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

Authors: Tianlong Shen; Jianhua Huang; Jin Li
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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. Brzez´niak 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:
(1)∂u(t,x)∂t=λΔαu(t,x)+f(t,x,ut)+g(t,x,ut)Zt,x,(t,x)∈[0,T]×ℝ,u(0,x)=u0(x),u(η,x)=ϕ(η,x),η∈[-r,0],
where Δα∶=-(-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 ut=u(t+η), and Zt,x is the one-dimensional Lévy process (see Section 2 for the definition). Recall that Dα 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 Section2, 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 Zt,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 Wt,x=(∂2W/∂t∂x)(t,x) (W(t,x) is a Brownian sheet on [0,∞)×ℝ) such that
(2)Zt,x=β1t+β2Wt,x+∫|z|<1‍zN~(t,x,dz)+∫|z|≥1‍zN(t,x,dz),
where
(3)N~(t,x,A)∶=N(t,x,A)-tν(A),
where ν(A)∶=E[N(1,A)] is the Lévy measure of Zt,x.Similar to [14], for any p, we denote
(4)c^p∶=(∫ℝ‍|z|pν(dz))1/p.
In what follows, we assume that
(5)c^∶=supp≥1c^p<∞.
Recalling that
(6)∫|z|<1zN(t,x,dz)=∫|z|<1zN~(t,x,dz)+t∫|z|<1zν(dz).
By absorbing β~∶=-∫|z|<1‍zν(dz) into β1, we can rewrite (2) into the following equation:
(7)Zt,x=β1t+β2Wt,x+∫ℝ‍zN(t,x,dz).
Let β1=0; then (1) can be written as
(8)du(t,x)=[λΔαu(t,x)+f(t,x,ut)]dt+h(t,x,ut)dWt,x+g(t,x,ut)dYt,x,(t,x)∈[0,T]×ℝ,u(0,x)=u0(x),u(η,x)=ϕ(η,x),η∈(r,0],
where Wt,x is a one-dimensional space-time white noise and Yt,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 datau0 and ϕ(η,x), f(ut), h(ut), and g(ut) to show the existence and uniqueness of the mild solution.  (H1)
The initial datau0 which is ℱ0-measurable and ϕ satisfy
(9)supx∈ℝ|u(0,x)|2<∞,supx∈ℝ∫-r0‍|ϕ(η,x)|2dη<∞.(H2)
There exists a constantK such that for all t≥0,
(10)|f(ut)|2+|h(ut)|2+(∫ℝ‍|g(ut)||z|ν(dz))2<K(|u|2+∫-r0‍|u(t+η)|2dη),|f(ut)-f(vt)|2+|h(ut)-h(vt)|2+(∫ℝ‍|g(ut)-g(vt)||z|ν(dz))2<K(|u-v|2+∫-r0‍|u(t+η)-v(t+η)|2dη).Let the Green kennelGα(t,x) be the fundamental solution of the Cauchy problem:
(11)∂υ∂t=λΔαυ,(t,x)∈(0,∞)×ℝ,υ(0,x)=δ0(x),x∈ℝ,
where δ0(x) denotes the Dirac function. By Fourier transform,
(12)Gα(t,x)=[ℱ-1(eλt|·|α)](x).
A higher order fractional Green kennel is introduced in [16].The following lemma gives some useful properties aboutGα(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 functionGα(t,x) satisfies the following properties.(1)
For anyt≥0Gα(t,x)=t-1/αGα(1,t-1/αx).(2)
Forn∈(1/(α+1),α+1), ∫0T‍∫ℝ‍|Gα(t,x)|ndtdx<∞.(3)
For anyx∈ℝ, ∫ℝ‍Gα(t,x)dx=1.(4)
For anyt,s∈ℝ, Gα(t,x)*Gα(s,x)=Gα(t+s,x).(5)
For anyx∈ℝ, there exists a constant C such that
(13)Gα(1,x)≤C1+|x|1+α,∂xmGα(1,x)≤C|x|α+m-11+|x|1+α.


## 3. Existence of the Mild Solution

In this section, we will first construct the proper working function space.LetT be a fixed positive time and 𝔹 the class of all ℱt-adapted càdlàg process {u(t,x),(t,x)∈[0,T]×ℝ} satisfying
(14)sup(t,x)∈[0,T]×ℝE[|u(t,x)|2]<∞.
