Nonlinear Pseudo-Differential Equations for Radial Real Functions on a Non-Archimedean Field

2019/07/25 by Kochubei, Anatoly N.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.11545

Abstract

In an earlier paper (A. N. Kochubei, \it Pacif. J. Math. 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator D^α, α>0, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse I^α that the change of an unknown function u=I^αv reduces the Cauchy problem for a linear equation with D^α (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we study nonlinear equations of this kind, find conditions of their local and global solvability.

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