On a new nonlocal boundary value problem for an equation of the mixed parabolic-hyperbolic type

In this work a new nonlocal boundary value problem for an equation of the mixed type is formulated. This equation is parabolic-hyperbolic and belongs to the first kind because the line of type change is not a characteristic of the equation. Non local condition binds points on boundaries of the parabolic and hyperbolic parts of the domain with each other. This problem is generalization of the well-known problems of Frankl type. A boundary value problem for the heat equation with conditions of the Samarskii-Ionlin type arises in solving this problem. Unlike the existing publications of the other authors related to the theme it is necessary to note that in this papers the nonlocal problems were considered in rectangular domains. But in our formulation of the problem the hyperbolic part of the domain coincides with a characteristic triangle. Unique strong solvability of the formulated problem is proved.