Boundary value problem with displacement for a third-order parabolic-hyperbolic equation

Zh. A. Balkizov ; A. G. Ezaova ; L. V. Kanukoeva

Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 2, pp. 5-18

Résumé

A boundary value problem with a shift is investigated for an inhomogeneous third order equation of parabolic-hyperbolic type when one of the boundary conditions is a linear combination of values of the sought function on independent characteristics. The following results are obtained in this work: the inequality of the characteristics $AC$ and $BC$, which bound the hyperbolic part $\Omega_{1}$ of the domain $\Omega$, as carriers of the data of the Tricomi problem for $0\le x\le\pi n$, $n \in N$ and the solvability of the Tricomi problem with data on the characteristic $BC$ in this case, in general, does not imply the solvability of the Tricomi problem with data on the characteristic $AC$; necessary and sufficient conditions for the existence and uniqueness of a regular solution of the problem are found. Under certain requirements for given functions, the solution to the problem is written out explicitly. It is shown that if the necessary conditions for the given functions found in the work are violated, the homogeneous problem corresponding to the problem has an infinite set of linearly independent solutions, and the set of solutions to the corresponding inhomogeneous problem can exist only with an additional requirement for the given functions.