–(2.3).

Recall that $X = L^{2} (O)$ , let ${φ_{n}}_{n = 1}^{\infty}$ be the orthonormal basis of $X$ consisting of the eigenfunctions of the Laplacian $- Δ$ subject to homogeneous Dirichlet boundary condition, that is

\begin{aligned} - Δ φ_{n} = λ_{n} φ_{n} in O, φ_{n} = 0 on \partial O, \end{aligned}

where we can assume that

{λ_{n}}_{n = 1}^{\infty}

is an increasing sequence,

λ_{n} > 0

and

λ_{n} \to + \infty

n \to \infty

. Then one can give a representation of solution to the linear problem

\begin{aligned} \partial_{t} u - (1 + θ \partial_{t}^{α}) Δ u = F in O, t > 0, \end{aligned}

(2.6)

\begin{aligned} u = 0 on \partial O, t ⩾ 0, \end{aligned}

(2.7)

\begin{aligned} u (\cdot, 0) = ϕ in O, \end{aligned}

(2.8)

where

F \in L_{loc~}^{1} (R^{+}; X)

and

ϕ \in X

. Indeed, let

\begin{aligned} u (x, t) = \sum_{n = 1}^{\infty} u_{n} (t) φ_{n} (x), \\ F (x, t) = \sum_{n = 1}^{\infty} F_{n} (t) φ_{n} (x), ϕ (x) = \sum_{n = 1}^{\infty} ϕ_{n} φ_{n} (x) . \end{aligned}

Then

\begin{aligned} u_{n}^{^{'}} (t) + λ_{n} (1 + θ \partial_{t}^{α}) u_{n} (t) = F_{n} (t), u_{n} (0) = ϕ_{n} . \end{aligned}

Employing Proposition 2.3, we get

\begin{aligned} u_{n} (t) = ω (t, λ_{n}) ϕ_{n} + \int_{0}^{t} ω (t - s, λ_{n}) F_{n} (s) d s . \end{aligned}

This implies

\begin{aligned} u (\cdot, t) = S (t) ϕ + \int_{0}^{t} S (t - s) F (\cdot, s) d s, \end{aligned}

(2.9)

where

S (t) : X \to X

is the resolvent operator defined by

\begin{aligned} S (t) ϕ = \sum_{\infty}^{n = 1} ω (t, λ_{n}) ϕ_{n} φ_{n} . \end{aligned}

(2.10)

We recall some properties of the resolvent operator in the following lemma.

Lemma 2.4.

For any $v \in X$ , $T > 0$ , we have: (1)

$S (\cdot) v \in C ([0, T]; X) \cap C ((0, T]; H^{2} (O) \cap H_{0}^{1} (O))$ .

(2)

$‖ S (t) v ‖ ⩽ ω (t, λ_{1}) ‖ v ‖$ , for all $t ⩾ 0$ . In particular, $‖ S (t) ‖ ⩽ 1$ for all $t ⩾ 0$ .

(3)

For $ψ : [0; T] \to L_{2}^{0} (Y, X)$ then

\begin{aligned} \int_{0}^{t} {‖ S (t - τ) ψ (τ) ‖}_{L_{2}^{0}}^{2} d τ ⩽ \int_{0}^{t} ω^{2} (τ, λ_{1}) {‖ ψ (t - τ) ‖}_{L_{2}^{0}}^{2} d τ . \end{aligned}

(4)

For $f \in C ([0, T]; X)$ then

\begin{aligned} {‖ \int_{0}^{t} S (t - τ) f (τ) d τ ‖}^{2} ⩽ β^{- 1} (1 - ω (t, λ_{1})) \int_{0}^{t} ω (t - τ, λ_{1}) {‖ f (τ) ‖}^{2} d τ . \end{aligned}

Proof.

The proofs of (1), (2) we could find in [2].

