UDC 517.946

MATHEMATICS

Kh. G. Bzhikhatlov, A. M. Nakhushev

ON A BOUNDARY-VALUE PROBLEM FOR AN EQUATION OF MIXED PARABOLIC-HYPERBOLIC TYPE

(Presented by Academician M. A. Lavrent’ev on 25 III 1968)

Consider the equation

\[ 0= \begin{cases} k(y)u_{xx}+u_{yy}+a(x,y)u_x+b(x,y)u_y+c(x,y)u, & y<0,\\ u_{xx}+\alpha(x,y)u_x+\beta(x,y)u_y+\gamma(x,y)u, & y>0, \end{cases} \tag{1} \]

in a simply connected mixed domain \(D\) of the plane of the independent variables \(x,y\), bounded by the segments \(AA_0\), \(A_0B_0\), \(BB_0\) of the straight lines \(x=0\), \(x=1\), \(y=y_0>0\), respectively, and by the real characteristics \(AC:\sqrt{-k}\,dy+dx=0\), \(BC:\sqrt{-k}\,dy-dx=0\) of equation (1), issuing from the points \(A(0,0)\), \(B(1,0)\). Let \(D_1(D_2)\) be the parabolic (hyperbolic) part of the mixed domain \(D\).

The following assumptions are made concerning the coefficients of equation (1): \(k(y)\) is a continuously differentiable and monotonically increasing function in \(D_2\), with \(k(0)=0\); \(a\), \(b\), and \(c\) belong to the space \(C^1(\overline{D}_2)\) and are connected by the relations (see \((^1)\))

\[ \delta(\sqrt{-k})+a+b\sqrt{-k}<0, \]

\[ \delta\left(\frac{\delta\sqrt{-k}}{\sqrt{-k}}+\frac{a+b\sqrt{-k}}{\sqrt{-k}}\right) +\frac{1}{2k}\,[\delta(\sqrt{-k})+a+b\sqrt{-k}] \]

\[ {}\times[\delta(\sqrt{-k})+a-b\sqrt{-k}]-2c\le 0,\qquad c\le 0, \]

where \(\delta \equiv \partial \backslash \partial y+\sqrt{-k}\,\partial \backslash \partial x\); \(\alpha,\beta,\gamma\in C(\overline{D}_1)\) and, moreover, \(\beta<0\), \(\gamma\le 0\).

By a solution of equation (1) that is regular in the domain \(D\) we shall mean a function
\[ u(x,y)\in C(\overline D)\cap C^1(D)\cap C^2(D_1\cup D_2) \]
and satisfying equation (1) in \(D_1\cup D_2\).

For equation (1), the direct analogue of the well-known Tricomi problem \((^2)\) is the following

Problem T. Find a solution \(u(x,y)\), regular in \(D\), of equation (1), satisfying the boundary conditions:

\[ u(0,y)=\varphi_0(y),\qquad u(1,y)=\varphi_1(y),\qquad 0\le y\le y_0, \tag{2} \]

\[ u\big|_{AC}=\psi(x), \tag{3} \]

where \(\varphi_0(y),\varphi_1(y)\in C\ (0\le y\le y_0)\), \(\psi(x)\in C^4\ (0\le x\le 1/2)\), \(\varphi_0(0)=\psi(0)=\varphi_1(1)=0\).

Lemma. Suppose: 1) \(u(x,y)\) is a regular solution of problem T when \(\psi(x)\equiv0\); 2) the derivative of the function \(u(x,y)\) in the direction of the characteristics of the family \(\sqrt{-k}\,dy+dx=0\) exists and is continuous in \(\overline D_2\setminus A\setminus B\). Then the positive maximum (negative minimum) of the function \(u(x,y)\) in \(\overline D_2\) is attained at some point \((\xi,0)\in AB\), and at this point \(u_y>0\) \((u_y<0)\).

This lemma is proved in the same way as in \((^{1,3})\), where the case \(k(y)=y\), \(a(x,y)\equiv0\) was considered, while \(b(x,y)\) and \(C(x,y)\) are connected in a special way.

