Abstract
In this paper we study a boundary value problem with the Poincare–Tricomi condition for a degenerate partial differential equation of elliptic-hyperbolic type of the second kind. In the hyperbolic part of a degenerate mixed differential equation of the second kind the line of degeneracy is a characteristic. For this type of differential equations a class of generalized solutions is introduced in the characteristic triangle. Using the properties of generalized solutions, the modified Cauchy and Dirichlet problems are studied. The solutions of these problems are found in the convenient form for further investigations. A new method has been developed for a differential equation of mixed type of the second kind, based on energy integrals. Using this method, the uniqueness of the considering problem is proved. The existence of a solution of the considering problem reduces to investigation of a singular integral equation and the unique solvability of this problem is proved by the Carleman–Vekua regularization method.
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1 INTRODUCTION
Degenerate partial differential equations occupy one of the central places in the theory of general partial differential equations and have numerous applications in various branches of science and technology. Partial differential equations of mixed type with degenerations have been systematically studied since the middle of the last century after the well-known works of F. I. Frankl, which are reflected in [1]. He showed applications of degenerate mixed type differential equations in solving problems of transonic and supersonic gas dynamics. Later, another applications were find of degenerate differential equations of mixed type in other fields of science and technology.
I. N. Vekua in [2] showed the importance of studying mixed-type differential equations in solving problems of the theory of infinitesimal bending of surfaces. The problem of the outflow of a supersonic jet from a vessel with flat walls reduces to the Tricomi problem for the Chaplygin equation. There are a number of works in which the problems of Tricomi, Gellerstedt, and Bitsadze are studied. It is easy to meet works, where new correct problems with bias are posed for equations of elliptic-hyperbolic and parabolic-hyperbolic types of the first kind, for which the line of degeneration is not a characteristic [3–19].
A feature of degenerate hyperbolic differential equations is that for this kind of equations the Cauchy problem with initial condition on the line of parabolic degeneracy does not always hold. For example, the Cauchy problem in the usual formulation may turn out to be unsolvable, if the hyperbolic equations degenerate along a line that is simultaneously a characteristic. Such type of differential equations are called degenerate equations of the second kind.
Note that the solution of the Cauchy problem with initial condition
for a differential equation
on the negative half-axis \(y<0\) has a representation [19, pages 259–260]:
where \(-1<2\beta<0\) for \(-1<m<0\) and
Definition 1. The function \(u\left(x,y\right)\) (represented by (2)) on the negative half-axis \(y<0\) is called a generalized solution from the class \(\mathbb{R}_{2}\) to the Cauchy problem for differential equation (1), if \(\tau\left(z\right)\) is representable in the following integral form \(\tau\left(z\right)=\int\limits_{0}^{z}\left(z-t\right)^{-2\beta}T\left(t\right)dt,\) where \(\nu\left(z\right)\) and \(T\left(z\right)\) are continuous and integrable functions on the interval \(\left(0,1\right)\).
In [20] a generalized solution was obtained from the class \(\mathbb{R}_{2}\) to the Cauchy problem for a hyperbolic differential equation of the second kind. This generalized solution has a form
where
Using this solution representation form, in [21–27] local and nonlocal boundary value problems for mixed differential equations of the second kind were studied. It is proved that such type of problems arise in studying some problems of mathematical biology [28] and physics [29].
In [30] for the differential equation \(u_{xx}+yu_{yy}+\left(\alpha+\frac{1}{2}\right)u_{y}=0\) built a special class of solutions in the case, when \(y<0,\quad\alpha\in\left(-\frac{1}{2},0\right)\cup\left(0,\frac{1}{2}\right)\).
In the hyperbolic part of the mixed domain, solvability of local and nonlocal boundary value problems are studied in the class \(\mathbb{R}_{2}\), and in the elliptic part of the domain the questions of classical solvability are studied. These problems are equivalently reduced to singular integral equations. Then by the regularization method they are reduced to study Fredholm integral equations of the second kind.
