About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 807459, 11 pages
http://dx.doi.org/10.1155/2013/807459
Research Article

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

Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received 27 June 2013; Accepted 18 August 2013

Academic Editor: Agacik Zafer

Copyright © 2013 Tianlong Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, 57] 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. Brzeniak 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: where is the fractional Laplacian operator with , the constants , are measurable, the function , and is the one-dimensional Lévy process (see Section 2 for the definition). Recall that reduces to be the Laplacian operator when .

In this paper, the existence, uniqueness, time regularity, and space regularity of the mild solution for (1) are shown for 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 be a complete probability space with filtration satisfying the usual condition. For one-dimensional Lévy process , it follows from Lévy-It decomposition that there exist a constant and a nonnegative constant , and a one-dimensional space-time white noise ( is a Brownian sheet on ) such that where where is the Lévy measure of .

Similar to [14], for any , we denote In what follows, we assume that Recalling that By absorbing into , we can rewrite (2) into the following equation: Let ; then (1) can be written as where is a one-dimensional space-time white noise and is a one-dimensional pure jump Lévy process with Lévy measure of . We suppose that generates a -martingale measure in the sense of Walsh [15].

The following assumptions are imposed to the initial data and , , , and to show the existence and uniqueness of the mild solution. (H1) The initial data which is -measurable and satisfy (H2) There exists a constant such that for all ,

Let the Green kennel be the fundamental solution of the Cauchy problem: where denotes the Dirac function. By Fourier transform, A higher order fractional Green kennel is introduced in [16].

The following lemma gives some useful properties about , 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 satisfies the following properties.(1)For any .(2)For , . (3)For any , . (4)For any , . (5)For any , there exists a constant such that

3. Existence of the Mild Solution

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

Let be a fixed positive time and the class of all -adapted càdlàg process satisfying Let be arbitrarily fixed; we define For any ,  . It is easy to verify that is a norm and is a Banach space.

Let and be given as in the previous section. Following the idea in [17], we represent a mild solution of (8) for .

Definition 2. An -adapted random field is said to be a mild solution of (8) with initial value satisfying (H1) if the following integral equation is fulfilled: where the stochastic integral with respect to is understood in the sense of that introduced by Walsh [15].

Theorem 3. For and , assume that (H1) and (H2) hold, then there exists a unique mild solution for (8).

Remark 4. In the following proof, 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 and denote where It follows from Hölder’s inequality, Lemma 1, (H1), and (H2) that
Applying Burkholder-Davis-Gundy inequality, Lemma 1, (H1), and (H2), we have Thus, combining (19) and (20) with (21), we derive Taking Laplace transform formula and (22), we deduce that that is, , which implies that operator .
Step 2. For any and , it follows from Hölder’s inequality, Lemma 1, and (H2) that By Burkholder-Davis-Gundy inequality, Lemma 1 and (H2), we have Thus, it follows that Finally, direct computation implies that where and .
Let that large enough such that which implies that the operator is contraction. By the Banach fixed point theorem, there exists a unique fixed point in . 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: where , 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 (H4) for , let and there exists some such that (H5).

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

Lemma 6. For , .
For , .

Theorem 7. Assume that the conditions (H1)–(H5) are satisfied; then for and , there exists a continuous modification , which is -Hölder continuous in , where .

Proof. For , it follows that, for any and , Next, we will estimate each term   (), respectively.
Combining Hölder inequality, Lemma 1 with (H3) yields Next, we consider . Let ; then, By Hölder inequality, Lemma 1, (H4), and (H5), we have
Taking the transformation , , by Burkholder-Davis-Gundy inequality, Lemmas 1 and 6, and (H2), we obtain Then, by the same method, we have Thus, from the previous estimates, let : Hence, it follows from Gronwall’s Lemma that Then, for , we have

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

Theorem 8. Assume that the conditions (H1)–(H3) are satisfied; then for and , there exists a continuous modification , which is -Hölder continuous in , where .

Proof. It follows that, for any and , By (H3) and Lemma 1, we have By (H2), Hölder’s inequality and Lemma 1, we set ( is small enough) and and we can derive By Lemma 1, recall that , and we have Then, by the mean value of the theorem and Lemma 1, we have that Choosing such that , then Combining (44) and (46), we have Taking the change of variable ,  , and by Burkholder-Davis-Gundy inequality, Lemmas 1 and 6, (H1), and (H2), we derive Combining (42)–(48), we have Then, we have, for , where .

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).

References

  1. C. Bardos, P. Penel, U. Frisch, and P.-L. Sulem, “Modified dissipativity for a nonlinear evolution equation arising in turbulence,” Archive for Rational Mechanics and Analysis, vol. 71, no. 3, pp. 237–256, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  2. L. Bo, K. Shi, and Y. Wang, “On a nonlocal stochastic Kuramoto-Sivashinsky equation with jumps,” Stochastics and Dynamics, vol. 7, no. 4, pp. 439–457, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  3. B. Guo, X. Pu, and F. Huang, Fractional Partial Differential Equations and Numerical Solution, Science Press, Beijing, China, 2011.
  4. J. Huang, Y. Li, and Y. Zheng, Introduction of Random Dynamical Systems, Science Press, Beijing, China, 2012.
  5. J. M. Angulo, M. D. Ruiz-Medina, V. V. Anh, and W. Grecksch, “Fractional diffusion and fractional heat equation,” Advances in Applied Probability, vol. 32, no. 4, pp. 1077–1099, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  6. P. Azerad and M. Mellouk, “On a stochastic partial differential equation with non-local diffusion,” Potential Analysis, vol. 27, no. 2, pp. 183–197, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  7. L. Debbi and M. Dozzi, “On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension,” Stochastic Processes and Their Applications, vol. 115, no. 11, pp. 1764–1781, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  8. A. Truman and J. Wu, Fractal Burgers Equations Driven by Lévy Noise, vol. 245 of Lecture Notes in Pure and Applied Mathematics, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2006.
  9. Z. Brze\'zniak and L. Debbi, “On stochastic Burgers equation driven by a fractional Laplacian and space-time white noise,” in Stochastic Differential Equations: Theory and Applications, vol. 2, pp. 135–167, World Scientific Publishing, Hackensack, NJ, USA, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Z. Brzeźniak, L. Debbi, and B. Goldys, “Ergodic properties of fractional stochastic Burgers equation,” Preprint, 2011, http://arxiv.org/pdf/1106.1918.pdf.
  11. T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes models with delays,” Journal of Differential Equations, vol. 205, no. 2, pp. 271–297, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  12. J. Wu, Theory and Applications of Partial Functional-Differential Equations, vol. 119 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  13. M. Niu and B. Xie, “Regularity of a fractional partial differential equation driven by space-time white noise,” Proceedings of the American Mathematical Society, vol. 138, no. 4, pp. 1479–1489, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  14. K. Kyeong-Hun and K. Panki, “An Lp-theory of stochastic parabolic equations with the random fractional Laplacian driven by Levy processes,” 2011, http://arxiv.org/pdf/1111.4712.pdf.
  15. J. B. Walsh, “An introduction to stochastic partial differential equations,” in École d'été de Probabilités de Saint-Flour, vol. 1180 of Lecture Notes in Mathematics, pp. 265–439, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  16. L. Debbi, “On some properties of a high order fractional differential operator which is not in general selfadjoint,” Applied Mathematical Sciences, vol. 1, no. 25-28, pp. 1325–1339, 2007. View at MathSciNet
  17. J. Walsh, An Introduction to Stochastic Partial Differential Equations, vol. 1180 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986.