Some Boundary Value Problems for the Hyperbolic : Hyperbolic type Equation with Two Line of Degeneration in Special Domain

Last years the increasing attention of mathematicians is involved with problems of correctness of the boundary value problems (BVP) for the degenerating equations of the mixed parabolic-hyperbolic, elliptichyperbolic and hyperbolic-hyperbolic types. It is closely connected with appendices of such problems to the decision of problems of mechanics, gas dynamics, biology and in other material sciences. The first basic researches under the theory of the degenerating equations of the mixed and mixed-compound type are Trikomi’s [1], Gellerstedt’s et al, [2], Bitsadze’s [3] and Salakhitdinov’s [1,4,5] works. The degenerating and singularity equations possess that nature, that for them the correctness of some classical problems not always takes place. This fact rather for the degenerating equations of elliptic type, in the first has been noticed of MKeldych [6], and concerning the degenerating equations of hyperbolic type of Gellerstedt. In this cases Bitsadze has suggested to study modify problems Cauchy for the degenerating equation of hyperbolic type because the problems Cauchy for such equations it is put incorrectly. Since Bitsadze’s [2] works, in the theory partial differential equations there was a new direction, in which the analogue of problem Tricomi for the first time is formulated and investigated in double connected domain for the modeling equations of the mixed type. After this work various problems for the equation of the mixed type on the second order in multiply and doubly connected domains are investigated, in works as Bers [3] and Salahitdinov, Urinov [1,5]. However, BVPs in double-connected domain are studied for the not degenerated modeling equations of the third order of elliptichyperbolic type [7], and uniqueness of solution of the problem for the degenerated hyperbolic-hyperbolic type equation in double-connected domain was proved by Islomov et al. [4].


Introduction
Last years the increasing attention of mathematicians is involved with problems of correctness of the boundary value problems (BVP) for the degenerating equations of the mixed parabolic-hyperbolic, elliptichyperbolic and hyperbolic-hyperbolic types. It is closely connected with appendices of such problems to the decision of problems of mechanics, gas dynamics, biology and in other material sciences. The first basic researches under the theory of the degenerating equations of the mixed and mixed-compound type are Trikomi's [1], Gellerstedt's et al, [2], Bitsadze's [3] and Salakhitdinov's [1,4,5] works. The degenerating and singularity equations possess that nature, that for them the correctness of some classical problems not always takes place. This fact rather for the degenerating equations of elliptic type, in the first has been noticed of MKeldych [6], and concerning the degenerating equations of hyperbolic type of Gellerstedt. In this cases Bitsadze has suggested to study modify problems Cauchy for the degenerating equation of hyperbolic type because the problems Cauchy for such equations it is put incorrectly. Since Bitsadze's [2] works, in the theory partial differential equations there was a new direction, in which the analogue of problem Tricomi for the first time is formulated and investigated in double connected domain for the modeling equations of the mixed type. After this work various problems for the equation of the mixed type on the second order in multiply and doubly connected domains are investigated, in works as Bers [3] and Salahitdinov, Urinov [1,5]. However, BVPs in double-connected domain are studied for the not degenerated modeling equations of the third order of elliptichyperbolic type [7], and uniqueness of solution of the problem for the degenerated hyperbolic-hyperbolic type equation in double-connected domain was proved by Islomov et al. [4].

The Statement of Problems
In the present work the analogues of problem A.V. Bitsadze [2] is formulated and investigated for the hyperbolic-hyperbolic type equation with two degenerating lines of the following kind: We introduce the following notations: Through Ω 1j and Ω 3j we will designate characteristic triangles A j B j E j and C 1 F j C 2 (j =1, 2), accordingly, and through Ω 2j we will designate characteristic quadrangles A j E j F j C 1 (j=1, 2). In the section 2 we have formulated and proved unique solvability of a problems I(I * ) and II (II * ) in the domain of , which, consist of four characteristic triangles and from two quadrangles. The result, which is obtained in this section shows that when we will investigate problems I(I * ) and II(II * ), in each sub domains, we find the solution of equation (1)in an explicit form In the section 3 we studying uniqueness and existence of solution of a problem III (III * ). Uniqueness of solution of problem III (III * ) are proven with principle an extremum. Existence of the solution of problem III (III * ) we have proved, by method integral equations. The main result of this section shows that when we will studying existence of the solution of problem III(III * ), we have singularity integral equation, which regularities by the method of Karleman's-Vekua [8], to the integral equation of Fredgolm of the second kind.
Proof: Is known, that the solution of problem Cauchy in domain Ω 11 for the equation (1) satisfying to conditions (2 1 ) and (6 1 ) looks like [8,9]: From here, by virtue condition (3), considering (51) ((71)) it is easily possible to define unknown function v 1 (x) (T 1 (x)), hence, owing to uniqueness of the solution of problem Cauchy, the solution of the problem I(I * ) in domain Ω 11 is uniquely defined.
Further, designating through h 1 (x) a trace of the solution of problem Cauchy-Gaursat 1 (Cauchy-Gaursat 2) from domain Ω 11 on the characteristic A 1 E 1 and considering the condition (3) taking into account (5 2 ), by the method of Riemann, we restore the solution of the problem I(I * ) in domain the of Ω 21 as the solution of problem Gaursat and this solution is given by the formula: Similarly, we find the unique solution of problem Gaursat for the equation (1) (51) and (10) ((7 j ) and (10)) are satisfied that the problem II (II * ) is unique solvability. The theorem 2 is proved similarly as the theorem 1.

