Regularity of a Stochastic Fractional Delayed Reaction-Diffusion Equation Driven by Lévy Noise

Abstract

The current paper is devoted to the regularity of the mild solution for a stochastic fractional delayed reaction-diffusion equation driven by Lévy space-time white noise. By the Banach fixed point theorem, the existence and uniqueness of the mild solution are proved in the proper working function space which is affected by the delays. Furthermore, the time regularity and space regularity of the mild solution are established respectively. The main results show that both time regularity and space regularity of the mild solution depend on the regularity of initial value and the order of fractional operator. In particular, the time regularity is affected by the regularity of initial value with delays.

1. Introduction

Recently, fractional partial differential equations attract more and more attention. They appear more and more frequently in different research areas and engineering applications. They have been applied to model various phenomena in image analysis, risk management, and statistical mechanics (see, e.g., [1, 2]). There are many papers concerning the existence and regularity of the solution for fractional Navier-Stokes, fractional Ginzburg-Landau equation, fractional Burgers equation, fractional Langevin equation, and so on (see [3, 4] and references therein).

Stochastic partial differential equations driven by Gaussian noise and non-Gaussian noise such as Lévy noise have also attracted a lot of attention. It seems more significant to investigate fractional partial differential equations with some random force, and some authors have investigated the existence and regularity of the solutions for stochastic fractional partial differential equations ([2, 5–7] and the references therein). The authors in [6, 7] proved the existence and uniqueness of the solution for a stochastic fractional partial differential equation driven by a space-time white noise in one dimension. Truman and Wu in [8] applied the Banach fixed point theorem to show the existence and uniqueness of the mild solution for fractal Burgers equations driven by Lévy noise on real line. Brzeźniak and Debbi in papers [9, 10] proved the existence and ergodicity of the solution for fractal Burgers equation driven by Gaussian space-time white noise, and we refer to [9, 10] for more details. In mathematical biology and other fields, delays are often considered in the model such as maturation time for population dynamics. Some efforts have been devoted to the development of the theory of PDEs with delay. Such equations are naturally more difficult since they are infinite dimensional both in time and space variables. We refer to the monographs [11, 12] for more details. To our knowledge, there is no paper to study the stochastic fractional reaction-diffusion equation with delays.

It is worth to point out that the authors in [8] study the existence of the mild solution for stochastic fractional Burgers equation driven by Lévy noise, but they could not provide the regularity of the mild solution. The authors in [7, 13] study the regularity of the mild solution for stochastic fractional partial differential equations driven by Gaussian white noise, but not Lévy noise. There is a natural question, how about the regularity of the mild solution for the stochastic fraction delayed reaction-diffusion equation driven by Lévy noise?

Motivated by [8], in the present paper, we will study the stochastic fractional reaction-diffusion equation with delays driven by Lévy process followed as: 𝜕𝑢(𝑡,𝑥)𝜕𝑡=𝜆Δ_𝛼𝑢(𝑡,𝑥)+𝑓𝑡,𝑥,𝑢_𝑡+𝑔𝑡,𝑥,𝑢_𝑡𝑍_𝑡,𝑥,()∈[]𝑢()𝑡,𝑥0,𝑇×ℝ,0,𝑥=𝑢₀(𝑥),𝑢𝜂,𝑥=𝜙𝜂,𝑥,𝜂∈[−𝑟,0], (1) where Δ_𝛼∶=−(−𝑑²/𝑑𝑥²)^𝛼/2 is the fractional Laplacian operator with 𝛼∈(0,2], the constants 𝜆∈ℝ, 𝑓,𝑔∶[0,∞)×ℝ×ℝ→ℝ are measurable, the function 𝑢_𝑡=𝑢(𝑡+𝜂), and 𝑍_𝑡,𝑥 is the one-dimensional Lévy process (see Section 2 for the definition). Recall that 𝐷_𝛼 reduces to be the Laplacian operator when 𝛼=2.

In this paper, the existence, uniqueness, time regularity, and space regularity of the mild solution for (1) are shown for 𝛼∈(1,2] in the proper working function space which is affected by the delays. The main results show that both time regularity and space regularity of the mild solution for (1) depend on the regularity of initial value and the order of fractional operator. In particular, the time regularity is affected by the the regularity of initial value with delays.

The rest of this paper is organized as follows. In Section 2, we introduce the definition of the Lévy space-time white noise. Then, some useful properties for the fractional Green kernel are presented. In Section 3, the proper working function space is constructed. Then the existence and uniqueness of the mild solution for (1) are proved by the Banach fixed point theorem in the proper working function space. Finally, the time regularity and space regularity of the mild solution are provided, respectively, in Section 4.

2. Preliminaries

In this section, we first introduce the Lévy space-time white noise. Then, some useful properties for the fractional Green kernel are presented.

Let (Ω,ℱ,{ℱ}_𝑡≥0,𝑃) be a complete probability space with filtration {ℱ}_𝑡≥0 satisfying the usual condition. For one-dimensional Lévy process 𝑍_𝑡,𝑥, it follows from Lévy-Ito decomposition that there exist a constant 𝛽₁ and a nonnegative constant 𝛽₂, and a one-dimensional space-time white noise 𝑊_𝑡,𝑥=(𝜕²𝑊/𝜕𝑡𝜕𝑥)(𝑡,𝑥) (𝑊(𝑡,𝑥) is a Brownian sheet on [0,∞)×ℝ) such that 𝑍_𝑡,𝑥=𝛽₁𝑡+𝛽₂𝑊_𝑡,𝑥+_|𝑧|<1𝑧𝑁()+𝑡,𝑥,𝑑𝑧_|𝑧|≥1𝑧𝑁(𝑡,𝑥,𝑑𝑧), (2) where 𝑁(𝑡,𝑥,𝐴)∶=𝑁(𝑡,𝑥,𝐴)−𝑡𝜈(𝐴), (3) where 𝜈(𝐴)∶=𝐸[𝑁(1,𝐴)] is the Lévy measure of 𝑍_𝑡,𝑥.

