Abstract

Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.

1. Introduction

Mathematical model of the movement of gas in a channel surrounded by a porous environment was described by parabolic-hyperbolic equation. This was done in the fundamental work of Gel’fand [1]. Modeling of heat transfer processes in composite environment with finite and infinite velocities leads to boundary value problems (BVPs) for parabolic-hyperbolic equations [2]. Omitting the huge amount of works devoted to studying these kinds of equations, we refer the readers to [3, 4].

We would like to note works [5–10], devoted to the studying of BVPs for parabolic-hyperbolic equations, involving fractional derivatives. In turn, applications of Fractional-order differential equations can be found in the monographs [11–15]. We also note some recent papers [16–18], related to the fractional diffusion and diffusion-wave equations.

BVP for parabolic-hyperbolic equations with integral gluing condition for the first time was investigated by Kapustin and Moiseev [19] and was generalized for this kind of equation, but with parameters, in the work [20]. Another motivation of the usage of integral gluing conditions comes from the appearance of them in heat exchange processes [21].

The consideration of equations with parameters was interesting because of the possibility of studying some multidimensional analogues of the main BVP via reducing them by Fourier transformation to the BVP for equations with parameters. On the other hand, consideration of equations with parameters will give possibility to study some spectral properties of BVPs for this kind of equations such as the existence of nontrivial solutions for corresponding homogeneous problem at some values of parameters [22].

2. Analog of the Tricomi Problem

Consider an equation𝑒π‘₯π‘₯βˆ’π·π›Όπ»(π‘₯)+2𝐻(βˆ’π‘₯)0𝑦𝑒=πœ†π‘’(2.1) in the domain Ξ©=Ξ©1βˆͺ𝐴𝐴0βˆͺΞ©2. Here Ξ©1={(π‘₯,𝑦)∢0<π‘₯<1,0<𝑦<1}, Ξ©2 is characteristic triangle with endpoints 𝐴(0,0), 𝐴0(0,1), 𝐢(βˆ’1/2,1/2), 𝐻(π‘₯) is Heaviside function, π·π›Όπ‘Žπ‘‘1𝑓(𝑑)=𝑑Γ(π‘›βˆ’π›Ό)ξ‚π‘‘π‘‘π‘›ξ€œπ‘‘π‘Ž(π‘‘βˆ’π‘ )βˆ’π›Ό+π‘›βˆ’1𝑓(𝑠)𝑑𝑠(2.2) is the 𝛼th Riemann-Liouville fractional-order derivative of a function 𝑓 given on interval [π‘Ž,𝑏], where 𝑛=[𝛼]+1 and [𝛼] is the integer part of 𝛼, and Ξ“(β‹…) is the Euler gamma function defined byξ€œΞ“(𝛼)=∞0π‘‘π›Όβˆ’1π‘’βˆ’π‘‘π‘‘π‘‘,𝛼>0.(2.3) For πœ†>0 and 0<𝛼≀1 given, we formulate the following problem called the analog of the Tricomi problem.

Problem AT
To find a solution of (2.1), which belongs to the class of functions π‘Š1=ξ‚†π‘’βˆΆπ·π›Όβˆ’10π‘¦ξ‚€π‘’βˆˆπΆΞ©1,𝑒π‘₯π‘₯,𝐷𝛼0π‘¦ξ€·Ξ©π‘’βˆˆπΆ1ξ€Έ,𝑒π‘₯ξ€·0Β±ξ€Έξ‚€,π‘¦βˆˆπ»(0;1),π‘’βˆˆπΆΞ©2ξ‚βˆ©πΆ2ξ€·Ξ©2,(2.4) satisfying the initial condition lim𝑦→0𝑦1βˆ’π›Όπ‘’(π‘₯,𝑦)=πœ”(π‘₯),0≀π‘₯≀1(2.5) together with the boundary conditions 𝑒(βˆ’π‘¦/2,𝑦/2)=πœ“1𝑒(𝑦),0≀𝑦≀1,(1,𝑦)=πœ“2(𝑦),0≀𝑦≀1,(2.6) and the gluing conditions 𝑒(0βˆ’1,𝑦)=ξ€œΞ“(1βˆ’π›Ό)𝑦0𝑒0+ξ€Έ,𝑑(π‘¦βˆ’π‘‘)βˆ’π›Όξ€œπ‘‘π‘‘,0<𝑦≀1,𝑦0𝑒π‘₯(0βˆ’,𝑑)𝐽0ξ‚ƒβˆšξ‚„1πœ†(π‘¦βˆ’π‘‘)𝑑𝑑=ξ€œΞ“(1βˆ’π›Ό)𝑦0𝑒π‘₯ξ€·0+ξ€Έ,𝑑(π‘¦βˆ’π‘‘)βˆ’π›Όπ‘‘π‘‘,0<𝑦<1.(2.7) Here πœ”(π‘₯), πœ“π‘–(𝑦) (𝑖=1,2) are given functions such as limyβ†’0𝑦1βˆ’π›Όπœ“1(𝑦)=πœ”(0).
Solution of the Cauchy problem for (2.1) in Ξ©2 defined as 1𝑒(π‘₯,𝑦)=2⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©πœβˆ’(𝑦+π‘₯)+πœβˆ’ξ€œ(π‘¦βˆ’π‘₯)+𝑦+π‘₯π‘¦βˆ’π‘₯πœˆβˆ’(𝑑)𝐽0ξ‚Έξ”πœ†ξ€Ί(π‘¦βˆ’π‘‘)2βˆ’π‘₯2ξ€»ξ‚Ήξ€œπ‘‘π‘‘+πœ†π‘₯𝑦+π‘₯π‘¦βˆ’π‘₯πœβˆ’π½(𝑑)1ξ‚Έξ”πœ†ξ€Ί(π‘¦βˆ’π‘‘)2βˆ’π‘₯2ξ€»ξ‚Ήξ”πœ†ξ€Ί(π‘¦βˆ’π‘‘)2βˆ’π‘₯2ξ€»βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭,𝑑𝑑(2.8) where π½π‘˜[β‹…] is the first-kind Bessel function of the order π‘˜, πœβˆ’(𝑦)=𝑒(0βˆ’,𝑦), πœˆβˆ’(𝑦)=𝑒π‘₯(0βˆ’,𝑦).
We calculate 𝑒(βˆ’π‘¦/2,𝑦/2) in order to use condition (2.5): =1𝑒(βˆ’π‘¦/2,𝑦/2)2⎧βŽͺ⎨βŽͺβŽ©πœβˆ’(0)+πœβˆ’ξ€œ(𝑦)βˆ’π‘¦0πœˆβˆ’(𝑑)𝐽0ξ‚ƒβˆšξ‚„π‘¦πœ†π‘‘(π‘‘βˆ’π‘¦)𝑑𝑑+πœ†2ξ€œπ‘¦0πœβˆ’π½(𝑑)1ξ‚ƒβˆšξ‚„πœ†π‘‘(π‘‘βˆ’π‘¦)√⎫βŽͺ⎬βŽͺ⎭.πœ†π‘‘(π‘‘βˆ’π‘¦)𝑑𝑑(2.9) Considering the condition (2.5) and the following integral operator [23] π΅βˆšπ‘›,πœ†π‘šπ‘₯[]ξ€œπ‘“(π‘₯)=𝑓(π‘₯)+π‘₯π‘šξ‚€π‘“(𝑑)π‘₯βˆ’π‘šξ‚π‘‘βˆ’π‘š1βˆ’π‘›πœ•π½πœ•π‘₯0ξ‚ƒβˆšξ‚„πœ†(π‘‘βˆ’π‘š)(π‘‘βˆ’π‘₯)𝑑𝑑,π‘š,𝑛=0,1,(2.10) equality (2.9) can be written as follows πœ“11(𝑦)=2ξ‚»πœ“1(0)+𝐡√0,πœ†0𝑦[πœβˆ’]βˆ’ξ€œ(𝑦)𝑦0𝐡√1,πœ†0𝑑[πœˆβˆ’]ξ‚Ό(𝑑)𝑑𝑑.(2.11) Now we use an integral operator π΄βˆšπ‘›,πœ†π‘šπ‘₯[]ξ€œπ‘“(π‘₯)=𝑓(π‘₯)βˆ’π‘₯π‘šξ‚€π‘“(𝑑)π‘‘βˆ’π‘šξ‚π‘₯βˆ’π‘šπ‘›πœ•π½πœ•π‘‘0ξ‚ƒβˆšξ‚„πœ†(π‘₯βˆ’π‘š)(π‘₯βˆ’π‘‘)𝑑𝑑,π‘š,𝑛=0,1,(2.12) which is mutually inverse with the operator (2.10). Applying the operator (2.12) to both sides of (2.11), we obtain 𝐴√0,πœ†0π‘¦ξ€Ίπœ“1ξ€»=1(𝑦)2ξ‚»πœ“1(0)+𝐴√0,πœ†0π‘¦ξ‚†π΅βˆš0,πœ†0𝑦[πœβˆ’](𝑦)βˆ’π΄βˆš0,πœ†0π‘¦ξ‚»ξ€œπ‘¦0𝐡√1,πœ†0𝑑[πœˆβˆ’](𝑑)𝑑𝑑.(2.13) Considering the following properties of operators (2.10) and (2.12) 𝐴√0,πœ†0π‘¦ξ‚†π΅βˆš0,πœ†0𝑦[]𝑓(𝑦)=𝑓(𝑦),𝐴√0,πœ†0π‘¦ξ‚»ξ€œπ‘¦0𝐡√1,πœ†0𝑑[]ξ‚Ό=ξ€œπ‘“(𝑑)𝑑𝑑𝑦0𝑓(𝑑)𝐽0ξ‚ƒβˆšξ‚„πœ†(π‘¦βˆ’π‘‘)𝑑𝑑,(2.14) we derive 2𝐴√0,πœ†0π‘¦ξ€Ίπœ“1ξ€»(𝑦)=πœ“1(0)+πœβˆ’ξ€œ(𝑦)βˆ’π‘¦0πœˆβˆ’(𝑑)𝐽0ξ‚ƒβˆšξ‚„πœ†(π‘¦βˆ’π‘‘)𝑑𝑑.(2.15) Taking gluing conditions (2.7) into account, we have π·π›Όβˆ’10π‘¦πœˆ+(𝑦)=π·π›Όβˆ’10π‘¦πœ+(𝑦)βˆ’2𝐴√0,πœ†0π‘¦ξ€Ίπœ“1(𝑦)+πœ“1(0).(2.16)
Applying operator 𝐷1βˆ’π›Ό0𝑦 to both sides of (2.16) and considering the following composition rule [11]: π·π›Όπ‘Žπ‘‘π·π›½π‘Žπ‘‘π‘“(𝑑)=𝐷𝛼+π›½π‘Žπ‘‘π‘“(𝑑),𝛽≀0,(2.17) we get 𝜏+(𝑦)=𝜈+(𝑦)+πœ“βˆ—1(𝑦),0<𝑦<1,(2.18) where πœ“βˆ—1(𝑦)=𝐷1βˆ’π›Ό0𝑦{2𝐴√0,πœ†0𝑦[πœ“1(𝑦)]βˆ’πœ“1(0)}.
Let us consider the following auxiliary problem: 𝑒π‘₯π‘₯βˆ’π·π›Ό0π‘¦π‘’π‘’βˆ’πœ†π‘’=0,π‘₯(0,𝑦)=𝜈+(𝑦),𝑒(1,𝑦)=πœ“2(𝑦),lim𝑦→0𝑦1βˆ’π›Όπ‘’(π‘₯,𝑦)=πœ”(π‘₯).(2.19) Solution of this problem can be defined as [24] ξ€œπ‘’(π‘₯,𝑦)=10ξ€œπœ”(πœ‰)𝐺(π‘₯,𝑦,πœ‰,0)π‘‘πœ‰βˆ’π‘¦0𝜈++ξ€œ(πœ‚)𝐺(π‘₯,𝑦,0,πœ‚)π‘‘πœ‚π‘¦0πœ“2(πœ‚)πΊπœ‰ξ€œ(π‘₯,𝑦,1,πœ‚)π‘‘πœ‚βˆ’πœ†10ξ€œπ‘¦0𝑒(πœ‰,πœ‚)𝐺(π‘₯,𝑦,πœ‰,πœ‚)π‘‘πœ‰π‘‘πœ‚,(2.20) where 𝐺(π‘₯,𝑦,πœ‰,πœ‚)=(π‘¦βˆ’πœ‚)π›½βˆ’12βˆžξ“π‘›=βˆ’βˆžξ‚Έπ‘’1,𝛽1,π›½ξ‚΅βˆ’||||π‘₯βˆ’πœ‰+2𝑛(π‘¦βˆ’πœ‚)𝛽+𝑒1,𝛽1,π›½ξ‚΅βˆ’||||π‘₯+πœ‰+2𝑛(π‘¦βˆ’πœ‚)𝛽(2.21) is the Green function of the problem (2.19), 𝑒1,𝛽1,𝛽(𝑧)=Ξ¦(βˆ’π›½,𝛽,𝑧)=βˆžξ“π‘›=0𝑧𝑛𝑛!Ξ“(βˆ’π›½π‘›+𝛽)(2.22) is the function of Wright [25], 𝛽=𝛼/2.
Considering (2.20) as an integral equation regarding the function 𝑒(π‘₯,𝑦), we write solution via resolvent of the kernel πœ†πΊ(π‘₯,𝑦,πœ‰,πœ‚): ξ€œπ‘’(π‘₯,𝑦)=𝑃(π‘₯,𝑦)βˆ’π‘¦0𝜈+(πœ‚)𝐾1(π‘₯,𝑦,πœ‚)π‘‘πœ‚,(2.23) where ξ€œπ‘ƒ(π‘₯,𝑦)=10ξ€œπœ”(πœ‰)𝐺(π‘₯,𝑦,πœ‰,0)π‘‘πœ‰+𝑦0ξ€œ10ξ€œ10+ξ€œπœ”(πœ‰)𝐺(𝑠,𝑑,πœ‰,0)𝑅(π‘₯,𝑦,πœ‰,0)π‘‘πœ‰π‘‘π‘ π‘‘π‘‘π‘¦0πœ“2ξ‚ΈπΊξ€œ(πœ‚)(π‘₯,𝑦,1,πœ‚)+π‘¦πœ‚ξ€œ10𝐺𝐾(𝑠,𝑑,1,πœ‚)𝑅(π‘₯,𝑦,1,πœ‚)π‘‘π‘ π‘‘π‘‘π‘‘πœ‚,1ξ€œ(π‘₯,𝑦,πœ‚)=𝐺(π‘₯,𝑦,0,πœ‚)+π‘¦πœ‚ξ€œ10𝐺(𝑠,𝑑,0,πœ‚)𝑅(π‘₯,𝑦,0,πœ‚)𝑑𝑠𝑑𝑑,(2.24)𝑅(π‘₯,𝑦,πœ‰,πœ‚) is a resolvent of the kernel πœ†πΊ(π‘₯,𝑦,πœ‰,πœ‚).
From (2.23), tending π‘₯ to 0+, we obtain 𝑒0+ξ€Έ,𝑦=𝜏+ξ€·0(𝑦)=𝑃+ξ€Έβˆ’ξ€œ,𝑦𝑦0πœˆβˆ’(πœ‚)𝐾1ξ€·0+ξ€Έ,𝑦,πœ‚π‘‘πœ‚.(2.25) Considering functional relation (2.18), from (2.25) we get 𝜈+ξ€œ(𝑦)+𝑦0𝜈+(πœ‚)𝐾1(𝑦,πœ‚)π‘‘πœ‚=πœ“βˆ—1(𝑦)βˆ’π‘ƒ(0,𝑦).(2.26) Equality (2.26) is the second-kind Volterra-type integral equation regarding the function 𝜈+(𝑦). Since kernel 𝐾1(𝑦,πœ‚) has weak singularity and functions on the right-hand side are continuous, we can conclude that (2.26) is uniquely solvable [26], and solution can be represented as 𝜈+ξ€œ(𝑦)=Ξ¨(𝑦)+𝑦0Ξ¨(πœ‚)𝐾2(𝑦,πœ‚)π‘‘πœ‚,(2.27) where Ξ¨(𝑦)=πœ“βˆ—1(𝑦)βˆ’π‘ƒ(0,𝑦), 𝐾2(𝑦,πœ‚) is the resolvent of the kernel 𝐾1(𝑦,πœ‚).
Once we have obtained 𝜈+(𝑦), considering (2.18) or (2.25) we find function 𝜏+(𝑦). Then using gluing conditions (2.7) we find functions πœβˆ’(y), πœˆβˆ’(y). Finally, we can define solution of the considered problem by the formula (2.23) in the domain Ξ©1, by formula (2.8) in the domain Ξ©2.

Hence, we prove the following theorem.

Theorem 2.1. If πœ”(π‘₯)∈𝐢2[]0,1,πœ“π‘–(𝑦)∈𝐢1[]0,1∩𝐢2(0,1)(𝑖=1,2),(2.28) then there exists unique solution of the Problem AT and is defined by formulas (2.23) and (2.8) in the domains Ξ©1, Ξ©2, respectively.

3. Analog of the Gellerstedt Problem

We would like to note some related works. Regarding the consideration of Gellerstedt problem for parabolic-hyperbolic equations with constant coefficients we refer the readers to [3] and for loaded parabolic-hyperbolic equations work by Khubiev [27], and also for Lavrent’ev-Bitsadze equation [28].

Consider an equation𝑒0=π‘₯π‘₯βˆ’π·π›Ό0π‘¦π‘’βˆ’πœ†π‘’,Ξ¦0𝑒π‘₯π‘₯βˆ’π‘’π‘¦π‘¦+πœ†π‘’,Φ𝑖,(𝑖=1,2)(3.1) in the domain ⋃Φ=(2π‘˜=0Ξ¦π‘˜)βˆͺ𝐼0, where Ξ¦0 is a domain, bounded by segments 𝐴𝐴0, 𝐡𝐡0, 𝐴0𝐡0 of straight lines π‘₯=0, π‘₯=1, 𝑦=1, respectively; Ξ¦1 is a domain, bounded by the segment AE of the axe π‘₯ and by characteristics of (3.1) 𝐴𝐢1∢π‘₯+𝑦=0, 𝐸𝐢1∢π‘₯βˆ’π‘¦=π‘Ÿ; Ξ¦2 is a domain, bounded by the segment 𝐸𝐡 of the axe π‘₯ and by characteristics of (3.1) 𝐸𝐢2∢π‘₯βˆ’π‘¦=π‘Ÿ, 𝐡𝐢2∢π‘₯βˆ’π‘¦=1; 𝐼0 is an interval 0<π‘₯<1, 𝐼1 is an interval 0<π‘₯<π‘Ÿ, and 𝐼2 is an interval π‘Ÿ<π‘₯<1.

Problem AG
To find a solution of (3.1) from the class of functions π‘Š2=ξ‚†π‘’βˆΆπ·π›Όβˆ’10π‘¦ξ‚€π‘’βˆˆπΆΞ¦0,𝑒π‘₯π‘₯,𝐷𝛼0π‘¦ξ€·Ξ¦π‘’βˆˆπΆ0ξ€Έ,𝑒𝑦π‘₯,0Β±ξ€Έξ€·πΌβˆˆπ»0ξ€Έξ‚€,π‘’βˆˆπΆΞ¦π‘–ξ‚βˆ©πΆ2Φ𝑖,(𝑖=1,2)(3.2) satisfying boundary conditions 𝑒(0,𝑦)=πœ‘1(𝑦),𝑒(1,𝑦)=πœ‘2(𝑦),0≀𝑦≀1,(3.3)π‘’βˆ£π΄πΆ1ξ‚€π‘₯=𝑒2π‘₯,βˆ’2=πœ‘3(π‘₯),0≀π‘₯β‰€π‘Ÿ,(3.4)π‘’βˆ£πΈπΆ2ξ‚΅=𝑒(π‘₯+π‘Ÿ)2,(π‘Ÿβˆ’π‘₯)2ξ‚Ά=πœ‘4(π‘₯),π‘Ÿβ‰€π‘₯≀1,(3.5) together with gluing conditions lim𝑦→0+𝑦1βˆ’π›Όπ‘’(π‘₯,𝑦)=lim𝑦→0+𝑒(π‘₯,𝑦),π‘₯∈𝐼0,(3.6)lim𝑦→0+𝑦1βˆ’π›Όξ€·π‘¦1βˆ’π›Όξ€Έπ‘’(π‘₯,𝑦)𝑦=ξ€œπ‘₯0lim𝑦→0+𝑒𝑦(𝑑,𝑦)𝐽0ξ‚ƒβˆšξ‚„π›Ό(π‘₯βˆ’π‘‘)𝑑𝑑,π‘₯∈𝐼0⧡{r}.(3.7) Here πœ‘π‘—(β‹…)(𝑗=1,4) are given functions such as lim𝑦→0+𝑦1βˆ’π›Όπœ‘1(𝑦)=πœ‘3(0), limyβ†’0+𝑦1βˆ’π›Όπ‘’(π‘Ÿ,𝑦)=πœ‘4(π‘Ÿ).

Theorem 3.1. If the following conditions πœ†β‰₯0,πœ‘π‘–(𝑦)∈𝐢1[]0,1∩𝐢2(0,1),πœ‘π‘—(π‘₯)∈𝐢1ξ‚€πΌπ‘–ξ‚βˆ©πΆ2𝐼𝑖(𝑖=1,2;𝑗=3,4)(3.8) are fulfilled, then the Problem AG has a unique solution.

Proof. Introduce the following designations: lim𝑦→0+𝑦1βˆ’π›Όπ‘’(π‘₯,𝑦)=𝜏+(π‘₯),lim𝑦→0βˆ’π‘’(π‘₯,𝑦)=πœβˆ’(π‘₯),π‘₯∈𝐼0,lim𝑦→0+𝑦1βˆ’π›Όξ€·π‘¦1βˆ’π›Όξ€Έπ‘’(π‘₯,𝑦)𝑦=𝜈+(π‘₯),lim𝑦→0βˆ’π‘’π‘¦(π‘₯,𝑦)=πœˆβˆ’(π‘₯),π‘₯∈𝐼0.(3.9)
Solution of the Cauchy problem for (3.1) in the domain Φ𝑖 (𝑖=1,2) in case, when πœ†β‰₯0 has a form 1𝑒(π‘₯,𝑦)=2⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©πœβˆ’(π‘₯+𝑦)+πœβˆ’ξ€œ(π‘₯βˆ’π‘¦)+π‘₯+𝑦π‘₯βˆ’π‘¦πœˆβˆ’(𝑑)𝐽0ξ‚Έξ”πœ†ξ€Ί(π‘₯βˆ’π‘‘)2βˆ’π‘¦2ξ€»ξ‚Ήξ€œπ‘‘π‘‘+πœ†π‘¦π‘₯+𝑦π‘₯βˆ’π‘¦πœβˆ’π½(𝑑)1ξ‚Έξ”πœ†ξ€Ί(π‘₯βˆ’π‘‘)2βˆ’π‘¦2ξ€»ξ‚Ήξ”πœ†ξ€Ί(π‘₯βˆ’π‘‘)2βˆ’π‘¦2ξ€»βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭.𝑑𝑑(3.10) Using boundary conditions (3.4), (3.5), and gluing conditions (3.6), (3.7), from (3.10) we obtain 𝜈+(π‘₯)=𝜏+(π‘₯)+πœ‘βˆ—3(π‘₯),π‘₯∈𝐼1,(3.11)𝜈+(π‘₯)=𝜏+(π‘₯)+πœ‘βˆ—4(π‘₯),π‘₯∈𝐼2,(3.12) where πœ‘βˆ—3(π‘₯)=πœ‘3(0)βˆ’π΄βˆš0,πœ†0π‘₯ξ€Ί2πœ‘3(ξ€»π‘₯),πœ‘βˆ—4(π‘₯)=πœ‘4(π‘Ÿ)βˆ’π΄βˆš0,πœ†π‘Ÿπ‘₯ξ€Ί2πœ‘4(ξ€»π‘₯).(3.13) According to [10], tending 𝑦 to +0, from (3.1) we get 𝜈+1(𝑦)=Ξ“ξ€Ίπœ(1+𝛼)+ξ…žξ…ž(π‘₯)βˆ’πœ†πœ+ξ€»(π‘₯).(3.14) In order to prove the uniqueness of the solution for the Problem AG, we need estimate the following integral: ξ€œπ•€=10𝜏+(π‘₯)𝜈+(π‘₯)𝑑π‘₯.(3.15) Considering homogeneous case of the condition (3.3) and taking designation (3.9) into account, after some evaluations we derive ξ€œπ•€=βˆ’10ξ‚†ξ€Ίπœ+ξ…žξ€»(π‘₯)2ξ€Ίπœ+πœ†+ξ€»(π‘₯)2𝑑π‘₯.(3.16) If πœ†β‰₯0, then 𝕀≀0. On the other hand, if we consider homogeneous cases of (3.11) and (3.12), one can easily be sure that 𝕀β‰₯0. Hence, we get that 𝕀≑0. Based on (3.16) we can conclude that 𝜏+(π‘₯)=0 for all π‘₯∈𝐼0. Due to the solution of the first boundary problem [24] we can conclude that 𝑒(π‘₯,𝑦)≑0 in Ξ¦0. Further, according to the gluing conditions and the solution of Cauchy problem, we have 𝑒(π‘₯,𝑦)≑0 in Ξ¦.
Considering functional relations (3.11)–(3.14) and conditions (3.3)–(3.5), we get the following problems: 𝜏+ξ…žξ…ž(π‘₯)βˆ’(πœ†+Ξ“(1+𝛼))𝜏+(π‘₯)=πœ‘βˆ—3𝜏(π‘₯)Ξ“(1+𝛼),+(0)=πœ‘3(0),𝜏+(π‘Ÿ)=πœ‘4(π‘Ÿ),π‘₯∈𝐼1,𝜏(3.17)+ξ…žξ…ž(π‘₯)βˆ’(πœ†+Ξ“(1+𝛼))𝜏+(π‘₯)=πœ‘βˆ—4𝜏(π‘₯)Ξ“(1+𝛼),+(π‘Ÿ)=πœ‘4(π‘Ÿ),𝜏+(1)=lim𝑦→+0𝑦1βˆ’π›Όπœ‘2(𝑦),π‘₯∈𝐼1.(3.18) The problems (3.17) and (3.18) are model problems and can be solved directly. After the finding function 𝜏+(π‘₯) for all π‘₯∈𝐼0, functions 𝜈+(π‘₯) and πœβˆ’(π‘₯), πœˆβˆ’(π‘₯) can be defined by formulas (3.14) and (3.6), (3.7), respectively. Finally, solution of the Problem AG can be recovered by formulas (3.10) and (2.23) in the domains Φ𝑖 (𝑖=1,2) and Ξ¦0, respectively, but only with some changes in (2.23), precisely, Green function 𝐺(π‘₯,𝑦,πœ‰,πœ‚) should be replaced by πΊβˆ—(π‘₯,𝑦,πœ‰,πœ‚)=(π‘¦βˆ’πœ‚)π›½βˆ’12βˆžξ“π‘›=βˆ’βˆžξ‚Έπ‘’1,𝛽1,π›½ξ‚΅βˆ’||||π‘₯βˆ’πœ‰+2𝑛(π‘¦βˆ’πœ‚)π›½ξ‚Άβˆ’π‘’1,𝛽1,π›½ξ‚΅βˆ’||||π‘₯+πœ‰+2𝑛(π‘¦βˆ’πœ‚)𝛽,(3.19) which is the Green function of the first boundary problem for the (3.1) in Ξ¦0 [24].
Theorem 3.1 is proved.

Acknowledgment

This paper is partially supported by the foundation ERASMUSMUNDUS (Project number 155778-EM-1-2009-1-BEERASMUNMUNDUS-ECW-LO9).