Existence and Uniqueness for Stochastic Age-Dependent Population with Fractional Brownian Motion

Abstract

A model for a class of age-dependent population dynamic system of fractional version with Hurst parameter ℎ∈(1/2,1] is established. We prove the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients. The proofs of our results combine techniques of fractional Brownian motion calculus. Ideas of the finite-dimensional approximation by the Galerkin method are used.

1. Introduction

Stochastic differential equations have been found in many applications in areas such as economics, biology, finance, ecology, and other sciences [1–3]. In recent years, existence, uniqueness, stability, invariant measures, and other quantitative and qualitative properties of solutions to stochastic partial differential equations have been extensively investigated by many authors. For example, it is well known that these topics have been developed mainly by using two different methods, that is, the semigroup approach [4, 5] (e.g., Taniguchi et al. [4] using semigroup methods discussed existence, uniqueness, pth moment, and almost sure Lyapunov exponents of mild solutions to a class of stochastic partial functional differential equations with finite delays) and the variational one (e.g., Krylov and Rozovskii [6] and Pardoux [7]). On the other hand, although stochastic partial functional differential equations also seem very important as stochastic models of biological, chemical, physical, and economical systems, the corresponding properties of these systems have not been studied in great detail (cf. [8, 9]). As a matter of fact, there exists extensive literature on the related topics for deterministic age-dependent population dynamic system. There has been much recent interest in application of deterministic age-structures mathematical models with diffusion. For example, Cushing [10] investigated hierarchical age-dependent populations with intraspecific competition or predation.

There has been much recent interest in application of stochastic population dynamics. For example, Qimin and Chongzhao gave a numerical scheme and showed the convergence of the numerical approximation solution to the true solution to stochastic age-structured population system with diffusion [11]. In papers [12, 13], Qi-Min et al. discussed the existence and uniqueness for stochastic age-dependent population equation, when diffusion coefficient 𝑘=0 and 𝑘≠0, respectively. Numerical analysis for stochastic age-dependent population equation has been studied by Zhang and Han [14]. In papers [11–14], the random disturbances are described by stochastic integrals with respect to Wiener processes.

However, the Wiener process is not suitable to replace a noise process if long-rang dependence is modeled. It is then desirable to replace the Wiener process by fractional Brownian motion. But this process is not a semimartingale, so that it is not possible to apply the Ito calculus. A stochastic analysis with respect to fractional Brownian motion is faced with difficulties.

Next, the stochastic continuous time age-dependent model is derived. In [12], the nonlinear age-dependent population dynamic with diffusion can be written in the following form: 𝜕𝑃(𝑟,𝑡,𝑥)+()𝜕𝑡𝜕𝑃𝑟,𝑡,𝑥𝜕𝑟−𝑘₁(𝑟,𝑡)Δ𝑃(𝑟,𝑡,𝑥)=−𝜇₁(𝑟,𝑡,𝑥)𝑃(𝑟,𝑡,𝑥)+𝑓₁(𝑟,𝑡,𝑥)+𝑔₁()(𝑡)𝑟,𝑡,𝑥𝑑𝑤𝑑𝑡,in𝑄_𝐴=()𝑃()=0,𝐴×𝑄,0,𝑡,𝑥^𝐴₀𝛽₁()𝑃()()𝑃()𝑟,𝑡,𝑥𝑟,𝑡,𝑥𝑑𝑟,in0,𝑇×Γ,𝑟,0,𝑥=𝑃₀()()𝑃()𝑟,𝑥,in0,𝐴×Γ,𝑟,𝑡,𝑥=0,onΣ_𝐴=()×()𝑦()=0,𝐴0,𝑇×𝜕Γ,𝑡,𝑥^𝐴₀𝑃(𝑟,𝑡,𝑥)𝑑𝑟,in𝑄, (1.1) where 𝑡∈(0,𝑇), 𝑟∈(0,𝐴), 𝑥∈Γ⊂𝑅^𝑁(1≤𝑁≤3), 𝑄=(0,𝑇)×Γ, 𝑃(𝑟,𝑡,𝑥) denotes the population density of age 𝑟 at time 𝑡 in spatial position, 𝑥, 𝛽₁(𝑟,𝑡,𝑥) denotes the fertility rate of females of age 𝑟 at time 𝑡, in spatial position 𝑥, 𝜇₁(𝑟,𝑡,𝑥) denotes the mortality rate of age 𝑟 at time 𝑡, in spatial position 𝑥, Δ denotes the Laplace operator with respect to the space variable, and 𝑘₁(𝑟,𝑡)>0 is the diffusion coefficient. 𝑓₁(𝑟,𝑡,𝑥)+𝑔₁(𝑟,𝑡,𝑥)(𝑑𝑤(𝑡)/𝑑𝑡) denotes effects of external environment for population system, such as emigration and earthquake have. The effects of external environment the deterministic and random parts which depend on 𝑟, 𝑡, and 𝑥. 𝑤(𝑡) is a standard Wiener process.