Similar to [14], for any 𝑝, we denote 𝑐_𝑝̂∶=_ℝ|𝑧|^𝑝𝜈(𝑑𝑧)^1/𝑝. (4) In what follows, we assume that 𝑐̂∶=sup_𝑝≥1𝑐_𝑝̂<∞. (5) Recalling that 𝑧𝑁(_|𝑧|<1𝑡,𝑥,𝑑𝑧)=_|𝑧|<1𝑧𝑁(𝑡,𝑥,𝑑𝑧)+𝑡_|𝑧|<1𝑧𝜈(𝑑𝑧). (6) By absorbing 𝛽∶=−∫_|𝑧|<1𝑧𝜈(𝑑𝑧) into 𝛽₁, we can rewrite (2) into the following equation: 𝑍_𝑡,𝑥=𝛽₁𝑡+𝛽₂𝑊_𝑡,𝑥+_ℝ𝑧𝑁(𝑡,𝑥,𝑑𝑧). (7) Let 𝛽₁=0; then (1) can be written as 𝑑𝑢(𝑡,𝑥)=𝜆Δ_𝛼𝑢(𝑡,𝑥)+𝑓𝑡,𝑥,𝑢_𝑡𝑑𝑡+ℎ𝑡,𝑥,𝑢_𝑡𝑑𝑊_𝑡,𝑥+𝑔𝑡,𝑥,𝑢_𝑡𝑑𝑌_𝑡,𝑥,()∈[]𝑢()𝑡,𝑥0,𝑇×ℝ,0,𝑥=𝑢₀(𝑥),𝑢𝜂,𝑥=𝜙𝜂,𝑥,𝜂∈(𝑟,0], (8) where 𝑊_𝑡,𝑥 is a one-dimensional space-time white noise and 𝑌_𝑡,𝑥∶=∫_ℝ𝑧𝑁(𝑡,𝑥,𝑑𝑧) is a one-dimensional pure jump Lévy process with Lévy measure of 𝜈. We suppose that 𝑊 generates a {ℱ}_𝑡≥0-martingale measure in the sense of Walsh [15].

The following assumptions are imposed to the initial data 𝑢₀ and 𝜙(𝜂,𝑥), 𝑓(𝑢_𝑡), ℎ(𝑢_𝑡), and 𝑔(𝑢_𝑡) to show the existence and uniqueness of the mild solution. (H1) The initial data 𝑢₀ which is ℱ₀-measurable and 𝜙 satisfy sup_𝑥∈ℝ|𝑢(0,𝑥)|²<∞,sup_𝑥∈ℝ⁰_−𝑟𝜙𝜂,𝑥²𝑑𝜂<∞. (9)(H2) There exists a constant 𝐾 such that for all 𝑡≥0, 𝑓𝑢_𝑡²+ℎ𝑢_𝑡²+_ℝ𝑔𝑢_𝑡|𝑧|𝜈(𝑑𝑧)²<𝐾|𝑢|²+⁰_−𝑟𝑢𝑡+𝜂²,𝑓𝑢𝑑𝜂_𝑡−𝑓𝑣_𝑡²+ℎ𝑢_𝑡−ℎ𝑣_𝑡²+_ℝ𝑔𝑢_𝑡−𝑔𝑣_𝑡|𝑧|𝜈(𝑑𝑧)²<𝐾|𝑢−𝑣|²+⁰_−𝑟𝑢𝑡+𝜂−𝑣𝑡+𝜂²𝑑𝜂. (10)

Let the Green kennel 𝐺_𝛼(𝑡,𝑥) be the fundamental solution of the Cauchy problem: 𝜕𝜐𝜕𝑡=𝜆Δ_𝛼()∈()𝜐()𝜐,𝑡,𝑥0,∞×ℝ,0,𝑥=𝛿₀(𝑥),𝑥∈ℝ, (11) where 𝛿₀(𝑥) denotes the Dirac function. By Fourier transform, 𝐺_𝛼(𝑡,𝑥)=ℱ⁻¹𝑒^{𝜆𝑡|⋅|𝛼}(𝑥). (12) A higher order fractional Green kennel is introduced in [16].

The following lemma gives some useful properties about 𝐺_𝛼(𝑡,𝑥), which are key technique tools to get the estimation for the existence and uniqueness of the mild solution.

Lemma 1 (see [7]). The Green kernel function 𝐺_𝛼(𝑡,𝑥) satisfies the following properties.(1)For any 𝑡≥0𝐺_𝛼(𝑡,𝑥)=𝑡^−1/𝛼𝐺_𝛼(1,𝑡^−1/𝛼𝑥).(2)For 𝑛∈(1/(𝛼+1),𝛼+1), ∫^𝑇₀∫_ℝ|𝐺_𝛼(𝑡,𝑥)|^𝑛𝑑𝑡𝑑𝑥<∞. (3)For any 𝑥∈ℝ, ∫_ℝ𝐺_𝛼(𝑡,𝑥)𝑑𝑥=1. (4)For any 𝑡,𝑠∈ℝ, 𝐺_𝛼(𝑡,𝑥)∗𝐺_𝛼(𝑠,𝑥)=𝐺_𝛼(𝑡+𝑠,𝑥). (5)For any 𝑥∈ℝ, there exists a constant 𝐶 such that 𝐺_𝛼()≤𝐶1,𝑥1+|𝑥|^1+𝛼,𝜕^𝑚_𝑥𝐺_𝛼()|𝑥|1,𝑥≤𝐶^{𝛼+𝑚−1}1+|𝑥|^1+𝛼. (13)

3. Existence of the Mild Solution

In this section, we will first construct the proper working function space.

Let 𝑇 be a fixed positive time and 𝔹 the class of all ℱ_𝑡-adapted càdlàg process {𝑢(𝑡,𝑥),(𝑡,𝑥)∈[0,𝑇]×ℝ} satisfying sup_{(𝑡,𝑥)∈[0,𝑇]×ℝ}𝐸|𝑢(𝑡,𝑥)|²<∞. (14) Let 𝜆>0 be arbitrarily fixed; we define |𝑢|²_𝜆=^𝑇₀𝑒^−𝜆𝑡sup_𝑥∈ℝ𝐸|𝑢(𝑡,𝑥)|²𝑑𝑡_𝑡≥0+⁰_−𝑟𝑒^−𝜆𝑡𝐸|𝑢(𝑡,𝑥)|²𝑑𝑡_{𝑡∈(−𝑟,0)}. (15) For any 𝑢∈𝔹,  |𝑢|²_𝜆<∞. It is easy to verify that |⋅|_𝜆 is a norm and (𝔹,|⋅|_𝜆) is a Banach space.

