We study the Vladimirov fractional differentiation operator \( {D}_N^{\alpha },\alpha >0,N\in \mathbb{Z}, \) on a p-adic ball BN = {x ∈ ℚp : |x|p ≤ pN}. To its known interpretations via the restriction of a similar operator to ℚp and via a certain stochastic process on BN, we add an interpretation in the form of a pseudodifferential operator in terms of the Pontryagin duality on the additive group of BN. We investigate the Green function of \( {D}_N^{\alpha } \) and a nonlinear equation on BN, an analog of the classical equation of porous medium.
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References
S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, “Theory of p-adic distributions,” Linear and Nonlinear Models, Cambridge Univ. Press (2010).
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York (2010).
M. Bonforte and J. L. Vázquez, “Fractional nonlinear degenerate diffusion equations on bounded domains,” Nonlin. Anal., 131, 363–398 (2016).
N. Bourbaki, Elements of Mathematics. Integration II, Springer, Berlin (2004).
H. Brézis, Functional Analysis, Sobolev Spaces, and Partial Differential Equations, Springer, New York (2011).
H. Brézis and W. Strauss, “Semilinear elliptic equations in L 1 ,” J. Math. Soc. Japan, 25, 15–26 (1973).
F. Bruhat, “Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes p-adiques,” Bull. Soc. Math. France, 89, 43–75 (1961).
O. F. Casas-Sánchez and J. J. Rodríguez-Vega, “Parabolic-type equations on p-adic balls,” Bol. Mat., 22, 97–106 (2015).
Ph. Clément, et al., One-Parameter Semigroups, North-Holland, Amsterdam (1987).
M. Crandall and M. Pierre, Regularizing effects for u t + A𝜓 (u) = 0 in L 1 ,” J. Funct. Anal., 45, 194–212 (1982).
A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, Amer. Math. Soc., Providence, RI (2006).
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis,), Vol. 2, Springer, Berlin (1979.
A. Khrennikov and A. N. Kochubei, p-Adic analogue of the porous medium equation,” J. Fourier Anal. Appl. (to appear), arXiv: 1611.08863.
A. Khrennikov, K. Oleschko, and M. J. Correa Lopez, “Application of p-adic wavelets to model reaction-diffusion dynamics in random porous media,” J. Fourier Anal. Appl., 22, 809–822 (2016).
A. Khrennikov, K. Oleschko, and M. J. Correa Lopez, “Modeling fluid’s dynamics with master equations in ultrametric spaces representing the treelike structure of capillary networks,” Entropy, 18, Art. 249 (2016), 28 p.
A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Marcel Dekker, New York (2001).
E. H. Lieb and M. Loss, Analysis, Amer. Math. Soc., Providence, RI (2001).
S. A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Cambridge Univ. Press (1977).
V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore (1994).
V. S. Vladimirov, Tables of Integrals of Complex-Valued Functions of p-Adic Arguments [in Russian], Steklov Math. Inst., Moscow (2003); English version: ArXiv: math-ph/9911027.
W. A. Zúñiga-Galindo, “Pseudodifferential equations over non-Archimedean spaces,” Lect. Notes Math., 2174 (2016).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 2, pp. 193–205, February, 2018.
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Kochubei, A.N. Linear and Nonlinear Heat Equations on a p-Adic Ball. Ukr Math J 70, 217–231 (2018). https://doi.org/10.1007/s11253-018-1496-x
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DOI: https://doi.org/10.1007/s11253-018-1496-x