On a Fractional SPDE Driven by Fractional Noise and a Pure Jump Lévy Noise in ℝ^𝑑

Abstract

We study a stochastic partial differential equation in the whole space 𝑥∈ℝ^𝑑, with arbitrary dimension 𝑑≥1, driven by fractional noise and a pure jump Lévy space-time white noise. Our equation involves a fractional derivative operator. Under some suitable assumptions, we establish the existence and uniqueness of the global mild solution via fixed point principle.

1. Introduction

Let 𝑑≥1, 𝛼=(𝛼₁,…,𝛼_𝑑), 𝛿=(𝛿₁,…,𝛿_𝑑), and 𝔻^𝛼_𝛿 be the nonlocal fractional differential operator defined by 𝔻^𝛼_𝛿=^𝑑_𝑖=1𝐷^𝛼𝑖_𝛿𝑖, (1) where 𝐷^𝛼𝑖_𝛿𝑖 denotes the fractional differential derivative with respect to the 𝑖th coordinate defined via its Fourier transform ℱ by ℱ𝐷^𝛼_𝛿𝜑(𝜉)=−𝜉^𝛼𝜋exp−𝑖𝛿2sgn(𝜉)ℱ𝜑(𝜉). (2)

In this paper, we are concerned with the following jump type stochastic partial differential equation (SPDE for abbreviation) with fractional noise: 𝜕𝑢()𝜕𝑡𝑡,𝑥=𝔻^𝛼_𝛿𝑢()((((𝐿̇()𝑡,𝑥+𝑏𝑡,𝑥,𝑢𝑡,𝑥))+𝜎𝑡,𝑥,𝑢𝑡,𝑥))𝑡,𝑥+𝐵^𝐻(),𝑢()𝑑𝑡,𝑑𝑥0,𝑥=𝑢₀(𝑥), (3) where (𝑡,𝑥)∈[0,𝑇]×ℝ^𝑑, 𝐿̇ is a pure jump Lévy space-time white noise on [0,𝑇]×ℝ^𝑑 defined on a complete probability space (Ω,ℱ,𝑃), and 𝐵^𝐻(𝑑𝑡,𝑑𝑥) denotes the fractional noise on [0,𝑇]×ℝ^𝑑 with multiparameter 𝐻=(ℎ₀,ℎ₁,…,ℎ_𝑑) for ℎ_𝑖∈(0,1) and 𝑖∈{0,1,2,…,𝑑} defined on a complete probability space (Ω,ℱ,𝑃) (see Section 2 for precise definitions). Actually, we understand this equation as Walsh [1] sense, and so we can rewrite (3) as follows: 𝑢()=𝑡,𝑥_ℝ𝑑𝔾_𝛼,𝛿𝑡,𝑥−𝑦𝑢₀𝑦+𝑑𝑦_𝑡0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦𝐵^𝐻+𝑑𝑦,𝑑𝑠_𝑡0ℝ𝑑𝔾_𝛼,𝛿𝑏+𝑡−𝑠,𝑥−𝑦𝑠,𝑦,𝑢𝑠,𝑦𝑑𝑦𝑑𝑠_𝑡0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦𝜎𝑠,𝑦,𝑢𝑠,𝑦𝐿̇𝑦,𝑠𝑑𝑦𝑑𝑠, (4) for all 𝑡∈[0,𝑇] and 𝑥∈ℝ^𝑑, where 𝔾_𝛼,𝛿(⋅,∗) denotes the Green function associated with (3).

In the present paper, we are interested in the study of (3) with respect to 𝑑-dimensional nonlocal fractional differential operator 𝔻^𝛼_𝛿. Such operator is initially introduced by Debbi and Dozzi [2] for 𝑑=1 and is generalized by Boulanba et al. [3] for the multidimensional space ℝ^𝑑. This operator is a generalization of various well-known operators, such as the Laplacian operator, the inverse of the generalized Riesz-Feller potential, and the Riemann-Liouville differential operator. In probabilistic terms, replacing the Laplacian by its fractional power (which is an integrodifferential operator) leads to interesting and largely open questions of extensions of results for Brownian motion driven stochastic equations to those driven by Lévy stable processes. In the physical literature, such fractal anomalous diffusions have been recently enthusiastically embraced by a slew of investigators in the context of hydrodynamics, acoustics, trapping effects in surface diffusion, statistical mechanics, relaxation phenomena, and biology. We refer the readers to [4–9] and references therein for more information about such fractional differential operator.

On the other hand, many researchers are interested in studying SPDES driven by a fractional noise. The heat equations with a multiparameter fractional noise of Hurst parameter 𝐻=(ℎ₀,…,ℎ_𝑑) on [0,∞]×ℝ^𝑑 were introduced by Hu [10], and he showed the existence and uniqueness of the solutions to the equation, via chaos expansion. Wei [11] considered a class of four-order stochastic partial differential equations driven by the multiparameter fractional noise; the author studied the regularity of the solution and the existence of the density of the law of the solution. Hu and Nualart [12] studied the 𝑑-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter ℎ∈(0,1) in time. More works for the fields can be found in Balan and Tudor [13, 14], Bo et al. [15, 16], Jiang et al. [17–19], and Shi and Wang [20] and the references therein. In the meanwhile, there also have been some works on SPDES involving Lévy space-time white noise (e.g., Albeverio et al. [21], Shi and Wang [22], Truman and Wu [23, 24], and Wu and Xie [25]). Løkka et al. [26] studied the stochastic partial differential equations driven by a 𝑑-parameter pure jump Lévy white noise. As an example they used this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension 𝑑. We notice that the fixed point principle and Picard iteration scheme work in Albeverio et al. [21] and Truman and Wu [23, 24], since Burkhölder-Davis-Gundy (B-D-G) inequality can be applied to estimate stochastic integral with respect to compensated Poisson random measure in 𝐿²-sense. Unfortunately, the usual B-D-G inequality cannot work in estimating stochastic integral with respect to compensated Poisson random measure in 𝐿^𝑝-sense (𝑝>2). Hence a new version of B-D-G inequality will be adopted for 𝐿^𝑝(𝑝≥2)-estimates on the solution to (3) (see Bo and Wang [27]).

Motivated by the above results, in this paper, we study a stochastic fractional partial differential equation in the whole space 𝑥∈ℝ^𝑑, with arbitrary dimension 𝑑≥1, driven by fractional noise and a pure jump Lévy space-time white noise. The main subject of this paper is to establish the existence and uniqueness of the solution of (3) via fixed point principle.

