Three-dimensional Problems for a Parabolic-Hyperbolic Equation with Two Planes of Change of Type

In this paper in infinite three-dimensional domains the analogues of the Gellerstedt problem (ProblemAG) are formulated and studied for a parabolic-hyperbolic equation with two type change planes.Applying the Fourier transform, the considering problem reduces to a plane analogue of the Gellerstedt problem (Problem AG(lambda)) with a spectral parameter and with the boundary value conditions. The uniqueness of the solutions of the Problems AG and AG(lambda) are proved by the aid of new extremum principle for the second order mixed type equations. The existence of solutions of the two Problems AG and AG(lambda) are proved by the method of integral equations. In addition, the asymptotic behavior of the solution of the Problem AG(lambda) is studied for large values of the spectral parameter. Sufficient conditions are found under which all operations in this work are legal.