Linear and Nonlinear Heat Equations on a p-Adic Ball

TL;DR

This paper explores the properties of the Vladimirov fractional differentiation operator on p-adic balls, providing new interpretations and analyzing solutions to linear and nonlinear heat equations in this setting.

Contribution

It introduces a novel interpretation of the fractional operator as a pseudo-differential operator using Pontryagin duality and studies the Green function and nonlinear equations on p-adic balls.

Findings

Derived the Green function for the fractional operator

Provided a new interpretation via Pontryagin duality

Analyzed nonlinear heat equations on p-adic domains

Abstract

We study the Vladimirov fractional differentiation operator $D^\alpha_N$, $\alpha >0, N\in \mathbb Z$, on a $p$-adic ball $B_N=\{ x\in \mathbb Q_p:\ |x|_p\le p^N\}$. To its known interpretations via restriction from a similar operator on $\mathbb Q_p$ and via a certain stochastic process on $B_N$, we add an interpretation as a pseudo-differential operator in terms of the Pontryagin duality on the additive group of $B_N$. We investigate the Green function of $D^\alpha_N$ and a nonlinear equation on $B_N$, an analog the classical porous medium equation.