(3) We have

\begin{aligned} \int_{0}^{t} {‖ S (t - τ) ψ (τ) ‖}_{L_{2}^{0}}^{2} d τ & = \int_{0}^{t} \sum_{k = 0}^{\infty} λ_{k}^{*} {‖ S (t - τ) ψ (τ) ξ_{k} ‖}^{2} d τ \\ = \sum_{k = 0}^{\infty} λ_{k}^{*} \sum_{n = 0}^{\infty} \int_{0}^{t} (ω (t - τ, λ_{n}) ψ_{n} (τ) φ_{n} ξ_{k})^{2} d τ \\ ⩽ \sum_{k = 0}^{\infty} λ_{k}^{*} \sum_{n = 0}^{\infty} \int_{0}^{t} ω^{2} (t - τ, λ_{n}) {| ψ_{n} (τ) φ_{n} ξ_{k} |}^{2} d τ \\ = \sum_{k = 0}^{\infty} λ_{k}^{*} \int_{0}^{t} ω^{2} (τ, λ_{1}) {‖ ψ (t - τ) ξ_{k} ‖}^{2} d τ \\ ⩽ \int_{0}^{t} ω^{2} (τ, λ_{1}) {‖ ψ (t - τ) ‖}_{L_{2}^{0}}^{2} d τ . \end{aligned}

The proof is complete.

(4) We see that

\begin{aligned} {‖ \int_{0}^{t} S (t - τ) f (τ) d τ ‖}^{2} & = \sum_{n = 1}^{\infty} {(\int_{0}^{t} ω (t - τ, λ_{n}) f_{n} (τ) d τ)}^{2} φ_{n} \\ ⩽ \sum_{n = 1}^{\infty} (\int_{0}^{t} ω (t - τ, λ_{n}) d τ \int_{0}^{t} ω (t - τ, λ_{n}) | f_{n} (τ) |^{2} d τ) φ_{n} \\ ⩽ β^{- 1} (1 - ω (t, λ_{1})) \sum_{n = 1}^{\infty} (\int_{0}^{t} ω (t - τ, λ_{n}) | f_{n} (τ) |^{2} d τ) φ_{n} \\ ⩽ β^{- 1} (1 - ω (t, λ_{1})) \int_{0}^{t} ω (t - τ, λ_{1}) (\sum_{n = 1}^{\infty} | f_{n} (τ) |^{2} φ_{n}) d τ \\ = β^{- 1} (1 - ω (t, λ_{1})) \int_{0}^{t} ω (t - τ, λ_{1}) {‖ f (τ) ‖}^{2} d τ, \end{aligned}

here we utilized Proposition 2.2(4). The proof is complete. □

Remark 2.1.

The first statement of Lemma 2.4 ensures that $S (t) : X \to X$ , $t > 0$ is a compact operator, due to the compactness of the embedding $H^{2} (O) \cap H_{0}^{1} (O) ↪ L^{2} (O)$ .

Consider the Cauchy operator $K : C ([0, T]; X) \to C ([0, T]; X)$ given by

\begin{aligned} K (v) (t) = \int_{0}^{t} S (t - s) v (s) d s . \end{aligned}

(2.11)

Denote by

{‖ \cdot ‖}_{\infty}

the sup norm in

C ([0, T]; X)

, i.e.

{‖ v ‖}_{\infty} = sup_{t \in [0, T]} ‖ g (t) ‖

. Lemma 2.5 shows the compactness of

K

Lemma 2.5 [2]

The Cauchy operator defined by (2.11) is compact.

We are in a position to prove a Halanay type inequality for the stability analysis in the next section.

Lemma 2.6.
Let w be a continuous and nonnegative function satisfying
$\begin{aligned} w (t) ⩽ ω (t, β) w_{0} + \int_{0}^{t} ω (t - s, β) [p sup_{μ \in [s - h (s), s]} w (μ) + q (s)] d s, t > 0, \end{aligned}$
(2.12)

