Generalized solutions of the degenerate hyperbolic equation of the second kind with a spectral parameter

In this paper, the Cauchy, Cauchy-Goursat, and Goursat problems for a degenerate second kind hyperbolic equation with a spectral parameter are studied. For these equations, depending on the degree of degeneracy, limit values of the sought solutions and its derivative on degeneration lines can have singularities. To provide the required smoothness of the solution outside the characteristic line of degeneration, it is necessary to require enhanced data smoothness. In order to ease this requirement, a definition of a class of generalized solutions is introduced and properties of this class are studied. In addition, on the basis of the well-known formula of the classical solution of the Cauchy problem for the above equation, a generalized solution of the Cauchy problem in the introduced class is obtained in an explicit form which is easy to use for further research. Properties of these solutions are studied. Some operators with Bessel functions in the nucleus are introduced and their basic properties are studied. The proved important identities of these operators and the above representation of the generalized solution of the Cauchy problem allow one to find an explicit representation of the generalized solutions of the Cauchy-Goursat and Goursat problems in the characteristic triangle. In addition, an example showing the importance of introducing such class is presented: if the solution does not belong to the newly introduced class, then the uniqueness of the solution of the Cauchy-Goursat problem can be broken. The resulting explicit integral representation of the generalized solution of the Cauchy-Goursat problem plays an important role in the study of problems for equations of the mixed type: it makes it easy to derive the basic functional relationship between the traces of the sought solution and of its derivative on the line of degeneration from the hyperbolic part of the mixed domain.

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