In this paper, suppose that 𝑓₁(𝑟,𝑡,𝑥) is stochastically perturbed, with 𝑓₁(𝑟,𝑡,𝑥)⟶𝑓₁(𝑟,𝑡,𝑥)+𝑔₁(𝑟,𝑡,𝑥)𝑑𝐵^ℎ(𝑡), (1.2) where 𝐵^ℎ(𝑡) is fractional Brownian motions with the Hurst constant ℎ. Then this environmentally perturbed system may be described by the Ito equation 𝜕𝑃(𝑟,𝑡,𝑥)+()𝜕𝑡𝜕𝑃𝑟,𝑡,𝑥𝜕𝑟−𝑘₁(𝑟,𝑡)Δ𝑃(𝑟,𝑡,𝑥)=−𝜇₁(𝑟,𝑡,𝑥)𝑃(𝑟,𝑡,𝑥)+𝑓₁(𝑟,𝑡,𝑥)+𝑔₁(𝑟,𝑡,𝑥)𝑑𝐵^ℎ(𝑡),in𝑄_𝐴=()𝑃()0,𝐴×𝑄,0,𝑡,𝑥=^𝐴₀𝛽₁()𝑃()()𝑃()𝑟,𝑡,𝑥𝑟,𝑡,𝑥𝑑𝑟,in0,𝑇×Γ,𝑟,0,𝑥=𝑃₀()()𝑃()𝑟,𝑥,in0,𝐴×Γ,𝑟,𝑡,𝑥=0,onΣ_𝐴=()×()𝑦()=0,𝐴0,𝑇×𝜕Γ,𝑡,𝑥^𝐴₀𝑃(𝑟,𝑡,𝑥)𝑑𝑟,in𝑄, (1.3)(1.4)(1.5)(1.6)(1.7) new stochastic differential equations (1.3)-(1.7) for an age-dependent population are derived. It is an extension of (1.1).

Our work differs from these references [11–14]. In papers [11–14], the random disturbances are described by stochastic integrals with respect to Wiener processes. In this paper, we study a stochastic age-dependent population dynamic system with an additive noise in the form of a stochastic integral with respect to a Hilbert space-valued fractional Borwnian motion. It is well known that a fractional Brownian motion 𝐵^ℎ is a semimartingale if and only if ℎ=1/2, that is, in the case of a classical Brownian motion. For ℎ=1/2, Qimin and Chongzhao discussed the existence and uniqueness for stochastic age-dependent population equation [12]. In this paper, we shall discuss the existence and uniqueness for a stochastic age-dependent population equation with fractional Brownian motions with ℎ∈[1/2,1]. The discussion uses ideas of the finite-dimensional approximation by the Galerkin method.

In Section 2, we begin with some preliminary results which are essential for our analysis and introduce the definition of a solution with respect to stochastic age-dependent populations. In Section 3, we shall prove existence and uniqueness of solution for stochastic age-dependent population equation (1.3).

2. Preliminaries

Consider stochastic age-structured population system with diffusion (1.3). 𝐴 is the maximal age of the population species, so 𝑃(𝑟,𝑡,𝑥)=0,∀𝑟≥𝐴. (2.1) By (1.7), integrating on [0,𝐴] to (1.3) and (1.5) with respect to 𝑟, we obtain the following system𝜕𝑦(𝑡)()()𝑦()()𝜕𝑡−𝑘Δ𝑦+𝜇𝑡,𝑥𝑦−𝛽𝑡,𝑥=𝑓𝑡,𝑥+𝑔𝑡,𝑥𝑑𝐵^ℎ(𝑡)()𝑦()𝑑𝑡,in𝑄=0,𝑇×Γ,0,𝑥=𝑦₀(𝑥)𝑦()(),inΓ,𝑡,𝑥=0,onΣ=0,𝑇×𝜕Γ, (2.2) where 𝛽(𝑡,𝑥)≡^𝐴₀𝛽₁(𝑟,𝑡,𝑥)𝑃(𝑟,𝑡,𝑥)𝑑𝑟^𝐴₀𝑃(𝑟,𝑡,𝑥)𝑑𝑟⁻¹, (2.3)