Note that boundary value problems for mixed differential and integro-differential equations in rectangular domains were studied in [31–34]. Boundary value problems with the Poincare–Tricomi condition for degenerate differential equations of elliptic and elliptic-hyperbolic types of the second kind have been studied, relatively little. We indicate here only the papers [35–36].
In this presented our paper by the aid of properties of generalized functions we study a boundary value problem with the Poincare–Tricomi condition for an elliptic-hyperbolic differential equation of the second kind.
2 FORMULATION OF THE PROBLEM
We consider the following mixed differential equation
in the domain \(D=D_{1}\cup D_{2}\), where \(D_{1}\) is domain bounded by a curve \(\sigma\) for \(y>0\) with the end points \(A(0,0)\), \(B(1,0)\) and with segment \(AB(y=0)\), \(D_{2}\) is domain bounded by segment \(AB\) and by characteristics
We introduce the notations
for
Problem \(\boldsymbol{PT}\). Find the function \(u(x,y)\), for which are true the following properties:
1) \(u(x,y)\in C\left(\overline{D}\right)\cup C^{1}\left(D\cup\sigma\cup J\right)\), there \(u_{x}\) can goes to infinity of order less than one units at the point \(A\left(0,0\right)\) and \(u_{y}\) can goes to infinity of order less than \(-2\beta\) at the point \(B\left(1,0\right)\);
2) function \(u(x,y)\in C^{2}\left(\overline{D}_{1}\right)\) is a regular solution of the differential equation (3) in the domain \(D_{1}\), and is generalized solution from the class \(\mathbb{R}_{2}\) in the domain \(D_{2}\);
3) is fulfilled the gluing condition \(u_{y}(x,-0)=-u_{y}(x,+0);\)
4) \(u(x,y)\) satisfies the boundary value conditions
where \(\frac{dx}{ds}=-\cos(n,y)\), \(\frac{dy}{ds}=\cos(n,x)\), \(n\) is external normal to the curve \(\sigma\), \(l\) is a length of the entire curve \(\sigma\), \(s\) is the arc length of the curve \(\sigma\), measured from point \(B\left(1,0\right)\); \(\delta(s)\), \(\rho(s)\), \(\varphi(s)\), \(\psi(x)\) are given sufficiently smooth functions and \(\psi(x)\in C^{1}\left[0,\frac{1}{2}\right]\cap C^{2}\left(0;\frac{1}{2}\right)\), \(\varphi(l)=\psi(0)=0\).
We note that if we set \(\delta(s)=0\) (\(\rho(s)=0\)), then the problem \(PT\)coincided with the problem \(T\left(T_{N}\right)\) for elliptic-hyperbolic equation of the first kind (see [37, pp. 177–185]). Therefore, we will assume in our work that \(\delta(s)\neq 0\), \(\rho(s)\neq 0\).
We assume that the curve \(\sigma\) satisfies the following conditions:
1) functions \(x(s)\) and \(y(s)\) are parametric equations of curve \(\sigma\), have continuous derivatives \(x^{\prime}(s)\), \(y^{\prime}(s)\) (these derivatives are nonzero simultaneously) and have second derivatives, which satisfy the Hölder condition of the order \(\kappa\left(0<\kappa<1\right)\) on the segment \(0\leq s\leq l\);
2) in the neighborhood of the end points of the curve \(\sigma\) the inequality is true:
3 UNIQUENESS OF THE SOLUTION OF THE PROBLEM \(PT\)
Theorem 1. If (4) and the following conditions are fulfilled
Then the solution of the problem \(PT\) is unique in the domain \(D\).
Proof. We prove the theorem by the method of energy integrals. Let \(u\left(x,y\right)\) be a twice continuously differentiable solution of equation (3) in the domain
where \(D_{1}^{\varepsilon_{1},\varepsilon_{2}}\) is domain with border
strictly lying in the domain \(D_{1}\), while \(D_{2}^{\varepsilon_{1},\varepsilon_{2}}\) is domain bounded with lines
where \(\varepsilon_{1},\varepsilon_{2}\) are small positive numbers.
