On a boundary value problem for a third-order parabolic-hyperbolic type equation with a displacement boundary condition in its hyperbolicity domain

In the article, we investigate a boundary-value problem with a thirdorder inhomogeneous parabolic-hyperbolic equation with a wave operator in a hyperbolicity domain. A linear combination with variable coefficients in terms of derivatives of the sought function on independent characteristics, as well as on the line of type and order changing is specified as a boundary condition. We have established necessary and sufficient conditions that guarantee existence and uniqueness of a regular solution to the problem under study. In some cases, a solution representation is written out explicitly.

where is a given function, = ( , ) is the desired function. Equation (1) as > 0 coincides with the equation which belongs to the class of third order equations with multiple characteristics [1, p. 9] of parabolic type [2, p. 69] and as < 0 equation (1) coincides with the inhomogeneous wave equation Thus equation (1) is parabolic-hyperbolic equation with type and order degeneration along the line = 0 and, as stated in [3], study of boundary value problems for the above equations brings a new aspect to the mixed type equations theory.
The problem with boundary conditions connecting values of a sought solution on the characteristics of the both families for Lavrent'ev-Bitsadze equation was first posed and investigated in [5].
In [6,7] the displacement boundary value problem was introduced, and a number of nonlocal boundary value problems with various types of displacements for hyperbolic, degenerate hyperbolic, and mixed-type equations have been investigated since. In particular, in [6] the existence of an unique solution to the nonlocal problem for equation (3) with the conditions and (5) as ( ) = ( ) ≡ 0, ( ) ̸ = ( ) ∀ ∈ [0, ] has been proved. In [7], the way of posing non-local boundary value problems with displacement for a degenerating hyperbolic equation of the form is offered employing the Riemann-Liouville fractional derivatives. The criteria for the unique solvability with conditions (6) and , are the affixes of the intersection points of the characteristics of equation (7), as above, and what is more 2( + 2) = .
Specific cases for displacement related problems include such nonlocal problems as the Bitsadze-Samarsky problem [8][9][10] [19][20][21][22][23][24], etc. In case of displacement problems, for mixed equations a nonlocal condition is imposed connecting values of desired solution or derivative of a certain order at two, three or more points lying on the boundary characteristics of different families and on the line of degeneracy or the type change line. If one or several coefficients of the displacement problem for mixed type equations is zero it becomes an ordinary Tricomi problem.
The displacement problems are well applicable in mathematical problems modeling in biology (synergetics), transonic gas dynamics. Similar nonlocal boundary conditions arise in the study of heat and mass transfer in capillary-porous media, in mathematical modeling of problems of gas dynamics and nonlocal physical processes, in the study of cell propagation, in the theory of electromagnetic wave propagation within an inhomogeneous media [2,18,25]. A bibliography of research papers devoted to the displacement boundary value problems is presented quite completely in monographs [4,14,[26][27][28][29][30][31][32][33][34].
In [35], the displacement boundary value problem is studied under condition (5) for mixed type equations of second order and a heat equation in the parabolicity domain; a necessary and sufficient condition for the existence of an unique solution is obtained. In this paper, we study the displacement boundary value problem for inhomogeneous parabolic-hyperbolic equation of the third order (1) and a third-order parabolic and wave equations in the hyperbolicity domain. One of the boundary conditions is a linear combination of the sought functions and their derivatives with variable coefficients in and , as well as in = lines of type and order change. Necessary and sufficient conditions for the existence and uniqueness of a regular solution to the problem under study have been obtained. The solution to the studied problem under certain conditions have been written out explicitly. We have shown that violation of the necessary conditions imposed on the specified functions leads to non-uniqueness of the studied problem. That is, the corresponding homogeneous problem has an infinite number of linear independent solutions. In addition, solutions to a non-homogeneous problem could exist only with additional requirements for the given functions.
Theorem on the existence and uniqueness of a solution The following theorem holds true.
Theorem 1. Assume the given functions 1 ( ), 2 ( ), 3 ( ); ( ), ( ), ( ), ( ), ( ) have the following properties: and one of the below conditions is satisfied: Therefore there is the unique regular solution for problem 1 in the domain Ω. P r o o f. Let there be a solution to problem 1 and assume that Passing to the limit as → +0 in equation (1) in view of notation used in (16) we obtain fundamental relation for the functions ( ) and ( ) moved from the parabolic part Ω 2 of the domain Ω to the line = 0: and with boundary conditions (4) obtain Now find fundamental relation for the functions ( ) and ( ) moved from the hyperbolic part Ω 1 of the domain Ω to the line = 0 of the type changing.
