On a non-local problem for mixed parabolic-hyperbolic type equation with non-smooth line of type changing

TL;DR

This paper studies a boundary problem for a mixed parabolic-hyperbolic equation with non-smooth type-changing lines, proving uniqueness and existence of solutions using energy methods and integral equations.

Contribution

It introduces a new approach to handle mixed equations with non-smooth type-changing lines, establishing uniqueness and existence results.

Findings

Proved uniqueness of solutions using energy integral method.

Reduced the problem to Volterra integral equations for existence proof.

Handled non-smooth lines of type change in the mixed domain.

Abstract

In the present article we investigate a boundary problem with non-local conditions for mixed parabolic-hyperbolic type equation with three lines of type changing. Considered mixed domain contains a rectangle as a parabolic part and three domains bounded by smooth curves and by type-changing lines as a hyperbolic part of the mixed domain. We prove the uniqueness applying energy integral method. The proof of the existence will be done by reducing the original problem into the system of the second kind Volterra integral equations.