On lacunae for some classes of equations with singularities

The Cauchy problem is considered for equations which decompose into Euler–Poisson–Darboux factors. A representation of the solution of this problem is obtained in terms of spherical means of the initial function. This makes it possible both to establish precise conditions for classical solvability and to investigate the question of the presence of lacunae. We note that a special feature of this class of problems is the appearance of lacunae in even-dimensional space and the appreciable effect of the parameter in the Bessel operator on the smoothness of the initial data and the order of the lacunae obtained.
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