Nonlinear pseudo-differential equations for radial real functions on a non-Archimedean field (Q2009327)

Let \(K\) be a non-Archimedean local field with absolute value \(|\cdot|_K.\) In this case \(P=\{x\in K:|x|_K<1\}\) is an ideal in the subring \(O=\{x\in K:|x|_K\le 1\}\) of \(K\) and the quotient ring \(O/P\) is a finite field of cardinality, say, \(q\). The Vladimirov fractional differential operator is defined for \(\alpha>0\) by \((D^\alpha\varphi)(x)=(1-q^\alpha)(1-q^{-\alpha-1})^{-1}\int_K|y|^{-\alpha-1}(\varphi(x-y)-\varphi(x))dy\) for \(\varphi\in\mathcal{D}(K)\) -- the space of all complex-valued locally finite functions with compact support defined on \(K\). This space, equipped with the double inductive limit topology generated by some finite dimensional subspaces, has as dual the space \(\mathcal{D}'(K)\) of Bruhat-Schwartz distributions. It is known that the Fourier transform acts by duality on this space, see, for instance, the book [\textit{A. N. Kochubei}, Pseudo-differential equations and stochastics over non-Archimedean fields. New York, NY: Marcel Dekker (2001; Zbl 0984.11063)]. The author considers the restriction of the operator \(D^\alpha\) to radial functions \(x\mapsto u(|x|_K)\) from \(K\) to \(\mathbb{C}\) and studies nonlinear equations of the form \((*)\;\;(D^\alpha u)(|t|_K)=f(|t|_K,u(|t|_K),\, 0\ne t\in K,\) with initial condition \(u(0)=0.\) The linear case was considered in a previous paper, [\textit{A. N. Kochubei}, Pac. J. Math. 269, No. 2, 355--369 (2014; Zbl 1396.35070)], where the study was based on the theory of compact operators. In the nonlinear case he finds conditions for the local and global solvability of the equation \((*)\) in terms of the properties of the function \(f\), by using direct constructions based on iteration processes.