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UDC 517.944
GLOBAL SOLUTION OF THE TRICOMI PROBLEM FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS OF MIXED TYPE
D. K. GVAZAVA
The differential equation
\[ \mathbf{T}[u]\equiv yu_{xx}+u_{yy}=f(x,y,u) \tag{1} \]
is elliptic for \(y>0\), hyperbolic for \(y<0\), and parabolically degenerates when \(y=0\).
Consider a simply connected mixed domain \(D\) in the plane of the variables \(x,y\), with boundary \(\Gamma=\sigma+AC+CB\), where \(\sigma\) is a simple Jordan curve with continuous curvature in the sense of Hölder, lying in the upper half-plane \(y>0\) and ending in arcs of the so-called normal contour \(\sigma_0\), given by the equation
\[ \left(x-\frac{1}{2}\right)^2+\frac{4}{9}y^3=\frac{1}{4} \]
at the points \(A(0,0)\) and \(B(1,0)\). It is assumed that the curve \(\sigma\) encloses the normal contour from above. The curves \(AC\) and \(CB\) are characteristics of equation (1), lying in the half-plane \(y<0\) and given respectively by the equations
\[ x-\frac{2}{3}(-y)^{3/2}=0,\qquad x+\frac{2}{3}(-y)^{3/2}=1 . \]
We denote by \(D_1\) the subdomain of \(D\) in which equation (1) is elliptic, and by \(D_2\) the subdomain in which it is hyperbolic.
Assume the function \(f(x,y,u)\) to be continuous and once continuously differentiable for all \((x,y)\in D\) and for all finite values of \(u\). For \((x,y)\in D_2\) it is assumed that \(f(x,y,u)\) has continuous derivatives with respect to all arguments up to second order inclusive, and that the following estimates hold:
\[ \left| \frac{\partial^i \mathcal{F}(\xi,\eta,u)}{\partial \eta^i} \right|<c, \]
\[ \left| \frac{\partial \mathcal{F}(\xi,\eta,u)}{\partial u} \right|<c(1-\eta)^{1/3}\eta^{1/3}, \]
\[ \left| \frac{\partial^2 \mathcal{F}(\xi,\eta,u)}{\partial u^2} \right|<c(1-\eta)^{1/3}\eta^{5/6}, \]
where \(c>0\) is some constant; \(\xi=x-\dfrac{2}{3}(-y)^{3/2}\), \(\eta=x+\dfrac{2}{3}(-y)^{3/2}\) are characteristic variables;
\[ \mathcal{F}(\xi,\eta,u) = f\{x(\xi,\eta),\,y(\xi,\eta),\,u[x(\xi,\eta),y(\xi,\eta)]\}. \]
Tricomi problem. Find a function \(u(x,y)\), continuous in the closed domain \(\overline D\), twice continuously differentiable and satisfying equation (1) in the domain \(D\), which assumes zero boundary values
\[ u(x,y)\big|_{\sigma}=0, \tag{2} \]
\[ u(x,y)\big|_{AC}=0. \tag{3} \]
It is assumed that the function \(u'_y(x,0)\equiv \nu(x)\), continuous on the interval \(0<x<1\), tends to infinity of order less than \(2/3\) as \(x\to 0\) or \(x\to 1\).
Theorem. There exists at least one solution of problem (1)—(3).
We shall regard the right-hand side of equation (1) as a known function of the arguments \(x,y\) and consider the linear equation
\[ \mathrm T[u]=f(x,y). \tag{4} \]
Let us determine the form of the functional relation between \(u(x,0)\equiv \tau(x)\) and \(\nu(x)\) on the segment \(0<x<1\), brought from the domain \(D_2\).
