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 (Problem $$AG$$ ) 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.

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