The rest of the paper is organized as follows. In Section 2, we begin by making some notations and by recalling some basic preliminaries about fractional noises, Lévy space-time white noise, and Green function which will be needed later. In Section 3, we will prove the existence and uniqueness of the mild solution to (4) in 𝐿^𝑝(𝑝≥2) sense under some approximate conditions. Most of the estimates of this paper contain unspecified constants. An unspecified positive and finite constant will be denoted by 𝐶, which may not be the same in each occurrence. Sometimes we will emphasize the dependence of these constants upon parameters.

2. Preliminaries

In this section, we will present the definitions and some results of the multiparameter fractional noises, Lévy space-time white noise, and Green function.

2.1. Multiparameter Fractional Noises

Recall that a fractional Brownian motion with Hurst parameter ℎ∈(0,1) on [0,𝑇] is a centered Gaussian process (𝐵_𝑡) with covariance 𝐸𝐵_𝑡𝐵_𝑠=12𝑡^2ℎ+𝑠^2ℎ−|𝑡−𝑠|^2ℎ. (5) Following Hu [10] and Wei [11], we introduce a multiparameter fractional Brownian field.

Definition 1. A multiparameter fractional Brownian field 𝐵^𝐻={𝐵^𝐻(𝑡,𝑥),(𝑡,𝑥)∈[0,𝑇]×ℝ^𝑑} with multiparameter 𝐻=(ℎ₀,ℎ₁,…,ℎ_𝑑) for ℎ_𝑖∈(0,1) and 𝑖∈{0,1,2,…,𝑑} is a centered Gaussian field defined on some probability space (Ω,ℱ,𝑃) with covariance 𝐸𝐵^𝐻(𝑡,𝑥)𝐵^𝐻=1𝑠,𝑦2^𝑑+1𝑡^2ℎ0+𝑠^2ℎ0−|𝑡−𝑠|^2ℎ0×^𝑑_𝑖=1𝑥^2ℎ𝑖_𝑖+𝑦^2ℎ𝑖_𝑖−𝑥_𝑖−𝑦_𝑖^2ℎ𝑖, (6) for all 𝑡,𝑠∈[0,𝑇] and 𝑥=(𝑥₁,…,𝑥_𝑑), 𝑦=(𝑦₁,…,𝑦_𝑑)∈ℝ^𝑑.

Throughout the paper, we limit our consideration on the multiparameter fractional Brownian field with parameter 𝐻=(ℎ₀,ℎ₁,…,ℎ_𝑑)=(ℎ,ℎ,…,ℎ) and ℎ∈(1/2,1).

Let Ψ_ℎ𝑡,𝑠,𝑥,𝑦∶=[ℎ(2ℎ+1)]^𝑑+1|𝑡−𝑠|^𝑑2ℎ−2_𝑖=1𝑥_𝑖−𝑦_𝑖^2ℎ−2,0≤𝑠≤𝑇,𝑥,𝑦∈ℝ^𝑑. (7) Introduce the following Hilbert space: 𝐿²_Ψ∶=𝑓∶[0,𝑇]×ℝ^𝑑𝑓⟶ℝ;_2Ψ=_{[0,𝑇]2ℝ2𝑑}Ψ_ℎ()𝑓,𝑡,𝑠,𝑥,𝑦×𝑓𝑡,𝑥𝑠,𝑦𝑑𝑦𝑑𝑥𝑑𝑠𝑑𝑡<∞ (8) where 𝑑𝑥=𝑑𝑥₁𝑑𝑥₂,…,𝑑𝑥_𝑑 and 𝑑𝑦=𝑑𝑦₁𝑑𝑦₂,…,𝑑𝑦_𝑑. Let 𝑓∈𝐿²_Ψ; one can define the following stochastic integral: ^𝑡_0ℝ𝑑𝑓(𝑠,𝑥)𝑑𝐵^𝐻(𝑡,𝑥);𝑡∈[0,𝑇], (9) (see, e.g., Hu [10]); it is easy to check that this integral has the usual properties as a stochastic integral.

Proposition 2. Let 𝑓,𝑔∈𝐿²_Ψ. Then(1)𝐸[∫^𝑇₀∫_ℝ𝑑𝑓(𝑠,𝑥)𝑑𝐵^𝐻(𝑠,𝑥)]=0,(2)𝐸[∫^𝑇₀∫_ℝ𝑑𝑓(𝑠,𝑥)𝑑𝐵^𝐻(𝑠,𝑥)∫^𝑇₀∫_ℝ𝑑𝑔(𝑠,𝑥)𝑑𝐵^𝐻(𝑠,𝑥)]=⟨𝑓,𝑔⟩_𝐿²_Ψ.

The following embedding proposition which is an extension to the results in Mémin et al. [28] had been proved in Wei [11].

Lemma 3. Consider the following:
𝐿^ℎ([0,𝑇]×ℝ^𝑑)⊂𝐿²_Ψ (ℎ>1/2).

2.2. Lévy Space-Time White Noise

Let (𝐸_𝑖,ℰ_𝑖,𝜇_𝑖),  𝑖=1,2, be two 𝜎-finite measurable spaces. We call 𝑁∶(𝐸₁,ℰ₁,𝜇₁)×(𝐸₂,ℰ₂,𝜇₂)×(Ω,ℱ,𝑃)→ℕ∪{0}∪{∞} a Poisson noise on (𝐸₁,ℰ₁,𝜇₁), if, for all 𝐴∈ℰ₁,𝐵∈ℰ₂, and 𝑛∈ℕ∪{0}∪{∞}, 𝑃(𝑁())=𝑒𝐴,𝐵=𝑛^{−𝜇1(𝐴)𝜇2(𝐵)[𝜇1(𝐴)𝜇2(𝐵)]𝑛}.𝑛! (10) In particular, if (𝐸₁,ℰ₁,𝜇₁)=([0,∞)×ℝ^𝑑,ℬ([0,∞)×ℝ^𝑑),𝑑𝑡×𝑑𝑥), then define a compensated random martingale measure by 𝑀(𝐵,𝐴,𝑡)=𝑁([0,𝑡]×𝐴,𝐵)−𝜇₁([0,𝑡]×𝐴)𝜇₂(𝐵), (11) by assuming that 𝜇₁([0,𝑡]×𝐴)𝜇₂(𝐵)<∞ for all (𝑡,𝐴,𝐵)∈[0,∞)×ℬ(ℝ^𝑑)×ℰ₂. Further, let 𝜙∶𝐸₁×𝐸₂×Ω→ℝ be an (ℱ_𝑡)_𝑡≥0-predictable function satisfying 𝔼^𝑡_0𝐴𝐵𝜙𝑡,𝑥,𝑦²𝜇₂𝑑𝑦𝑑𝑥𝑑𝑠<∞, (12) for all 𝑡>0 and (𝐴,𝐵)∈ℰ₁×ℰ₂. We can define a stochastic integral process 𝑅_𝑡=₀^𝑡+_𝐴𝐵𝜙𝑠,𝑥,𝑦𝑀𝑑𝑦,𝑑𝑥,𝑑𝑠 (13) which is a square integrable (𝑃,(ℱ_𝑡)_𝑡≥0)-martingale. It is well known that a (pure jump) Lévy space-time white noise possesses the following structure: 𝐿̇(𝑥,𝑡)=_𝑈0ℎ₁𝑀̇+𝑡,𝑥,𝑦𝑑𝑦,𝑥,𝑡_{𝐸2⧵𝑈0}ℎ₂𝑡,𝑥,𝑦𝑁̇𝑑𝑦,𝑥,𝑡, (14) for some 𝑈₀∈𝐸₂ such that 𝜇₂(𝐸₂⧵𝑈₀)<∞ and ∫_𝑈0𝑧²𝜇₂(𝑑𝑧)<+∞. Here ℎ₁,ℎ₂∶[0,∞)×ℝ×𝐸₂→ℝ are some measurable functions; 𝑀̇ and 𝑁̇ are the Radon-Nikodym derivatives given by 𝑀̇𝑑𝑦,𝑥,𝑡=𝑀𝑑𝑦,𝑑𝑥,𝑑𝑡,𝑁̇=𝑁𝑑𝑡×𝑑𝑥𝑑𝑦,𝑥,𝑡𝑑𝑡×𝑑𝑥,𝑑𝑦,𝑑𝑡×𝑑𝑥 (15) with (𝑡,𝑥,𝑦)∈[0,∞)×ℝ^𝑑×𝐸₂.