Let \(u(x,y)\in C^{1}(D_{1}\cup A_{0}B_{0})\) and satisfy the conditions of the lemma. Then the positive maximum and negative minimum of \(u(x,y)\) in \(\overline{D}_{1}\) are attained on \(\overline{AA_{0}}\cup \overline{BB_{0}}\).

The validity of this extremum principle follows from the lemma just given and the known extremum principle for parabolic equations \((^{4})\).

From the extremum principle follows the uniqueness of the solution \(u(x,y)\) of problem \(T\), if \(u(x,y)\in C^{1}(D_{1}\cup A_{0}B_{0})\).

We shall prove the existence of a solution of problem \(T\), restricting ourselves to the case
\(a=b=c=\alpha=\gamma=0,\ \beta=-1,\ k(y)=y\).

Let there exist a solution \(u(x,y)\) of problem \(T\), and let \(u(x,0)=\tau(x)\), \(u_{y}(x,0)=\nu(x)\). Then from equation (1) it follows directly that \(\tau(x)\) and \(\nu(x)\) on \(AB\) will be connected by the following relation, brought from the domain \(D_{1}\):

\[ \tau''(x)=\nu(x),\qquad 0<x<1, \]

or

\[ \tau(x)=\int_{0}^{x}(x-t)\nu(t)\,dt+x\int_{0}^{1}(t-1)\nu(t)\,dt . \tag{4} \]

Solving the Darboux problem for equation (1) in the domain \(D_{2}\) with data (3) and \(\nu(x)\) on \(AB\), we find \((^{5})\) that these functions are also connected by the relation

\[ \tau(x)=\gamma_{0}\int_{0}^{x}\frac{\nu(t)\,dt}{(x-t)^{1/3}}+\Psi(x),\qquad 0<x<1, \tag{5} \]

where \(4\pi^{2}\gamma_{0}=3^{2/3}\Gamma^{3}(1/3)\),

\[ \Psi(x)= \frac{\Gamma(1/6)}{\Gamma(5/6)\Gamma(1/3)} x^{-1/6}\int_{0}^{x} \left[ \psi'\!\left(\frac{\eta}{2}\right)+\frac{\psi(\eta/2)}{6\eta} \right]\eta^{1/3}(x-\eta)^{-1/6}\,d\eta . \tag{6} \]

Since

\[ \psi(x)=x\int_{0}^{1}\psi'(tx)\,dt,\qquad 0\le x\le \frac12 , \]

from (6) it is easy to see that

\[ \Psi(x)\in C^{3}(0\le x\le 1),\qquad \Psi(x)=O(1)x. \tag{7} \]

Eliminating \(\tau(x)\) from the system (4)—(5), we obtain

\[ \int_{0}^{x}\frac{\gamma_{0}-(x-t)^{4/3}}{(x-t)^{1/3}}\nu(t)\,dt = x\int_{0}^{1}(t-1)\nu(t)\,dt-\Psi(x). \tag{8} \]

Obviously, equation (8) is equivalent to problem \(T\).

Applying to both sides of the integral equation (8) the operator (see \((^{6})\))

\[ A\mu(x)\equiv \frac{\partial}{\partial x}\int_{0}^{x}\frac{\mu(t)\,dt}{(x-t)^{2/3}}, \tag{9} \]

after some transformations we find

\[ \frac{2\pi}{\sqrt{3}}\gamma_{0}\nu(x)-3\int_{0}^{x}(x-t)^{1/3}\nu(t)\,dt = \int_{0}^{1}(t-1)\nu(t)\,dt\,Ax-A\Psi(x), \]

or

\[ \nu(x)-\lambda\int_{0}^{x}(x-t)^{1/3}\nu(t)\,dt = \lambda x^{1/3}\int_{0}^{1}(t-1)\nu(t)\,dt+\Phi(x), \tag{10} \]

where

\[ \lambda=\frac{3\sqrt{3}}{2\pi\gamma_0}, \qquad \Phi(x)=-\frac{\lambda}{3}A\Psi(x). \tag{11} \]

