Radial solutions of a pseudo-differential equations associated with a p-adic non-local ultradiffusion operator

Abstract

AbstractThis paper is dedicated to study some pseudo-differential equations related to the operator $$D^{\alpha }_{d_1,d_2}$$ D d 1 , d 2 α with symbol $$\mathcal {A}^{\alpha }_{d_1,d_2}(x)=\max \left\{ |x|_p^{d_1},\ |x|_p^{d_2}\right\} ^\alpha $$ A d 1 , d 2 α ( x ) = max | x | p d 1 , | x | p d 2 α , acting on a radial function $$\varphi (|x|_p)$$ φ ( | x | p ) . The importance of this symbol is that it is a generalization of Vladimirov and Bessel operators. Our research provides conditions for the existence of its right inverse and the explicit formulae for both operators, as well as a practical application. The inverse allows solving a Cauchy problem with the operator, which can be reduced to an integral equation whose properties are similar to those of the classical Volterra equation. We also find conditions for the existence and uniqueness of a mild solution for a degenerate nonlinear Cauchy problem.