The following B-D-G inequality is given by Bo and Wang [27], which is useful to estimate the higher order moments of mild solution to (4).

Lemma 4. Let 𝜙∶[0,∞)×ℝ^𝑑×𝐸₂×Ω→ℝ be (ℱ_𝑡)_𝑡≥0-predictable and satisfy (13). Define the integral process by 𝑋_𝑡=₀^𝑡+_ℝ𝑑𝐸2𝜙𝑠,𝑦,𝑧𝑀𝑑𝑧,𝑑𝑦,𝑑𝑠,𝑡≥0. (16) Then, for any 𝑇>0 and 𝑝>1, there exists a constant 𝐶_𝑝,𝑇>0 such that sup_{[]𝑡∈0,𝑇}𝔼𝑋_𝑡^𝑝≤𝐶_𝑝,𝑇^𝑇_{0ℝ𝑑𝐸2}𝔼𝜙𝑠,𝑦,𝑧^𝑝2/𝑝𝜇₂(𝑑𝑧)𝑑𝑦𝑑𝑠^𝑝/2. (17)

2.3. Green Function

In this subsection, we will introduce the nonlocal fractional differential operator defined by 𝔻^𝛼_𝛿=^𝑑_𝑖=1𝐷^𝛼𝑖_𝛿𝑖, (18) where 𝛼=(𝛼₁,…,𝛼_𝑑), 𝛿=(𝛿₁,…,𝛿_𝑑), and 𝐷^𝛼𝑖_𝛿𝑖 denotes the fractional differential derivative with respect to the 𝑖th coordinate defined via its Fourier transform ℱ by ℱ𝐷^𝛼_𝛿𝜑(𝜉)=−𝜉^𝛼𝜋exp−𝑖𝛿2sgn(𝜉)ℱ𝜑(𝜉). (19) In this paper, we will assume that |𝛿_𝑖|≤min{𝛼_𝑖−[𝛼_𝑖]₂,2+[𝛼_𝑖]₂−𝛼_𝑖}, 𝑖=1,…,𝑑,[𝛼_𝑖]₂ is the largest even integer less than or equal to 𝛼_𝑖 (even part of 𝛼_𝑖), and 𝛿_𝑖∈2ℕ+1.

In one space dimension, the operator 𝐷^𝛼_𝛿 is a closed, densely defined operator on 𝐿²(ℝ) and it is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction. This operator is a generalization of various well-known operators, such as the Laplacian operator (when 𝛼=2), the inverse of the generalized Riesz-Feller potential (when 𝛼>2), and the Riemann-Liouville differential operator (when |𝛿|=2+[𝛼]₂ or |𝛿|=𝛼−[𝛼]). It is self-adjoint only when 𝛿=0 and, in this case, it coincides with the fractional power of the Laplacian. We refer the readers to Debbi [29], Debbi and Dozzi [2], and Komatsu [30] for more details about this operator.

According to Komatsu [30], 𝐷^𝛼_𝛿 can be represented, for 1<𝛼<2, by 𝐷^𝛼_𝛿𝜑(𝑥)=_ℝ𝜑𝑥+𝑦−𝜑(𝑥)−𝑦𝜑^′(𝑥)𝑦^1+𝛼×𝜅^𝛿₋1_(−∞,0)𝑦+𝜅^𝛿₊1_(−0,+∞)𝑦𝑑𝑦 (20) and, for 0<𝛼<1, by 𝐷^𝛼_𝛿𝜑(𝑥)=_ℝ𝜑𝑥+𝑦−𝜑(𝑥)𝑦^1+𝛼×𝜅^𝛿₋1_(−∞,0)𝑦+𝜅^𝛿₊1_(−0,+∞)𝑦𝑑𝑦, (21) where 𝜅^𝛿₋ and 𝜅^𝛿₊ are two nonnegative constants satisfying 𝜅^𝛿₋+𝜅^𝛿₊>0 and 𝜑 is a smooth function for which the integral exists, and 𝜑^′ is its derivative. This representation identifies it as the infinitesimal generator for a nonsymmetric 𝛼-stable Lévy process.

Let 𝐺_𝛼,𝛿(𝑡,𝑥) be the fundamental solution of the following Cauchy problem: 𝜕𝑢()𝜕𝑡𝑡,𝑥=𝐷^𝛼_𝛿(),𝑢()𝑡,𝑥0,𝑥=𝛿₀(𝑥),𝑡>0,𝑥∈ℝ, (22) where 𝛿₀(⋅) is the Dirac distribution. By Fourier transform, we see that 𝐷^𝛼_𝛿(𝑡,𝑥) is given by 𝔽: 𝐺_𝛼,𝛿()=1𝑡,𝑥2𝜋_ℝexp−𝑖𝑧𝑥−𝑡|𝑧|^𝛼𝜋exp−𝑖𝛿2sgn(𝑧)𝑑𝑧. (23) The relevant parameters 𝛼 called the index of stability and 𝛿 (related to the asymmetry) improperly referred to as the skewness are real numbers satisfying |𝛿|≤min{𝛼−[𝛼]₂,2+[𝛼]₂−𝛼}, and 𝛿=0 when 𝛿∈2ℕ+1.

