Existence of a solution of a mixed problem for a hyperbolic vector equation of second order (Q1080073)

We consider a mixed problem in the region \(t>0\), \(x>0\), \(-\infty <y<\infty\), for a system of n equations, in which the principal part of each equation consists of the same homogeneous strongly hyperbolic operator \({\mathcal L}_ 0\) of second order with smooth real coefficients, and the boundary conditions are given in the form of n relations whose principal parts are linear combinations of the first derivatives of the unknown functions. We consider only those operators \({\mathcal L}_ 0\) for which the boundary \(x=0\) is not characteristic of equations for any t, \(x=0\), y, and exactly n boundary conditions are required in the mixed problem. We use the general theorem on the existence of solutions of mixed problems of symmetric hyperbolic systems of equations with dissipative boundary conditions and show how to prove the existence of solutions of our problem using this general theorem. In our consideration, we suppose that the uniform Lopatinskij condition is fulfilled.