Nonlinear pseudo-differential equations for radial real functions on a non-Archimedean field

Abstract

In an earlier paper (Kochubei, 2014, [4]), the author considered a restriction of Vladimirov's fractional differentiation operator Dα, α>0, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse Iα that the change of an unknown function u=Iαv reduces the Cauchy problem for a linear equation with Dα (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.