where ∫^𝐴₀𝑃(𝑟,𝑡,𝑥)𝑑𝑟=𝑦(𝑡,𝑥) is the total population, and the birth process is described by the nonlocal boundary conditions ∫^𝐴₀𝛽₁(𝑟,𝑡,𝑥)𝑃(𝑟,𝑡,𝑥)𝑑𝑟 clearly, 𝛽(𝑡,𝑥) denotes the fertility rate of total population at time 𝑡 and in spatial position 𝑥. 𝜇(𝑡,𝑥)≡^𝐴₀𝜇₁(𝑟,𝑡,𝑥)𝑃(𝑟,𝑡,𝑥)𝑑𝑟^𝐴₀𝑃(𝑟,𝑡,𝑥)𝑑𝑟⁻¹, (2.4) where 𝜇(𝑡,𝑥) denotes the mortality rate at time 𝑡 and in spatial position 𝑥𝑓(𝑡,𝑥)≡^𝐴₀𝑓₁()𝑔()≡𝑟,𝑡,𝑥𝑑𝑟,𝑡,𝑥^𝐴₀𝑔₁(𝑟,𝑡,𝑥)𝑑𝑟. (2.5) Let 𝑉=𝐻¹(Γ)≡𝜑∣𝜑∈𝐿²(Γ),𝜕𝜑𝜕𝑥_𝑖∈𝐿²(Γ),where𝜕𝜑𝜕𝑥_𝑖aregeneralizedpartialderivatives. (2.6) Then 𝑉′=𝐻⁻¹(Γ) the dual space of 𝑉. We denote by |⋅| and ‖⋅‖ the norms in 𝑉 and 𝑉^′ respectively, by ⟨⋅,⋅⟩ the duality product between 𝑉, 𝑉^′, and by (⋅,⋅) the scalar product in 𝐻.

We consider stochastic age-structured population system with diffusion of the form 𝑑_𝑡𝑦(𝑡)(𝑡)()𝑦(𝑡)()𝑦(𝑡)()()−𝑘Δ𝑦𝑑𝑡+𝜇𝑡,𝑥𝑑𝑡−𝛽𝑡,𝑥𝑑𝑡=𝑓𝑡,𝑥𝑑𝑡+𝑔𝑡,𝑥𝑑𝐵^ℎ(𝑡)()𝑦(),in𝑄=0,𝑇×Γ,0,𝑥=𝑦₀(𝑥)𝑦()(),inΓ,𝑡,𝑥=0,onΣ=0,𝑇×𝜕Γ, (2.7) where 𝑑_𝑡𝑦(𝑡) is the differential of 𝑦(𝑡,𝑥) relative to 𝑡, that is, (𝑑_𝑡𝑦(𝑡)=𝜕𝑦(𝑡)/𝜕𝑡)𝑑𝑡, 𝑦(𝑡)∶=𝑦(𝑡,𝑥). 𝑇>0,𝐴>0.

The integral version of (2.7) is given by the equation 𝑦(𝑡)−𝑦(0)−^𝑡₀𝑘Δ𝑦(𝑠)𝑑𝑠−^𝑡₀𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)𝑦(𝑠)𝑑𝑠=^𝑡₀𝑓(𝑠,𝑥)𝑑𝑠+^𝑡₀𝑔(𝑠,𝑥)𝑑𝐵^ℎ(𝑠), (2.8) here 𝑦(𝑡,𝑥)=0, on Σ=(0,𝑇)×𝜕Γ.

Let 𝐵^ℎ_𝑗(𝑡)_𝑡≥0 (𝑗=1,2,…) be independent centered Gaussian processes with 𝐵^ℎ_𝑗(0)=0 on a given probability space (Ω,ℱ,𝑃), where we assume that 𝐸𝐵^ℎ_𝑗(𝑡)−𝐵^ℎ_𝑗(𝑠)²=|𝑡−𝑠|^2ℎ𝜇_𝑗,𝜇𝑗=1,2,…_𝑗>0,^∞_𝑗=1𝜇_𝑗<∞, (2.9) and ℎ∈[1/2,1].

The processes 𝐵^ℎ_𝑗(𝑡)_𝑡≥0 are independent fractional Brownian motions with the Hurst constant ℎ and 𝐸(𝐵^ℎ_𝑗(1))²=𝜇_𝑗(𝑗=1,2,…).