Let λ>0 be arbitrarily fixed; we define
(15)|u|λ2={∫0T‍e-λtsupx∈ℝE|u(t,x)|2dt}t≥0+{∫-r0‍e-λtE|u(t,x)|2dt}t∈(-r,0).
For any u∈𝔹,  |u|λ2<∞. It is easy to verify that |·|λ is a norm and (𝔹,|·|λ) is a Banach space.Let(Ω,ℱ,(ℱt)t≥0,P) and Gα(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.
Anℱt-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:
(16)u(t,x)=∫ℝ‍Gα(t,x-y)u0(y)dy+∫0t‍∫ℝ‍Gα(t-s,x-y)f(us)dyds+∫0t‍∫ℝ‍Gα(t-s,x-y)h(us)W(dy,ds)+∫0t‍∬ℝGα(t-s,x-y)g(us)zN(ds,dy,dz),
where the stochastic integral with respect to W(t,x) is understood in the sense of that introduced by Walsh [15].Theorem 3.
Fort≥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
(17)𝒯u(t,x)=∫ℝ‍Gα(t,x-y)u0(y)dy+𝒯1u(t,x)+𝒯2u(t,x)+𝒯3u(t,x),
where
(18)𝒯1u(t,x)=∫0t‍∫ℝ‍Gα(t-s,x-y)f(us)dyds,𝒯2u(t,x)=∫0t‍∫ℝ‍Gα(t-s,x-y)h(us)W(dy,ds),𝒯3u(t,x)=∫0t‍∬ℝGα(t-s,x-y)𝒯3u(t,x)×g(us)zN(ds,dy,dz).
It follows from Hölder’s inequality, Lemma 1, (H1), and (H2) that
(19)E|𝒯1u(t,x)|2=E|∫0t‍∫ℝ‍Gα(t-s,x-y)f(us)dyds|2≤C∫0t‍∫ℝ‍Gα(t-s,x-y)dyds×∫0t‍∫ℝ‍Gα(t-s,x-y)E|f(us)|2dyds≤CK∫0t‍∫ℝ‍Gα(t-s,x-y)×E(∫-r0‍|u(s,y)|2+∫-r0‍|u(s+η,y)|2dη)dyds≤C∫0t‍supy∈ℝE(∫-r0‍|u(s,y)|2+∫-r0‍|u(s+η,y)|2dη)ds≤Ct+C∫0t‍∫-r0‍supy∈ℝE|u(s+η,y)|2dηds≤Ct+C∫-r0‍∫-rt‍supy∈ℝE|u(s,y)|2dsdη≤Ct+Cr∫-r0‍supy∈ℝE|u(s,y)|2ds+Cr∫0t‍supy∈ℝE|u(s,y)|2ds≤Ct(r+1)+Cr<∞.
Applying Burkholder-Davis-Gundy inequality, Lemma1, (H1), and (H2), we have
(20)E|𝒯2u(t,x)|2=E|∫0t‍∫ℝ‍Gα(t-s,x-y)h(us)W(dsdy)|2≤C∫0t‍∫ℝ‍Gα2(t-s,x-y)E|h(us)|2dsdy≤CK∫0t‍(t-s)-1/αsupy∈ℝE(∫-r0‍|u(s,y)|2+∫-r0‍|u(s+η,y)|2dη)ds≤Ct1-(1/α)+C∫-r0‍∫-rt‍|(t+η-s)|-1/α×supy∈ℝE|u(s,y)|2dsdη=Ct1-(1/α)+C(∫-rt+η‍∫-r0‍(t+η-s)-1/αdη×supy∈ℝE|u(s,y)|2ds+∫t+ηt‍∫-r0‍(s-η-t)-1/αdη×supy∈ℝE|u(s,y)|2ds)≤Ct1-(1/α)+C(∫-rt+η‍(t-s)1-(1/α)×supy∈ℝE|u(s,y)|2ds+∫t+ηt‍(s+r-t)1-(1/α)×supy∈ℝE|u(s,y)|2ds)=Ct1-(1/α)+C(∫-r0‍(t-s)1-(1/α)×supy∈ℝE|u(s,y)|2ds+∫0t+η‍(t-s)1-(1/α)×supy∈ℝE|u(s,y)|2ds+∫t+ηt‍(s+r-t)1-(1/α)×supy∈ℝE|u(s,y)|2ds)≤C(t1-(1/α)+(t+r)1-(1/α)+t2-(1/α)+r2-(1/α))<∞,(21)E|𝒯3u(t,x)|2=E|∫0t‍∬ℝGα(t-s,x-y)g(us)zN(ds,dy,dz)|2≤C∫0t‍∫ℝ‍Gα2(t-s,x-y)×E(∫ℝ‍|g(us)||z|ν(dz))2dyds≤CK∫0t‍(t-s)-1/αsupy∈ℝE(|u(s,y)|2+∫-r0‍|u(s+η,y)|2dη)ds≤C(t1-(1/α)+(t+r)1-(1/α)+t2-(1/α)+r2-(1/α))<∞.