Let us list some known facts on 𝐺_𝛼,𝛿(𝑡,𝑥) which will be used later on (see, e.g., Debbi [29] and Debbi and Dozzi [2]).

Lemma 5. Let 𝛼∈(0,∞)/{ℕ}; one has the following. (1)The function 𝐺_𝛼,𝛿(𝑡,⋅) is not in general symmetric relatively to 𝑥 and it is not everywhere positive.(2)For any 𝑠,𝑡∈(0,∞) and 𝑥∈ℝ, 𝜕^𝑛𝜕𝑥^𝑛𝐺_𝛼,𝛿(𝑡,𝑥)=(𝑠)^{−(𝑛+1)/𝛼}𝐺_𝛼,𝛿𝑠⁻¹𝑡,𝑠^−1/𝛼𝑥, (24) or equivalently 𝜕^𝑛𝜕𝑥^𝑛𝐺_𝛼,𝛿(𝑡,𝑥)=(𝑡)^{−(𝑛+1)/𝛼}𝐺_𝛼,𝛿1,(𝑡)^−1/𝛼𝑥. (25)(3)𝐺_𝛼(𝑠,⋅)∗𝐺_𝛼,𝛿(𝑡,⋅)=𝐺_𝛼,𝛿(𝑠+𝑡,⋅) for any 𝑠,𝑡∈(0,∞).(4)For 𝑛≥1, there exist some constants 𝐶 and 𝐶_𝑛>0 such that, for all 𝑥∈ℝ, 𝐺_𝛼,𝛿(11,𝑥)≤𝐶1+|𝑥|^1+𝛼,𝜕^𝑛𝜕𝑥^𝑛𝐺_𝛼,𝛿(1,𝑥)≤𝐶_𝑛|𝑥|^{𝛼+𝑛−1}1+|𝑥|^𝛼+𝑛2. (26)(5)∫^𝑇₀∫_ℝ|𝐺_𝛼,𝛿(𝑡,𝑥)|^𝜆𝑑𝑡𝑑𝑥<∞ if and only if 1/𝛼<𝜆<𝛼.

For 𝑑≥1 and any multi-index 𝛼=(𝛼₁,…,𝛼_𝑑) and 𝛿=(𝛿₁,…,𝛿_𝑑), let 𝔾_𝛼,𝛿(𝑡,𝑥) be the Green function of the deterministic equation 𝜕𝑢()𝜕𝑡𝑡,𝑥=𝔻^𝛼_𝛿(),𝑢()𝑡,𝑥0,𝑥=𝛿₀(𝑥),𝑡>0,𝑥∈ℝ^𝑑. (27) Clearly 𝔾_𝛼,𝛿()=𝑡,𝑥_𝑑𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}𝑡,𝑥_𝑖=1(2𝜋)^𝑑×_ℝ𝑑exp−𝑖⟨𝜉,𝑥⟩−𝑡^𝑑_𝑖=1𝜉_𝑖^𝛼𝑖𝜋exp−𝑖𝛿2sgn𝜉_𝑖𝑑𝜉. (28)

3. Existence and Uniqueness

In this section, we are going to prove the existence and uniqueness of the global mild solution to (3). Recall (4) and (14). Then, for all (𝑡,𝑥)∈[0,𝑇]×ℝ^𝑑, 𝑢()=𝑡,𝑥_ℝ𝑑𝔾_𝛼,𝛿𝑡,𝑥−𝑦𝑢₀𝑦+𝑑𝑦_𝑡0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦𝐵^𝐻+𝑑𝑦,𝑑𝑠_𝑡0ℝ𝑑𝔾_𝛼,𝛿𝑏+𝑡−𝑠,𝑥−𝑦𝑠,𝑦,𝑢𝑠,𝑦𝑑𝑦𝑑𝑠_𝑡0ℝ𝑑𝔾_𝛼,𝛿𝜎𝜓+𝑡−𝑠,𝑥−𝑦𝑠,𝑦,𝑢𝑠,𝑦𝑠,𝑦𝑑𝑦𝑑𝑠_{0𝑡+ℝ𝑑𝐸2}𝔾_𝛼,𝛿ℎ𝑀,𝑡−𝑠,𝑥−𝑦×𝜎𝑠,𝑦,𝑢𝑠,𝑦𝑠,𝑦,𝑧𝑑𝑧,𝑑𝑦,𝑑𝑠 (29) with the mappings 𝜓𝑡,𝑦=_{𝐸2⧵𝑈0}ℎ₂𝑡,𝑦,𝑧𝜇₂(),ℎ𝑑𝑧𝑡,𝑦,𝑧=ℎ₁𝑡,𝑦,𝑧1_𝑈0(𝑧)+ℎ₂𝑡,𝑦,𝑧1_{𝐸2⧵𝑈0}(𝑧), (30) with indicator 1_𝐴(⋅) of the set 𝐴∈ℰ₂.

In the following, we will show that such a mild solution indeed exists and is unique, which is stated as follows.

Theorem 6. Let 1+∑^𝑑_𝑖=1(1/𝛼_𝑖−1/ℎ𝛼_𝑖)>0. Suppose the following conditions hold.(1)𝑏,𝜎 are uniformly Lipschitzian; that is, there exists a constant 𝐶>0 such that for (𝑡,𝑥)∈[0,𝑇]×ℝ^𝑑 and 𝑢,𝑣∈ℝ|𝑏(𝑡,𝑥,𝑢)−𝑏(𝑡,𝑥,𝑣)|+|𝜎(𝑡,𝑥,𝑢)−𝜎(𝑡,𝑥,𝑣)|≤𝐶|𝑢−𝑣|. (31)(2) 𝑏 is linear growth; that is, there exists a constant 𝐶>0 such that for 𝑢∈ℝ|𝑏(𝑡,𝑥,𝑢)|≤𝐶(1+|𝑢|). (32)(3) For 𝑝∈(2(1+∑^𝑑_𝑖=1(1/𝛼_𝑖))/(∑^𝑑_𝑖=1(1/𝛼_𝑖)−1),+∞) with 𝛼>1, sup_{𝑡∈[0,𝑇]}𝜓(𝑡,⋅)^𝑝_𝑝<∞,sup_{𝑡∈[0,𝑇]𝐸2}|ℎ(𝑡,⋅,𝑧)|²𝜇₂(𝑑𝑧)^𝑝/2_𝑝/2<∞. (33)(34) Then, for all ℱ₀-measurable 𝑢₀∶ℝ^𝑑×Ω→ℝ satisfying 𝔼[‖𝑢₀(⋅)‖^𝑝_𝑝]<∞, there exists a unique mild solution (𝑢(𝑡,𝑥))_{(𝑡,𝑥)∈[0,𝑇]×ℝ}^𝑑 to (3) and, for all 𝑝∈(2(1+∑^𝑑_𝑖=1(1/𝛼_𝑖))/(∑^𝑑_𝑖=1(1/𝛼_𝑖)−1),+∞), sup_{𝑡∈[0,𝑇]}𝔼‖𝑢(𝑡,⋅)‖^𝑝_𝑝<∞. (35)

In order to prove the above theorem, we need the following lemmas.