It follows from Kleptsyna et al. (cf. [15]) that 𝐵^ℎ_𝑗(𝑡)=⁰_−∞|𝑡−𝑟|^ℎ−1/2−|𝑟|^ℎ−1/2𝑑𝑊_𝑗(𝑟)+^𝑡₀|𝑡−𝑟|^ℎ−1/2𝑑𝑊_𝑗(𝑟), (2.10) where (𝑊_𝑗(𝑡))_𝑡≥0 (𝑗 = 1, 2, …) are real independent Wiener processes with 𝐸𝑊²_𝑗(𝑡)=𝜇_𝑗𝑡.

Let 𝐾 be a separable Hilbert space with the scalar product (⋅,⋅)_𝐾, and (𝑒_𝑗)_𝑗=1,2,… denotes a complete orthogonal system in 𝐾, Then ^∞_𝑗=1𝐸‖𝐵^ℎ_𝑗(𝑡)𝑒_𝑗‖²_𝐾=𝑡^∞2ℎ_𝑗=1𝜇_𝑗<∞, (2.11) and 𝐵^ℎ(𝑡)=∑^∞_𝑗=1𝐵^ℎ_𝑗(𝑡)𝑒_𝑗 is called a 𝐾-valued fractional Brownian motion where the sum is defined mean square.

Definition 2.1. A 𝐻-valued continuous stochastic process (𝑦(𝑡))_{𝑡∈[0,𝑇]} with 𝑦(𝑡)∈𝑉 (𝑃-a.s) is a solution of (2.7) if it holds for 𝑣∈𝑉 and all 𝑡∈[0,𝑇] that 𝑦(𝑡),𝑣_𝐻=𝑦(0),𝑣_𝐻+^𝑡₀𝑘Δ𝑦(𝑠),𝑣𝑑𝑠+^𝑡₀𝛽(𝑠,𝑥)𝑦(𝑠)−𝜇(𝑠,𝑥)𝑦(𝑠),𝑣_𝐻+𝑑𝑠_𝑡0𝑓(𝑠,𝑥),𝑣_𝐻𝑑𝑠+^𝑡₀𝑔(𝑠,𝑥)𝑑𝐵^ℎ(𝑠),𝑣_𝐻,𝑃-a.s. (2.12)
The objective in this paper is that we hopefully find a unique process 𝑦(𝑡) such that (2.7) holds For this objective, we assume that the following conditions are satisfied:
(1)𝜇(𝑡,𝑥), 𝛽(𝑡,𝑥) and 𝑘(𝑟,𝑡) are nonnegative measurable, and0≤𝑘₀≤𝑘(𝑡)<∞in(0,𝐴)×(0,𝑇),0≤𝜇₀()()()≤𝜇𝑡,𝑥<∞in0,𝐴×Γ,0≤𝛽𝑡,𝑥≤𝛽₀<∞in(0,𝐴)×Γ. (2.13)(2)Let 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) be measurable functions which are defined on 𝑄 with 𝑓(𝑡,𝑥)𝑔(𝑡,𝑥)≤𝐾, (2.14) where 𝐾 is a positive constant.

3. Existence and Uniqueness of Solutions

Consider also the 𝐾-valued fractional Brownian motion 𝐵^ℎ,𝑛(𝑡)=∑^𝑛_𝑖=1𝐵^ℎ_𝑖(𝑡)𝑒_𝑖. Obviously, the following lemma holds.

If the process (𝑦(𝑡))_{𝑡∈[0,𝑇]} is a solution of (2.7), then the process 𝑍(𝑡)=𝑦(𝑡)−∫^𝑡₀𝑔(𝑠)𝑑𝐵^ℎ(𝑠) solves 𝑑_𝑡𝑍(𝑡)−𝑘Δ𝑍(𝑡)𝑑𝑡+𝜇(𝑡,𝑥)𝑍(𝑡)+^𝑡₀𝑔(𝑠)𝐵^ℎ(𝑠)𝑑𝑠−𝛽(𝑡,𝑥)𝑍(𝑡)+^𝑡₀𝑔(𝑠)𝑑𝐵^ℎ(𝑠)()𝑑𝑡=𝑓𝑡,𝑥𝑑𝑡+𝑘Δ^𝑡₀𝑔(𝑠)𝑑𝐵^ℎ(𝑠)()𝑍()𝑑𝑡,in𝑄=0,𝑇×Γ,0,𝑥=𝑍₀(𝑥)𝑍()(),inΓ,𝑡,𝑥=0,onΣ=0,𝑇×𝜕Γ, (3.1) where 𝑍(𝑡)∶=𝑍(𝑡,𝑥). If 𝑍(𝑡) is a solution of (3.1), then exists a process 𝑦(𝑡)_{𝑡∈[0,𝑇]} so that 𝑍(𝑡) can be written as 𝑍(𝑡)=𝑦(𝑡)−∫^𝑡₀𝑔(𝑠,𝑥)𝑑𝐵^ℎ(𝑠), and consequently 𝑦(𝑡) solves (2.7).

