The Cauchy Problem for the Iterated Klein–Gordon Equation with the Bessel Operator

Abstract

An analogue of the Cauchy problem for an iterated multidimensional Klein–Gordon equation with a time-dependent Bessel operator is investigated. Applying the generalized Erdelyi–Kober operator of fractional order, we reduce the formulated problem to the Cauchy problem for the polywave equation. Applying a spherical mean method, we construct an explicit formula to solve this problem for the polywave equation; then, basing on this solution, we find an integral representation of the solution of the formulated problem. The obtained formula allows one to immediately discern the character of the dependence of the solution on the initial functions and, in particular, to establish conditions for the smoothness of the classical solution. The paper will be useful for specialists engaged in the resolving of problems of higher spin theory.