Lemma 7. Suppose 𝑝∈[1,∞), 𝜌∈[1,𝑝], and 𝑟∈[1,∞) such that 1𝑟=1𝑝−1𝜌+1∈[0,1]. (36) Let 𝔾_𝛼,𝛿=𝔾_𝛼,𝛿(𝑡,𝑥−𝑦) be the Green kernel, 𝐻=𝔾_𝛼,𝛿, or 𝔾²_𝛼,𝛿 with (𝑡,𝑥,𝑦)∈[0,𝑇]×ℝ^𝑑×ℝ^𝑑. Define an operator 𝒜 by 𝒜(𝑣)(𝑡,𝑥)=^𝑡_0ℝ𝑑𝐻𝑡−𝑠,𝑥−𝑦𝑣𝑠,𝑦𝑑𝑦𝑑𝑠, (37) with 𝑣∈𝐿¹([0,𝑇];𝐿^𝜌()). Then 𝒜∶𝐿¹([0,𝑇];𝐿^𝜌())→𝐿^∞([0,𝑇];𝐿^𝜌()) is bounded linear operator and satisfies the following.(a)If 𝐻=𝔾_𝛼,𝛿, then there exists a constant 𝐶>0 such that for all 𝑟∈[1,1+min𝛼_𝑖)‖𝒜(𝑣)(𝑡,⋅)‖_𝑝≤𝐶^𝑡₀(𝑡−𝑠)^{∑𝑑𝑖=1((𝑟−1)/𝛼𝑖)}‖𝑣(𝑥,⋅)‖_𝜌𝑑𝑠,∀𝑡∈[0,𝑇]. (38)(b)If 𝔾²_𝛼,𝛿, then there exists a constant 𝐶>0 such that for all 𝑟∈[1,(1+min𝛼_𝑖)/2)‖𝒜(𝑣)(𝑡,⋅)‖_𝑝≤𝐶^𝑡₀(𝑡−𝑠)^{∑𝑑𝑖=1((𝑟−2)/𝛼𝑖)}‖𝑣(𝑥,⋅)‖_𝜌𝑑𝑠,∀𝑡∈[0,𝑇]. (39)

Proof. We only need to prove the case (a), since the proof of (b) is similar. Together with Minkowski’s inequality, (3) of Lemma 5, and Young inequality, one can get ‖𝒜(𝑣)(𝑡,⋅)‖_𝑝=^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,⋅−𝑦𝑣𝑠,𝑦𝑑𝑦𝑑𝑠_𝑝≤^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,⋅−𝑦𝑣(𝑠,𝑦)𝑑𝑦_𝑝𝑑𝑠≤𝐶^𝑡₀(𝑡−𝑠)^{−∑𝑑𝑖=1(1/𝛼𝑖)}×_ℝ𝑑^𝑑_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,(𝑡−𝑠)^{−1/𝛼𝑖}⋅−𝑦_𝑖𝑣𝑠,𝑦𝑑𝑦_𝑝𝑑𝑠≤𝐶^𝑡₀(𝑡−𝑠)^{−∑𝑑𝑖=1(1/𝛼𝑖)}×^𝑑_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,(𝑡−𝑠)^{−1/𝛼𝑖}(⋅)∗|𝑣(𝑠,⋅)|_𝑝𝑑𝑠≤𝐶^𝑡₀(𝑡−𝑠)^{−∑𝑑𝑖=1(1/𝛼𝑖)}×^𝑑_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,(𝑡−𝑠)^{−1/𝛼𝑖}(⋅)_𝑟⋅‖𝑣(𝑠,⋅)‖_𝜌𝑑𝑠≤𝐶^𝑡₀(𝑡−𝑠)^{−∑𝑑𝑖=1(1/𝛼𝑖)}‖𝑣(𝑠,⋅)‖_𝜌𝑑𝑠, (40) where we have used the fact that for 𝑟∈[1,1+min𝛼_𝑖)^𝑑_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,(𝑡−𝑠)^{−1/𝛼𝑖}(⋅)_𝑟=^𝑑_𝑖=1ℝ𝐺_{𝛼𝑖,𝛿𝑖}1,(𝑡−𝑠)^{−1/𝛼𝑖}𝑦_𝑖^𝑟𝑑𝑦_𝑖^1/𝑟=(𝑡−𝑠)^{∑𝑑𝑖=1(𝑟/𝛼𝑖)𝑑}_𝑖=1ℝ𝐺_{𝛼𝑖,𝛿𝑖}1,𝑧_𝑖^𝑟𝑑𝑧_𝑖^1/𝑟≤𝐶(𝑡−𝑠)^{∑𝑑𝑖=1(𝑟/𝛼𝑖)}. (41) The proof of this lemma is completed.

In the following, we mainly adopt the fixed point principle to prove Theorem 6. Let 𝔹 be the space of all 𝐿^𝑝(ℝ)-valued ℱ_𝑡-adapted processes 𝑢(𝑡,⋅)_{0≤𝑡≤𝑇}∶[0,𝑇]×ℝ^𝑑×Ω→ℝ with the norm ‖𝑎‖_𝔹∶=sup_{0≤𝑡≤𝑇}𝑒^−𝜂𝑡𝔼‖𝑢(𝑡,⋅)‖^𝑝_𝑝^1/𝑝,𝜂>0. (42) Then (𝔹,‖⋅‖_𝔹) is a Banach space. Now, for 𝑢∈𝔹, let us define an operator 𝒯 by 𝒯(𝑡,𝑥)=⁵_𝑖=1𝒯_𝑖(𝑢)(𝑡,𝑥), (43) where 𝒯₁(𝑢)(𝑡,𝑥)=_ℝ𝑑𝔾_𝛼,𝛿𝑡,𝑥−𝑦𝑢₀𝑦𝒯𝑑𝑦,₂(𝑢)(𝑡,𝑥)=^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝑏𝒯𝑡−𝑠,𝑥−𝑦𝑠,𝑦,𝑢𝑠,𝑦𝑑𝑦𝑑𝑠,₃(𝑢)(𝑡,𝑥)=^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦𝐵^𝐻,𝒯𝑑𝑦,𝑑𝑠₄(𝑢)(𝑡,𝑥)=^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝜎𝒯𝑡−𝑠,𝑥−𝑦𝑠,𝑦,𝑢𝑠,𝑦×𝜓𝑠,𝑦𝑑𝑦𝑑𝑠,₅(𝑢)(𝑡,𝑥)=^𝑡_{0ℝ𝑑𝐸2}𝔾_𝛼,𝛿𝜎𝑀.𝑡−𝑠,𝑥−𝑦𝑠,𝑦,𝑢𝑠,𝑦×ℎ𝑠,𝑦,𝑧𝑑𝑧,𝑑𝑦,𝑑𝑠 (44) According to (43), we have the following.