Let (Ω,ℱ,(ℱ_𝑡)_𝑡≥0,𝑃) and 𝐺_𝛼(𝑡,𝑥) be given as in the previous section. Following the idea in [17], we represent a mild solution of (8) for 𝑡≥0.

Definition 2. An ℱ_𝑡-adapted random field {𝑢(𝑡,𝑥),𝑡≥0,𝑥∈ℝ} is said to be a mild solution of (8) with initial value 𝑢₀ satisfying (H1) if the following integral equation is fulfilled: 𝑢(𝑡,𝑥)=_ℝ𝐺_𝛼𝑡,𝑥−𝑦𝑢₀𝑦𝑑𝑦+^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑓𝑢_𝑠+𝑑𝑦𝑑𝑠_𝑡0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦ℎ𝑢_𝑠𝑊+𝑑𝑦,𝑑𝑠_𝑡0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑔𝑢_𝑠𝑧𝑁𝑑𝑠,𝑑𝑦,𝑑𝑧, (16) where the stochastic integral with respect to 𝑊(𝑡,𝑥) is understood in the sense of that introduced by Walsh [15].

Theorem 3. For 𝑡≥0 and 𝛼∈(1,2], assume that (H1) and (H2) hold, then there exists a unique mild solution 𝑢∈𝔹 for (8).

Remark 4. In the following proof, 𝐶 is a local constant which may change from line to line.