$\begin{aligned} w (s) = ψ (s), s \in [- ζ, 0], \end{aligned}$
(2.13)
where $0 < p < β$ , $ψ \in C ([- ζ, 0]; R^{+})$ and $q \in L_{l o c}^{1} (R^{+})$ which is nondecreasing. Then
$\begin{aligned} w (t) ⩽ \frac{β}{β - p} [w_{0} + \int_{0}^{t} ω (t - s, β) q (s) d s] + sup_{s \in [- ζ, 0]} ψ (s), \forall t > 0. \end{aligned}$
(2.14)
In addition, if $ω (\cdot, β) * q$ is bounded on $R^{+}$ then
$\begin{aligned} \underset{t \to \infty}{\lim \sup} w (t) ⩽ sup_{t \in R^{+}} \int_{0}^{t} ω (t - s, β) q (s) d s . \end{aligned}$
(2.15)
In particular, if $q = 0$ then $w (t) \to 0$ as $t \to \infty$ .
Proof.
We use the following result [21]: if $w \in C ([- τ, \infty); R^{+})$ is a nonnegative function satisfying
$\begin{aligned} w (t) ⩽ ζ (t) + ℓ sup_{ζ \in [- τ, t]} w (ζ), t > 0, \\ w (s) = ψ (s), s \in [- τ, 0], \end{aligned}$
where $ζ (\cdot)$ is a nondecreasing function and $ℓ \in (0, 1)$ , then
$\begin{aligned} w (t) ⩽ {(1 - ℓ)}^{- 1} ζ (t) + sup_{s \in [- τ, 0]} ψ (s), \forall t > 0. \end{aligned}$
(2.16)
We can conclude from (2.12) that
$\begin{aligned} w (t) & ⩽ w_{0} + ω (\cdot, μ) * q (t) + p sup_{ζ \in [- h, t]} w (ζ) \int_{0}^{t} ω (t - s, μ) d s \\ ⩽ w_{0} + ω (\cdot, μ) * q (t) + \frac{p}{μ} sup_{ζ \in [- h, t]} w (ζ) (1 - ω (t, μ)) \\ ⩽ w_{0} + ω (\cdot, μ) * q (t) + \frac{p}{μ} sup_{ζ \in [- h, t]} w (ζ), \end{aligned}$
here we employed Proposition 2.2(4). Since $q (\cdot)$ is nondecreasing, it is evident that the function $ω (\cdot, μ) * q$ is nondecreasing as well. Applying inequality (2.16) for $ζ (\cdot) = ω (\cdot, μ) * q$ and $ℓ = p / μ$ , we get (2.14) as desired. Now assume that $ω (\cdot, μ) * q$ is bounded on $R^{+}$ . Then by (2.14), $w (\cdot)$ is bounded by
$\begin{aligned} M := \frac{μ}{μ - p} [w_{0} + sup_{t \in R^{+}} \int_{0}^{t} ω (t - s, μ) q (s) d s] + sup_{s \in [- τ, 0]} ψ (s), \end{aligned}$
and therefore the limit $L = lim_{t \to \infty} sup_{ζ \in [t, \infty)} w (ζ)$ exists. Since $t - ρ (t) \to \infty$ as $t \to \infty$ , for any $ε > 0$ , one can find $T_{1} > 0$ such that
$\begin{aligned} sup_{ζ \in [t - ρ (t), t]} w (ζ) ⩽ sup_{ζ \in [t - ρ (t), \infty]} w (ζ) ⩽ L + ε, \forall t ⩾ T_{1} . \end{aligned}$
According to the last estimate, we see that
$\begin{aligned} w (t) & ⩽ ω (t, μ) w_{0} + ω (\cdot, μ) * q (t) \\ + (\int_{0}^{T_{1}} + \int_{T_{1}}^{t}) ω (t - s, μ) p sup_{ζ \in [s - ρ (s), s]} w (ζ) d s \\ ⩽ ω (t, μ) w_{0} + ω (\cdot, μ) * q (t) \\ + p M \int_{0}^{T_{1}} ω (t - s, μ) d s + p (L + ε) \int_{T_{1}}^{t} ω (t - s, μ) d s \\ ⩽ ε w_{0} + ω (\cdot, μ) * q (t) \\ + p M \int_{t - T_{1}}^{t} ω (t - s, μ) d s + p (L + ε) \int_{0}^{t} ω (t - s, μ) d s \\ ⩽ ε w_{0} + ω (\cdot, μ) * q (t) + p M ε + p (L + ε) μ^{- 1}, \end{aligned}$
(2.17)
provided t chosen such that
$\begin{aligned} ω (t, μ) ⩽ ε, \int_{t - T_{1}}^{t} ω (t - s, μ) d s ⩽ ε, \end{aligned}$
which is possible since $ω (t, μ) \to 0$ as $t \to \infty$ and $ω (\cdot, μ) \in L^{1} (R^{+})$ . Equation (2.17) yields that
$\begin{aligned} L = lim_{t \to \infty} sup_{ζ \in [t, \infty]} w (ζ) ⩽ p L μ^{- 1} + sup_{t \in R^{+}} ω (\cdot, μ) * q (t) + (w_{0} + p M + p μ^{- 1}) ε, \end{aligned}$
which implies that
$\begin{aligned} L ⩽ \frac{μ}{μ - p} sup_{t \in R^{+}} ω (\cdot, μ) * q (t) + \frac{μ}{μ - p} (w_{0} + p M + p μ^{- 1}) ε . \end{aligned}$
Hence
$\begin{aligned} \underset{t \to \infty}{\lim \sup} w (t) ⩽ L ⩽ \frac{μ}{μ - p} sup_{t \in R^{+}} ω (\cdot, μ) * q (t), \end{aligned}$
thanks to the fact that $ε$ is an arbitrarily positive number.