Proposition 8. Suppose 1+∑^𝑑_𝑖=1(1/𝛼_𝑖−1/ℎ𝛼_𝑖)>0 and the assumptions (1), (2), and (3) of Theorem 6 are satisfied. Then, for each 𝑝>2(1+∑^𝑑_𝑖=1(1/𝛼_𝑖))/∑^𝑑_𝑖=1(1/𝛼_𝑖)−1 and 𝑢∈𝔹, it holds that 𝒯∈𝔹.

Proof. From Lemma 3 and Young inequality, it follows that 𝒯₁(𝑢)(𝑡,𝑥)_𝑝=_ℝ𝑑𝔾_𝛼,𝛿(𝑡,𝑥−𝑦)𝑢₀(𝑦)𝑑𝑦_𝑝=𝑡^{−∑𝑑𝑖=1(1/𝛼𝑖)}_ℝ𝑑^𝑑_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,𝑡^{−1/𝛼𝑖}(⋅−𝑦_𝑖)𝑢₀(𝑦)𝑑𝑦_𝑝≤𝑡^{−∑𝑑𝑖=1(1/𝛼𝑖)𝑑}_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,𝑡^{−1/𝛼𝑖}⋅∗𝑢₀(⋅)(⋅)_𝑝≤𝑡^{−∑𝑑𝑖=1(1/𝛼𝑖)𝑑}_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,𝑡^{−1/𝛼𝑖}⋅₁⋅𝑢₀(⋅)_𝑝≤𝐶𝑢₀(⋅)_𝑝<∞, (45) where we have used the fact that 𝔼[‖𝑢₀(⋅)‖^𝑝_𝑝]<∞. Now we deal with 𝒯₂(𝑢). Applying (a) of Lemma 7 with 1/𝑟=1/𝑝−1/𝑝+1=1 and condition (1) of Theorem 6, we conclude that 𝔼𝒯₂(𝑢)(𝑡,𝑥)^𝑝_𝑝≤𝐶𝔼^𝑡₀(𝑡−𝑠)^{∑𝑑𝑖=1((𝑟−1)/𝛼𝑖)}‖𝑏(𝑠,⋅,𝑢(𝑠,⋅))‖_𝑝𝑑𝑠^𝑝≤𝐶𝔼^𝑡₀1+‖𝑢(𝑠,⋅)‖_𝑝𝑑𝑠^𝑝≤𝐶_𝑝,𝑇1+sup_{0≤𝑡≤𝑇}𝔼‖𝑢(𝑠,⋅)‖^𝑝_𝑝≤𝐶_𝑝,𝑇1+‖𝑢(𝑠,⋅)‖^𝑝_𝔹<∞, (46) since 𝑢∈𝔹.
Now, let us consider 𝒯₃(𝑢)(𝑡,𝑥). By Lemma 3, we deduce that 𝔼𝒯₃(𝑢)(𝑡,𝑥)^𝑝_𝑝=_ℝ𝑑𝔼^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦𝐵^𝐻𝑑𝑦,𝑑𝑠^𝑝=𝑑𝑥_ℝ𝑑𝔼^𝑡_0ℝ𝑑𝐾^∗_𝐻𝐺_𝛼,𝛿(𝑡−⋅,𝑥−⋅)𝑠,𝑦𝑊𝑑𝑦,𝑑𝑠^𝑝𝑑𝑥≤𝐶_𝑝ℝ𝑑𝔾_𝛼,𝛿(𝑡−⋅,𝑥−⋅)^𝑝_{𝐿1/ℎ([0,𝑇]×ℝ𝑑)}𝑑𝑥, (47) where 𝑊(𝑠,𝑦) is a space-time white noise on [0,𝑇]×ℝ^𝑑.
Note that 𝔾_𝛼,𝛿(𝑡−⋅,𝑥−∗)^𝑝_{𝐿1/ℎ[]0,𝑇×ℝ𝑑}=^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦^1/ℎ𝑑𝑦𝑑𝑠^𝑝ℎ=^𝑡₀(𝑡−𝑠)^{−∑𝑑𝑖=1(1/ℎ𝛼𝑖)}×_ℝ𝑑^𝑑_𝑖=1𝐺_{𝛼𝑖,𝛿𝑖}1,(𝑡−𝑠)^{−1/𝛼𝑖}𝑥_𝑖−𝑦_𝑖^1/ℎ𝑑𝑦𝑑𝑠^𝑝ℎ=^𝑡₀(𝑡−𝑠)^{∑𝑑𝑖=1(1/𝛼𝑖−1/ℎ𝛼𝑖)}×^𝑑_𝑖=1ℝ𝐺_{𝛼𝑖,𝛿𝑖}1,𝑧_𝑖^1/ℎ𝑑𝑧_𝑖𝑑𝑠^𝑝ℎ≤𝐶_𝛼,𝐻^𝑡₀(𝑡−𝑠)^{∑𝑑𝑖=1(1/𝛼𝑖−1/ℎ𝛼𝑖)}𝑑𝑠^𝑝ℎ<∞, (48) under the assumption 1+∑^𝑑_𝑖=1(1/𝛼_𝑖−1/ℎ𝛼_𝑖)>0. So we have 𝒯₃(𝑢)(𝑡,𝑥)∈𝔹 for 𝑝≥2.