The solution of equation (4) satisfying the conditions
\[ u_y^*(x,y)\big|_{y=-0}=\nu(x), \qquad u(x,y)\big|_{AC}=0, \]
where \(\nu(x)\) is a once continuously differentiable function on the interval \(0<x<1\), is written in characteristic variables \(\xi,\eta\) explicitly as follows:
\[ \begin{aligned} U(\xi,\eta) &=\left(\frac{4}{3}\right)^{1/3}\gamma_0 \int_0^\xi \nu(\xi_1)(\xi-\xi_1)^{-1/6}(\eta-\xi_1)^{-1/6}\,d\xi_1 \\ &\quad -\left(\frac{1}{6}\right)^{2/3} \int_0^\xi d\xi_1 \int_{\xi_1}^{\eta} \mathcal F(\xi_1,\eta_1)(\eta_1-\xi_1)^{-2/3} V(\xi_1,\eta_1;\xi,\eta)\,d\eta_1, \tag{5} \end{aligned} \]
where \(V\), the Green–Hadamard function, is given by the formula
\[ V(\xi_1,\eta_1;\xi,\eta)= \begin{cases} \left(\dfrac{\eta_1-\xi_1}{\eta-\xi}\right)^{1/6} F\left(\dfrac16,\dfrac56,1,s\right), & \text{for } \eta_1\ge \xi, \\[1.2em] \dfrac{\gamma_0(\eta_1-\xi_1)^{1/3}} {(\xi-\xi_1)^{1/6}(\eta-\eta_1)^{1/6}} F\left(\dfrac16,\dfrac16,\dfrac13,\dfrac1s\right), & \text{for } \eta_1\le \xi, \end{cases} \]
\[ s=\frac{(\xi-\xi_1)(\eta-\eta_1)} {(\eta_1-\xi_1)(\eta-\xi)}, \qquad \gamma_0=\frac{\Gamma\left(\dfrac16\right)} {\Gamma\left(\dfrac56\right)\Gamma\left(\dfrac13\right)}, \]
\[ U(\xi,\eta)\equiv u[x(\xi,\eta),\,y(\xi,\eta)]. \]
Passing to the limit as \(\xi-\eta\to 0\), \(0<\xi<1\), we obtain
\[ \tau(x)=\left(\frac{4}{3}\right)^{1/3}\gamma_0 \int_0^x \frac{\nu(t)\,dt}{(x-t)^{1/3}}-\, \]
\[ -\left(\frac{1}{6}\right)^{2/3}\gamma_{0}\int_{0}^{x}d\xi \int_{\xi}^{x} \mathfrak{F}(\xi,\eta)(\eta-\xi)^{-1/3}(x-\xi)^{-1/6}(x-\eta)^{-1/6}\,d\eta . \tag{6} \]
Let \(g(x,x_1,y_1)\) be a solution, regular in the domain \(D_1\), of the homogeneous Tricomi equation \(\mathbf T[u]=0\) with respect to the variables \(x_1,y_1\), satisfying the boundary conditions
\[ g(x,x_1,y_1)\big|_{\sigma} = -\left[(x_1-x)^2+\frac{4}{9}y_1^3\right]^{-1/6}, \qquad \left.\frac{\partial g(x,x_1,y_1)}{\partial y_1}\right|_{AB}=0. \]
Such a function exists (see [1]) and, in the case of a normal domain, is written explicitly as
\[ g(x,x_1,y_1) = -\left[(x_1+x-2xx_1)^2+\frac{4}{9}(1-2x)^2y_1^3\right]^{-1/6}. \]
It is clear that for the function
\[ G(x,x_1,y_1) = \left[(x_1-x)^2+\frac{4}{9}y_1^3\right]^{-1/6} + g(x,x_1,y_1) \]
we have \(\mathbf T G(x,x_1,y_1)=0\) with respect to the variables \(x_1,y_1\); on the curve \(\sigma\) it assumes zero values, and on \(AB\) its normal derivative \(G_{y_1}(x,x_1,y_1)\) vanishes.