In the domain \(D_{2}\) differential equation (3) takes the form \((-y)^{m}u_{xx}-u_{yy}=0.\) It is easy to check that the following identity holds:
Integrating this identity over the domain \(D_{2}^{\varepsilon_{1},\varepsilon_{2}}\), we derive
Applying Green formula [35] to the first integral on the right-hand side of (10), we obtain
Calculating the first integral on the right-hand side of the last equality, taking into account the condition (6) on the characteristic \(AC\), we obtain
Consequently, we have
Passing to the limits as \(\varepsilon_{1}\to 0\) and \(\varepsilon_{2}\to 0\), taking the gluing condition \(u_{y}(x,-0)=-u_{y}(x,+0)\) into account we obtain
On the characteristic \(BC\) we have \(dx=(-y)^{\frac{m}{2}}dy\). Therefore we have
Integrating in parts the last integral of (11) with conditions \(\left.u\right|_{AC}=0\) and (9), we obtain
For \(-1<m<0\) from (11) yields
Now we show that the first integral on the right-hand side of equality (10) is not positive. Passing to characteristic variables
we obtain
where \(\Delta_{1}=\left\{\left(\xi,\eta\right):0<\xi<1,\ \xi<\eta<1\right\}\) is image of the domain \(D_{2}\) in coordinates \(\left(\xi,\eta\right)\).
In domain \(\Delta_{1}\) the differential equation (3) takes the form \(u_{\xi\eta}-\frac{\beta}{\eta-\xi}(u_{\eta}-u_{\xi})=0.\) Multiplying both sides of this equation by \(u_{\eta}\), we have
Substitute (14) in (13). As a result yields
Integrating the last integral in parts, we have
Since the last term in the square bracket is zero for \(\eta=\xi\), then we obtain
Here by virtue of
for \(-1<m<0\) yields
By virtue of inequalities (12) and (15), from (10) we obtain
Differential equation (3) in domain \(D_{1}\) has the form \(y^{m}u_{xx}+u_{yy}=0\). Integrating the following identity
over the domain \(D_{1}^{\varepsilon_{1},\varepsilon_{2}}\subset D_{1}\), we derive
Applying Green formula [35] to the first integral on the right-hand side of the last equality, we obtain
Hence by virtue of \(dy=0\) on \(AB\) and \(dy=\cos(n,x)ds\), \(dx=-\cos(n,y)ds\), we obtain
where \(x_{1}\), \(x_{2}\) are abscissas of the intersection points of line \(y=\varepsilon_{2}\) with curve \(\sigma_{\varepsilon_{1}}\).
By virtue of condition 1) of the problem \(PT\), zero values \(\varphi(s)\equiv b(x)\equiv 0\) and (5), from (17) as \(\varepsilon_{1}\to 0\), \(\varepsilon_{2}\to 0\) we obtain
By virtue of inequalities (8) and (16), from (18) implies that \(u_{x}=u_{y}=0\) in \(D_{1}\), i.e. \(u(x,y)={\textrm{const}}\) for all \((x,y)\in D_{1}\). Since, each term of equality (18) is non-negative, then \(u(x,y)=0\) on \(\overline{\sigma}\). By the aid of the Hopf principle [19, pages 44–48], we conclude that \(u(x,y)\equiv 0\) in \(\bar{D}_{1}\) for \(\delta(s)\neq 0\). Now from the uniqueness of the solution of Cauchy problem it follows that \(u(x,y)\equiv 0\) in \(\bar{D}_{2}\). Consequently, \(u(x,y)\equiv 0\) in \(\overline{D}\). This is ended the proof of uniqueness of the solution of the problem \(PT\). Theorem 1 is proved. \(\Box\)
Remark. Uniqueness of the solution of the problem \(PT\) for \(\rho(s)\neq 0\), \(\forall s\in\left[0,l\right]\) be proved by the principle of extremum.
4 BASIC FUNCTIONAL RELATIONSHIPS
We introduce the following notations:
In studying this problem \(PT\) an important role are played functional relations between \(\nu^{\pm}\left(x\right)\) and \(\tau\left(x\right)\), which were bring from elliptical and hyperbolic parts of the domain \(D\).