As a result, we shall consider (3.1) instead of (2.7). It is noted that, for fixed 𝜔∈Ω, (3.1) is a deterministic problem.

Lemma 3.1. Problem (3.1) has, for fixed 𝜔∈Ω, a unique solution 𝑍(𝑡), and there exists a nonnegative random variable 𝜂 with finite expectation such that sup_{0≤𝑠≤𝑇}|𝑍(𝑠)|²+𝑘₀^𝑇₀‖𝑍(𝑠)‖²𝑑𝑠≤𝜂, (3.2) where for fixed 𝜔∈Ω, 𝑍(𝑡) is continuous with respect to 𝑡 in 𝐻.

Proof. The Galerkin approximations are defined by 𝑍_𝑛(𝑡)=∑^𝑛_𝑖=1𝑍_𝑛,𝑖(𝑡)𝑣_𝑖, where 𝑍_𝑛,𝑖(𝑡) solves the stochastic equations 𝑍_𝑛,𝑖(𝑡)=𝑦(0),𝑣_𝑖𝐻+^𝑡₀𝑘Δ^𝑛_𝑘=1𝑍_𝑛,𝑘(𝑠)𝑣_𝑘,𝑣_𝑖+𝑑𝑠_𝑡0𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)^𝑛_𝑘=1𝑍_𝑛,𝑘(𝑠)𝑣_𝑘,𝑣_𝑖+𝑑𝑠_𝑡0𝑓(𝑠,𝑥),𝑣_𝑖𝐻𝑑𝑠+^𝑡₀𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢),𝑣_𝑖𝑑𝑠+^𝑡₀𝑘Δ^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢),𝑣_𝑖𝑑𝑠.(𝑖=1,2,…,𝑛). (3.3) It follows from the assumption (2) that (3.3) can be solved for every 𝜔 by the method of successive approximation, and the iterates are measurable with respect to 𝜔. Consequently, (𝑍_𝑛,𝑖(𝑡))_{𝑡∈[0,𝑇]}(𝑖=1,2,…,𝑛) are stochastic processes since 𝑦₀ is a random 𝐻-valued variable and (𝐵^ℎ,𝑛(𝑡))_{𝑡∈[0,𝑇]} is a stochastic process. It follows from (3.3) that 𝑍_𝑛(𝑡)=^𝑛_𝑖=1𝑦(0),𝑣_𝑖𝐻𝑣_𝑖+^𝑡₀^𝑛_𝑖=1𝑘Δ𝑍_𝑛(𝑠),𝑣_𝑖𝑣_𝑖+𝑑𝑠_{𝑡0𝑛𝑖=1}𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)^𝑛_𝑖=1𝑍_𝑛(𝑠),𝑣_𝑖𝑣_𝑖+𝑑𝑠_𝑡0𝑓(𝑠,𝑥),𝑣_𝑖𝐻𝑣_𝑖𝑑𝑠+^𝑡₀𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢),𝑣_𝑖𝑣_𝑖+𝑑𝑠_𝑡0𝑘Δ^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢),𝑣_𝑖𝑣_𝑖𝑑𝑠. (3.4) Using the chain rule, we get the following 