The Appell’s function F₃ and the solutions of a PDE with singular coefficients

ABSTRACT

We consider the Cauchy problem for the following partial differential equation

where and are complex constants and the initial data is analytic or has singularities along one or two hyperplanes of the initial surface . We give an explicit representation of the solutions and show that the propagation of the singularities of the coefficients of and or generates singularities, among others, on the characteristic cone

More precisely, depending on various of values of and , the solutions might have singularities on

where

KEYWORDS:

• Cauchy problem
• Appell’s function
• singular coefficients
• singular solutions

AMS SUBJECT CLASSIFICATIONS:

• 35Q05
• 33C65
• 35C05
• 35C10
• 35A20

Acknowledgements

The author would like to thank the referees for helpful comments and valuable suggestions.

Notes

No potential conflict of interest was reported by the author.