Removing the point \((x,0)\) from \(D_1\), together with the neighborhood
\((x_1-x)^2+\frac{4}{9}y_1^3=\varepsilon^2\), applying Green’s formula to the remaining part of the domain \(D_1\), and passing to the limit as \(\varepsilon\to0\), we obtain
\[ \tau(x)+\gamma\int_{0}^{1}G(x,x_1,0)v(x_1)\,dx_1 = -\gamma\iint_{D_1}G(x,x_1,y_1)f(x_1,y_1)\,dx_1dy_1. \tag{7} \]
Taking into account the functional relations obtained above and the fact that, according to the formulation of the problem, the limiting values
\[ u(x,+0)=u(x,-0)=\tau(x), \qquad u'_y(x,+0)=u'_y(x,-0)=v(x), \]
we obtain an Abel integral equation, whose inversion leads us to a singular integral equation with respect to the function \(v(x)\):
\[ Z[v(x)]\equiv v(x)+\frac{1}{\pi\sqrt{3}}\int_{0}^{1} \left(\frac{t}{x}\right)^{2/3} \left(\frac{1}{t-x}-\frac{1}{t+x-2tx}\right)v(t)\,dt = \]
\[ = -\int_{0}^{1}K(x,t)v(t)\,dt - \frac{1}{\pi\sqrt{3}}\frac{d}{dx}\int_{0}^{x}(x-t)^{-2/3} \left\{ \iint_{D_1}G(t,x_1,y_1)f(x_1,y_1)\,dx_1dy_1 \right\}dt - \]
\[ -\frac{1}{\pi\sqrt{3}}\frac{d}{dx}\int_{0}^{x}(x-t)^{-2/3}\,dt \int_{0}^{t}(t-\xi)^{-1/6}\,d\xi \int_{\xi}^{t}\mathfrak{F}(\xi,\eta)(\eta-\xi)^{-1/3}\times \]
\[ \times (t-\eta)^{-1/6}\,d\eta \equiv -\int_0^1 K(x,t)v(t)\,dt+M(x)+\mathscr L(x)\equiv \]
\[ \equiv -\int_0^1 K(x,t)v(t)\,dt+M(f)+L(f), \tag{8} \]
where \(K(x,t)\) is a kernel continuously differentiable on the interval \(0\le x\le 1\) (see [1, 7]).
Integrating by parts the second term on the right-hand side of the integral equation (8), and taking into account that \(G(0,x_1,y_1)=0\), we shall have
\[ M(x)=-\frac{1}{\pi\sqrt3}\int_0^x (x-t)^{-2/3} \left\{\iint_{D_1} G_t'(t,x_1,y_1) f(x_1,y_1)\,dx_1dy_1\right\}\,dt. \]
The third term \(\mathscr L(x)\), by means of interchanging the order of integration, assumes the form
\[ \mathscr L(x)=-\gamma_2\int_0^x d\eta\int_0^\eta \mathscr F(\xi,\eta)(\eta-\xi)^{1/3}(x-\xi)^{-5/6}(x-\eta)^{-5/6}\,d\xi = \]
\[ =\gamma_2 x^{2/3}\int_0^1 (1-z)^{-1/3}\,dz \int_0^1 f\left[ \frac{2z+t-tz}{2}\,x,\, -\left(\frac{3t}{4}\right)^{2/3} \right. \]
\[ \left. \times (1-z)^{2/3}x^{2/3} \right] t^{1/3}(1-t)^{5/6}\,dt, \]
where
\[ \gamma_2=\left(\frac{3}{16}\right)^{1/3}\frac{1}{\pi\sqrt3}\frac{1}{\gamma_0}. \]
By virtue of the conditions imposed on the function \(f\), the function \(M(x)+\mathscr L(x)\) is continuous for \(0\le x\le 1\) and continuously differentiable on the interval \(0<x<1\). It itself has a zero of order \(1/3\) at \(x=0\), and its first derivative may tend to infinity of order not higher than \(2/3\) as \(x\to 0\) and not higher than order \(1/3\) as \(x\to 1\).
