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Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables

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Abstract

We study the Cauchy problem for p-adic non-linear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.

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References

  1. 1.

    Albeverio, S., Khrennikov, AYu., Shelkovich, V.M.: Theory of p-Adic Distributions. Linear and Nonlinear Models. Cambridge University Press, Cambridge (2010)

  2. 2.

    Avetisov, V.A., Bikulov, A.H., Kozyrev, S.V.: Application of p-adic analysis to models of spontaneous breaking of replica symmetry. J. Phys. A 32(50), 8785–8791 (1999)

  3. 3.

    Avetisov, V.A., Bikulov, A.H., Kozyrev, S.V., Osipov, V.A.: p-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes. J. Phys. A 35(2), 177–189 (2002)

  4. 4.

    Barbu, V.: Nonlinear Differential Equations of Monotone Type in Banach Spaces. Springer, New York (2010)

  5. 5.

    Bendikov, A.D., Grigor’yan, A.A., Pittet, Ch., Woess, W.: Isotropic Markov semigroups on ultra-metric spaces. Russ. Math. Surv. 69, 589–680 (2014)

  6. 6.

    Brézis, H., Strauss, W.: Semilinear elliptic equations in \(L^1\). J. Math. Soc. Jpn. 25, 15–26 (1973)

  7. 7.

    Crandall, M., Pierre, M.: Regularizing effects for \(u_t+A\psi (u)=0\) in \(L^1\). J. \({\tilde{{\rm F}}}\)unct. Anal. 45, 194–212 (1982)

  8. 8.

    Dynkin, E.B.: Markov Processes, vol. I. Springer, Berlin (1965)

  9. 9.

    Edwards, R.E.: Functional Analysis. Theory and Applications. Holt, Rinehart and Winston, New York (1965)

  10. 10.

    Evans, S.N.: Local properties of Lévy processes on a totally disconnected group. J. Theor. Probab. 2(2), 209–259 (1989)

  11. 11.

    Fischenko, S., Zelenov, E.: p-Adic models of turbulence. In: Branko Dragovich, Zoran Rakic (Eds.), \(p\)-Adic Mathematical Physics, 2nd International Conference, Belgrade, Serbia and Montenegro, 15–21 September 2005, In: AIP Conf. Proc., Vol. 826, Melville, New York, pp. 174–191 (2006)

  12. 12.

    Gihman, I.I., Skorohod, A.V.: The Theory of Stochastic Processes, in 3 vol. Springer (1979)

  13. 13.

    Gelfand, I.M., Graev, M.I., Piatetskii-Shapiro, I.I.: Generalized Functions. Vol 6: Representation Theory and Automorphic Functions. Nauka, Moscow, 1966. Translated from the Russian by K.A. Hirsch, Published in 1990, Academic Press (Boston)

  14. 14.

    Heyer, H.: Probability Measures on Locally Compact Groups. Springer, Berlin (1977)

  15. 15.

    Khrennikov, A.: p-Adic Valued Distributions in Mathematical Physics. Kluwer Academic Publ, Dordrecht (1994)

  16. 16.

    Khrennikov, A.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publisher, Dordrecht (1997)

  17. 17.

    Kochubei, A.N.: Pseudo-Differential Equations and Stochastics Over Non-Archimedean Fields. Marcel Dekker, New York (2001)

  18. 18.

    Khrennikov, A., Kochubei, A.N.: \(p\)-Adic analogue of the porous medium equation. J. Fourier Anal. Appl. 24, 1401–1424 (2018)

  19. 19.

    Kozyrev, S.V.: p-Adic pseudodifferential operators: methods and applications. Proc. Steklov Inst. Math. 245, 154–165 (2004)

  20. 20.

    Kozyrev, S.V.: Methods and Applications of Ultrametric and p-Adic Analysis: From Wavelet Theory to Biophysics, Sovrem. Probl. Mat., Vol. 12, Steklov Inst. Math. Moscow (2008). http://www.mi.ras.ru/spm/pdf/012.pdf

  21. 21.

    [20] Kozyrev, S.V.: Towards ultrametric theory of turbulence. Teoret. Mat. Fiz. 157(3), 413–424 (2008); translation in: Theoret. Math. Phys. 157(3), 1713–1722 (2008)

  22. 22.

    Parthasarathy, K.R.: Propbability Measures on Metric Spaces. Acac. Press, New York and London (1967)

  23. 23.

    Taibleson, M.H.: Fourier Analysis on Local Fields, Mathematical Notes. Princeton University Press, Princeton (1975)

  24. 24.

    Vladimirov, V.S.: Tables of integrals of complex values functions of \(p\)-adic arguments

  25. 25.

    Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-Adic Analysis and Mathematical Physics, p. xx+319. World Scientific Publishing Co. Inc, River Edge (1994)

  26. 26.

    Zùñiga-Galindo, W.A.: Pseudodifferential Equations over Non-Archimedean Spaces, Lect. Notes Math. Vol. 2174 (2016), XVI+175 p

  27. 27.

    Zùñiga-Galindo, W.A.: Non-Archimedean reaction-ultradiffusion equations and complex hierarchic systems. Nonlinearity 31(6), 2590–2616 (2018)

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Acknowledgements

The work by the first- and third-named authors was funded in part under the budget program of Ukraine No. 6541230 “Support to the development of priority research trends”. The third-named author was also supported in part in the framework of the research work “Markov evolutions in real and p-adic spaces” of the Dragomanov National Pedagogical University of Ukraine.

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Correspondence to Anatoly N. Kochubei.

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Antoniouk, A.V., Khrennikov, A.Y. & Kochubei, A.N. Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables. J. Pseudo-Differ. Oper. Appl. 11, 311–343 (2020). https://doi.org/10.1007/s11868-019-00320-3

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Keywords

  • p-adic numbers
  • Porous medium equation
  • Markov process
  • m-accretive operator

Mathematics Subject Classification

  • Primary 35S10
  • 47J35
  • Secondary 11S80
  • 60J25
  • 76S05