A generalized solution from class \(\mathbb{R}_{2}\) to the Cauchy problem with conditions (19), (20) for differential equation (3) in the domain \(D_{2}\) is given by the formula [37, p. 230, form. 27.5]:
where
\(T\left(x\right)\) and \(\nu^{-}\left(x\right)\) are functions of continuous in \(\left(0,1\right)\) and integrable on \(\left[0,1\right]\), \(\tau\left(x\right)\) is zero of the order not less then \(-2\beta\) as \(x\to 0\). Putting \(\xi=0\) in (21) and taking into account (6), (22), we obtain
This is first kind Volterra integral equation with respect to \(N(\zeta)\). We set
Then we have
We apply a fractional order differential operator \(D_{0x}^{\alpha}f(x)\) to (26):
We apply the differential operator \(D_{0\eta}^{1-\beta}\) to both sides of equation (27) taking into account \(D_{0\eta}^{1-\beta}D_{0\eta}^{\beta-1}\Phi\left(\eta\right)=\Phi\left(\eta\right)\). Then we have
By direct verification it is easy to check that function (28) is a solution of the equation (26). Taking into account (25) and (28) from (23) we obtain the first functional relation between \(T(x)\) and \(\nu^{-}(x)\), which are bringing from the \(D_{2}\) to the domain \(J\):
where \(\gamma_{3}=2\gamma_{2}\cos\pi\beta\).
In the positive half-plane \(y>0\) the differential equation (3) takes the form
Consider the following auxiliary problem.
Problem \(\boldsymbol{PT}^{\mathbf{+}}\). Find in domain \(D_{1}\) a solution \(u(x,y)\in C(\overline{D}_{1})\cap C^{1}\big{(}D_{1}\cup\sigma\cup J\big{)}\cap C^{2}\big{(}D_{1}\big{)}\) of the equation (30), satisfying to the boundary value conditions (5) and (19).
Solution of the problem \(PT^{+}\) with conditions (5) and (19) for differential equation (3) in domain \(D_{1}\) exists, unique and has the form (see [37, page 179]):
where \(G_{2}(\xi,\eta;x,y)\) is Green function of the problem \(PT^{+}\) for equation (3) and
\(G_{02}(\xi,\eta;x,y)\) is the Green function of the problem \(PT^{+}\) for equation (3) in normal domain \(D_{0}\), which bounded with segment \(\overline{AB}\) and normal curve
\(\lambda_{2}\left(s;\xi,\eta\right)\) is solution of the integral equation
\(q_{2}\left(x,y,x_{0},y_{0}\right)\) is fundamental solution of differential equation (3) and
\(F\left(a,b,c;z\right)\) is Gauss hypergeometric function.
We differentiate (31) with respect to \(y\). Then taking (32) and (33) into account as \(y\to 0\) we obtain the functional relation between \(\tau(x)\) and \(\nu(x)\), which are bringing from the domain \(D_{1}\) to \(J\):
where \(\chi(s)\) is solution of the integral equation
Substituting (24) into 34 and taking into account the identities:
we obtain the functional relationship between \(T(x)\) and \(\nu^{+}(x)\), which are bringing from \(D_{1}\) to domain \(J\):
5 EXISTENCE OF THE SOLUTION OF PROBLEM \(PT\)
Definition 2 [38, pages 255–259]. We say that the solution \(\omega(z)\) of a singular integral equation
belongs to class \(h(0)\), if this function \(\omega(z)\) is bounded as \(z\to 0\) and unbounded as \(z\to 1\).
Theorem 2. If conditions (4) and (7) are satisfied, then a solution of problem \(PT\) exists in the domain \(D\).
