Radial solutions of non-Archimedean pseudodifferential equations

Anatoly N. Kochubei

Vol. 269 (2014), No. 2, 355–369

DOI: 10.2140/pjm.2014.269.355

Abstract

We consider a class of equations with the fractional differentiation operator $D^{\alpha }$, $\alpha >0$, for complex-valued functions $x\mapsto f\left(|x{|}_K\right)$ on a non-Archimedean local field $K$ depending only on the absolute value $|\cdot {|}_K$. We introduce a right inverse $I^{\alpha }$ to $D^{\alpha }$, such that the change of an unknown function $u=I^{\alpha }v$ reduces the Cauchy problem for an equation with $D^{\alpha }$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. This contrasts much more complicated behavior of $D^{\alpha }$ on other classes of functions.

Keywords

fractional differentiation operator, non-Archimedean local field, radial functions, Cauchy problem

Mathematical Subject Classification 2010

Primary: 11S80, 35S10

Milestones

Received: 20 February 2013

Accepted: 3 July 2013

Published: 26 July 2014