On the analogue of the cauchy problem for a system of second order partial differential equations

We consider the system of n Euler – Poisson – Darboux Equations in matrix notation and study the case when matrix coefficient has one eigenvalue lying in (1/2,1). The singularity of equation makes the classical formulation of the Cauchy problem ill-posed.We formulate well-posed analogue of Cauchy problem. The singular behavior can be compensated by adding weight to both of conditions.We perform the change of variables to reduce the coefficient matrix to Jordan normal form. The coefficient is one Jordan block of order n for the case of real eigenvalues and a real analogue of the Jordan block of order n/2 for the case of complex conjugate eigenvalues. We construct the solutions using the Riemann method and properties of matrix functions and formulate the well-posedness theorems.