To prove the main results, we use the following hypothesis throughout this work Assumption G

The map $g : R^{+} \to L_{2}^{0} (Y, X)$ satisfies

\begin{aligned} \underset{t \to + \infty}{lim sup} t^{2 H - 1} \int_{0}^{t} {ω^{2} (t - τ, λ_{1}) ‖ g (τ) ‖}_{L_{2}^{0}}^{2} d τ < + \infty . \end{aligned}

(2.18)

3. Solvability results

Based on representation (2.9), we give Definition 3.1.

Definition 3.1.
Let $ϕ \in C ([- ζ, 0]; L^{2} (Ω, X))$ be given. An $X$ -valued stochastic process $u (t)$ is said to be a mild solution to the problem (1.1)–(1.3) on the interval $[- ζ, T]$ iff (i)
$u \in C ([- ζ, T]; L^{2} (Ω, X))$ ,
(ii)
$u (\cdot, s) = ϕ (\cdot, s)$ for $s \in [- ζ, 0]$ ,
(iii)
$u (\cdot, t) = S (t) ϕ (\cdot, 0) + \int_{0}^{t} S (t - s) f (s, u_{h} (\cdot, s)) d s + \int_{0}^{t} S (t - s) g (s) d W^{H} (s)$ , $t \in [0, T]$ .

For given $ϕ \in C ([- ζ, 0]; L^{2} (Ω, X)$ , denote $C_{ϕ} ([0, T]; L^{2} (Ω, X) := {u \in C ([0, T]; L^{2} (Ω, X) : u (\cdot, 0) = ϕ (\cdot, 0)}$ .

For $u \in C_{ϕ} ([0, T]; L^{2} (Ω, X)$ , we define $u [ϕ] \in C ([- ζ, T]; L^{2} (Ω, X)$ as follows
$\begin{aligned} u [ϕ] (\cdot, t) = {\begin{cases} u (\cdot, t) & if t \in [0, T], \\ ϕ (\cdot, t) & if t \in [- ζ, 0] . \end{cases} \end{aligned}$
Hence, we have
$\begin{aligned} u [ϕ]_{h} (\cdot, t) = {\begin{cases} u (\cdot, t - h (t)) & if t - h (t) \in [0, T], \\ ϕ (\cdot, t - h (t)) & if t - h (t) \in [- ζ, 0] \end{cases} \end{aligned}$

In what follows, we use the notation $‖ \cdot ‖_{\infty}$ for the sup norm in the spaces $C ([- ζ, 0]; L^{2} (Ω, X), C ([- ζ, T]; L^{2} (Ω, X)$ and $C ([0, T]; L^{2} (Ω, X)$ . For example if $v \in C ([- ζ, 0]; L^{2} (Ω, X)$ then $‖ v ‖_{\infty} = (sup_{t \in [- ζ, 0]} E ‖ v (t) ‖^{2})^{\frac{1}{2}}$ .

Let $Σ : C_{ϕ} ([0, T]; L^{2} (Ω, X) \to C_{ϕ} ([0, T]; L^{2} (Ω, X)$ be the operator defined by
$\begin{aligned} Σ (u) (\cdot, t) = S (t) ϕ (\cdot, 0) + \int_{0}^{t} S (t - s) f (s, u [ϕ]_{h} (\cdot, s)) d s + \int_{0}^{t} S (t - s) g (s) d W^{H} (s), \end{aligned}$
which will be referred to as the solution operator. This operator is continuous if f is a continuous map. Obviously, u is a fixed point of Σ iff $u [ϕ]$ is a mild solution of (1.1)–(1.3).

We show global existence for (1.1)–(1.3) in the next theorems.
Theorem 3.1.
Let $f : [0, T] \times L^{2} (Ω, X) \to L^{2} (Ω, X)$ be a continuous mapping such that