𝑍_𝑛(𝑡)²=^𝑛_𝑗=1𝑦(0),𝑣_𝑗²_𝐻+2^𝑡₀𝑘Δ𝑍_𝑛(𝑠),𝑍_𝑛(𝑠)𝑑𝑠+2^𝑡₀𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)𝑍_𝑛(𝑠),𝑍_𝑛(𝑠)𝑑𝑠+2^𝑡₀𝑓(𝑠,𝑥),𝑍_𝑛(𝑠)𝑑𝑠+2^𝑡₀𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢)𝑑𝑠,𝑍_𝑛(𝑠)𝑑𝑠+2^𝑡₀𝑘Δ^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢),𝑍_𝑛(𝑠)𝑑𝑠. (3.5) If we set 𝐵(𝑡)≡0 in of Qimin and Chongzhao [11], under assumptions (1)-(2), then this result implies that sup_{0≤𝑠≤𝑇}𝑍_𝑛(𝑠)²+𝑘₀^𝑡₀𝑍_𝑛(𝑠)²𝑑𝑠≤𝜂. (3.6) The following result is an analogous to that of Theorem  4 in [1]. In the Galerkin approximation, we have 𝐸𝑍_𝑛(𝑡)−𝑍(𝑡)²+𝐸^𝑡₀𝑍_𝑛(𝑠)−𝑍(𝑠)²𝑑𝑠⟶0 (3.7) for all 𝑡∈[0,𝑇] and for 𝑛→∞. 𝑍(𝑡) is a 𝐻-valued continuous process with 𝑍(𝑡)∈𝑉 for all 𝑡∈[0,𝑇]𝑃-a.s., and 𝑍(𝑡) is a 𝑃-a.s. unique solution.
Now let (𝐵^ℎ(𝑡))_{𝑡∈[0,𝑇]} be a 𝐻-valued fractional Brownian motion with ∑^∞_𝑗=1𝜆_𝑗𝜇_𝑗<∞ and ∑^∞_𝑗=1𝜆_𝑗𝜇_𝑗^1/2<∞. We consider the finite-dimensional approximation ^𝑛_𝑗=1^𝑡₀𝑔(𝑠,𝑥)𝑑𝐵^ℎ_𝑗(𝑠)𝑣_𝑗𝑑𝑠 (3.8)
in mean square of the stochastic integral ∫^𝑡₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ(𝑢). Obviously this is a stochastic integral with respect to the 𝑉-valued Brownain motion 𝐵^ℎ,𝑛(𝑢)=∑^𝑛_𝑗=1𝐵^ℎ_𝑗(𝑢)𝑣_𝑗. Consequently, the corresponding Galerkin equations for (2.7) are given by 𝑑_𝑡𝑦^𝑚(𝑡)−𝑘Δ𝑦^𝑚(𝑡)𝑑𝑡+𝜇(𝑡,𝑥)𝑦^𝑚(𝑡)𝑑𝑡−𝛽(𝑡,𝑥)𝑦^𝑚(𝑡)()()𝑑𝑡=𝑓𝑡,𝑥𝑑𝑡+𝑔𝑡,𝑥𝑑𝐵^ℎ,𝑚(𝑡)()𝑦,in0,𝑇×Γ,_𝑚(0)=^𝑚_𝑗=1𝑦₀,𝑣_𝑗𝑣_𝑗𝑦,inΓ,_𝑚()()𝑑𝑡,𝑥=0,onΣ=0,𝑇×𝜕Γ,_𝑡𝑦^𝑛(𝑡)−𝑘Δ𝑦^𝑛𝑑𝑡+𝜇(𝑡,𝑥)𝑦^𝑛(𝑡)𝑑𝑡−𝛽(𝑡,𝑥)𝑦^𝑛(𝑡)()()𝑑𝑡=𝑓𝑡,𝑥𝑑𝑡+𝑔𝑡,𝑥𝑑𝐵^ℎ,𝑛(𝑡)()𝑦,in0,𝑇×Γ,_𝑛(0)=^𝑛_𝑗=1𝑦₀,𝑣_𝑗𝑣_𝑗𝑦,inΓ,_𝑛(𝑡,𝑥)=0,onΣ=(0,𝑇)×𝜕Γ. (3.9)(3.10) Lemma 3.1 shows that these problems have solutions.