Applying the known inversion formula for the integral equation (8) (see [1, 7]), we obtain
\[ v(x)+\int_0^1 K_1(x,t)v(t)\,dt = Z^{-1}[M(x)+\mathscr L(x)], \tag{9} \]
where
\[ Z^{-1}[p(x)]\equiv \frac{3}{4}\,p(x) -\frac{\sqrt3}{4\pi}\int_0^1 \left[\frac{t(1-t)}{x(1-x)}\right]^{1/3} \times \]
\[ \times\left(\frac{1}{t-x}-\frac{1}{t+x-2tx}\right)p(t)\,dt. \]
The right-hand side of equation (9) is continuous for \(0\le x<1\), and as \(x\to 1\) it may have a singularity of order not higher than \(1/3\). Moreover, it is continuously differentiable for \(0<x<1\), and as \(x\to 0\) its derivative admits a singularity not higher than of first order.
The kernel \(K_1(x,t)\) is continuous in the rectangle \(0<x,t<1\) and may have fixed integrable singularities at \(x=0\), \(x=1\), \(t=0\), \(t=1\). Consequently, (9) is a Fredholm equation of the second kind.
Thus, problem (2)—(4) has been reduced to the equivalent integral equation (9).
The uniqueness of the solution of equation (4) satisfying the homogeneous boundary conditions (2), (3) is known. Consequently, equation (9) is solvable. We shall denote by \(\mathfrak R(x,t)\) the resolvent corresponding to the kernel \(K_1(x,t)\).
If we introduce the notation
\[ \mathbf R[p(x)] \equiv p(x)-\int_0^1 \mathfrak R(x,t)p(t)\,dt, \]
then we may write
\[ \nu(x)=\mathbf R Z^{-1}[M(x)+\mathscr L(x)]. \tag{10} \]
If in (10) we substitute the values of the functions \(M(x)\) and \(\mathscr L(x)\), it is easy to verify that \(\nu(x)\) is a function continuous on the half-open interval \(0\le x<1\), and for it the estimate
\[ |\nu(x)|<c_1 x^{1/3}(1-x)^{-1/3}, \]
holds; i.e., as \(x=0\), \(\nu(x)\) vanishes with order no higher than \(1/3\), while as \(x\to 1\) it may tend to infinity with order no higher than \(1/3\). The constant \(c_1>0\) depends on the right-hand side of equation (4) and on the domain \(D\). In the interval \(0<x<1\) the function \(\nu(x)\) is continuously differentiable, and for its derivative the following estimate holds:
\[ |\nu'(x)|<c_1 x^{-2/3}(1-x)^{-4/3}. \]
By formula (7) we define the function \(\tau(x)\):
\[ \tau(x)=-\gamma \iint_{D_1} G(x;x_1,y_1)f(x_1,y_1)\,dx_1dy_1- \]
\[ -\gamma\int_0^1 G(x;x_1,0)\mathbf R Z^{-1}[\mathscr L(x_1)+M(x_1)]\,dx_1. \tag{11} \]
It can be shown that \(\tau(x)\) is continuous on the segment \(0\le x\le 1\) and continuously differentiable in the interval \(0<x<1\). From (11) it is seen that \(\tau(0)=\tau(1)=0\).
The solution of the Dirichlet problem for equation (4) in the domain \(D_1\), taking zero values on \(\sigma\), and on \(AB\) the values \(\tau(x)\), is written explicitly with the aid of the Green function. We shall denote this solution by \(u^+(x,y)\).
In the domain \(D_2\), by formula (5), we write the solution \(u^-(x,y)\) of the Darboux problem for equation (4), which vanishes on the characteristic \(AC\), and whose normal derivative on \(AB\) takes the values \(\nu(x)\) given by formula (10).
