The Cauchy problem for the Euler–Poisson–Darboux equation in a symmetric space

The classical solvability of the singular Cauchy problem for the Euler–Poisson–Darboux equation in a homogeneous, globally symmetric space of rank 1 is studied. Starting out from the mean value theorem for spaces of the indicated type, the Darboux and the Euler–Poisson–Darboux equations are introduced. For the Cauchy problem with specific singularity conditions, analogs of Kirchhoff's formulas are derived, i.e. a representation of the solution in terms of spherical means of the initial data is given. The representations so obtained permitted the establishment of necessary and sufficient conditions for the problems under consideration to satisfy Huygens' principle. In particular, Kirchhoff's formulas for the wave equation have been obtained.
Bibliography: 27 titles.