Proof. Eliminating \(\nu^{-}(x)\) and \(\nu^{+}(x)\) from relations (29) and (35), taking into account (20) and the gluing condition \(\nu^{-}(x)=-\nu^{+}(x)\), we have
where \(\gamma_{4}=\frac{\cos\beta\pi}{\pi\left(\sin\beta\pi-1\right)}\), \(\tilde{T}(x)=x^{1-2\beta}T(x)\),
We study the kernel and the right side of the singular integral equation (36). For \(0<x<1\) and \(0<z<1\) it is true the inequality [37, page 181]:
where \(C_{1}={\textrm{const}}\) in \(D_{1}\). By virtue of (39), from (37) we obtain
By changing the variable \(z=t\) \((1-\sigma)\) and using the integral representation of the hypergeometric function [37, § 2, form. 2.10], from (40) we obtain
Since \(c-a-b=2-2\beta-2+4\beta=2\beta<0\), then by the aid of the formula:
and by the estimate
from (41) for \(0\leq t\leq 1\) we come to an estimate
Now we estimate the right-hand side of equation (36). Differentiating (33) with respect to \(y\), then putting \(y=0\), we obtain
Substituting (43) into (38), we have
By virtue of properties of the functions \(\psi\left(x\right)\) and \(\varphi\left(s\right)\), it follows from (44) that the function \(F\left(x\right)\) has derivatives of any order in the interval \((0,1)\). Let us find out the behavior of the function \(F\left(x\right)\) and its derivative as \(x\to 0\) and as \(x\to 1\). Consider the expression
We estimate the expression (45). By virtue of \(\delta(s),\rho(s),\varphi(s)\in C\left[0,l\right]\), for enough small \(x>0\) there hold inequalities
Hence, by virtue of (26), for sufficiently small \(\varepsilon>0\) we obtain
By the change \(\mu^{2}=\omega\) in (46), using the integral representation of the hypergeometric function [37, § 2, form. 2.10] and taking into account (42) we obtain
If \(1-x\) is sufficiently small, by the similarly way we obtain
Carrying out the same reasoning, we obtain
By virtue of (47)–(49), from (44) we deduce that \(F(x)\in C(\overline{J})\cap C^{2}(J)\) and the function \(F^{\prime}(x)\) turns to infinity order less then \(2\beta+1\) as \(x\to 1\), and as \(x\to 0\) this derivative \(F^{\prime}(x)\) is bounded.
By the change of variables \(\zeta=\frac{t^{2}}{1-2t+2t^{2}},\) \(z=\frac{x^{2}}{1-2x+2x^{2}},\) we represent the equation (36) as follows
where
Since \(1-\gamma_{4}^{2}\neq 0\), then equation (50) is a normal type equation. Its index is zero in the class \(h(0)\), i.e. in the class of functions, which bounded as \(z\to 0\) and unbounded as \(z\to 1\).
Thus, we studied the solution \(\omega(z)\) of singular integral equation (50) in the class \(h(0)\).
We reduce the singular integral equation (50) by the well-known Carleman–Vekua regularization method [38] to an equivalent Fredholm equation of the second kind, the solvability of which implies from the uniqueness of the solution of problem \(PT\).
From the equality \(\tilde{T}(x)=x^{1-2\beta}T(x)\) and \(\omega(z)=(1-2x+2x^{2})\tilde{T}(x)\) we find the function \(T(x)\), which is continuous in \((0,1)\) and integrable on \(\left[0,1\right]\).
Substituting \(T(x)\) into (24), we find \(\tau(x)\). Then from (29) and (35) we find \(\nu^{\pm}(x)\). When \(\tau(x)\) is known function, then the solution of the problem \(PT\) for differential equation (3) in the domain \(D_{1}\) we restore as a solution of the problem \(PT^{+}\) for differential equation (3) with conditions (5) and (19). The solution of the problem \(PT\) for differential equation (3) in the domain \(D_{2}\) we restore as a generalized solution of the Cauchy problem with conditions (19) and (20) for differential equation (3).
Thus, in the domain \(D\) a solution of the problem \(PT\) exists. Theorem 2 is proved. \(\Box\)
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Yuldashev, T.K., Islomov, B.I. & Abdullaev, A.A. On Solvability of a Poincare–Tricomi Type Problem for an Elliptic–Hyperbolic Equation of the Second Kind. Lobachevskii J Math 42, 663–675 (2021). https://doi.org/10.1134/S1995080221030239
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DOI: https://doi.org/10.1134/S1995080221030239