Mixed problem for a third order parabolic-hyperbolic model equation

In 1978, the journal Differential Equations published an article by A.M. Nakhushev, that presented a method for correctly formulating a boundary value problem for a class of second-order parabolic-hyperbolic equations in an arbitrarily bounded domain with a smooth or piecewise smooth boundary. In that work, a boundary value problem was formulated and investigated using the method of a priori estimates, which is currently called the first boundary value problem for a second-order mixed parabolic-hyperbolic equation. In this work, a boundary value problem for a third-order model parabolic-hyperbolic equation is formulated and investigated in a mixed domain, following the approach of A.M. Nakhushev for second-order mixed parabolic-hyperbolic equations. In one part of the mixed domain, the equation under consideration is a degenerate hyperbolic equation of the first kind of the second order, and in the other part, it is a nonhomogeneous equation of the third order with multiple characteristics and reverse-time parabolic type. For various values of the parameter, existence and uniqueness theorems for a regular solution are proved. The uniqueness theorem is proved using the method of energy integrals combined with A.M. Nakhushev’s method. The existence theorem is proved by the method of integral equations. In terms of the MittagLeffler function, the solution of the problem is found and written out explicitly. Sufficient smoothness conditions for the given functions are found, which ensure the regularity of the obtained solution.