(F1) $E ‖ f (t, v) ‖^{2} ⩽ κ (t) E (L (‖ v ‖))^{2}$ for all $t \in [0, T]$ and $v \in L^{2} (Ω, X)$ , where $κ \in L^{1} (0, T)$ is a nonnegative function and $L$ is a continuous and nonnegative function obeying that
$\begin{aligned} \underset{r \to 0}{\lim \sup} \frac{L (r)}{r} \cdot sup_{t \in [0, T]} \int_{0}^{t} ω (t - s, λ_{1}) κ (s) d s < 1. \end{aligned}$
(3.1)
Then there exists $δ > 0$ such that the problem (1.1)–(1.3) has at least one mild solution on $[- ζ, T]$ , provided $‖ ϕ ‖_{\infty} ⩽ δ$ .
Proof.
Let
$\begin{aligned} γ = \underset{r \to 0}{\lim \sup} \frac{L (r)}{r}, M = sup_{t \in [0, T]} ω (\cdot, λ_{1}) * κ (t) . \end{aligned}$
Then by assumption, one can take $ϵ > 0$ such that $(γ + ϵ) M < 1$ . In addition, there is $τ > 0$ such that
$\begin{aligned} \frac{L (r)}{r} ⩽ γ + ϵ, \forall r \in [0, 2 τ] \end{aligned}$
and
$\begin{aligned} 6 H t^{2 H - 1} \int_{0}^{t} ω^{2} (t - s, λ_{1}) {‖ g (s) ‖}_{L_{2}^{0}}^{2} d s ⩽ \frac{τ^{2}}{2} . \end{aligned}$
Let
$\begin{aligned} δ_{0}^{2} & = \frac{τ^{2}}{2} inf_{t \in [0, T]} {{[3 ω^{2} (t, λ_{1}) + 6 (γ + ϵ)^{2} β^{- 1} ω (\cdot, λ_{1}) * κ (t)]}^{- 1} \\ \times [1 - 12 (γ + ϵ)^{2} β^{- 1} ω (\cdot, λ_{1}) * κ (t)]}, \end{aligned}$
then $δ_{0}^{2} > 0$ provided that $12 (γ + ϵ) β^{- 1} ⩽ 1$ . Indeed, we see that
$\begin{aligned} {[3 ω^{2} (t, λ_{1}) + 6 (γ + ϵ)^{2} β^{- 1} ω (\cdot, λ_{1}) * κ (t)]}^{- 1} ⩾ {[3 + 6 (γ + ϵ)^{2} β^{- 1} M]}^{- 1}, \end{aligned}$
then
$\begin{aligned} δ_{0}^{2} & = \frac{τ^{2}}{2} {[3 + 6 (γ + ϵ)^{2} β^{- 1} M]}^{- 1} inf_{t \in [0, T]} [1 - 12 (γ + ϵ)^{2} β^{- 1} ω (\cdot, λ_{1}) * κ (t)] \\ ⩾ \frac{τ^{2}}{2} {[3 + 6 (γ + ϵ)^{2} β^{- 1} M]}^{- 1} [1 - 12 (γ + ϵ)^{2} β^{- 1} sup_{t \in [0, T]} ω (\cdot, λ_{1}) * κ (t)] \\ = \frac{τ^{2}}{2} {[3 + 6 (γ + ϵ)^{2} β^{- 1} M]}^{- 1} [1 - 12 (γ + ϵ)^{2} β^{- 1} M] > 0. \end{aligned}$
Denote by $B_{τ}$ the closed ball in $C_{ϕ} ([0, T]; L^{2} (Ω, X))$ centered at origin with radius τ. Considering $Σ : B_{τ} \to C_{ϕ} ([0, T]; L^{2} (Ω, X))$ , we have
$\begin{aligned} E {‖ Σ (u) (\cdot, t) ‖}^{2} & ⩽ 3 E {‖ S (t) ϕ (\cdot, 0) ‖}^{2} + 3 E {‖ \int_{0}^{t} S (t - s) f (s, u [ϕ]_{h} (\cdot, s)) d s ‖}^{2} \\ + 3 E {‖ \int_{0}^{t} S (t - s) g (s) d W^{H} (s) ‖}^{2} \\ ⩽ I_{1} (t) + I_{2} (t) + I_{3} (t), \end{aligned}$
where
$\begin{aligned} I_{1} (t) = 3 E {‖ S (t) ϕ (\cdot, 0) ‖}^{2}; \\ I_{2} (t) = 3 E {‖ \int_{0}^{t} S (t - s) f (s, u [ϕ]_{h} (\cdot, s)) d s ‖}^{2}; \\ I_{3} (t) = 3 E {‖ \int_{0}^{t} S (t - s) g (s) d W^{H} (s) ‖}^{2} . \end{aligned}$
Put $δ = min {δ_{0}, τ}$ . If $ϕ \in C ([- ζ, 0]; L^{2} (Ω, X))$ such that $‖ ϕ ‖_{\infty} ⩽ δ$ , then
$\begin{aligned} E {‖ u [ϕ]_{h} (\cdot, s) ‖}^{2} ⩽ 2 ‖ u ‖_{\infty}^{2} + 2 ‖ ϕ ‖_{\infty}^{2} ⩽ 2 τ^{2} + 2 δ^{2} for all s \in [0, T] . \end{aligned}$
So
$\begin{aligned} I_{1} (t) ⩽ 3 ω^{2} (t, λ_{1}) ‖ ϕ ‖^{2} ⩽ 3 ω^{2} (t, λ_{1}) δ^{2} . \end{aligned}$
(3.2)
And
$\begin{aligned} I_{2} (t) & ⩽ 3 β^{- 1} (1 - ω (t, λ_{1})) \int_{0}^{t} ω (t - s, λ_{1}) E {‖ f (s, u [ϕ]_{h} (\cdot, s)) ‖}^{2} d s \\ ⩽ 3 β^{- 1} (1 - ω (t, λ_{1})) \int_{0}^{t} ω (t - s, λ_{1}) κ (s) E (L (‖ u [ϕ]_{h} (\cdot, s) ‖)^{2} d s \\ ⩽ 3 (γ + ϵ)^{2} β^{- 1} (1 - ω (t, λ_{1})) \int_{0}^{t} ω (t - s, λ_{1}) κ (s) E {‖ u [ϕ]_{h} (\cdot, s) ‖}^{2} d s \\ ⩽ 3 (γ + ϵ)^{2} (2 τ^{2} + 2 δ^{2}) β^{- 1} (1 - ω (t, λ_{1})) \int_{0}^{t} ω (t - s, λ_{1}) κ (s) d s \\ ⩽ 3 (γ + ϵ)^{2} (2 τ^{2} + 2 δ^{2}) β^{- 1} \int_{0}^{t} ω (t - s, λ_{1}) κ (s) d s, \end{aligned}$
(3.3)
thanks to Lemma 2.4(4) and assumption (F1).

