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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 635690, 4 pages
-Adic Fractional Pseudodifferential Equations and Sobolev Type Spaces over -Adic Fields
Department of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China
Received 23 April 2013; Accepted 13 June 2013
Academic Editor: Thabet Abdeljawad
Copyright © 2013 Bo Wu. 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.
In this paper, we study the solutions of the pseudodifferential equations of type over -adic field , where is a -adic fractional pseudodifferential operator. If is a Bruhat-Schwartz function, then there exists a distribution , a fundamental solution, such that is a solution. We also show that the solution belongs to a certain Sobolev space. Furthermore, we give conditions for the continuity and uniqueness of .
In recent years, -adic analysis has received a lot of attention due to its applications in mathematical physics; see, for example, [1–11] and references therein. As a consequence, new mathematical problems have emerged, among them, the study of -adic pseudodifferential equations; see, for example, [10–16] and references therein. In this paper, we study the solutions of -adic fractional pseudodifferential equations on the Sobolev type spaces.
A -adic fractional pseudodifferential operator , which is defined by Su in 1992 , is an operator of the form for , where and denote the Fourier transform and the inverse Fourier transform, denotes the -vector space of the Bruhat-Schwartz functions over -adic fields , and is a positive real number. A -adic fractional pseudodifferential equation is an equation of type If , then there exists a distribution , a fundamental solution, such that is a solution.
We will also show that (2) has a solution belonging to a certain Sobolev space. Furthermore, we give conditions for the continuity and uniqueness of .
We use the notations indicated in Taibleson’s book . Let us fix a prime number . The field of -adic numbers is defined as the completion of the field of rational numbers with respect to -adic norm , which is defined as follows: ; if an arbitrary rational number is represented as , where and integers and are not divisible by .
The -adic norm satisfies the strong triangle inequality .
Any -adic number can be uniquely presented as series convergent in -adic norm (the canonical presentation of ).
Define the bitwise operation of addition and multiplication of in (either from left to right carry, or not carry), then is a locally compact, nondiscrete, complete, and totally disconnected topological field.
Denote by the ring of integers in , . Let denote the Haar measure on normalized by the condition . Denote by respectively, the ball and the sphere of radius with the center at center . Obviously, , .
A complex-valued function defined on is called locally constant if for any there exists an integer satisfying Denote by the linear space of all the locally constant functions. is defined as the linear space of all locally constant functions with compact support in .
The convergence of the point in has the following definition: , if and only if for any compact subset , , hold uniformly on . The convergence in has the following meaning: , if and only if there exist the indices and which do not depend on , such that the functions with supports in the ball and with constant on the coset of , , hold uniformly in . Then, and are complete topological linear spaces. Also, denote by the Bruhat-Schwartz function space.
Denote by the distribution space of Bruhat-Schwartz function space . is a complete topological linear space under the dual topology. The convergence of the point in has the following definition: if and only if , hold for any .
The Fourier transform of is defined by the following formula: and the inverse Fourier transform by where is an additive character of the field , with value in and . The Fourier transform and inverse Fourier transform map from into .
The Fourier transform of a distribution is defined by the standard relation , .
In 1992, Su  has given definitions of the derivative for the -adic local fields , including derivatives of the fractional orders and real orders.
Let and . Its role is played by the operator of pseudodifferential operator which is defined as for . It is easy to see that . With the defined domain of can be extended to the space . Thus, we also have with and .
Definition 1. If , then is defined as -adic derivative of the order on . And if , then is defined as -adic integral of the order on . If , for any , then is called the identity operator.
In [19, 20], Qiu and Su constructed the convolution kernel of the operator . Consider where
Here, is the indicative function of set satisfying , and is a distribution defined as
3. Solutions of the Pseudodifferential Equations on Sobolev Type Spaces
We now consider the following pseudodifferential equation: We say that is a fundamental solution of (13) if is a solution.
Lemma 3. If is a fundamental solution of (13), then for any constant , is also a fundamental solution.
Proof. Let be a fundamental solution for (13), then because and the constant function, , are in the domain of .
Theorem 4. A fundamental solution of (13) is
Proof. We use the definition of (8), then
The existence of a fundamental solution is equivalent to the existence of a distribution satisfying
as distributions. By Lemma 2, we have
Then, . By (11), Theorem 4 is proved.
In what follows we will introduce some relevant spaces over -adic fields (see ).
The Holder type space , :(1)for , we define as the continuous function space on ;(2)for , we define as the set of all distributions with the Littlewood-Paley decomposition with respect to the following norm: Thus, becomes a Banach space with the above norm.