Proof. We will prove the theorem by the following two steps.
Step 1. Suppose that 𝑢∈𝔹 and denote 𝒯𝑢(𝑡,𝑥)=_ℝ𝐺_𝛼𝑡,𝑥−𝑦𝑢₀𝑦𝑑𝑦+𝒯₁𝑢(𝑡,𝑥)+𝒯₂𝑢(𝑡,𝑥)+𝒯₃𝑢(𝑡,𝑥), (17) where 𝒯₁𝑢(𝑡,𝑥)=^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑓𝑢_𝑠𝒯𝑑𝑦𝑑𝑠,₂𝑢(𝑡,𝑥)=^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦ℎ𝑢_𝑠𝑊,𝒯𝑑𝑦,𝑑𝑠₃𝑢(𝑡,𝑥)=^𝑡_0ℝ𝐺_𝛼𝑢𝑡−𝑠,𝑥−𝑦×𝑔_𝑠𝑧𝑁𝑑𝑠,𝑑𝑦,𝑑𝑧. (18) It follows from Hölder’s inequality, Lemma 1, (H1), and (H2) that 𝐸𝒯₁𝑢(𝑡,𝑥)²=𝐸^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑓𝑢_𝑠𝑑𝑦𝑑𝑠²≤𝐶^𝑡_0ℝ𝐺_𝛼×𝑡−𝑠,𝑥−𝑦𝑑𝑦𝑑𝑠_𝑡0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝐸𝑓𝑢_𝑠²𝑑𝑦𝑑𝑠≤𝐶𝐾^𝑡_0ℝ𝐺_𝛼𝑢𝑡−𝑠,𝑥−𝑦×𝐸𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑦𝑑𝑠≤𝐶^𝑡₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑠≤𝐶𝑡+𝐶^𝑡₀⁰_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑠≤𝐶𝑡+𝐶⁰_−𝑟^𝑡_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑑𝑠𝑑𝜂≤𝐶𝑡+𝐶𝑟⁰_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑑𝑠+𝐶𝑟^𝑡₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²()𝑑𝑠≤𝐶𝑡𝑟+1+𝐶𝑟<∞. (19)
Applying Burkholder-Davis-Gundy inequality, Lemma 1, (H1), and (H2), we have 𝐸𝒯₂𝑢(𝑡,𝑥)²=𝐸^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦ℎ𝑢_𝑠𝑊𝑑𝑠𝑑𝑦²≤𝐶^𝑡_0ℝ𝐺²_𝛼𝑡−𝑠,𝑥−𝑦𝐸ℎ𝑢_𝑠²𝑑𝑠𝑑𝑦≤𝐶𝐾^𝑡₀(𝑡−𝑠)^−1/𝛼sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑠≤𝐶𝑡^1−(1/𝛼)+𝐶⁰_−𝑟^𝑡_−𝑟𝑡+𝜂−𝑠^−1/𝛼×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑑𝑠𝑑𝜂=𝐶𝑡^1−(1/𝛼)+𝐶^𝑡+𝜂_−𝑟⁰_−𝑟𝑡+𝜂−𝑠^−1/𝛼𝑑𝜂×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑑𝑠+^𝑡_𝑡+𝜂⁰_−𝑟𝑠−𝜂−𝑡^−1/𝛼𝑑𝜂×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑑𝑠≤𝐶𝑡^1−(1/𝛼)+𝐶^𝑡+𝜂_−𝑟(𝑡−𝑠)^1−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²+𝑑𝑠_{𝑡𝑡+𝜂}(𝑠+𝑟−𝑡)^1−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑑𝑠=𝐶𝑡^1−(1/𝛼)+𝐶⁰_−𝑟(𝑡−𝑠)^1−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²+𝑑𝑠_0𝑡+𝜂(𝑡−𝑠)^1−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²+𝑑𝑠_{𝑡𝑡+𝜂}(𝑠+𝑟−𝑡)^1−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑡𝑑𝑠≤𝐶^1−(1/𝛼)+(𝑡+𝑟)^1−(1/𝛼)+𝑡^2−(1/𝛼)+𝑟^2−(1/𝛼)𝐸𝒯<∞,₃𝑢(𝑡,𝑥)²=𝐸^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑔𝑢_𝑠𝑧𝑁𝑑𝑠,𝑑𝑦,𝑑𝑧²≤𝐶^𝑡_0ℝ𝐺²_𝛼𝑡−𝑠,𝑥−𝑦×𝐸_ℝ𝑔𝑢_𝑠|𝑧|𝜈(𝑑𝑧)²𝑑𝑦𝑑𝑠≤𝐶𝐾^𝑡₀(𝑡−𝑠)^−1/𝛼sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦²𝑡𝑑𝜂𝑑𝑠≤𝐶^1−(1/𝛼)+(𝑡+𝑟)^1−(1/𝛼)+𝑡^2−(1/𝛼)+𝑟^2−(1/𝛼)<∞. (20)(21) Thus, combining (19) and (20) with (21), we derive 𝐸|𝑢(𝑡,𝑥)|²≤𝐶𝑡(𝑟+1)+𝑟+𝑡^1−(1/𝛼)+(𝑡+𝑟)^1−(1/𝛼)+𝑡^2−(1/𝛼)+𝑟^2−(1/𝛼). (22) Taking Laplace transform formula and (22), we deduce that |𝒯𝑢(𝑡,𝑥)|²_𝜆=^𝑇₀𝑒^−𝜆𝑡sup_𝑥∈ℝ|𝒯𝑢(𝑡,𝑥)|²𝑑𝑡≤𝐶^∞₀𝑒^−𝜆𝑡𝑡(𝑟+1)+𝑟+𝑡^1−(1/𝛼)+(𝑡+𝑟)^1−(1/𝛼)+𝑡^2−(1/𝛼)+𝑟^2−(1/𝛼))Γ(2)𝜆𝑑𝑡≤𝐶(𝑟+1⁻²+𝑟𝜆⁻¹+𝑒^𝑟Γ1+12−𝛼𝜆^{−(2−(1/𝛼))}1+Γ3−𝛼𝜆^{−(3−(1/𝛼))}+𝑟^2−(1/𝛼)𝜆⁻¹≤𝐶𝜆⁻²+𝜆⁻¹+𝜆^{−(2−(1/𝛼))}+𝜆^{−(3−(1/𝛼))}<∞; (23) that is, 𝒯𝑢∈𝔹, which implies that operator 𝒯∶𝔹→𝔹.
Step 2. For any 𝑢,𝑣∈𝔹 and 𝑡≥0, it follows from Hölder’s inequality, Lemma 1, and (H2) that 𝐸𝒯₁𝑢(𝑡,𝑥)−𝒯₁𝑣(𝑡,𝑥)²=𝐸^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑓𝑢_𝑠−𝑓𝑣_𝑠𝑑𝑦𝑑𝑠²≤𝐶^𝑡_0ℝ𝐺_𝛼×𝑡−𝑠,𝑥−𝑦𝑑𝑦𝑑𝑠_𝑡0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝐸𝑓𝑢_𝑠−𝑓𝑣_𝑠²𝑑𝑦𝑑𝑠≤𝐶𝐾^𝑡₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦−𝑣𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑠≤𝐶^𝑡₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_{0−𝑟𝑡−𝑟}sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²()𝑑𝑠𝑑𝜂=𝐶𝑟+1^𝑡₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟⁰_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠. (24) By Burkholder-Davis-Gundy inequality, Lemma 1 and (H2), we have 