For $I_{3} (t)$ , we have
$\begin{aligned} I_{3} (t) & ⩽ 3 E {‖ \int_{0}^{t} S (t - s) g (s) d W^{H} (s) ‖}^{2} \\ ⩽ 6 H t^{2 H - 1} \int_{0}^{t} {‖ S (t - s) g (s) ‖}_{L_{2}^{0}}^{2} d s \\ ⩽ 6 H t^{2 H - 1} \int_{0}^{t} ω^{2} (t - s, λ_{1}) {‖ g (s) ‖}_{L_{2}^{0}}^{2} d s \end{aligned}$
(3.4)
Combining (3.2), (3.3), (3.4) we arrive at
$\begin{aligned} E {‖ Σ (u) (\cdot, t) ‖}^{2} & ⩽ [3 ω^{2} (t, λ_{1}) + 6 (γ + ϵ)^{2} β^{- 1} ω (\cdot, λ_{1}) * κ (t)] δ_{0}^{2} \\ + 6 τ^{2} (γ + ϵ)^{2} β^{- 1} ω (\cdot, λ_{1}) * κ (t) \\ + 6 H t^{2 H - 1} \int_{0}^{t} ω^{2} (t - s, λ_{1}) {‖ g (s) ‖}_{L_{2}^{0}}^{2} d s \\ ⩽ τ^{2}, \forall t \in [0, T] . \end{aligned}$
We have shown that $Σ (B_{τ}) \subset B_{τ}$ , provided $‖ ϕ ‖_{\infty} ⩽ δ$ . Consider $Σ : B_{τ} \to B_{τ}$ . In order to apply the Schauder fixed point theorem, it remains to check that Σ is a compact operator. It should be noted that, Σ admits the representation
$\begin{aligned} Σ (u) (\cdot, t) = S (t) ϕ (\cdot, 0) + K \circ N_{f} (u) (\cdot, t) + \int_{0}^{t} S (t - s) g (s) d W^{H} (s), \end{aligned}$
where $N_{f} (u) (\cdot, t) = f (t, u [ϕ]_{h} (\cdot, t))$ . According to the compactness of $K$ stated in Lemma 2.4, we conclude that Σ is compact. The proof is complete.