Thus, combining (19) and (20) with (21), we derive
(22)E|u(t,x)|2≤C[r2-(1/α)t(r+1)+r+t1-(1/α)+(t+r)1-(1/α)+t2-(1/α)+r2-(1/α)].
Taking Laplace transform formula and (22), we deduce that
(23)|𝒯u(t,x)|λ2=∫0T‍e-λtsupx∈ℝ|𝒯u(t,x)|2dt≤C∫0∞‍e-λt[r2-(1/α)t(r+1)+r+t1-(1/α)+(t+r)1-(1/α)+t2-(1/α)+r2-(1/α)]dt≤C[r2-(1/α)(r+1)Γ(2)λ-2+rλ-1+(er+1)Γ(2-1α)λ-(2-(1/α))+Γ(3-1α)λ-(3-(1/α))+r2-(1/α)λ-1]≤C[λ-2+λ-1+λ-(2-(1/α))+λ-(3-(1/α))]<∞;
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
(24)E|𝒯1u(t,x)-𝒯1v(t,x)|2=E|∫0t‍∫ℝ‍Gα(t-s,x-y)(f(us)-f(vs))dyds|2≤C∫0t‍∫ℝ‍Gα(t-s,x-y)dyds×∫0t‍∫ℝ‍Gα(t-s,x-y)E|f(us)-f(vs)|2dyds≤CK∫0t‍supy∈ℝE(∫-r0‍|u(s,y)-v(s,y)|2+∫-r0‍|u(s+η,y)-v(s+η,y)|2dη)ds≤C(∫0t‍supy∈ℝE|u(s,y)-v(s,y)|2ds+∫-r0‍∫-rt‍supy∈ℝE|u(s,y)-v(s,y)|2dsdη)=C((r+1)∫0t‍supy∈ℝE|u(s,y)-v(s,y)|2ds+r∫-r0‍supy∈ℝE|u(s,y)-v(s,y)|2ds).
By Burkholder-Davis-Gundy inequality, Lemma 1 and (H2), we have
(25)E|𝒯2u(t,x)-𝒯2v(t,x)|2=E|∫0t‍∫ℝ‍Gα(t-s,x-y)(h(us)-h(vs))W(dyds)|2≤C∫0t‍∫ℝ‍Gα2(t-s,x-y)E|h(us)-h(vs)|2dyds≤CK∫0t‍(t-s)-(1/α)×supy∈ℝE(∫-r0‍|u(s,y)-v(s,y)|2+∫-r0‍|u(s+η,y)-v(s+η,y)|2dη)ds≤C(∫0t‍(t-s)-1/αsupy∈ℝE|u(s,y)-v(s,y)|2ds+∫-r0‍∫-rt‍|t+η-s|-1/α×supy∈ℝE|u(s,y)-v(s,y)|2dsdη∫0t‍)≤C(∫0t‍(t-s)-1/αsupy∈ℝE|u(s,y)-v(s,y)|2ds+∫-rt+η‍∫-r0‍(t+η-s)-1/α×supy∈ℝE|u(s,y)-v(s,y)|2dηds+∫t+ηt‍∫-r0‍(s-η-t)-1/α×supy∈ℝE|u(s,y)-v(s,y)|2dηds)≤C(∫0t‍(t-s)-1/αsupy∈ℝE|u(s,y)-v(s,y)|2ds+∫-r0‍(t-s)1-(1/α)supy∈ℝE|u(s,y)-v(s,y)|2ds+∫0t+η‍(t-s)1-(1/α)supy∈ℝE|u(s,y)-v(s,y)|2ds+r1-(1/α)∫t+ηt‍supy∈ℝE|u(s,y)-v(s,y)|2ds),E|𝒯3u(t,x)-𝒯3v(t,x)|2=E|∫0t‍∬ℝGα(t-s,x-y)×(g(us)-g(vs))zN(ds,dy,dz)∫0t‍|2≤C∫0t‍∫ℝ‍Gα2(t-s,x-y)×E(∫ℝ‍|g(us)-g(vs)||z|ν(dz))2dyds≤CK∫0t‍(t-s)-1/α×supy∈ℝE(∫-r0‍|u(s,y)-v(s,y)|2+∫-r0‍|u(s+η,y)-v(s+η,y)|2dη)ds≤C(∫0t‍(t-s)-1/αsupy∈ℝE|u(s,y)-v(s,y)|2ds+∫-r0‍(t-s)1-(1/α)supy∈ℝE|u(s,y)-v(s,y)|2ds+∫0t+η‍(t-s)1-(1/α)supy∈ℝE|u(s,y)-v(s,y)|2ds+r1-(1/α)∫t+ηt‍supy∈ℝE|u(s,y)-v(s,y)|2ds).
