The first with displacement problem for a third-order parabolic-hyperbolic equation and the effect of inequality of characteristics as data carriers of the Tricomi problem

Abstract

As part of this scientific work, we study a displacement boundary value problem for a third - order parabolichyperbolic type equation with a third - order parabolic equation backward in time and a wave equation in the domain of hyperbolicity. As one of the boundary conditions we have a linear combination including variable coefficients of the sought function on the characteristic lines AC and BC . The present paper reports following results: inequality between characteristics of AC and BC lines limiting the hyperbolic part Ω₁ of the domain Ω as carriers of data for the Tricomi problem as 0≤ x ≤2 π , as a matter of fact, the solvability of the Tricomi problem with data on the characteristic line BC does not imply the solvability of the Tricomi problem with data on the AC ; necessary and sufficient conditions for the existence and uniqueness of a regular solution to the problem under study are found. Under certain conditions for the given functions, the solution to the problem under study is written out explicitly. It is shown that under violation of the necessary conditions established in this paper the homogeneous problem has innumerable linearly independent solutions, while the set of solutions to the corresponding inhomogeneous problem can exist only with additional conditions