In the following theorem, we prove the existence and uniqueness of the solution. Theorem 3.2.
Let $f : [0, T] \times L^{2} (Ω, X) \to L^{2} (Ω, X)$ be a continuous mapping such that

(F2) $f (\cdot, 0) = 0$ and $E ‖ f (t, v_{1}) - f (t, v_{2}) ‖^{2} ⩽ κ (t) η (r) E ‖ v_{1} - v_{2} ‖^{2}$ for all $t \in [0, T]$ and $v_{1}, v_{2} \in L^{2} (Ω, X)$ such that $E ‖ v_{1} ‖^{2}, E ‖ v_{2} ‖^{2} ⩽ r^{2}$ , where $κ \in L^{1} (0, T)$ is a nonnegative function and η is a continuous function obeying that
$\begin{aligned} \underset{r \to 0}{\lim \sup} η (r) \cdot sup_{t \in [0, T]} \int_{0}^{t} ω (t - s, λ_{1}) κ (s) d s < 1. \end{aligned}$
(3.5)
Then there exists $δ > 0$ such that the problem (1.1)–(1.3) has a unique mild solution on $[- ζ, T]$ , provided $‖ ϕ ‖_{\infty} ⩽ δ$ .
Proof.
The existence result can be obtained by applying Theorem 3.1 with $L (r) = η (r) r$ . It remains to prove the uniqueness. Assume that $u_{1} [ϕ]$ , $u_{2} [ϕ]$ are solutions of (1.1)–(1.3). Let $R = max {‖ u_{1} [ϕ] ‖_{\infty}, ‖ u_{2} [ϕ] ‖_{\infty}}$ , then
$\begin{aligned} E {‖ u_{1} (\cdot, t) - u_{2} (\cdot, t) ‖}^{2} & ⩽ β^{- 1} \int_{0}^{t} ω (t - s, λ_{1}) E {‖ f (u_{1} [ϕ]_{h} (\cdot, s)) - f (u_{2} [ϕ]_{h} (\cdot, s)) ‖}^{2} d s \\ ⩽ \int_{0}^{t} ω (t - s, λ_{1}) \frac{κ (s) η (R)}{β} sup_{μ \in [0, s]} E {‖ u_{1} (\cdot, μ) - u_{2} (\cdot, μ) ‖}^{2} d s, \end{aligned}$
thanks to Lemma 2.4(4) and the fact that $u_{1} (\cdot, s) = u_{2} (\cdot, s)$ for $s \in [- ζ, 0]$ . It is evident that the last integral does not decrease in t then
$\begin{aligned} sup_{μ \in [0, t]} E {‖ u_{1} (\cdot, μ) - u_{2} (\cdot, μ) ‖}^{2} \\ ⩽ \int_{0}^{t} ω (t - s, λ_{1}) \frac{κ (s) η (R)}{β} sup_{μ \in [0, s]} E {‖ u_{1} (\cdot, μ) - u_{2} (\cdot, μ) ‖}^{2} d s, t \in [0, T], \end{aligned}$
by means of the Halanay inequality in Lemma 2.6 with $q (\cdot) = 0$ , $p = \frac{‖ κ ‖_{\infty} η (R)}{β}$ implies that $u_{1} = u_{2}$ . a.e. The proof is complete.

4. Mean square moment asymptotic stability

Our goal of this section is to prove the mean square moment asymptotic stability of zero solution to (1.1).

Theorem 4.1.

Let $f : R^{+} \times L^{2} (Ω, X) \to L^{2} (Ω, X)$ be a continuous mapping such that

(F3) $f (\cdot, 0) = 0$ and $E {‖ f (t, v_{1}) - f (t, v_{2}) ‖}^{2} ⩽ κ (t) η (r) E {‖ v_{1} - v_{2} ‖}^{2}$ for all $t \in R^{+}$ and $v_{1}, v_{2} \in L^{2} (Ω, X)$ such that $E ‖ v_{1} ‖^{2}, E ‖ v_{2} ‖^{2} ⩽ r^{2}$ , where $κ \in L^{\infty} (R^{+})$ is a nonnegative function and η is a continuous function satisfying that

\begin{aligned} ‖ κ ‖_{\infty} \cdot \underset{r \to 0}{\lim \sup} η (r) < β λ_{1} . \end{aligned}

Then the zero solution of (1.1) is mean square asymptotically stable.

Proof.

Let $γ = lim sup_{r \to 0} η (r)$ . Choosing $θ > 0$ such that $‖ κ ‖_{\infty} (γ + θ) < β λ_{1}$ , we can find $τ > 0$ such that $η (r) ⩽ γ + θ$ for all $r \in [0, 2 τ]$ . Reasoning as in the proof of Theorems 3.1 and 3.2, there exists $δ > 0$ such that the problem (1.1)–(1.3) has a unique mild solution $u \in B_{τ}$ as long as $‖ ϕ ‖_{\infty} ⩽ δ$ , which is defined on $[- ζ, T]$ for all $T > 0$ .

