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

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.

arXiv admin note: text overlap with arXiv:1302.4850