Combining (a) of Lemma 7 with 1/𝑟=1/𝑝−2/𝑝+1=1−1/𝑝∈(0,1] and assumption (33), we obtain that, for 𝑝∈[2,∞), 𝔼𝒯₄(𝑢)(𝑡,𝑥)^𝑝_𝑝≤𝐶𝔼^𝑡₀(𝑡−𝑠)^{−(∑𝑑𝑖=1(1/𝛼𝑖))(1−1/𝑟)}‖1+|𝑢(𝑠,⋅)|‖_𝑝⋅𝜓_𝑝𝑑𝑠^𝑝≤𝐶_𝑝1+‖𝑢(⋅)‖^𝑝_𝔹sup_{0≤𝑡≤𝑇}𝜓(𝑡,⋅)^𝑝_𝑝⋅^𝑡₀(𝑡−𝑠)^{−(∑𝑑𝑖=1(1/𝛼𝑖))(1−1/𝑟)(𝑝/(𝑝−1))}𝑑𝑠^𝑝−1≤𝐶_𝑝,𝑇sup_{0≤𝑡≤𝑇}𝜓(𝑡,⋅)^𝑝_𝑝1+‖𝑢(⋅)‖^𝑝_𝔹. (49) with 1−(1/(∑^𝑑_𝑖=1(1/𝛼_𝑖))(𝑝−1))>0; that is, 𝑝>1+(1/∑^𝑑_𝑖=1(1/𝛼_𝑖)). Finally, let us estimate 𝒯₅(𝑢)(𝑡,𝑥). This is a key step in the proof of Proposition 8. Together with the condition (34), (b) of Lemma 7 with 1/𝑟=1/𝑝−4/𝑝+1=1−2/𝑝∈(0,1] and Lemma 4, we conclude that, for 𝑝>2(1+∑^𝑑_𝑖=1(1/𝛼_𝑖))/(∑^𝑑_𝑖=1(1/𝛼_𝑖)−1), 𝔼𝒯₅(𝑢)(𝑡,⋅)^𝑝_𝑝=𝔼₀^𝑡+_ℝ𝑑𝐸2𝔾_𝛼,𝛿𝜎𝑀𝑡−𝑠,⋅−𝑦𝑠,𝑦,𝑢𝑠,𝑦×ℎ𝑠,𝑦,𝑧𝑑𝑦,𝑑𝑧,𝑑𝑠^𝑝_𝑝≤𝐶_𝑝ℝ𝑑^𝑡_{0ℝ𝑑𝐸2}𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦ℎ𝑠,𝑦,𝑧²×𝔼1+𝑢𝑠,𝑦^𝑝2/𝑝×𝜇₂(𝑑𝑧)𝑑𝑦𝑑𝑠^𝑝/2𝑑𝑥=𝐶_𝑝^𝑡_0ℝ𝑑𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦²×_𝐸2ℎ𝑠,𝑦,𝑧²𝜇₂(×|𝑢(𝑑𝑧)1+𝔼𝑠,⋅)|^𝑝2/𝑝𝑑𝑦𝑑𝑠^𝑝/2≤𝐶_𝑝^𝑡₀(𝑡−𝑠)^{−(𝑝+2)/𝑝(∑𝑑𝑖=1(1/𝛼𝑖))}×_𝐸2ℎ𝑠,𝑦,𝑧²𝜇₂(×|𝑢(𝑑𝑧)1+𝔼𝑠,⋅)|^𝑝2/𝑝_𝑝/4𝑑𝑠^𝑝/2≤𝐶_𝑝^𝑡₀(𝑡−𝑠)^{−(𝑝+2)/𝑝(∑𝑑𝑖=1(1/𝛼𝑖))}×_𝐸2ℎ𝑠,𝑦,𝑧²𝜇₂(𝑑𝑧)_𝑝/2×1+𝔼|𝑢(𝑠,⋅)|^𝑝2/𝑝_𝑝/2𝑑𝑠^𝑝/2≤𝐶_𝑝sup_{0≤𝑡≤𝑇𝐸2}ℎ𝑠,𝑦,𝑧²𝜇₂(𝑑𝑧)^𝑝/2_𝑝/2⋅sup_{0≤𝑡≤𝑇}1+𝔼|𝑢(𝑠,⋅)|^{𝑝2/𝑝𝑝/2}_𝑝/2⋅^𝑡₀(𝑡−𝑠)^{−(𝑝+2)/(𝑝−2)(∑𝑑𝑖=1(1/𝛼𝑖))(𝑝−2)/2}≤𝐶_𝑝,𝑇sup_{0≤𝑡≤𝑇𝐸2}ℎ𝑠,𝑦,𝑧²𝜇₂(𝑑𝑧)^𝑝/2_𝑝/2×1+‖𝑢(⋅)‖^𝑝_𝔹<∞. (50) Thus we have proved that the operator 𝒯 defined by (43) is an operator from 𝔹 to itself. On the other hand, from the similar argument as in (45)–(50), let 𝜂>0 sufficiently large; then 𝒯∈𝔹. Thus we complete the proof of the proposition.