The Sobolev type space , : where denotes the set of measurable functions on satisfying the condition .
Lemma 6. For any and , the mapping is a well-defined continuous mapping between Banach spaces.
Proof. Let . We have that The result follows from the fact that is dense in .
Theorem 7. Let be a -adic fractional pseudodifferential operator. Let be a positive real number satisfying . Then, the equation has a unique uniformly continuous solution .
Proof. Let , as is dense in and we have
In this way, we established the existence of . By Lemma 5, is uniformly continuous for .
Finally, we show that is unique. Indeed, if , then And, thus, . Then, almost everywhere, and a fortiori almost everywhere, and by the continuity of , for any . Theorem 7 is proved.
In this work, the pseudodifferential equations of type over -adic field were investigated, where is a -adic fractional pseudodifferential operator. The fundamental solution of the equation was obtained. And the continuity and uniqueness of solution which belongs to the Sobolev type space were obtained when , using the explicit calculation of fundamental solutions approach.
The author would like to thank the Academic Editor Thabet Abdeljawad and all anonymous reviewers for their kind support which helped the author to improve the paper considerably. This work was supported by the National Natural Science Foundation of China under NSFC nos. 11071109 and 11001119 and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
- S. Albeverio and W. Karwoski, “Diffusion in p-adic numbers,” in Gaussian Random Fields, K. Ito and H. Hida, Eds., pp. 86–99, World Scientific, Singapore, 1991.
- S. Albeverio and W. Karwowski, “A random walk on p-adics: the generator and its spectrum,” Stochastic Processes and Their Applications, vol. 53, no. 1, pp. 1–22, 1994.
- S. Albeverio and P. Kurasov, “Singular perturbations of differential operators,” in Solvable Schrodinger Type Operators, vol. 271 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 2000.
- S. Albeverio, A. Y. Khrennikov, and V. M. Shelkovich, “Harmonic analysis in the p-adic Lizorkin spaces: fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems,” The Journal of Fourier Analysis and Applications, vol. 12, no. 4, pp. 393–425, 2006.
- S. Albeverio and S. Kuzhel, “One-dimensional Schrödinger operators with p-symmetric zero-range potentials,” Journal of Physics A, vol. 38, no. 22, pp. 4975–4988, 2005.
- A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics, vol. 309, Kluwer Academic, Dordrecht, The Netherlands, 1994.
- A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic, Dordrecht, The Netherlands, 1997.
- V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev, and V. A. Osipov, “p-adic models of ultrametric diffusion constrained by hierarchical energy landscapes,” Journal of Physics A, vol. 35, no. 2, pp. 177–189, 2002.
- V. A. Avetisov, A. Kh. Bikulov, and V. Al. Osipov, “p-adic description of characteristic relaxation in complex systems,” Journal of Physics A, vol. 36, no. 15, pp. 4239–4246, 2003.
- A. N. Kochubeĭ, “Parabolic equations over the field of p-adic numbers,” Mathematics of the USSR-Izvestiya, vol. 39, pp. 1263–1280, 1992.
- A. N. Kochubeĭ, “Fundamental solutions of pseudodifferential equations associated with p-adic quadratic forms,” Izvestiya, vol. 62, pp. 1169–1188, 1998.
- A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Marcel Dekker, New York, NY, USA, 2001.
- A. N. Kochubei, “A non-Archimedean wave equation,” Pacific Journal of Mathematics, vol. 235, no. 2, pp. 245–261, 2008.
- N. M. Chuong and N. V. Co, “The Cauchy problem for a class of pseudodifferential equations over p-adic field,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 629–645, 2008.
- V. S. Varadarajan, “Path integrals for a class of p-adic Schrödinger equations,” Letters in Mathematical Physics, vol. 39, no. 2, pp. 97–106, 1997.
- V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994.
- W. Y. Su, “Pseudo-differential operators and derivatives on locally compact Vilenkin groups,” Science in China A, vol. 35, no. 7, pp. 826–836, 1992.
- M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ, USA, 1975.
- H. Qiu and W. Y. Su, “Pseudo-differential operators over p-adic fields,” Science in China A, vol. 41, no. 4, pp. 323–336, 2011.
- W. Y. Su and H. Qiu, “p-adic calculus and its applications to fractal analysis and medical science,” Facta Universitatis, vol. 1, no. 3, pp. 339–348, 2008.
- W. Su and Q. Xu, “Function spaces on local fields,” Science in China A, vol. 49, no. 1, pp. 66–74, 2006.
- W. Y. Su, Harmonic Analysis and Fractal Analysis and Its Application on Local Fields, Science Press, Beijing, China, 2011.