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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2011 Volume 75, Issue 6, Pages 163–194 (Mi im4281)

This article is cited in 4 papers

$p$-adic evolution pseudo-differential equations and $p$-adic wavelets

V. M. Shelkovich

St. Petersburg State University of Architecture and Civil Engineering

Abstract: In the theory of $p$-adic evolution pseudo-differential equations (with time variable $t\in\mathbb{R}$ and space variable $x\in \mathbb{Q}_p^n$), we suggest a method of separation of variables (analogous to the classical Fourier method) which enables us to solve the Cauchy problems for a wide class of such equations. It reduces the solution of evolution pseudo-differential equations to that of ordinary differential equations with respect to the real variable $t$. Using this method, we solve the Cauchy problems for linear evolution pseudo-differential equations and systems of the first order in $t$, linear evolution pseudo-differential equations of the second and higher orders in $t$, and semilinear evolution pseudo-differential equations. We derive a stabilization condition for solutions of linear equations of the first and second orders as $t\to \infty$. Among the equations considered are analogues of the heat equation and linear or non-linear Schrödinger equations. The results obtained develop the theory of $p$-adic pseudo-differential equations and can be used in applications.

Keywords: $p$-adic pseudo-differential operator, $p$-adic fractional operator, $p$-adic wavelet bases, $p$-adic pseudo-differential equations.

UDC: 517.983.37+517.984.57+512.625.5

MSC: Primary 47G30, 42C40, 11F85; Secondary 26A33

Received: 31.12.2009
Revised: 12.07.2010

DOI: 10.4213/im4281


 English version:
Izvestiya: Mathematics, 2011, 75:6, 1249–1278

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