In what follows, we will prove that the operator 𝒯∶𝔹↦𝔹 is a contract operator.

Proposition 9. Suppose the assumptions (1), (2), and (3) of Theorem 6 are satisfied. Then, for each 𝑝>2(1+∑^𝑑_𝑖=1(1/𝛼_𝑖))/(∑^𝑑_𝑖=1(1/𝛼_𝑖)−1), the operator 𝒯_𝛼 is a contraction on 𝔹. In other words, there exists a constant 𝜚∈(0,1) such that ‖𝒯(𝑢)−𝒯(𝑣)‖_𝔹≤𝜚‖𝑢−𝑣‖_𝔹,for𝑢,𝑣∈𝔹. (51)

Proof. Let 𝑢₀ and 𝑣₀ be initials of (ℱ_𝑡)_𝑡≥0-adapted random fields 𝑢,𝑣∈𝔹 such that 𝑢₀=𝑣₀. Let us consider 𝒯₁(𝑢) firstly. Note that, for 𝜌=𝑝/3, together with (a) of Lemma 7 with 1/𝑟=(1/𝑝)−(1/𝜌)+1=1−(2/𝑝) and condition (1) of Theorem 6, one can get 𝔼𝒯₂(𝑢)(𝑡,⋅)−𝒯₂(𝑣)(𝑡,⋅)^𝑝_𝑝≤𝐶𝔼^𝑡₀(𝑡−𝑠)^{∑𝑑𝑖=1((𝑟−1)/𝛼𝑖)}×𝑏𝑠,𝑦,𝑢𝑠,𝑦−𝑏𝑠,𝑦,𝑣𝑠,𝑦_𝑝𝑑𝑠^𝑝≤𝐶_𝑝^𝑡₀(𝑡−𝑠)^{∑𝑑𝑖=1((𝑟−1)/𝛼𝑖)}𝔼‖𝑢(𝑠,⋅)−𝑣(𝑠,⋅)‖_𝑝𝑑𝑠^𝑝. (52) Then 𝒯₂(𝑢)(𝑡,⋅)−𝒯₂(𝑣)(𝑡,⋅)^𝑝_𝔹=sup_{0≤𝑡≤𝑇}𝑒^−𝜂𝑡𝔼𝒯₂(𝑢)(𝑡,⋅)−𝒯₂(𝑣)(𝑡,⋅)^𝑝_𝑝≤𝐶sup_{0≤𝑡≤𝑇}𝔼^𝑡₀𝑒^{−(𝜂/𝑝)(𝑡−𝑠)}(𝑡−𝑠)^{∑𝑑𝑖=1((𝑟−1)/𝛼𝑖)}𝑒^{−(𝜂/𝑝)𝑠}×‖𝑢(𝑠,⋅)−𝑣(𝑠,⋅)‖_𝑝𝑑𝑠^𝑝≤𝐶_𝑝sup_{0≤𝑡≤𝑇}𝔼^𝑡₀𝑒^−𝜂𝑠‖𝑢(𝑠,⋅)−𝑣(𝑠,⋅)‖^𝑝_𝑝⋅𝑑𝑠_𝑡0𝑒^{−(𝜂/𝑝)(𝑡−𝑠)}(𝑡−𝑠)^{∑𝑑𝑖=1((𝑟−1)/𝛼𝑖)𝑝/(𝑝−1)}𝑑𝑠^𝑝−1≤𝐶_𝑝sup_{0≤𝑡≤𝑇}^𝑡₀𝑒^−𝜂𝑠𝔼‖𝑢(𝑠,⋅)−𝑣(𝑠,⋅)‖^𝑝_𝑝⋅𝑑𝑠_𝑡0𝑒^{−(𝜂/𝑝)(𝑡−𝑠)}(𝑡−𝑠)^{−(1/𝛼)(1−𝑟)𝑝/(𝑝−1)}𝑑𝑠^𝑝−1≤𝜚‖𝑢−𝑣‖^𝑝_𝔹, (53) with 𝜚∈(0,1) by choosing 𝜂>0 large enough. Now we are going to consider the term 𝒯₅(𝑢)(𝑡,𝑥). From a similar argument as in (50), thanks to the generalized B-D-G inequality, we derive from the conditions of Theorem 6 that 𝒯₅(𝑢)(𝑡,⋅)−𝒯₅(𝑣)(𝑡,⋅)^𝑝_𝔹=sup_{0≤𝑡≤𝑇}𝑒^−𝜂𝑡𝔼𝒯₅(𝑢)(𝑡,⋅)−𝒯₅(𝑣)(𝑡,⋅)^𝑝_𝑝=sup_{0≤𝑡≤𝑇}𝑒^−𝜂𝑡_ℝ𝑑𝔼₀^𝑡+_ℝ𝑑𝐸2𝔾_𝛼,𝛿ℎ⋅𝜎𝑡−𝑠,𝑥−𝑦𝑠,𝑦,𝑧𝑠,𝑦,𝑢𝑠,𝑦−𝜎𝑠,𝑦,𝑣𝑠,𝑦×𝑀𝑑𝑦,𝑑𝑧,𝑑𝑠^𝑝𝑑𝑥≤𝐶_𝑝sup_{0≤𝑡≤𝑇}𝑒^−𝜂𝑡×_ℝ𝑑^𝑡_{0ℝ𝑑𝐸2}𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦ℎ𝑠,𝑦,𝑧²⋅𝔼𝑢𝑠,𝑦−𝑣𝑠,𝑦^𝑝2/𝑝×𝜇₂(𝑑𝑧)𝑑𝑦𝑑𝑠^𝑝/2𝑑𝑥≤𝐶_𝑝sup_{0≤𝑡≤𝑇}𝑒^−𝜂𝑡×_ℝ𝑑^𝑡_{0ℝ𝑑𝐸2}𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦ℎ𝑠,𝑦,𝑧²𝑒^{2𝜂𝑠/𝑝}⋅𝑒^−𝜂𝑠𝔼𝑢𝑠,𝑦−𝑣𝑠,𝑦^𝑝2/𝑝×𝜇₂(𝑑𝑧)𝑑𝑦𝑑𝑠^𝑝/2𝑑𝑥≤𝐶_𝑝sup_{0≤𝑡≤𝑇}×_ℝ𝑑^𝑡_{0ℝ𝑑𝐸2}𝑒^−𝜂𝑠𝔼𝑢𝑠,𝑦−𝑣𝑠,𝑦^𝑝𝜇₂()⋅𝑑𝑧𝑑𝑦𝑑𝑠_{𝑡0ℝ𝑑𝐸2}𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦ℎ𝑠,𝑦,𝑧^{2𝑝/(𝑝−2)}×𝑒^{−(2𝜂(𝑡−𝑠)/(𝑝−2))}𝜇₂(𝑑𝑧)𝑑𝑦𝑑𝑠^(𝑝−2)/2𝑑𝑥≤𝐶_𝑝,𝑇𝜇₂𝐸₂‖𝑢−𝑣‖^𝑝_𝔹⋅sup_{0≤𝑡≤𝑇ℝ𝑑}^𝑡_{0ℝ𝑑𝐸2}𝔾_𝛼,𝛿𝑡−𝑠,𝑥−𝑦ℎ𝑠,𝑦,𝑧^𝑝×𝑒^{−𝜂(𝑡−𝑠)}𝜇₂()‖‖𝑑𝑧𝑑𝑦𝑑𝑠𝑑𝑥≤𝜚𝑢−𝑣^𝑝_𝔹. (54) Then 𝒯₅(𝑢) is a contraction on 𝔹.
A similar procedure as the above arguments yields that 𝒯₄(𝑢) is a contraction on 𝔹 by letting 𝜂>0 large enough. Therefore, it follows from (49) that 𝒯(⋅) is a contraction on 𝔹 if 𝜂>0 large enough. Thus the proof of Proposition 9 is complete.

Based on Propositions 8 and 9 and fixed point principal on the set {𝑢∈𝔹∶𝑢(0)=𝑢₀}, we conclude that (3) admits a unique solution 𝑢∈𝔹. Thus the conclusion of Theorem 6 follows.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere gratitude to the editor and the anonymous referees for their valuable comments and error corrections. Xichao Sun is partially supported by Natural Science Foundation of Anhui Province (no. 1408085QA10), Key Natural Science Foundation of Anhui Education Commission (no. KJ2013A183), and Mathematical Tianyuan Foundation of China (no. 11226198). Zhi Wang is partially supported by the Research Project of Education of Zhejiang Province (Y201326507) and the Natural Science Foundation of Ningbo Municipality (201401A6111010). Jing Cui is partially supported by the National Natural Science Foundation of China (11326171), the Natural Science Foundation of Anhui Province (1308085QA14), the Key Natural Science Foundation of Anhui Educational Committee (KJ2013A133), and the PhD Start-up Fund of Anhui Normal University.