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

## TL;DR

This paper extends the study of fractional differentiation operators on non-Archimedean fields to nonlinear equations, establishing conditions for their local and global solvability, thus creating a non-Archimedean analogue of classical differential equations.

## Contribution

It introduces and analyzes nonlinear pseudo-differential equations in non-Archimedean settings, providing solvability conditions and expanding the framework beyond linear cases.

## Key findings

- Established local and global solvability conditions for nonlinear equations
- Extended the non-Archimedean fractional calculus to nonlinear contexts
- Demonstrated the existence of solutions under specific conditions

## 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 $D^\alpha$, $\alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^\alpha$ that the change of an unknown function $u=I^\alpha v$ reduces the Cauchy problem for a linear equation with $D^\alpha$ (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.