Uniqueness and Existence of Solutions of the Problem III (III * )
Problem III (III * ): Find a function u(x, y) in domain Ω satisfies to all conditions problem I(I * ),except (3) which are replaced with conditions: where φ * (x) , Ψ * (x) -given functions, and Theorem 3: If conditions (5 1 ) and (15) ((7 j ) are satisfied and (15)) that solution of a problem III (III * ) exists and is unique.   and (14) taking into account properties of integro-differential operators of fractional order [8,9] accordingly we will receive: At the proof of the theorem 3 takes place Lemma: The solution u(x, y) of the problem III (III * ) at in the domain the positive maximum and negative minimum reaches only in points C 1 and C 2 .
Proof: By virtue (20 j ) and considering solutions of problem Cauchy-Gaursat (in the domain Ω 11 ) and Gaursat (in the domain 21) for the equation (1) we will receive, that u(x, y)=0 on the characteristic C 1 F 1 . From here, owing to a principle of extremum for the hyperbolic equations [4,9,10] function u(x, y) reaches the positive maximum and the negative minimum in the domain 31 only on the piece C1C2. Similarly, owing to the principle of an extremum for the hyperbolic equations [10] with the account Ψ * (x)≡0, we have, that the function u(x, y) reaches the positive maximum and the negative minimum in domain Ω 32 only on the piece C 1 C 2 .
Let function u(x, y) reaches the positive maximum (the negative minimum) in some point y 0 of the interval C 1 C 2 (i.e. y 0 € (−1,−q)) then owing to (18) and (19) taking into account a principle of extremum for the integro-differential operators of fractional order [9], accordingly we will receive u x (+0, y 0 ) <0, y 0 € (−1,−q), and u x (−0, y 0 ) >0, y 0 € (−1,−q). From here owing to continuity of solution u(x, y) have received the contradiction, i.e., the function u(x, y) does not reach the positive maximum (the negative minimum) in the interval C 1 C 2 . Hence, function u(x, y) can reach the positive maximum (the negative minimum) only on points C 1 and C 2 . The lemma is proved. As, u(x, y)=0 on the characteristic C 1 F 1 and C 2 F 2 , we have that, u(x, y)=0 on the points C 1 and C 2 . From here, owing to continuity of solution u(x, y) in the domain of, the problem III (III * ) with zeroes dates, has only trivial solution, i.e. uniqueness of the solution of problem III (III * ) is roved. Existence of the solution of the problem III (III * ) is proved, by method integral equations.

(y)
τ and considering properties of the integro-differential operators, we will receive singular integral equation.
Entering designations, a(y * )=1 + cos π(1-2β) , b (y * )=−i sin π(1 − 2 β) and we will copy the equation (23) in the form of integral Cauchy [11,12]: We will estimate the function Ф(y * ), for this considering properties integro-differential operators, we have from (24) As, a 2 (y * )-b 2 (y * ) ≠ 0 the integral equation (23) is singular integral equation of the normal type, and by virtue (27) and |K(t, y * )| ≤ const·t 1−2β (1−y * ) 2β −1 we have that, index of integral equation (26), is equal to zero. Hence, by virtue of the theory singular integral equations and by the method regularities of Karlemens-Vekua [11], the integral equation (26) will be reduced to the integral equation of Fredgolm of the second kind with weak singularity. Thus, by virtue uniqueness of solution of the problem III(III * ) the function v 3 (y) € C(−1;−q] C 2 (−1;−q) will be unequivocally find from the equation (26), and this function can have singularity of an order less than 1 − 2β at the y→ −1 and continuous at y→ −q. After it is found v 3 (y), from (21) and (22) we will found v 3 (y) accordingly in domains Ω 31 and Ω 32 , hence in the domains of Ω 31 and Ω 32 the solution of problem III (III * ) is restored as the solution of problem Cauchy, and in the domains of Ω 21 and Ω 22 as the solution of problem Cauchy-Gaursat. The theorem 3 is proved.