Some boundary-value problems for linear multidimensional second-order hyperbolic equations

Abstract

For the linear hyperbolic equations

$$\sum\limits_{i,j = 1}^{m + 1} {a_{ij} \left( {x,x_{m + 1} } \right)u_{x_i x_j } + \sum\limits_{i = 1}^{m + 1} {a_i \left( {x,x_{m + 1} } \right)u_{x_i } + c\left( {x,x_{m + 1} } \right)u = 0,x = \left( {x_1 ,...,x_m } \right)} ,} m \geqslant 2,$$

the correctness of multidimensional analogues of the problems of Darboux and Goursat is established and a theorem on the uniqueness of a solution of the Cauchy characteristic problem is proved.