Thus, it follows that
(26)E|𝒯u(t,x)-𝒯v(t,x)|2≤C((r+1)∫0t‍supy∈ℝE|u(s,y)-v(s,y)|2ds+r∫-r0‍supy∈ℝE|u(s,y)-v(s,y)|2ds+∫0t‍(t-s)-1/αsupy∈ℝE|u(s,y)-v(s,y)|2ds+∫-r0‍(t-s)1-(1/α)supy∈ℝE|u(s,y)-v(s,y)|2ds+∫0t+η‍(t-s)1-(1/α)supy∈ℝE|u(s,y)-v(s,y)|2ds+r1-(1/α)∫t+ηt‍supy∈ℝE|u(s,y)-v(s,y)|2ds).
Finally, direct computation implies that
(27)|𝒯u(t,x)-𝒯v(t,x)|λ2=∫0T‍e-λtsupy∈ℝE|𝒯u(t,x)-𝒯v(t,x)|2≤C∫0T‍e-λt(∫0T‍(r+1)×∫0t‍supy∈ℝE|u(s,y)-v(s,y)|2ds+r∫-r0‍supy∈ℝE|u(s,y)-v(s,y)|2ds+∫0t‍(t-s)-1/α×supy∈ℝE|u(s,y)-v(s,y)|2ds+∫-r0‍(t-s)1-(1/α)×supy∈ℝE|u(s,y)-v(s,y)|2ds+∫0t+η‍(t-s)1-(1/α)×supy∈ℝE|u(s,y)-v(s,y)|2ds+r1-(1/α)×∫t+ηt‍supy∈ℝE|u(s,y)-v(s,y)|2ds)dt≤C[∫0∞‍e-λtdt×((r+1)∫0T‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+r∫-r0‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds)+∫0∞‍e-λtt-1/αdt×∫0T‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+∫0∞‍e-λtt1-(1/α)dt×∫-r0‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+∫0∞‍e-λtt1-(1/α)dt×∫0T+η‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+∫0∞‍e-λtdt×r1-(1/α)×∫T+ηT‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds]≤C[(r+1λ+Γ(1-(1/α))λ1-(1/α))×∫0T‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+(rλ+Γ(2-(1/α))λ2-(1/α))×∫-r0‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+Γ(2-(1/α))λ2-(1/α)×∫0T+η‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+r1-(1/α)λ∫T+ηT‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds]≤C1λK1(∫0T‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds+∫-r0‍e-λssupy∈ℝE|u(s,y)-v(s,y)|2ds)=C1λK1|u-v|λ2,
where C1=2C·max{r+1,Γ(1-(1/α)),Γ(2-(1/α)),r1-(1/α)} and K1=1-(1/α).