𝐸𝒯₂𝑢(𝑡,𝑥)−𝒯₂𝑣(𝑡,𝑥)²=𝐸^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥−𝑦ℎ𝑢_𝑠−ℎ𝑣_𝑠𝑊𝑑𝑦𝑑𝑠²≤𝐶^𝑡_0ℝ𝐺²_𝛼𝑡−𝑠,𝑥−𝑦𝐸ℎ𝑢_𝑠−ℎ𝑣_𝑠²𝑑𝑦𝑑𝑠≤𝐶𝐾^𝑡₀(𝑡−𝑠)^−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦−𝑣𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑠≤𝐶^𝑡₀(𝑡−𝑠)^−1/𝛼sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_{0−𝑟𝑡−𝑟}𝑡+𝜂−𝑠^−1/𝛼×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠𝑑𝜂≤𝐶^𝑡₀(𝑡−𝑠)^−1/𝛼sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+^𝑡+𝜂_−𝑟⁰_−𝑟𝑡+𝜂−𝑠^−1/𝛼×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝜂𝑑𝑠_{𝑡𝑡+𝜂0−𝑟}𝑠−𝜂−𝑡^−1/𝛼×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝜂𝑑𝑠≤𝐶^𝑡₀(𝑡−𝑠)^−1/𝛼sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0−𝑟(𝑡−𝑠)^1−(1/𝛼)sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0𝑡+𝜂(𝑡−𝑠)^1−(1/𝛼)sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟^{1−(1/𝛼)𝑡}_𝑡+𝜂sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²,𝐸𝒯𝑑𝑠₃𝑢(𝑡,𝑥)−𝒯₃𝑣(𝑡,𝑥)²=𝐸^𝑡_0ℝ𝐺_𝛼×𝑔𝑢𝑡−𝑠,𝑥−𝑦_𝑠−𝑔𝑣_𝑠𝑧𝑁𝑑𝑠,𝑑𝑦,𝑑𝑧²≤𝐶^𝑡_0ℝ𝐺²_𝛼𝑡−𝑠,𝑥−𝑦×𝐸_ℝ𝑔𝑢_𝑠−𝑔𝑣_𝑠|𝑧|𝜈(𝑑𝑧)²𝑑𝑦𝑑𝑠≤𝐶𝐾^𝑡₀(𝑡−𝑠)^−1/𝛼×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦−𝑣𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑠≤𝐶^𝑡₀(𝑡−𝑠)^−1/𝛼sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0−𝑟(𝑡−𝑠)^1−(1/𝛼)sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+₀^𝑡+𝜂(𝑡−𝑠)^1−(1/𝛼)sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟^{1−(1/𝛼)𝑡}_𝑡+𝜂sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠. (25) Thus, it follows that 𝐸|𝒯𝑢(𝑡,𝑥)−𝒯𝑣(𝑡,𝑥)|²≤𝐶(𝑟+1)^𝑡₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟⁰_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_𝑡0(𝑡−𝑠)^−1/𝛼sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0−𝑟(𝑡−𝑠)^1−(1/𝛼)sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0𝑡+𝜂(𝑡−𝑠)^1−(1/𝛼)sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟^{1−(1/𝛼)𝑡}_𝑡+𝜂sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠. (26) Finally, direct computation implies that |𝒯𝑢(𝑡,𝑥)−𝒯𝑣(𝑡,𝑥)|²_𝜆=^𝑇₀𝑒^−𝜆𝑡sup_𝑦∈ℝ𝐸|𝒯𝑢(𝑡,𝑥)−𝒯𝑣(𝑡,𝑥)|²≤𝐶^𝑇₀𝑒^−𝜆𝑡()×𝑟+1_𝑡0sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟⁰_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_𝑡0(𝑡−𝑠)^−1/𝛼×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0−𝑟(𝑡−𝑠)^1−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0𝑡+𝜂(𝑡−𝑠)^1−(1/𝛼)×sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟^1−(1/𝛼)×^𝑡_𝑡+𝜂sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠𝑑𝑡≤𝐶^∞₀𝑒^−𝜆𝑡×()𝑑𝑡𝑟+1^𝑇₀𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+𝑟⁰_−𝑟𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠+^∞₀𝑒^−𝜆𝑡𝑡^−1/𝛼×𝑑𝑡_𝑇0𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_∞0𝑒^−𝜆𝑡𝑡^1−(1/𝛼)×𝑑𝑡_0−𝑟𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_∞0𝑒^−𝜆𝑡𝑡^1−(1/𝛼)×𝑑𝑡_0𝑇+𝜂𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_∞0𝑒^−𝜆𝑡𝑑𝑡×𝑟^1−(1/𝛼)×^𝑇_𝑇+𝜂𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²𝑑𝑠≤𝐶𝑟+1𝜆+Γ(1−(1/𝛼))𝜆^1−(1/𝛼)×^𝑇₀𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑟𝑑𝑠𝜆+Γ(2−(1/𝛼))𝜆^2−(1/𝛼)×⁰_−𝑟𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+Γ((𝑑𝑠2−1/𝛼))𝜆^2−(1/𝛼)×₀^𝑇+𝜂𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑟𝑑𝑠_1−(1/𝛼)𝜆^𝑇_𝑇+𝜂𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²≤𝐶𝑑𝑠₁𝜆^𝐾1𝑇0𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²+𝑑𝑠_0−𝑟𝑒^−𝜆𝑠sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦−𝑣𝑠,𝑦²=𝐶𝑑𝑠₁𝜆^𝐾1|𝑢−𝑣|²_𝜆, (27) where 𝐶₁=2𝐶⋅max{𝑟+1,Γ(1−(1/𝛼)),Γ(2−(1/𝛼)),𝑟^1−(1/𝛼)} and 𝐾₁=1−(1/𝛼).
Let 𝜆 that large enough such that 𝐶₁𝜆^𝐾1<1, (28) which implies that the operator 𝒯∶𝔹→𝔹 is contraction. By the Banach fixed point theorem, there exists a unique fixed point in 𝔹. Moreover, the fixed point is the unique mild solution of (8).