Theorem 3.2. If ∑^∞_𝑗=1𝜆_𝑗𝜇_𝑗<∞ and ∑^∞_𝑗=1𝜆_𝑗𝜇_𝑗^1/2<∞, then there exists a 𝑃-𝑎.𝑠 unique solution (𝑦(𝑡))_{𝑡∈[0,𝑇]} of (2.7) with 𝐸𝑦(𝑡)²+𝑘₀𝐸^𝑡₀‖𝑍(𝑠)‖²𝑑𝑠≤𝑀_𝑡,ℎ,∀𝑡∈[0,𝑇], (3.11) where 𝑀_𝑡,ℎ is a positive constant.

Proof. We choose 𝑛>𝑚 with 𝑛=𝑚+𝑝 and define 𝑍^𝑚,𝑝(𝑡)=𝑦^𝑚+𝑝(𝑡)−𝑦^𝑚(𝑡)−^𝑡₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑚(𝑢)+^𝑡₀𝑔(𝑢,𝑥)𝑑𝐵^{ℎ,𝑚+𝑝}(𝑢). (3.12) Then 𝑍^𝑚,𝑝(𝑡)²≤𝑦^𝑚+𝑝(0)−𝑦^𝑚(0)²+2^𝑡₀𝑘Δ𝑍^𝑚,𝑝(𝑠),𝑍^𝑚,𝑝(𝑠)𝑑𝑠+2^𝑡₀𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)𝑦^𝑚+𝑝(𝑠)−𝑦^𝑚(𝑠),𝑍^𝑚,𝑝(𝑠)𝑑𝑠+2^𝑡₀𝑓(𝑠,𝑥),𝑍^𝑚,𝑝(𝑠)𝑑𝑠+2^𝑡₀𝑘Δ^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^{ℎ,𝑚+𝑝}(𝑢)−𝐵^ℎ,𝑚(𝑢),𝑍^𝑚,𝑝(𝑠)𝑑𝑠. (3.13) However, by Lemma 2.2 [14] and assumptions (1)-(2), we have ^𝑡₀𝑘Δ^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^{ℎ,𝑚+𝑝}(𝑢)−𝐵^ℎ,𝑛(𝑢),𝑍^𝑚,𝑝𝑑𝑠≤𝑘₀^𝑛_𝑗=𝑚+1𝜆_𝑗𝐸^𝑡₀^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ_𝑗(𝑢)𝑣_𝑗,𝑍^𝑚,𝑝(𝑠)≤1𝑑𝑠2𝑘₀^𝑛_𝑗=𝑚+1𝜆_𝑗^𝑡₀𝐸^𝑠₀𝑔(𝑢,𝑥)𝑣_𝑗𝑑𝐵^ℎ_𝑗(𝑢)²1𝑑𝑠+2^𝑡₀𝐸𝑍^𝑚,𝑝(𝑠)²≤1𝑑𝑠2𝑘₀𝑇^2ℎ𝐾²𝑇^𝑛_𝑗=𝑚+1𝜆_𝑗𝜇_𝑗+12^𝑡₀𝐸𝑍^𝑚,𝑝(𝑠)²𝑑𝑠. (3.14) Further, 𝐸𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)𝑦^𝑚+𝑝(𝑠)−𝑦^𝑚(𝑠),𝑍^𝑚,𝑝(𝑠)𝛽𝑑𝑠≤𝐸₀−𝜇₀𝑍^𝑚,𝑝(𝑠)²+𝛽₀−𝜇₀^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^{ℎ,𝑚+𝑝}(𝑢)−𝐵^ℎ,𝑛(𝑢)𝑍^𝑚,𝑝(𝑠𝛽)≤2₀−𝜇₀𝐸𝑍^𝑚,𝑝(𝑠)²+𝛽₀−𝜇₀𝐾²𝑇^𝑚+𝑝_𝑗=𝑚+1𝜇_𝑗. (3.15) Consequently, in view of (3.13), 𝐸𝑍^𝑛(𝑠)²+2𝑘₀𝐸^𝑡₀𝑍^𝑛(𝑠)²𝑑𝑠≤2𝛽₀−𝜇₀+𝐾²+1^𝑡₀𝐸𝑍^𝑛(𝑠)²𝑑𝑠+𝑘₀𝐶_ℎ𝐾²𝑇^𝑚+𝑝_𝑗=𝑚+1𝜆_𝑗𝜇_𝑗+2𝛽₀−𝜇₀𝑇𝐾^2𝑚+𝑝_𝑗=𝑚+1𝜇_𝑗. (3.16) Then, the Gronwall’s lemma implies that 𝐸𝑍^𝑚,𝑝(𝑠)²⟶0,𝐸^𝑡₀𝑍^𝑚,𝑝(𝑠)²⟶0 (3.17) for 𝑚,𝑝→∞ for all 𝑡∈[0,𝑇]. In particular, there exists a process (𝑍(𝑡))_{𝑡∈[0,𝑇]} with 𝐸|𝑍^𝑚(𝑡)−𝑍(𝑡)|²→0 for 𝑚→∞, and consequently, there exists a process 𝑦(𝑡) with 𝐸|𝑦^𝑚(𝑡)−𝑦(𝑡)|²→0 for 𝑚→∞. We must now show that (𝑦(𝑡))_{𝑡∈[0,𝑇]} is solution of (2.7). We have 