Letλ that large enough such that
(28)C1λK1<1,
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, Theorem3 can be solved in the following working function space:
(29)|u|λ2=∫0T‍e-λtsupx∈ℝE|u(t,x)|2dt,
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 someγ<1/2(30)supx∈ℝE(|u0(x+z)-u0(x)|2)<c|z|2γ;(H4)
forθ>0, let |ϕ(η,x)|<∞ and there exists some π<1 such that
(31)supx∈ℝ∫-r-θ‍E|u(η+θ,x)-u(η,x)|2dη<c|θ|2π;(H5)
|f(t1,y,ut1)-f(t2,y,ut2)|2≤c(|t1-t2|2+|u(t1)-u(t2)|2+∫-r0‍|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|Gα(1+v,z)-Gα(v,z)|ndzdv<∞.
(
2
) For (α+1)/2<n<α+1, ∫0∞‍∫ℝ‍|Gα(v,z+1)-Gα(v,z)|ndzdv<∞.Theorem 7.
Assume that the conditions (H1)–(H5) are satisfied; then forα∈(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.
Fort≥0, it follows that, for any x∈ℝ and θ>0,
(32)|u(t+θ,x)-u(t,x)|≤|∫ℝ‍(Gα(t+θ,x-y)-Gα(t,x-y))u0(y)dy|+|∫0t+θ‍∫ℝ‍(Gα(t+θ-s,x-y)-Gα(t-s,x-y))f(us)dyds∫0t+θ‍|+|∫0t‍∫ℝ‍(Gα(t+θ-s,x-y)-Gα(t-s,x-y))h(us)W(dsdy)∫0t‍|+|∫tt+θ‍∫ℝ‍Gα(t+θ-s,x-y)h(us)W(dsdy)|+|∫0t‍∬ℝ(Gα(t+θ-s,x-y)-Gα(t-s,x-y))×g(us)zN(ds,dz)dy∫0t‍|+|∫tt+θ‍∬ℝGα(t+θ-s,x-y)g(us)zN(ds,dz)dy|=ϕθ0+ϕθ1+ϕθ2+ϕθ3+ϕθ4+ϕθ5.
Next, we will estimate each term ϕθj  (j=0,1,…,5), respectively.
Combining Hölder inequality, Lemma1 with (H3) yields
(33)E|ϕθ0|2=E|∫ℝ‍Gα(θ,z)[∫ℝ‍Gα(t,x-y)×(u0(y-z)-u0(y))dy∫ℝ‍]dz|2≤CE(∫ℝ‍Gα(θ,z)[∫ℝ‍Gα(t,x-y)×(u0(y-z)-u0(y))dy∫ℝ‍]2dz)×(∫ℝ‍Gα(θ,z)dz)≤C∫ℝ‍Gα(θ,z)supy∈ℝE|u0(y-z)-u0(y)|2dz≤C∫ℝ‍Gα(θ,z)|z|2γdz=C∫ℝ‍θ2γ/αGα(1,z)z2γdz≤Cθ2γ/α∫ℝ‍|z|2γ1+|z|1+αdz≤Cθ2γ/α.
Next, we consider ϕθ1. Let s′=s-θ; then,
(34)|ϕθ1|≤|∫0t‍∫ℝ‍Gα(t-s,x-y)×(f(s+θ,y,us+θ)-f(s,y,us))dyds|+|∫0θ‍∫ℝ‍Gα(t+θ-s,x-y)×f(s,y,us)dyds∫0θ‍|=ϕθ1.1+ϕθ1.2.
By Hölder inequality, Lemma 1, (H4), and (H5), we have
(35)E|ϕθ1.1|2≤C∫0t‍∫ℝ‍Gα(t-s,x-y)dyds×∫0t‍∫ℝ‍Gα(t-s,x-y)×E|f(s+θ,y,us+θ)-f(s,y,us)|2dyds≤C∫0t‍(θ2+supy∈ℝE|u(s+θ,y)-u(s,y)|2+∫-r0‍supy∈ℝE|u(s+θ+η,y)-u(s+η,y)|2dη∫0t‍)ds≤C(∫-θ0‍Tθ2+(r+1)×∫0t‍supy∈ℝE|u(s+θ,y)-u(s,y)|2ds+r∫-r-θ‍supy∈ℝE|u(s+θ,y)-u(s,y)|2ds+∫-θ0‍supy∈ℝE|u(s+θ,y)-u(s,y)|2ds)≤C(∫0t‍θ2+θ2π+θ+∫0t‍supy∈ℝE|u(s+θ,y)-u(s,y)|2ds),E|ϕθ1.2|2≤C∫0θ‍∫ℝ‍Gα(t+θ-s,x-y)dyds×∫0θ‍∫ℝ‍Gα(t+θ-s,x-y)E|f(us)|2dyds≤Cθ∫0θ‍∫ℝ‍Gα(t+θ-s,x-y)E|f(us)|2dyds≤Cθ∫0θ‍supy∈ℝE(|u(s,y)|2+∫-r0‍|u(s+η,y)|2dη)ds≤Cθ((r+1)∫0θ‍supy∈ℝE|u(s,y)|2ds+r∫-r0‍supy∈ℝE|u(s,y)|2ds)≤C(θ2+θ).