Let \(\Gamma(x,t,\lambda)\) be the resolvent of the kernel \((x-t)^{1/3}\) of the Volterra operator in the left-hand side of equation (10). It is not difficult to verify that

\[ \Gamma(x,t,\lambda)=\sum_{n=1}^{\infty} \frac{\lambda^{\,n-1}\Gamma^n(4/3)}{\Gamma(4n/3)} (x-t)^{(4n-3)/3}. \tag{12} \]

After the Volterra integral operator is inverted, which is legitimate by virtue of (7) and (11), equation (10) takes the form

\[ v(x)-\lambda\int_0^1 k(x,t)v(t)\,dt=f(x), \tag{13} \]

where

\[ k(x,t)=(t-1)\left[x^{1/3}+\lambda\int_0^x s^{1/3}\Gamma(x,s,\lambda)\,ds\right], \tag{14} \]

\[ f(x):=\Phi(x)+\lambda\int_0^x \Gamma(x,t,\lambda)\Phi(t)\,dt. \tag{15} \]

From (11), taking (7) into account, we have

\[ \Phi(x)=-\frac{\lambda}{3}\int_0^x \frac{\Psi'(t)\,dt}{(x-t)^{2/3}}, \]

therefore \(\Phi(x)\in C(0\le x\le 1)\cap C^2(0<x<1)\), and at the point \(x=0\) it vanishes to order not less than \(1/3\). Obviously, by virtue of (12) and (15), the right-hand side \(f(x)\) of equation (13) possesses the same properties.

From (14) and (12) we conclude:

1) \(k(x,t)\in C(0\le x,t\le 1)\), \(k(x,t)=x^{1/3}k^*(x,t)\), where \(k^*(x,t)\in C(0\le x,t\le 1)\);

2) \(k(x,t)\) is twice continuously differentiable with respect to \(x\) in the square \(0<x,t<1\).

Thus, if solutions \(v(x)\) of the Fredholm integral equation of the second kind (13) exist, then they belong to the class
\(C(0\le x\le 1)\cap C^2(0<x<1)\) and at \(x=0\) vanish to order not less than \(1/3\). From relation (5) it follows that, for such \(v(x)\), the function
\(\tau(x)\in C(0\le x\le 1)\cap C^2(0<x<1)\) and at \(x=0\) vanishes to order not less than one.

The solutions \(u(x,y)\) of problem T in the domain \(D_2\) (if they exist) are representable by Darboux’s formula

\[ u(x,y)= \frac{\Gamma(1/3)}{\Gamma^2(1/6)} \int_0^1 \tau\!\left[x+\frac{2}{3}(-y)^{3/2}(2t-1)\right] [t(1-t)]^{-5/6}\,dt+ \]

\[ +\frac{\Gamma(5/3)}{\Gamma^2(5/6)}\,y \int_0^1 v\!\left[x+\frac{2}{3}(-y)^{3/2}(2t-1)\right] [t(1-t)]^{-1/6}\,dt, \]

whence it follows that \(u(x,y)\in C^1(\overline{D}_2)\).

It is now not difficult to see the equivalence (in the sense of solvability) of problem T to the integral equation (13), and that the solution \(u(x,y)\) of this problem belongs to the class of functions for which the extremum principle has been proved.

From the uniqueness of the solution of problem T there follows the unconditional solvability of the Fredholm integral equation of the second kind (13), and, consequently, of problem T.

Let now \(D_1\) be a simply connected domain of the half-plane \(y>0\), whose boundary contains the segment \(AB\) of the axis \(y=0\), and let \(D=D_1\cup D_2\cup AB\). In the general case, in proving the existence of a solution of problem T in the domain \(D\), no fundamental difficulties arise if, in addition, the domain \(D_1\) and the functions \(a, b, c, \alpha, \beta, \gamma\) are such that in the domain \(D_1\) the basic boundary-value problem for equation (1) (4) is solvable, and in \(D_2\) the Darboux problem (5) is solvable.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
20 III 1968

REFERENCES

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