𝐸𝑦^𝑛(𝑡)−𝑦^𝑚(𝑡)+^𝑠₀𝑔(𝑢,𝑎)𝑑𝐵^{ℎ,𝑚+𝑝}(𝑢)−𝐵^ℎ,𝑛(𝑢)²+2𝑘₀𝐸^𝑡₀𝑦^𝑛(𝑠)−𝑦^𝑚(𝑠)²𝑑𝑠≤2𝐸^𝑡₀𝑘Δ𝑦^𝑛(𝑠)−𝑦^𝑚(𝑠),^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢)−𝐵^ℎ,𝑚(𝑢)𝑑𝑠+2𝐸^𝑡₀𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)𝑦^𝑛−𝑦^𝑚(𝑠),𝑦^𝑛(𝑠)−𝑦^𝑚(𝑠)+^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢)−𝐵^ℎ,𝑚(𝑢)𝑑𝑠+2𝐸^𝑡₀𝑓(𝑠,𝑥),𝑦^𝑛(𝑠)−𝑦^𝑚(𝑠)+^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^{ℎ,𝑚+𝑝}(𝑢)−𝐵^ℎ,𝑛(𝑢)𝑑𝑠. (3.18) Let 𝜀>0 be chosen arbitrary. Then there exists 𝑝₀>0 so that ∑^𝑚+𝑝_𝑗=𝑝+1𝜆_𝑗𝜇_𝑗^1/2<𝜀 for all 𝑝>𝑝₀. Let 𝑦^𝑛,𝑟(𝑡)=∑^𝑟_𝑗=1𝑦_𝑗^𝑛,𝑟(𝑡)𝑣_𝑗 and 𝑦^𝑚,𝑟(𝑡)=∑^𝑟_𝑗=1𝑦_𝑗^𝑚,𝑟(𝑡)𝑣_𝑗 be the 𝑟th Galerkin approximation of 𝑦^𝑛(𝑡) and 𝑦^𝑚(𝑡), respectively. For 𝑟=𝑚+𝑝, we have 𝐸^𝑡₀𝑘Δ𝑦^𝑛,𝑟(𝑠)−𝑦^𝑚,𝑟(𝑠),^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ,𝑛(𝑢)−𝐵^ℎ,𝑚(𝑢)𝑑𝑠≤𝑘₀𝐸^𝑚+𝑝_𝑖=𝑝+1^𝑡₀𝜆_𝑖𝑦_𝑖^𝑛,𝑟(𝑠)−𝑦_𝑖^𝑚,𝑟(𝑠)^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ_𝑖(𝑢)𝑑𝑠≤𝑘₀𝐸^𝑚+𝑝_𝑖=𝑝+1^𝑡₀𝑦_𝑖^𝑛,𝑟(𝑠)−𝑦_𝑖^𝑚,𝑟(𝑠)²𝑑𝑠^1/2𝜆_𝑖𝐸^𝑡₀^𝑠₀𝑔(𝑢,𝑥)𝑑𝐵^ℎ_𝑖(𝑢)²𝑑𝑠^1/2≤const.𝑘₀𝐸^𝑚+𝑝_𝑖=𝑝+1𝜆_𝑖𝜇_𝑖^1/2<const.×𝜀. (3.19) Consequently, the first term on the right-hand side of (3.10) is also less than const.×𝜀. It is clear that the second term and third term on the right-hang side of (3.18) tends to zero. Then (3.18) gives 𝐸^𝑡₀‖𝑦^𝑚+𝑝(𝑠)−𝑦^𝑝(𝑠)‖²⟶0 (3.20) for 𝑚,𝑝→∞, there is (𝑦^𝑚(𝑡)) is also a Cauchy sequence in 𝐿²_𝑉(Ω×[0,𝑇]) for all 𝑡∈[0,𝑇]. Let 𝑦 be the limit a of this sequence. Then it follows from the properties of a Gelfand triple that 𝐸^𝑡₀𝑦_𝑛(𝑠)−𝑦(𝑠)²≤𝑀𝐸^𝑡₀𝑦_𝑛(𝑠)−𝑦(𝑠)²⟶0 (3.21) for 𝑛→∞, where 𝑀 is a positive constant. Consequently, 𝑦(𝑠)=𝑦(𝑠)(a.s) and it follows from (3.9) that 𝑑_𝑡𝑦(𝑡)−𝑘Δ𝑦(𝑡)𝑑𝑠−𝛽(𝑠,𝑥)−𝜇(𝑠,𝑥)𝑦(𝑡)𝑑𝑠=𝑓(𝑠,𝑥)𝑑𝑠+𝑔(𝑠,𝑥)𝑑𝐵^ℎ(𝑡), (3.22) hence, we have proved Theorem 3.2.

Acknowledgments

The authors would like to thank the referees for their very helpful comments which greatly improved this paper. The research was supported by The National Natural Science Foundation (no. 11061024) (China).