Taking the transformations=θv, y=θ1/αz, by Burkholder-Davis-Gundy inequality, Lemmas 1 and 6, and (H2), we obtain
(36)E|ϕθ2|2≤C∫0t‍∫ℝ‍E(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2h2(us)dsdy≤CK∫0t‍∫ℝ‍(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2×E(|u(s)|2+∫-r0‍|u(s+η)|2dη)dyds≤C∫0t‍∫ℝ‍(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2×E(|u(s)|2+∫-r0‍|u(η)|2dη+∫0t‍|u(η′)|2dη′)dyds≤C((T+1)sup[0,T]×ℝE|u(s,y)|2+supy∈ℝ∫-r0‍|u(η,y)|2dη)×∫0t‍∫ℝ‍(Gα(s+θ,y)-Gα(s,y))2dyds≤C(θ-1/αθ∫0∞‍∫ℝ‍(Gα(v+1,z)-Gα(v,z))2dzdv)≤Cθ1-(1/α),E|ϕθ3|2≤C∫tt+θ‍∫ℝ‍E(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2|h(us)|2dsdy≤C∫tt+θ‍∫ℝ‍E(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2×(|u(s,y)|2+∫-r0‍|u(s+η)|2dη)dyds≤C∫tt+θ‍∫ℝ‍E(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2×(|u(s,y)|2+∫-rt+θ‍|u(η′)|2dη′)dyds≤C((T+1+θ)sup[0,T+θ]×ℝE|u(s,y)|2+supy∈ℝ∫-r0‍|u(η,y)|2dη)×∫0θ‍∫ℝ‍E(Gα(s+θ,y)-Gα(s,y))2dyds≤C(1+θ)θ1-(1/α)×∫01‍∫ℝ‍E(Gα(1+v,z)-Gα(v,z))2dzdv≤C(θ1-(1/α)+θ2-(1/α)).
Then, by the same method, we have
(37)E|ϕθ4|2≤∫0t‍∫ℝ‍(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2×E(∫ℝ‍|g(us)||z|ν(dz))2dsdy≤CK∫0t‍∫ℝ‍(Gα(t+θ-s,x-y)-Gα(t-s,x-y))2×E(|u(s)|2+∫-r0‍|u(s+η)|2dη)dyds≤Cθ1-(1/α),E|ϕθ5|2≤C(θ1-(1/α)+θ2-(1/α)).
Thus, from the previous estimates, let β=min{γ/α,π,(1/2)-(1/2α)}:
(38)E|u(t+θ,x)-u(t,x)|2≤C[θ2β+∫0t‍supy∈ℝE|u(s+θ,y)-u(s,y)|2ds].
Hence, it follows from Gronwall’s Lemma that
(39)E|u(t+θ,x)-u(t,x)|2≤Cθ2β.
Then, for t≥0, we have
(40)|u(t+θ,x)-u(t,x)|λp=(∫0T‍e-λtsupx∈ℝE|u(t+θ,x)-u(t,x)|2ds)p/2≤Cθβp.Finally, we study the space regularity of the mild solution for (8).Theorem 8.
Assume that the conditions (H1)–(H3) are satisfied; then forα∈(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 anyt∈[0,T] and ζ>0,
(41)|u(t,x+ζ)-u(t,x)|≤|∫ℝ‍(Gα(t,x+ζ-y)-Gα(t,x-y))u0(y)dy|+|∫0t‍∫ℝ‍(Gα(t-s,x+ζ-y)-Gα(t-s,x-y))f(us)dyds∫0t‍|+|∫0t‍∫ℝ‍(Gα(t-s,x+ζ-y)-Gα(t-s,x-y))h(us)W(dsdy)∫0t‍|+|∫0t‍∬ℝ(Gα(t-s,x+ζ-y)-Gα(t-s,x-y))g(us)zN(ds,dz)dy∫0t‍|∶=∑j=03‍ϕζj.