The function
\[ u(x,y)= \begin{cases} u^+(x,y), & \text{for } y>0,\\ u^-(x,y), & \text{for } y<0 \end{cases} \tag{12} \]
is a solution of the Tricomi problem (2)—(4).
If we assume that the function \(f\), besides the arguments \(x,y\), also depends on the sought function \(u(x,y)\), we obtain that the Tricomi problem (1)—(3) is equivalent to the following system:
\[ u^{+}(x,y)=-\iint\limits_{D_1}G(x,y;x_1,y_1)f[x_1,y_1,u(x_1,y_1)]\,dx_1dy_1+ \]
\[ +\left(\frac{1}{3}\right)^{5/3}y\int\limits_0^1G(x,y;x_1,0)\tau(x_1)\,dx_1, \tag{13} \]
\[ u^{-}(x,y)=U[\xi(x,y),\eta(x,y)]= \]
\[ =\left(\frac{4}{3}\right)^{1/3}\gamma_0 \int\limits_0^\xi \nu(\xi_1)(\xi-\xi_1)^{-1/6}(\eta-\xi_1)^{-1/6}\,d\xi_1- \]
\[ -\left(\frac{1}{6}\right)^{2/3} \int\limits_0^\xi d\xi_1\int\limits_{\xi_1}^{\eta} \mathcal F[\xi_1,\eta_1,U(\xi_1,\eta_1)](\eta_1-\xi_1)^{-2/3}\times \]
\[ \times V(\xi_1,\eta_1;\xi,\eta)\,d\eta_1, \tag{14} \]
where \(G(x,y;x_1,y_1)\) is Green’s function (see [6, 9]),
\[ \tau(x)=-\gamma\iint\limits_{D_1}G(x,x_2,y_2)f(x_2,y_2,u(x_2,y_2))\,dx_2dy_2- \]
\[ -\gamma\int\limits_0^1G(x;x_2,0)\mathbf{RZ}^{-1}\{\mathbf{L}(f)+\mathbf{M}(f)\}\,dx_2, \]
\[ \nu(x)=\mathbf{RZ}^{-1}\{\mathbf{M}(f)+\mathbf{L}(f)\}. \]
If in this system we substitute the values of the functions \(\tau(x)\) and \(\nu(x)\), expressed through the right-hand side of equation (1), and take into account the condition \(|f(x,y,u)|<c\), we become convinced of the uniform boundedness of the set of all functions defined by formulas (13) and (14),
\[ |u(x,y)|\leq c_2, \]
where the constant \(c_2\) depends only on the numbers \(c\), \(c_1\), and on the domain \(D\).
Let us prove that the set of functions defined by formulas (13), (14) is equicontinuous.
Taking into account the fact that
\[ G(x,y;x_1,y_1)=-\frac{1}{18\pi}(yy_1)^{3/4}\log\left[(x-x_1)^2+\right. \]
\[ \left.+\frac{4}{9}\bigl(\sqrt{y^3}-\sqrt{y_1^3}\bigr)^2\right]^{1/2}+P(x,y;x_1,y_1), \]
where \(P(x,y;x_1,y_1)\) is a function regular at the point \(x=x_1,\ y=y_1\) (see [6, 9]), for the first term \(J_1(x,y)\) of formula (13), by a method analogous to that given in [2], we obtain
\[ |J_1(x',y')-J_1(x'',y'')|\leq \]
\[ \leq c\iint\limits_{D_1}|G(x',y';x_1,y_1)-G(x'',y'';x_1,y_1)|\,dx_1dy_1\leq \]
\[ \leq c_3\left[(x'-x'')^2+(y'-y'')^2\right]^{1/2}, \]
where the constant \(c_3\) depends on \(c\) and on the domain \(D_1\).
