NONLOCAL BOUNDARY VALUE PROBLEM FOR ONE MIXED THIRD ORDER EQUATION

The paper considers a nonlocal boundary value problem for a mixed hyperbolic-parabolic equation of the third order. The equation is considered in a finite simply connected domain consisting of a hyperbolic and a parabolic part. The solution to the problem posed is considered for various cases of the parameter λ, which is in the original equation. In the case when (1-2m)/2< <λ<1, the solution of the problem is reduced to a singular integral equation, which is reduced by the well-known Carleman-Vekua method to the Fredholm integral equation of the third kind. In the case when λ=(1-2m)/2, a theorem on the existence and uniqueness of a solution to the problem posed is formulated and proved. To prove the uniqueness of the solution, the method of energy integrals is used and inequalities of the type are derived on the given functions that are in the boundary condition. It is shown that the homogeneous problem corresponding to the original problem, under the conditions of the uniqueness theorem, has only a trivial solution in the entire considered domain. From which we can conclude that the original problem has only a single solution. If the obtained conditions for the given functions are violated, the problem posed does not have a unique solution. When investigating the question of the existence of a solution to the problem posed, a system of two equations is considered, consisting of the basic functional relations between the trace of the desired function and the traces of the derivative of the desired function, brought to the line of degeneration y = 0. Eliminating from the system the function τ (x) - the trace of the desired solution on the line of degeneration, we arrive at an equation for the trace of the derivative of the desired function. Under the condition of the existence and uniqueness theorem, the problem posed is equivalently reduced to the Fredholm integral equation of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problem posed.