By (H3) and Lemma 1, we have
(42)E|ϕζ0|2=E|∫ℝ‍Gα(t,x-y)[u0(y+ζ)-u0(y)]dy|2≤supy∈ℝE(|u0(y+ζ)-u0(y)|2)×(∫ℝ‍Gα(t,x-y)dy)2≤Cζ2γ.
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
(43)E|ϕζ1|2≤C((1+T)sup[0,T]×ℝE|u(t,x)|2+supy∈ℝ∫-r0‍|u(η,y)|2dη)×|∫0t‍∫ℝ‍(Gα(s,y+ζ)-Gα(s,y))dyds|2≤C|∫0t‍∫ℝ‍s-1/α(Gα(1,s-1/α(y+ζ))-Gα(1,s-1/αy))dyds∫0t‍|2=C|∫0t‍∫ℝ‍s-ϵ/α(Gα(1,s-1/α(y+ζ))-Gα(1,s-1/αy))(1-ϑ)×s-((1-ϵ)/α)(Gα(1,s-1/α(y+ζ))-Gα(1,s-1/αy))ϑdyds|2≤C(∫0t‍∫ℝ‍s-2ϵ/α|Gα(1,s-1/α(y+ζ))-Gα(1,s-1/αy)|2(1-ϑ)dyds)×(∫0t‍∫ℝ‍s-2(1-ϵ)/α|Gα(1,s-1/α(y+ζ))-Gα(1,s-1/αy)|2ϑdyds)∶=cI×II.
By Lemma 1, recall that 2ϵ<α+1, and we have
(44)I≤C(∫0t‍∫ℝ‍s-(2ϵ-1)/α(Gα(1,y))2(1-ϑ)dyds)≤C∫0t‍s-(2ϵ-1)/αds<∞.
Then, by the mean value of the theorem and Lemma 1, we have that
(45)II=∫0t‍∫ℝ‍s-2(1-ϵ)/α(∂∂yGα(1,s-1/α(y+ε))×s-1/αζ)2ϑdyds(ε∈(0,ζ))=ζ2ϑ∫0t‍∫ℝ‍s-(2(1-ϵ+ϑ)-1)/α×(∂∂yGα(1,y))2ϑdyds≤Cζ2ϑ∫0t‍s-(2(1-ϵ+ϑ)-1)/αds.
Choosing ϑ<α-δ such that (2(1-ϵ+ϑ)-1)/α<1, then
(46)II<Cζ2ϑ.
Combining (44) and (46), we have
(47)E|ϕζ1|2≤Cζ2ϑ.
Taking the change of variable s=ζαv,  y=zζ, and by Burkholder-Davis-Gundy inequality, Lemmas 1 and 6, (H1), and (H2), we derive
(48)E|ϕζ2|2≤(∫0t‍∫ℝ‍E(∫0t‍Gα(t-s,x+ζ-y)-Gα(t-s,x-y)∫0t‍)2×E|h(us)|2dyds∫0t‍)≤C((T+1)sup[0,T]×ℝE|u(t,x)|2+supy∈ℝ∫-r0‍|u(η,y)|2dη)×(∫0t‍∫ℝ‍(Gα(s,y+ζ)-Gα(s,y))2dyds)≤C(ζ-2ζζα∫0∞‍∫ℝ‍(Gα(v,z+1)-Gα(v,z))2dzdv∫0∞‍)≤Cζ2(α-1),E|ϕζ3|2≤C∫0t‍∫ℝ‍(Gα(t-s,x+ζ-y)-Gα(t-s,x-y))2×E(∫ℝ‍|g(us)|zν(dz))2dsdy≤Cζ2(α-1).
Combining (42)–(48), we have
(49)E|u(t,x+ζ)-u(t,x)|2≤C(ζ2γ+ζ2ϑ+ζ2(α-1)).
Then, we have, for t∈[0,T],
(50)|u(t,x+ζ)-u(t,x)|λp≤C(ζγp+ζϑp+ζ(α-1)p)≤Cζρp,
where ρ=min{γ,ϑ,α-1}.Remark 9.
Theorems7 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.



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