It is clear that for any preassigned \(\varepsilon>0\) there is a \(\delta(\varepsilon)=\dfrac{\varepsilon}{2c_3}\) such that, when \(\rho[(x',y'),(x'',y'')]<\delta\),
\[ |J_1(x',y')-J_1(x'',y'')|<\frac{\varepsilon}{2}. \]
For the second term \(J_2(x,y)\) of formula (13) we have
\[ |J_2(x',y')-J_2(x'',y'')|< \]
\[ <\left(\frac{1}{3}\right)^{5/3}\int_0^1 \bigl[y'G(x',y';x_1,0)-y''G(x'',y'';x_1,0)\bigr]\cdot|\tau(x_1)|\,dx_1. \]
In the same way as for the preceding case, we obtain
\[ |J_2(x',y')-J_2(x'',y'')|<c_4\rho[(x',y'),(x'',y'')], \]
where the constant \(c_4\) depends only on \(c\) and on the domain \(D_1\), and in this case, if \(\rho<\delta(\varepsilon)=\dfrac{\varepsilon}{2c_4}\), then
\[ |J_2(x',y')-J_2(x'',y'')|<\frac{\varepsilon}{2}. \]
Thus,
\[ |u^+(x',y')-u^+(x'',y'')|<\varepsilon \quad\text{for}\quad \rho[(x',y'),(x'',y'')]<\delta(\varepsilon). \]
We represent the first term on the right-hand side of equality (14), \(J_3(\xi,\eta)\), in the form
\[ \left(\frac{4}{3}\right)^{1/3}\gamma_0 \int_0^\xi \frac{v(t)(\eta-t)^{-1/6}(\xi-t)^{5/6}\,dt}{\xi-t}. \]
Since \(\eta\geq \xi\), the function
\(v(t)(\eta-t)^{-1/6}(\xi-t)^{5/6}\) satisfies a Hölder condition on the interval
\(0\leq t\leq \xi\leq \eta\) with some exponent \(\alpha\). According to the
Plemelj–Privalov theorem (see [5]), the function \(J_3(\xi,\eta)\) satisfies a Hölder condition with exponent \(\alpha_1\leq\alpha\), whence it follows that
\[ |J_3(\xi',\eta')-J_3(\xi'',\eta'')| <c_5\{\rho[(\xi',\eta');(\xi'',\eta'')]\}^{\alpha_1}, \]
and for \(\rho<\delta(\varepsilon)\) we shall have that this difference in absolute value is less than the preassigned \(\varepsilon>0\).
For the last term we have
\[ \left(\frac{1}{6}\right)^{2/3} \int_0^{\xi'} d\xi_1 \int_{\xi_1}^{\eta'} \mathscr{F}[\xi_1,\eta_1,U](\eta_1-\xi_1)^{-2/3} V(\xi_1,\eta_1;\xi',\eta')\,d\eta_1 - \]
\[ -\left(\frac{1}{6}\right)^{2/3} \int_0^{\xi''} d\xi_1 \int_{\xi_1}^{\eta''} \mathscr{F}(\xi_1,\eta_1,U)(\eta_1-\xi_1)^{-2/3} V(\xi_1,\eta_1;\xi'',\eta'')\,d\eta_1. \]
Breaking this difference into a sum of several integrals and taking into account the properties of the function $\mathcal F(\xi,\eta,U)$ and of the Green–Hadamard function, we obtain that for any prescribed $\varepsilon>0$ there is a $\delta(\varepsilon)$, depending only on $\varepsilon$, such that, when the distance between the points $(\xi',\eta')$ and $(\xi'',\eta'')$ is less than $\delta(\varepsilon)$, this difference is less than $\varepsilon$.
Thus the equicontinuity of the set of functions under consideration is proved.
The operation (13), (14) maps the space of all functions continuous in the domain $\overline D$ into a uniformly bounded and equicontinuous subset of the same space. Consequently, by Schauder’s theorem, there exists at least one solution of the system (13), (14). This proves the solvability of the Tricomi problem (1), (3).
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Received by the editors
April 5, 1966
Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR