# On the Volterra property of a boundary problem with integral gluing condition for a mixed parabolic-hyperbolic equation

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## Abstract

In the present work, we consider a boundary value problem with gluing conditions of an integral form for the parabolic-hyperbolic type equation. We prove that the considered problem has the Volterra property. The main tools used in the work are related to the method of the integral equations and functional analysis.

## Keywords

Strong Solution Unique Solvability Mixed Type Equation Tricomi Problem Hyperbolic Part## Introduction

The theory of mixed type equations is one of the principal parts of the general theory of partial differential equations. The interest for these kinds of equations arises intensively because of both theoretical and practical uses of their applications. Many mathematical models of applied problems require investigations of this type of equations.

The actuality of the consideration of mixed type equations has been mentioned, for the first time, by S. A. Chaplygin in 1902 in his famous work ‘On gas streams’ [1]. The first fundamental results in this direction was obtained in 1920-1930 by Tricomi [2] and Gellerstedt [3]. The works of Lavrent’ev [4], Bitsadze [5, 6], Frankl [7], Protter [8, 9] and Morawetz [10], have had a great impact in this theory, where outstanding theoretical results were obtained and pointed out important practical values of them. Bibliography of the main fundamental results on this direction can be found, among others, in the monographs of Bitsadze [6], Berezansky [11], Bers [12], Salakhitdinov and Urinov [13] and Nakhushev [14].

In most of the works devoted to the study of mixed type equations, the object of study was mixed elliptic-hyperbolic type equations. Comparatively, few results have been obtained on the study of mixed parabolic-hyperbolic type equations. However, this last type of equations have also numerous applications in the real life processes (see [15] for an interesting example in mechanics). The reader can find a nice example given, for the first time, by Gelfand in [16], and connect with the movement of the gas in a channel surrounded by a porous environment. Inside the channel, the movement of gas was described by the wave equation and outside by the diffusion one. Mathematic models of this kind of problems arise in the study of electromagnetic fields, in a heterogeneous environment, consisting of dielectric and conductive environment for modeling the movement of a little compressible fluid in a channel surrounded by a porous medium [17]. Here, the wave equation describes the hydrodynamic pressure of the fluid in the channel, and the equation of filtration-pressure fluid in a porous medium. Similar problems arise in the study of the magnetic intensity of the electromagnetic field [17].

In the last few years, the investigations on local boundary value problems, for mixed equations in domains with non-characteristic boundary data, were intensively increased. We point out that the studies made on boundary value problems for equations of mixed type, in domains with deviation from the characteristics (with a non-characteristic boundary), have originated with the fundamental works of Bitsadze [5], where the generalized Tricomi problem (Problem M) for an equation of mixed type is discussed.

In the works [18] and [19], the analog to the Tricomi problem for a modeled parabolic-hyperbolic equation, was investigated in a domain with a non-characteristic boundary in a hyperbolic part. Moreover, the uniqueness of solution and the Volterra property of the formulated problem was proved. We also refer to the recent works devoted to the study of parabolic-hyperbolic equations [20, 21, 22, 23].

In the last years, the interest for considering boundary value problems of parabolic-hyperbolic type, with integral gluing condition on the line of type changing, is increasing [24, 25].

In the present work, we study the analog to the generalized Tricomi problem with an integral gluing condition on the line of type changing. We prove that the formulated problem has the Volterra property. The obtained result generalizes some previous ones from Sadybekov and Tajzhanova given in [28].

## Formulation of the problem

Now we state the problem that we will consider along the paper:

**Problem B**To find a solution of Eq. (1), satisfying boundary conditions

where *Q* is a given function such that $Q\in {C}^{1}([0,1]\times [0,1])$, and $\alpha ,\beta \in \mathbb{R}$ satisfy ${\alpha}^{2}+{\beta}^{2}>0$.

When the curve *AC* coincides with the characteristic one $x+y=0$, $\alpha =1$ and $\beta =0$, the Problem B is just the Tricomi problem for parabolic-hyperbolic equation with a non-characteristic line of type changing, which has been studied in [26].

Regular solvability of the Problem B with continuous gluing conditions ($\alpha =1$, $\beta =0$) have been proved, for the first time, in [27], and strong solvability of this problem was proved in the work [28].

Several properties, including the Volterra property of boundary problems for mixed parabolic-hyperbolic equations, have been studied in the works [29, 30, 31, 32, 33].

We denote the parabolic part of the mixed domain Ω as ${\mathrm{\Omega}}_{0}$ and the hyperbolic part by ${\mathrm{\Omega}}_{1}$.

that satisfies Eq. (1) in the domains ${\mathrm{\Omega}}_{0}$ and ${\mathrm{\Omega}}_{1}$, the boundary conditions (3)-(4), and the gluing condition (5).

Regarding the curve *AC*, we assume that $x+\gamma (x)$ is monotonically increasing. Then, rewriting it by using the characteristic variables $\xi =x+y$ and $\eta =x-y$, we have that the equation of the curve *AC* can be expressed as $\xi =\lambda (\eta )$, $0\le \eta \le 1$.

## Main result

**Theorem 1** *Let* $\gamma \in {C}^{1}[0,l]$ *and* $Q\in {C}^{1}([0,1]\times [0,1])$. *Then for any function* $f\in {C}^{1}(\overline{\mathrm{\Omega}})$, *there exists a unique regular solution of the Problem * B.

*Proof*By a regular solution of the Problem B in the domain ${\mathrm{\Omega}}_{1}$ we look for a function that fulfills the following expression:

*AB*from the parabolic part of the domain, implies that

Hence, the Problem B is equivalent, in the sense of unique solvability, to the second kind Volterra integral equation (14).

The restrictions imposed on the functions *γ*, *Q*, and the right-hand side of Eq. (1) guarantees that, by virtue of (11) and (15), the kernel ${k}_{1}(x,t)$ is a kernel with weak singularity. So, we have that Eq. (14) has a unique solution and ${\tau}^{\prime}\in {C}^{1}(0,1)$. Since $\tau (0)=0$, we deduce the uniqueness of the function *τ*. Equation (8) gives us the uniqueness of function ${\nu}_{1}$ and, as consequence, we deduce, from Eq. (6), the uniqueness of solution of Problem B when $\alpha \ne 0$.

Consider now the other case, *i.e.* $\alpha =0$ and $\beta \ne 0$.

where $R(x,z)$ is the resolvent kernel of (18).

As a consequence, arguing as in the case $\alpha \ne 0$, we deduce, from Eq. (6), the uniqueness of solution of Problem B for $\alpha =0$ and $\beta \ne 0$, and the result is proved. □

In the sequel, we will deduce the exact expression of the integral kernel related to the unique solution of Problem B.

where ${\mathrm{\Gamma}}_{1}(x,t)=1+{\int}_{t}^{x}\mathrm{\Gamma}(z,t)\phantom{\rule{0.2em}{0ex}}dz$.

Thus, we have partially proved the following lemma.

**Lemma 1**

*The unique regular solution of Problem*B

*can be represented as follows*:

*where*$K(x,y,{x}_{1},{y}_{1})\in {L}_{2}(\mathrm{\Omega}\times \mathrm{\Omega})$

*and*

*Proof* Expression (25) has been proved before. Let us see that $K(x,y;{x}_{1},{y}_{1})\in {L}_{2}(\mathrm{\Omega}\times \mathrm{\Omega})$.

is bounded.

As a consequence, $K(x,y;{x}_{1},{y}_{1})\in {L}_{2}(\mathrm{\Omega}\times \mathrm{\Omega})$ and Lemma 1 is completely proved. □

We have the following regularity result for this function.

**Lemma 2** *If* $f\in {C}^{1}(\overline{\mathrm{\Omega}})$, $f(0,0)=0$ *and* $Q\in {C}^{1}([0,1]\times [0,1])$, *then* ${F}_{\alpha \beta}\in {C}^{1}[0,1]$ *and* ${F}_{\alpha \beta}(0)=0$.

*Proof* Using the explicit form of the Green’s function given in (10), it is not complicated to prove that function ${F}_{\alpha \beta}$, defined by formulas (16) and (19), belongs to the class of functions ${C}^{1}[0,1]$ and ${F}_{\alpha \beta}(0)=0$.

Lemma 2 is proved. □

**Lemma 3**

*Suppose that*$Q\in {C}^{1}([0,1]\times [0,1])$

*and*$f\in {L}_{2}(\mathrm{\Omega})$,

*then*${F}_{\alpha \beta}\in {L}_{2}(\mathrm{\Omega})$

*and*

*Proof*Consider the following problem in ${\mathrm{\Omega}}_{0}$:

Obviously, we have that ${F}_{0}(x)={lim}_{y\to 0}{\omega}_{y}(x,y)$.

Now, by virtue of the conditions of Lemma 3 and the representations (16) and (19), from expression (30) and the Cauchy-Bunjakovskii inequalities, we get the estimate (26) and conclude the proof. □

Denote now ${\parallel \cdot \parallel}_{l}$ as the norm of the Sobolev space ${H}^{l}(\mathrm{\Omega})\equiv {W}_{2}^{l}(\mathrm{\Omega})$ with ${W}_{2}^{0}(\mathrm{\Omega})\equiv {L}_{2}(\mathrm{\Omega})$.

**Lemma 4**

*Let*

*u*

*be the unique regular solution of Problem*B.

*Then the following estimate holds*:

*Here*, *c* *is a positive constant that does not depend on* *u*.

*Proof*By virtue of Lemma 3, and from (20) and (21), we deduce that

The result follows from expression (22). □

**Definition 1** We define the set *W* as the set of all the regular solutions of Problem B.

A function $u\in {L}_{2}(\mathrm{\Omega})$ is said to be a strong solution of Problem B, if there exists a functional sequence $\{{u}_{n}\}\subset W$, such that ${u}_{n}$ and $L{u}_{n}$ converge in ${L}_{2}(\mathrm{\Omega})$ to *u* and *f*, respectively.

Define $\mathbb{L}$ as the closure of the differential operator $\mathbb{L}:W\to {L}_{2}(\mathrm{\Omega})$, given by expression (2).

Note that, according to the definition of the strong solution, the function *u* will be a strong solution of Problem B if and only if $u\in D(\mathbb{L})$.

Now we are in a position to prove the following uniqueness result for strong solutions.

**Theorem 2** *For any function* $Q\in {C}^{1}([0,1]\times [0,1])$ *and* $f\in {L}_{2}(\mathrm{\Omega})$, *there exists a unique strong solution* *u* *of Problem * B. *Moreover*, $u\in {W}_{2}^{1}(\mathrm{\Omega})\cap {W}_{x,y}^{1,2}({\mathrm{\Omega}}_{1})\cap C(\overline{\mathrm{\Omega}})$, *satisfies inequality* (31) *and it is given by the expression* (25).

*Proof* Let ${C}_{0}^{1}(\overline{\mathrm{\Omega}})$ be the set of the ${C}^{1}(\overline{\mathrm{\Omega}})$ functions that vanish in a neighborhood of *∂* Ω (*∂* Ω is a boundary of the domain Ω). Since ${C}_{0}^{1}(\overline{\mathrm{\Omega}})$ is dense in ${L}_{2}(\mathrm{\Omega})$, we have that for any function $f\in {L}_{2}(\mathrm{\Omega})$, there exist a functional sequence ${f}_{n}\in {C}_{0}^{1}(\overline{\mathrm{\Omega}})$, such that $\parallel {f}_{n}-f\parallel \to 0$, as $n\to \mathrm{\infty}$.

It is not difficult to verify that if ${f}_{n}\in {C}_{0}^{1}(\overline{\mathrm{\Omega}})$ then ${F}_{\alpha \beta n}\in {C}^{1}([0,1])$ (with obvious notation). Therefore, Eqs. (14) and (18) can be considered as a second kind Volterra integral equation in the space ${C}^{1}([0,1])$. Consequently, we have that ${\tau}_{n}^{\prime}(x)={u}_{nx}(x,0)\in {C}^{1}[0;1]$. Due to the properties of the solutions of the boundary value problem for the heat equation in ${\mathrm{\Omega}}_{0}$ and the Darboux problem, by using the representations (6) and (9), we conclude that ${u}_{n}\in W$ for all ${f}_{n}\in {C}_{0}^{1}(\overline{\mathrm{\Omega}})$.

Consequently, $\{{u}_{n}\}$ is a sequence of strong solutions, hence, Problem B is strongly solvable for all right hand $f\in {L}_{2}(\mathrm{\Omega})$, and the strong solution belongs to the space ${W}_{2}^{1}(\mathrm{\Omega})\cap {W}_{x,y}^{1,2}({\mathrm{\Omega}}_{1})\cap C(\overline{\mathrm{\Omega}})$. Thus, Theorem 2 is proved. □

and *K* defined in Lemma 1.

**Lemma 5**

*For the iterated kernels*${K}_{n}(x,y;{x}_{1},{y}_{1})$

*we have the following estimate*:

*where* Open image in new window *and* Γ *is the Gamma*-*function of Euler*.

*Proof* The proof will be done by induction in *n*.

is automatically deduced.

which proves Lemma 5. □

Now we are in a position to prove the final result of this paper, which gives us the Volterra property for the inverse of operator $\mathbb{L}$.

**Theorem 3**

*The integral operator defined in the right hand of*(25),

*i*.

*e*.

*has the Volterra property* (*it is almost continuous and quasi*-*nilpotent*) *in* ${L}_{2}(\mathrm{\Omega})$.

*Proof*Since the continuity of this operator follows from the fact that $K\in {L}_{2}(\mathrm{\Omega}\times \mathrm{\Omega})$. To prove this theorem, we only need to verify that operator ${\mathbb{L}}^{-1}$, defined by (33), is quasi-nilpotent,

*i.e.*

From the last equality, one can state the validity of the equality (34) and Theorem 3 is proved. □

**Consequence 1** Problem B has the Volterra property.

**Consequence 2**For any complex number

*λ*, the equation

is uniquely solvable for all $f\in {L}_{2}(\mathrm{\Omega})$.

which is a second kind of Volterra equation. This proves Consequence 2 of Theorem 3.

## Notes

### Acknowledgements

This research was partially supported by Ministerio de Ciencia e Innovación-SPAIN, and FEDER, project MTM2010-15314 and KazNPU Rector’s grant for 2013.

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