Remark 5. If there are no delays, Theorem 3 can be solved in the following working function space: |𝑢|²_𝜆=^𝑇₀𝑒^−𝜆𝑡sup_𝑥∈ℝ𝐸|𝑢(𝑡,𝑥)|²𝑑𝑡, (29) where 𝑢∈𝔹, which implies that the delays affect the working function space.

4. The Regularity of the Mild Solution

In this section, we will show the time regularity and space regularity of the mild solution for (8). In order to prove the regularity, we need the following assumptions: (H3) there exists some 𝛾<1/2sup_𝑥∈ℝ𝐸𝑢₀(𝑥+𝑧)−𝑢₀(𝑥)²<𝑐|𝑧|^2𝛾; (30)(H4) for 𝜃>0, let |𝜙(𝜂,𝑥)|<∞ and there exists some 𝜋<1 such that sup_𝑥∈ℝ^−𝜃_−𝑟𝐸𝑢𝜂+𝜃,𝑥−𝑢𝜂,𝑥²𝑑𝜂<𝑐|𝜃|^2𝜋; (31)(H5)|𝑓(𝑡₁,𝑦,𝑢_𝑡1)−𝑓(𝑡₂,𝑦,𝑢_𝑡2)|²≤𝑐(|𝑡₁−𝑡₂|²+|𝑢(𝑡₁)−𝑢(𝑡₂)|²+∫⁰_−𝑟|𝑢(𝑡₁+𝜂)−𝑢(𝑡₂+𝜂)|²𝑑𝜂).

To the end, we will give an important lemma from [7].

Lemma 6. (1) For 1<𝑛<𝛼+1, ∫^∞₀∫_ℝ|𝐺_𝛼(1+𝑣,𝑧)−𝐺_𝛼(𝑣,𝑧)|^𝑛𝑑𝑧𝑑𝑣<∞.
(2) For (𝛼+1)/2<𝑛<𝛼+1, ∫^∞₀∫_ℝ|𝐺_𝛼(𝑣,𝑧+1)−𝐺_𝛼(𝑣,𝑧)|^𝑛𝑑𝑧𝑑𝑣<∞.

Theorem 7. Assume that the conditions (H1)–(H5) are satisfied; then for 𝛼∈(1,2] and 𝑡≥0, there exists a continuous modification 𝑢(𝑡,𝑥), which is 𝛽-Hölder continuous in 𝑡, where 𝛽=min{𝛾/𝛼,𝜋,(1/2)−(1/2𝛼)}.

Proof. For 𝑡≥0, it follows that, for any 𝑥∈ℝ and 𝜃>0, |𝑢()(≤𝑡+𝜃,𝑥−𝑢𝑡,𝑥)|_ℝ𝐺_𝛼𝑡+𝜃,𝑥−𝑦−𝐺_𝛼𝑡,𝑥−𝑦𝑢₀𝑦+𝑑𝑦_{0𝑡+𝜃ℝ}𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑓𝑢_𝑠+𝑑𝑦𝑑𝑠_𝑡0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦ℎ𝑢_𝑠𝑊+𝑑𝑠𝑑𝑦_{𝑡𝑡+𝜃ℝ}𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦ℎ𝑢_𝑠𝑊+𝑑𝑠𝑑𝑦_𝑡0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑢𝑡−𝑠,𝑥−𝑦×𝑔_𝑠()+𝑧𝑁𝑑𝑠,𝑑𝑧𝑑𝑦_{𝑡𝑡+𝜃ℝ}𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦𝑔𝑢_𝑠𝑧𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑦=𝜙⁰_𝜃+𝜙¹_𝜃+𝜙²_𝜃+𝜙³_𝜃+𝜙⁴_𝜃+𝜙⁵_𝜃. (32) Next, we will estimate each term 𝜙^𝑗_𝜃  (𝑗=0,1,…,5), respectively.
Combining Hölder inequality, Lemma 1 with (H3) yields 𝐸𝜙⁰_𝜃²=𝐸_ℝ𝐺_𝛼(𝜃,𝑧)_ℝ𝐺_𝛼×𝑢𝑡,𝑥−𝑦₀𝑦−𝑧−𝑢₀𝑦𝑑𝑦𝑑𝑧²≤𝐶𝐸_ℝ𝐺_𝛼(𝜃,𝑧)_ℝ𝐺_𝛼×𝑢𝑡,𝑥−𝑦₀𝑦−𝑧−𝑢₀𝑦𝑑𝑦²×𝑑𝑧_ℝ𝐺_𝛼(𝜃,𝑧)𝑑𝑧≤𝐶_ℝ𝐺_𝛼(𝜃,𝑧)sup_𝑦∈ℝ𝐸𝑢₀𝑦−𝑧−𝑢₀𝑦²𝑑𝑧≤𝐶_ℝ𝐺_𝛼(𝜃,𝑧)|𝑧|^2𝛾𝑑𝑧=𝐶_ℝ𝜃^2𝛾/𝛼𝐺_𝛼(1,𝑧)𝑧^2𝛾𝑑𝑧≤𝐶𝜃^2𝛾/𝛼_ℝ|𝑧|^2𝛾1+|𝑧|^1+𝛼𝑑𝑧≤𝐶𝜃^2𝛾/𝛼. (33) Next, we consider 𝜙¹_𝜃. Let 𝑠^′=𝑠−𝜃; then, 𝜙¹_𝜃≤^𝑡_0ℝ𝐺_𝛼×𝑓𝑡−𝑠,𝑥−𝑦𝑠+𝜃,𝑦,𝑢_𝑠+𝜃−𝑓𝑠,𝑦,𝑢_𝑠+𝑑𝑦𝑑𝑠_𝜃0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦×𝑓𝑠,𝑦,𝑢_𝑠𝑑𝑦𝑑𝑠=𝜙_𝜃^1.1+𝜙_𝜃^1.2. (34) By Hölder inequality, Lemma 1, (H4), and (H5), we have 𝐸𝜙_𝜃^1.12≤𝐶^𝑡_0ℝ𝐺_𝛼×𝑡−𝑠,𝑥−𝑦𝑑𝑦𝑑𝑠_𝑡0ℝ𝐺_𝛼𝑓𝑡−𝑠,𝑥−𝑦×𝐸𝑠+𝜃,𝑦,𝑢_𝑠+𝜃−𝑓𝑠,𝑦,𝑢_𝑠²𝑑𝑦𝑑𝑠≤𝐶^𝑡₀𝜃²+sup_𝑦∈ℝ𝐸𝑢𝑠+𝜃,𝑦−𝑢𝑠,𝑦²+⁰_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠+𝜃+𝜂,𝑦−𝑢𝑠+𝜂,𝑦²𝑑𝜂𝑑𝑠≤𝐶𝑇𝜃²+()×𝑟+1_𝑡0sup_𝑦∈ℝ𝐸𝑢𝑠+𝜃,𝑦−𝑢𝑠,𝑦²𝑑𝑠+𝑟^−𝜃_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠+𝜃,𝑦−𝑢𝑠,𝑦²+𝑑𝑠_0−𝜃sup_𝑦∈ℝ𝐸𝑢𝑠+𝜃,𝑦−𝑢𝑠,𝑦²𝜃𝑑𝑠≤𝐶²+𝜃^2𝜋++𝜃_𝑡0sup_𝑦∈ℝ𝐸𝑢𝑠+𝜃,𝑦−𝑢𝑠,𝑦²,𝐸𝜙𝑑𝑠_𝜃1.22≤𝐶^𝜃_0ℝ𝐺_𝛼×𝑡+𝜃−𝑠,𝑥−𝑦𝑑𝑦𝑑𝑠_𝜃0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦𝐸𝑓𝑢_𝑠²𝑑𝑦𝑑𝑠≤𝐶𝜃^𝜃_0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦𝐸𝑓𝑢_𝑠²𝑑𝑦𝑑𝑠≤𝐶𝜃^𝜃₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂,𝑦²()𝑑𝜂𝑑𝑠≤𝐶𝜃𝑟+1^𝜃₀sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝑑𝑠+𝑟⁰_−𝑟sup_𝑦∈ℝ𝐸𝑢𝑠,𝑦²𝜃𝑑𝑠≤𝐶²+𝜃. (35)
Taking the transformation 𝑠=𝜃𝑣, 𝑦=𝜃^1/𝛼𝑧, by Burkholder-Davis-Gundy inequality, Lemmas 1 and 6, and (H2), we obtain 𝐸𝜙²_𝜃²≤𝐶^𝑡_0ℝ𝐸𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²ℎ²𝑢_𝑠𝑑𝑠𝑑𝑦≤𝐶𝐾^𝑡_0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝐸|𝑢(𝑠)|²+⁰_−𝑟𝑢𝑠+𝜂²𝑑𝜂𝑑𝑦𝑑𝑠≤𝐶^𝑡_0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝐸|𝑢(𝑠)|²+⁰_−𝑟𝑢𝜂²+𝑑𝜂_𝑡0𝑢𝜂^′2𝑑𝜂^′()𝑑𝑦𝑑𝑠≤𝐶𝑇+1sup_{[]0,𝑇×ℝ}𝐸𝑢𝑠,𝑦²+sup_𝑦∈ℝ⁰_−𝑟𝑢𝜂,𝑦²×𝑑𝜂_𝑡0ℝ𝐺_𝛼𝑠+𝜃,𝑦−𝐺_𝛼𝑠,𝑦²𝜃𝑑𝑦𝑑𝑠≤𝐶^−1/𝛼𝜃^∞_0ℝ𝐺_𝛼(𝑣+1,𝑧)−𝐺_𝛼(𝑣,𝑧)²𝑑𝑧𝑑𝑣≤𝐶𝜃^1−(1/𝛼),𝐸𝜙³_𝜃²≤𝐶_𝑡^𝑡+𝜃_ℝ𝐸𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²ℎ𝑢_𝑠²𝑑𝑠𝑑𝑦≤𝐶_𝑡^𝑡+𝜃_ℝ𝐸𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝑢𝑠,𝑦²+⁰_−𝑟𝑢𝑠+𝜂²𝑑𝜂𝑑𝑦𝑑𝑠≤𝐶_𝑡^𝑡+𝜃_ℝ𝐸𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝑢𝑠,𝑦²+^𝑡+𝜃_−𝑟𝑢𝜂^′2𝑑𝜂^′()𝑑𝑦𝑑𝑠≤𝐶𝑇+1+𝜃sup_{[0,𝑇+𝜃]×ℝ}𝐸𝑢𝑠,𝑦²+sup_𝑦∈ℝ⁰_−𝑟𝑢𝜂,𝑦²×𝑑𝜂_𝜃0ℝ𝐸𝐺_𝛼𝑠+𝜃,𝑦−𝐺_𝛼𝑠,𝑦²()𝜃𝑑𝑦𝑑𝑠≤𝐶1+𝜃^1−(1/𝛼)×¹_0ℝ𝐸𝐺_𝛼(1+𝑣,𝑧)−𝐺_𝛼(𝑣,𝑧)²𝜃𝑑𝑧𝑑𝑣≤𝐶^1−(1/𝛼)+𝜃^2−(1/𝛼). (36) Then, by the same method, we have 𝐸𝜙⁴_𝜃²≤^𝑡_0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝐸_ℝ𝑔𝑢_𝑠|𝑧|𝜈(𝑑𝑧)²𝑑𝑠𝑑𝑦≤𝐶𝐾^𝑡_0ℝ𝐺_𝛼𝑡+𝜃−𝑠,𝑥−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝐸|𝑢(𝑠)|²+⁰_−𝑟𝑢𝑠+𝜂²𝑑𝜂𝑑𝑦𝑑𝑠≤𝐶𝜃^1−(1/𝛼),𝐸𝜙⁵_𝜃²≤𝐶𝜃^1−(1/𝛼)+𝜃^2−(1/𝛼). (37) Thus, from the previous estimates, let 𝛽=min{𝛾/𝛼,𝜋,(1/2)−(1/2𝛼)}: 𝐸|𝑢(𝑡+𝜃,𝑥)−𝑢(𝑡,𝑥)|²≤𝐶𝜃^2𝛽+^𝑡₀sup_𝑦∈ℝ𝐸𝑢𝑠+𝜃,𝑦−𝑢𝑠,𝑦²𝑑𝑠. (38) Hence, it follows from Gronwall’s Lemma that 𝐸|𝑢(𝑡+𝜃,𝑥)−𝑢(𝑡,𝑥)|²≤𝐶𝜃^2𝛽. (39) Then, for 𝑡≥0, we have |𝑢(𝑡+𝜃,𝑥)−𝑢(𝑡,𝑥)|^𝑝_𝜆=^𝑇₀𝑒^−𝜆𝑡sup_𝑥∈ℝ𝐸|𝑢(𝑡+𝜃,𝑥)−𝑢(𝑡,𝑥)|²𝑑𝑠^𝑝/2≤𝐶𝜃^𝛽𝑝. (40)

Finally, we study the space regularity of the mild solution for (8).

Theorem 8. Assume that the conditions (H1)–(H3) are satisfied; then for 𝛼∈(1,2] and 𝑡≥0, there exists a continuous modification 𝑢(𝑡,𝑥), which is 𝜌-Hölder continuous in 𝑥, where 𝜌=min{𝛾,𝜗,𝛼−1}.

Proof. It follows that, for any 𝑡∈[0,𝑇] and 𝜁>0, 𝑢()(≤𝑡,𝑥+𝜁−𝑢𝑡,𝑥)_ℝ𝐺_𝛼𝑡,𝑥+𝜁−𝑦−𝐺_𝛼𝑡,𝑥−𝑦𝑢₀𝑦+𝑑𝑦_𝑡0ℝ𝐺_𝛼𝑡−𝑠,𝑥+𝜁−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑓𝑢_𝑠+𝑑𝑦𝑑𝑠_𝑡0ℝ𝐺_𝛼𝑡−𝑠,𝑥+𝜁−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦ℎ𝑢_𝑠𝑊+𝑑𝑠𝑑𝑦_𝑡0ℝ𝐺_𝛼𝑡−𝑠,𝑥+𝜁−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦𝑔𝑢_𝑠𝑧𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑦∶=³_𝑗=0𝜙^𝑗_𝜁. (41) By (H3) and Lemma 1, we have 𝐸𝜙⁰_𝜁²=𝐸_ℝ𝐺_𝛼𝑡,𝑥−𝑦𝑢₀𝑦+𝜁−𝑢₀𝑦𝑑𝑦²≤sup_𝑦∈ℝ𝐸𝑢₀𝑦+𝜁−𝑢₀𝑦²×_ℝ𝐺_𝛼𝑡,𝑥−𝑦𝑑𝑦²≤𝐶𝜁^2𝛾. (42) By (H2), Hölder’s inequality and Lemma 1, we set 𝜖=(1/2)(𝛼+1)−𝛿 (𝛿 is small enough) and 𝜗∈(0,1) and we can derive 𝐸𝜙¹_𝜁²≤𝐶(1+𝑇)sup_{[]0,𝑇×ℝ}𝐸|𝑢(𝑡,𝑥)|²+sup_𝑦∈ℝ⁰_−𝑟𝑢𝜂,𝑦²×𝑑𝜂_𝑡0ℝ𝐺_𝛼𝑠,𝑦+𝜁−𝐺_𝛼𝑠,𝑦𝑑𝑦𝑑𝑠²≤𝐶^𝑡_0ℝ𝑠^−1/𝛼𝐺_𝛼1,𝑠^−1/𝛼𝑦+𝜁−𝐺_𝛼(1,𝑠^−1/𝛼𝑦)𝑑𝑦𝑑𝑠²=𝐶^𝑡_0ℝ𝑠^−𝜖/𝛼𝐺_𝛼1,𝑠^−1/𝛼𝑦+𝜁−𝐺_𝛼1,𝑠^−1/𝛼𝑦^(1−𝜗)×𝑠^{−((1−𝜖)/𝛼)}𝐺_𝛼1,𝑠^−1/𝛼𝑦+𝜁−𝐺_𝛼1,𝑠^−1/𝛼𝑦^𝜗𝑑𝑦𝑑𝑠²≤𝐶^𝑡_0ℝ𝑠^{−2𝜖/𝛼}𝐺_𝛼1,𝑠^−1/𝛼𝑦+𝜁−𝐺_𝛼1,𝑠^−1/𝛼𝑦^2(1−𝜗)×𝑑𝑦𝑑𝑠_𝑡0ℝ𝑠^{−2(1−𝜖)/𝛼}𝐺_𝛼1,𝑠^−1/𝛼𝑦+𝜁−𝐺_𝛼1,𝑠^−1/𝛼𝑦^2𝜗𝑑𝑦𝑑𝑠∶=𝑐𝐼×𝐼𝐼. (43) By Lemma 1, recall that 2𝜖<𝛼+1, and we have 𝐼≤𝐶^𝑡_0ℝ𝑠^{−(2𝜖−1)/𝛼}𝐺_𝛼1,𝑦^2(1−𝜗)𝑑𝑦𝑑𝑠≤𝐶^𝑡₀𝑠^{−(2𝜖−1)/𝛼}𝑑𝑠<∞. (44) Then, by the mean value of the theorem and Lemma 1, we have that 𝐼𝐼=^𝑡_0ℝ𝑠^{−2(1−𝜖)/𝛼}𝜕𝐺𝜕𝑦^𝛼1,𝑠^−1/𝛼𝑦+𝜀×𝑠^−1/𝛼𝜁^2𝜗𝑑𝑦𝑑𝑠(𝜀∈(0,𝜁))=𝜁^2𝜗𝑡_0ℝ𝑠^{−(2(1−𝜖+𝜗)−1)/𝛼}×𝜕𝐺𝜕𝑦^𝛼1,𝑦^2𝜗𝑑𝑦𝑑𝑠≤𝐶𝜁^2𝜗𝑡₀𝑠^{−(2(1−𝜖+𝜗)−1)/𝛼}𝑑𝑠. (45) Choosing 𝜗<𝛼−𝛿 such that (2(1−𝜖+𝜗)−1)/𝛼<1, then 𝐼𝐼<𝐶𝜁^2𝜗. (46) Combining (44) and (46), we have 𝐸𝜙¹_𝜁²≤𝐶𝜁^2𝜗. (47) Taking the change of variable 𝑠=𝜁^𝛼𝑣,  𝑦=𝑧𝜁, and by Burkholder-Davis-Gundy inequality, Lemmas 1 and 6, (H1), and (H2), we derive 𝐸𝜙²_𝜁²≤^𝑡_0ℝ𝐸𝐺_𝛼𝑡−𝑠,𝑥+𝜁−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝐸ℎ𝑢_𝑠²()𝑑𝑦𝑑𝑠≤𝐶𝑇+1sup_{[0,𝑇]×ℝ}𝐸|𝑢(𝑡,𝑥)|²+sup_𝑦∈ℝ⁰_−𝑟𝑢𝜂,𝑦²×𝑑𝜂_𝑡0ℝ𝐺_𝛼𝑠,𝑦+𝜁−𝐺_𝛼𝑠,𝑦²𝜁𝑑𝑦𝑑𝑠≤𝐶⁻²𝜁𝜁^𝛼∞_0ℝ𝐺_𝛼(𝑣,𝑧+1)−𝐺_𝛼(𝑣,𝑧)²𝑑𝑧𝑑𝑣≤𝐶𝜁^2(𝛼−1),𝐸𝜙³_𝜁²≤𝐶^𝑡_0ℝ𝐺_𝛼𝑡−𝑠,𝑥+𝜁−𝑦−𝐺_𝛼𝑡−𝑠,𝑥−𝑦²×𝐸_ℝ𝑔𝑢_𝑠𝑧𝜈(𝑑𝑧)²𝑑𝑠𝑑𝑦≤𝐶𝜁^2(𝛼−1). (48) Combining (42)–(48), we have 𝐸𝑢(𝑡,𝑥+𝜁)−𝑢(𝑡,𝑥)²≤𝐶𝜁^2𝛾+𝜁^2𝜗+𝜁^2(𝛼−1). (49) Then, we have, for 𝑡∈[0,𝑇], 𝑢(𝑡,𝑥+𝜁)−𝑢(𝑡,𝑥)^𝑝_𝜆≤𝐶𝜁^𝛾𝑝+𝜁^𝜗𝑝+𝜁^{(𝛼−1)𝑝}≤𝐶𝜁^𝜌𝑝, (50) where 𝜌=min{𝛾,𝜗,𝛼−1}.

Remark 9. Theorems 7 and 8 show that the regularity of initial value and the order of fractional operator can affect both time regularity and space regularity of the mild solution for (1). In particular, the time regularity is affected by the regularity of initial value with delays.

Acknowledgments

This work is supported by the NSF of China (nos. 10971225 and 11371367) and NSF of Hunan Province (no. 11JJ3004).