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 Erdélyi–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.

中文翻译：

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具有贝塞尔算子的迭代Klein-Gordon方程的Cauchy问题

摘要

研究了具有时间依赖性贝塞尔算子的迭代多维Klein-Gordon方程的柯西问题的类似物。应用分数阶的广义Erdélyi–Kober算符，我们将多波方程的拟定问题简化为Cauchy问题。应用球均值法，我们构造了一个明确的公式来解决多波方程的这一问题。然后，基于该解决方案，我们找到了所提出问题的解决方案的完整表示。所获得的公式使人们能够立即辨别出解对初始函数的依赖性的特性，尤其是为经典解的光滑性建立条件。该论文对于从事解决高自旋理论问题的专家很有用。