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Izv. Akad. Nauk SSSR Ser. Mat., 1991, Volume 55, Issue 6, Pages 1312–1330 (Mi izv975)  

This article is cited in 29 scientific papers (total in 29 papers)

Parabolic equations over the field of $p$-adic numbers

A. N. Kochubei

Abstract: The author constructs and investigates a fundamental solution of Cauchy's problem for a parabolic equation with a $p$-adic space variable and a real time variable. The question of existence and uniqueness of solutions to Cauchy's problem in classes of bounded and increasing functions is considered, and conditions for nonnegativity of the fundamental solution are found. The problem of determining if the solution stabilizes as $t\to\infty$ is solved for a model equation with constant coefficients.

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English version:
Mathematics of the USSR-Izvestiya, 1992, 39:3, 1263–1280

Bibliographic databases:

UDC: 517.944+512.625.5
MSC: Primary 35K30; Secondary 11S80, 46F12, 35K05
Received: 27.02.1990

Citation: A. N. Kochubei, “Parabolic equations over the field of $p$-adic numbers”, Izv. Akad. Nauk SSSR Ser. Mat., 55:6 (1991), 1312–1330; Math. USSR-Izv., 39:3 (1992), 1263–1280

Citation in format AMSBIB
\by A.~N.~Kochubei
\paper Parabolic equations over the field of $p$-adic numbers
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1991
\vol 55
\issue 6
\pages 1312--1330
\jour Math. USSR-Izv.
\yr 1992
\vol 39
\issue 3
\pages 1263--1280

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    This publication is cited in the following articles:
    1. A. N. Kochubei, “The differentiation operator on subsets of the field of $p$-adic numbers”, Russian Acad. Sci. Izv. Math., 41:2 (1993), 289–305  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. N. Kochubei, “On $p$-adic Green's functions”, Theoret. and Math. Phys., 96:1 (1993), 854–865  mathnet  crossref  mathscinet  zmath  isi
    3. A. N. Kochubei, “Gaussian integrals and spectral theory over a local field”, Russian Acad. Sci. Izv. Math., 45:3 (1995), 495–503  mathnet  crossref  mathscinet  zmath  isi
    4. A. N. Kochubei, “Fundamental solutions of pseudodifferential equations connected with $p$-adic quadratic forms”, Izv. Math., 62:6 (1998), 1169–1188  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. A. Kh. Bikulov, “Stochastic $p$-adic equations of mathematical physics”, Theoret. and Math. Phys., 119:2 (1999), 594–604  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. V.S. Varadarajan, “Some remarks on arithmetic physics”, Journal of Statistical Planning and Inference, 103:1-2 (2002), 3  crossref
    7. V. S. Vladimirov, “Tables of Integrals of Complex-Valued Functions of $p$-Adic Arguments”, Proc. Steklov Inst. Math., 284, suppl. 2 (2014), S1–S59  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. Hiroshi Kaneko, Xuelei Zhao, “Stochastic processes on ℚ<sub>p</sub> induced by maps and recurrence criteria”, form, 16:1 (2004), 69  crossref  mathscinet  zmath  isi
    9. Proc. Steklov Inst. Math., 245 (2004), 105–116  mathnet  mathscinet  zmath
    10. S. V. Kozyrev, “$p$-Adic Pseudodifferential Operators: Methods and Applications”, Proc. Steklov Inst. Math., 245 (2004), 143–153  mathnet  mathscinet  zmath
    11. S. V. Kozyrev, “$p$-Adic Pseudodifferential Operators and $p$-Adic Wavelets”, Theoret. and Math. Phys., 138:3 (2004), 322–332  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Hiroshi Kaneko, Xuelei Zhao, “Transition Semi–groups on a Local Field Induced by Galois Group and their Representation”, J Theoret Probab, 19:1 (2006), 221  crossref  mathscinet  zmath  isi
    13. Hiroshi Kaneko, “Sobolev Space and Dirichlet Space Associated with Symmetric Markov Process on a Local Field”, Potential Anal, 24:1 (2006), 87  crossref  mathscinet  zmath  isi
    14. Hiroshi Kaneko, “(r, p)-capacity and Hausdorff measure on a local field”, Indagationes Mathematicae, 17:2 (2006), 251  crossref
    15. W. A. Zúñiga-Galindo, “Parabolic Equations and Markov Processes Over p-Adic Fields”, Potential Anal, 28:2 (2008), 185  crossref  mathscinet  isi  elib
    16. Chuong N.M., Van Co N., “The Cauchy problem for a class of pseudodifferential equations over p-adic field”, Journal of Mathematical Analysis and Applications, 340:1 (2008), 629–645  crossref  mathscinet  zmath  isi  elib
    17. S. V. Kozyrev, “Methods and Applications of Ultrametric and $p$-Adic Analysis: From Wavelet Theory to Biophysics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S1–S84  mathnet  crossref  crossref  zmath  isi  elib
    18. Chuong N.M., Van Co N., “P-adic pseudodifferential operators and wavelets”, Frames and Operator Theory in Analysis and Signal Processing, Contemporary Mathematics Series, 451, 2008, 33–45  crossref  isi
    19. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, “On p-adic mathematical physics”, P-Adic Num Ultrametr Anal Appl, 1:1 (2009), 1  crossref  mathscinet
    20. J. Galeano-Peñaloza, “On the regularity of solutions of p-adic parabolic equations”, P-Adic Num Ultrametr Anal Appl, 3:4 (2011), 288  crossref
    21. V. M. Shelkovich, “$p$-adic evolution pseudo-differential equations and $p$-adic wavelets”, Izv. Math., 75:6 (2011), 1249–1278  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    22. Hiroshi Kaneko, Yoichi Tsuzuki, “A lift of spatially inhomogeneous Markov process to extensions of the field of -adic numbers”, Stochastic Processes and their Applications, 121:2 (2011), 394  crossref
    23. Khrennikov A.Y., Kosyak A.V., Shelkovich V.M., “Wavelet Analysis on Adeles and Pseudo-Differential Operators”, J. Fourier Anal. Appl., 18:6 (2012), 1215–1264  crossref  isi
    24. Wu B., “P-Adic Fractional Pseudodifferential Equations and Sobolev Type Spaces Over P-Adic Fields”, Discrete Dyn. Nat. Soc., 2013, 635690  crossref  isi
    25. Bo Wu, “Existence and Continuity of Solutions to a Class of Pseudodifferential Equations over -Adic Field”, Mathematical Problems in Engineering, 2013 (2013), 1  crossref
    26. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196  mathnet  crossref  crossref  isi  elib  elib
    27. A. D. Bendikov, A. A. Grigor'yan, Ch. Pittet, W. Woess, “Isotropic Markov semigroups on ultra-metric spaces”, Russian Math. Surveys, 69:4 (2014), 589–680  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. O. M. Sizova, “Ultrametricheskaya diffuziya v silnom tsentralno-simmetrichnom pole”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 87–104  mathnet  crossref  zmath  elib
    29. Dragovich B. Khrennikov A.Yu. Kozyrev S.V. Volovich I.V. Zelenov E.I., “P-Adic Mathematical Physics: the First 30 Years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121  crossref  isi
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