The final step involves proving that the obtained solution is asymptotically stable. Let $\hat{u}$ be the solution of (1.1) with respect to the initial datum $\hat{ϕ}$ such that $‖ \hat{ϕ} ‖_{\infty} ⩽ δ$ . Moreover, we have

\begin{aligned} E {‖ \hat{u} (\cdot, t) - u (\cdot, t) ‖}^{2} & ⩽ 2 ω^{2} (t, λ_{1}) E {‖ \hat{ϕ} - ϕ ‖}^{2} \\ + 2 β^{- 1} \int_{0}^{t} ω (t - s, λ_{1}) E {‖ f (\hat{u} {[\hat{ϕ}]}_{h} (\cdot, s)) - f (u [ϕ]_{h} (\cdot, s)) ‖}^{2} d s \\ ⩽ 2 ω (t, λ_{1}) E {‖ \hat{ϕ} - ϕ ‖}^{2} \\ + 2 \int_{0}^{t} ω (t - s, λ_{1}) \frac{κ (s) (γ + θ)}{β} sup_{ζ \in [s - h (s), s]} E {‖ \hat{u} (\cdot, ζ) - u (\cdot, ζ) ‖}^{2} d s \end{aligned}

Employing Lemma 2.6 again with

q (\cdot) = 0

p = \frac{‖ κ ‖_{\infty} (γ + θ)}{β}

, we obtain

\begin{aligned} E {‖ \hat{u} (t) - u (t) ‖}^{2} & ⩽ (\frac{λ_{1}}{λ_{1} - \frac{{‖ κ ‖}_{\infty} (γ + θ)}{β}} + 1) E ‖ \hat{ϕ} - ϕ ‖^{2} \end{aligned}

and

\begin{aligned} lim_{t \to \infty} E {‖ \hat{u} (t) - u (t) ‖}^{2} & = 0, \end{aligned}

which implies the mean square asymptotic stability of the zero solution of (1.1). The proof is complete.

Remark 4.1.

To show that a function g exists such that it satisfies the Assumptions (G) in (2.18), we consider function $g : R^{+} \to L_{2}^{0} (X, X)$ define by

\begin{aligned} g (t) x = \frac{1}{\sqrt{1 + t^{m}}} x, \forall x \in X \end{aligned}

with

m ⩾ 2 H - 1

. For

t ⩾ T > 0

, we have

\begin{aligned} t^{2 H - 1} \int_{0}^{t} ω^{2} (t - τ, λ_{1}) {‖ g (τ) ‖}_{L_{2}^{0}}^{2} d τ & = t^{2 H - 1} \int_{0}^{t} ω^{2} (τ, λ_{1}) {‖ g (t - τ) ‖}_{L_{2}^{0}}^{2} d τ \\ ⩽ t^{2 H - 1} \int_{0}^{t / 2} ω^{2} (τ, λ_{1}) {‖ g (t - τ) ‖}_{L_{2}^{0}}^{2} d τ \\ + t^{2 H - 1} \int_{t / 2}^{t} ω^{2} (τ, λ_{1}) {‖ g (t - τ) ‖}_{L_{2}^{0}}^{2} d τ \\ ⩽ \frac{t^{2 H - 1}}{1 + (\frac{t}{2})^{m}} \int_{0}^{t / 2} ω^{2} (τ, λ_{1}) d τ \\ + t^{2 H - 1} \int_{t / 2}^{t} \frac{1}{λ_{1}^{2} τ^{2}} d τ \\ = \frac{t^{2 H - 1}}{1 + (\frac{t}{2})^{m}} \int_{0}^{t / 2} ω^{2} (τ, λ_{1}) d τ + \frac{t^{2 H - 2}}{λ_{1}^{2}} < + \infty, \end{aligned}

thanks to the fact that

2 H - 2 < 0

‖ g (t) ‖_{L_{2}^{0}} ⩽ 1

\forall t ⩾ 0

and Proposition 2.2 (1), (3). Hence the Assumptions (G) is fulfilled.

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The existences and asymptotic behavior of solutions to stochastic semilinear generalized Rayleigh–Stokes equation with delays

Abstract

Keywords

1. Introduction

2.1. Fractional Brownian motion

References