35M12Boundary value problem for a mixed-type equation with a higher order elliptic operator

In this paper, we consider a boundary value problem for a mixed-type equation with a positive,formally self-adjoint, high order elliptic operator. The results of the work were obtained usingthe Fourier method. Theorems on the existence and uniqueness of the classical solution of theproblem are proved. In this case, the positivity of elliptic operator turned out to be essential.At the end of the paper, a mixed-type equation with a non-negative elliptic operator is considered,and it is shown that the solution of the corresponding problem is not unique.