Existence and uniqueness for $p$-adic counterpart of the porous medium equation

TL;DR

This paper establishes existence and uniqueness of solutions for a $p$-adic analogue of the fractional porous medium equation, addressing unique topological challenges using Pontryagin duality and fractional Sobolev spaces.

Contribution

It introduces a novel framework for nonlinear evolution equations over $p$-adic fields, including the construction of fractional Sobolev spaces and a nonlinear semigroup.

Findings

Proved existence of generalized solutions

Established uniqueness of solutions

Constructed a nonlinear semigroup for the equation

Abstract

We develop a theory of generalized solutions of the nonlinear evolution equations for complex-valued functions of a real positive time variable and $p$-adic spatial variable, which can be seen as non-Archimedean counterparts of the fractional porous medium equation. In this case, we face the problem that a $p$-adic ball is simultaneously open and closed, thus having an empty boundary. To address this issue, we use the algebraic structure of the field of $p$-adic numbers and apply the Pontryagin duality theory to construct the appropriate fractional Sobolev type spaces. We prove the existence and uniqueness results for the corresponding nonlinear equation and define an associated nonlinear semigroup.