Pseudo-Differential Equations with Weak Degeneration for Radial Functions of $p$-adic Argument

TL;DR

This paper extends the theory of pseudo-differential equations for radial functions over non-Archimedean fields to include weakly degenerate cases, establishing local and global solution properties.

Contribution

It introduces a new class of pseudo-differential equations with weak degeneration and proves existence, uniqueness, and regularity of solutions in this setting.

Findings

Proved local existence and uniqueness of solutions.

Established conditions for global extension of solutions.

Demonstrated regularity properties of solutions.

Abstract

In earlier papers (A. N. Kochubei, Pacif. J. Math., 269 (2014), 355-369; J. Math. Anal. Appl.483 (2020), Article 123609), one of the authors developed a theory of pseudo-differential equations for radial real-valued functions on a non-Archimedean local field, with some features resembling those of classical ordinary differential equations. Here we consider equations of this kind, but with a weak degeneration. Under various assumptions, we prove the local existence and uniqueness of mild solutions, existence of their global extensions and a regularity property.