On solution uniqueness for Dezin problem for parabolic-hyperbolic type equation with boundary conditions of the first kind

In this paper considered an inhomogeneous second-order parabolic-hyperbolic mixed type equation, represented as one-dimensional heat equation in the parabolic part and the onedimensional wave equation in the hyperbolic part. For the equation, a Dezin problem with boundary conditions of the first kind is investigated, which means to find a solution to the equation that satisfies inner-boundary condition, relating the value of the desired function on the equation type change line to the value of the normal derivative on the hyperbolicity region boundary, and boundary conditions of the first kind. It is established a criterion for the solution uniqueness to the problem. In case when the uniqueness criterion is violated, an example of a nontrivial solution to a homogeneous problem is given, and is obtained a necessary and sufficient condition for the existence of a solution to an inhomogeneous problem.