p-Adic Analogue of the Porous Medium Equation

TL;DR

This paper introduces a p-adic analogue of the porous medium equation, establishing an $L^1$-theory for Vladimirov's fractional operator, proving existence and uniqueness of solutions, and providing an explicit example.

Contribution

It develops a novel $L^1$-theory for p-adic fractional differentiation and proves well-posedness of the p-adic porous medium equation, a non-Archimedean counterpart.

Findings

Proved m-accretivity of the nonlinear operator

Established existence and uniqueness of mild solutions

Provided an explicit solution example

Abstract

We consider a nonlinear evolution equation for complex-valued functions of a real positive time variable and a p-adic spatial variable. This equation is a non-Archimedean counterpart of the fractional porous medium equation. Developing, as a tool, an $L^1$-theory of Vladimirov's p-adic fractional differentiation operator, we prove m-accretivity of the appropriate nonlinear operator, thus obtaining the existence and uniqueness of a mild solution. We give also an example of an explicit solution of the p-adic porous medium equation.