The First Boundary Value Problem for a Model Equation of Parabolic-Hyperbolic Type of the Third Order

In 1978, the journal Differential Equations published an article by A. M. Nakhushev, which provided a technique for correctly formulating a boundary value problem for a class of second-order parabolic-hyperbolic equations in an arbitrary bounded domain  $\Omega$ with a smooth or piecewise smooth boundary $\Sigma$. The boundary value problem investigated in the above-mentioned work is currently called the first boundary value problem for a second-order mixed parabolic-hyperbolic equation. Within the framework of this work, the first boundary value problem for a third-order model parabolic-hyperbolic equation in a mixed domain is formulated and investigated in the sense in which it was formulated and investigated by A. M. Nakhushev for second-order equations. In one part of the mixed domain, the equation under consideration coincides with a degenerate hyperbolic equation of the first kind of the second order, and in the other part it is an inhomogeneous third-order equation with multiple characteristics of parabolic type. For various values of the parameter $\lambda$ included in the equation under consideration, theorems of existence and uniqueness of a regular solution of the problem under study are proved. To prove the uniqueness theorem, the method of energy integrals is used in conjunction with the method of A.M. Nakhushev. To prove the existence theorem, the method of integral equations is used. In terms of the Mittag-Leffler function, the solution to the problem is found and written out in explicit form.