Next, assume that theorem conditions (8), (9), (12) are satisfied. Then by inequality (20) we arrive at the identity Exclude from relations (17) and (24) the function ( ). Then for the function ( ) we arrive at the boundary problem for the equation with conditions (18).
The solution of problem (25), (18) is written out explicitly by the formula  (17) and (20) for ( ) we arrive at the boundary problem for the equation subject to conditions (18). The solution to the problem (18) for equation (27) is written out with respect to the sign of the value of ( + + )( − − ) using one of the following formulae: Thus, under the assumption of (8), (9) and also under the assumption of at least one of the conditions of (10), (11), (12), (13)  Now we can pass to studying the general case. Let (8), (9) and (15) be satisfied for the given functions 1 ( ), 2 ( ), 3 ( ); ( ), ( ), ( ), ( ), ( ). First we prove the uniqueness of a regular solution to the problem 1. Consider the homogeneous problem corresponding to problem 1, that is, we let ( ) ≡ 0, Consider the integral By relation (17) in view of (18) obtain while by (20) with (15) and (18) we have and (29) imply the identity * = 0, which by (15) can hold if and only if ( ) ≡ 0. At that by relations (17) and (20) we can see that ( ) ≡ 0. Thus, let us show that under the assumption (15) of theorem 1 the functions ( ) and ( ) are identically zero for the homogeneous problem corresponding to problem 1. At the same time formula (19) implies immediately that ( , ) ≡ 0 inΩ 1 . In the domain Ω 2 we arrive at the problem on finding a solution to the homogeneous equation − = 0, ( , ) ∈ Ω 2 satisfying the homogeneous initial ( , 0) = 0 (0 ) and boundary (0, ) = 0, (0, ) = 0, ( , ) = 0 (0 ) conditions. This problem as it stated in [1, p. 144] has only a trivial solution ( , ) ≡ 0 ∀ ( , ) ∈Ω 2 . Therefore the solution ( , ) to the homogeneous problem corresponding to the problem under problem 1 is identically zero in the whole domainΩ; this implies the uniqueness of a regular solution to problem (1), (4), (5). Now we prove the existence of a regular solution to problem 1 subjected to conditions (8), (9) and (15). By relations (17) and (20) we arrive at the problem of finding a solution to the equation satisfying condition (18).
By means of repeated integration 3 times of equation (30) within the limit from 0 to in view of boundary conditions (18) the solution to problem (30) (18) is equivalently reduced to the solution of the integral equation where By properties (8) (17) or (20) we can find function = ( ) as well.
Once the functions ( ) and ( ) are found the solution to problem 1 in Ω 1 is defined as a solution to problem (16) for equation (3) and is written out by formula (19) while in the domain Ω 2 we arrive at the problem of finding a regular solution to equation (2) satisfying the initial ( , 0) = ( ) and boundary (4) conditions; the solution to the above problem is written out in [1]. Theorem 1 is proved.
Assume that for the coefficients ( ), ( ), ( ), ( ) condition (9) is violated, i.e. the identity In the domain Ω 1 a set of solutions to problem 1 subjected to condition (33) is written out by the formula while in Ω 2 for the set of solutions to problem 1 holds the representation [1] ( , ) = Conclusion. The paper studies a displacement boundary value problem for inhomogeneous parabolic-hyperbolic equation of the third order (1) with a thirdorder parabolic and wave equations in the hyperbolicity domain. A linear combination of the sought functions is given as one of the boundary conditions. Their derivatives with variable coefficients are in and , and in = lines of type and order change. A necessary and sufficient conditions for the existence and uniqueness of a regular solution to the problem under study are obtained. In some special cases, the representation of the solution to the studied problem is written out explicitly. We have shown that violation of the obtained necessary conditions imposed on the specified functions leads to non-uniqueness of the studied problem. That is, the corresponding homogeneous problem has an infinite number of linear independent solutions. In addition, solutions to a non-homogeneous problem could exist only with additional requirements for the given functions.
Thus, in contrast to the results obtained in [35], necessary and sufficient conditions (8), (9) for the functions specified become insufficient if in the parabolicity domain consider the third-order equation with multiple characteristics (2) instead of the heat equation.