UDC 517.946

MATHEMATICS

A. B. NERSESYAN

ON THE CAUCHY PROBLEM FOR DEGENERATE HYPERBOLIC EQUATIONS OF SECOND ORDER

(Presented by Academician M. A. Lavrent’ev on 9 VI 1965)

In works devoted to the study of the correctness of the Cauchy problem for a hyperbolic equation of second order with initial data on a line of parabolic degeneration, much attention has been paid to the following problem (see \((^1,^2)\)):

\[ -\Delta^2(x,y)u_{xx}+u_{yy}=au_x+bu_y+cu+f \quad (y>0,\ 0\leq x\leq 1); \tag{1} \]

\[ u(x,+0)=\mu(x),\qquad u_y(x,+0)=v(x) \quad (0\leq x\leq 1), \tag{2} \]

where

\[ \Delta(x,y)=\omega\Delta(y),\qquad 1\leq \omega\leq \mathrm{const},\qquad \Delta(y)>0\ (y>0), \]

\[ \Delta(+0)=0,\qquad \Delta'\geq 0. \tag{3} \]

L. Bers showed \((^3)\) that problem (1)—(2) is correct in the classical sense if the lower-order terms are absent from equation (1). Later it was noted that restrictions need be imposed only on the coefficient \(a(x,y)\). The following (apparently, up to now the most general) condition for correctness was established by Protter \((^4)\)*:

\[ a_0(x)=\overline{\lim}_{y\to +0}\frac{y\,|a(x,y)|}{\Delta(y)}=0 \quad (0\leq x\leq 1). \tag{4} \]

However, even in the case of a power-law decrease of the function \(\Delta(y)\), this condition turns out to be stringent.

In particular, for \(\Delta(x,y)=y\) Hellwig showed \((^5)\) that the estimate \(a_0<2\) is sufficient, and for \(\Delta(x,y)=y^\alpha\) \((\alpha>0)\) Chi Min-yu \((^6)\) indicates the estimate \(a_0<\alpha\). From recent results of S. A. Tersenov \((^7)\) it follows that if \(\Delta(x,y)=y^2\) \((\alpha>0)\), \(a_0\leq \mathrm{const}\), and all the parameters of problem (1)—(2) are differentiable with respect to \(x\) a sufficient number of times (depending on \(a_0\)), then this problem is posed correctly. As Tellerstedt \((^8)\) and I. S. Berezin \((^9)\) showed, problem (1)—(2) may turn out to be incorrectly posed (stability of the solution in the uniform metric is violated) if the coefficient \(a\) is not subjected to restrictions. Works \((^{11-13})\) are also devoted to this circle of questions.

In the present note the above-mentioned criteria of correctness are refined and generalized.

\(1^\circ\). Consider the Cauchy problem

\[ Au_{xx}+2Bu_{xy}+u_{yy}=au_x+bu_y+cu+f \quad (y>0,\ 0\leq x\leq 1); \tag{5} \]

\[ u(x,+0)=\mu(x),\qquad u_y(x,+0)=v(x) \quad (0\leq x\leq 1), \tag{6} \]

where

\[ \Delta^2(x,y)=B^2-A>0\ (y>0),\qquad \Delta(x,+0)\geq 0, \tag{7} \]

i.e., on the line \(y=0\) degeneration of type is possible.

* As analysis shows, in paper \((^4)\), contrary to the author’s assertion, criterion (4) is proved only in the case \(\Delta(y)\geq \mathrm{const}\cdot y^\alpha\) \((\alpha>0)\).

By \(D\) we shall denote the open characteristic triangle based on the segment \((y=0,\ 0\le x\le 1)\). For an arbitrary function \(\varphi(x,y)\), continuous for \((x,y)\in D\), we adopt the notation

\[ \varphi^*(y)=\max_x |\varphi(x,y)|. \tag{8} \]

We shall also denote

\[ \alpha_1=a-bB+B_y+\Delta_y+B(B_x+\Delta_x),\qquad \alpha_2=\alpha_1-2(\Delta_y+B\Delta_x), \]

\[ 2\Delta a=|\alpha_1|+|\alpha_2|. \tag{9} \]

Theorem 1. Let the functions \(A,B\), and \(f\) be continuously differentiable in \(\overline D\) with respect to \(x\) \(2p+3\) times \((p\ge 0)\), and let the functions \(\nu\) and \(\mu\) be, respectively, \(2p+4\) and \(2p+5\) times; in addition, let \(A\) and \(B\) be differentiable in \(D\) with respect to \(y\), and let the functions

\[ \left(\frac{\partial^k}{\partial x^k}a\right)^*,\qquad \left(\frac{\partial^k}{\partial x^k}b\right)^* \quad\text{and}\quad y\left(\frac{\partial^k}{\partial x^k}C\right)^* \quad (k=0,1,\ldots,2p+2) \]

be integrable for \(y\ge 0\).

If for some \(i\;(=1,2)\)

\[ \int_0^y t\Delta^*(t)\,d \left\{ f_p^i(t)\exp\left(\int_t^y \alpha^*(\tau)\,d\tau\right) \right\} <+\infty \quad (y>0), \tag{10} \]

where

\[ f_p^i(y)=\int_0^y \alpha_i^*(t_1)(y-t_1) \int_0^{t_1}\alpha_i^*(t_2)(t_1-t_2)\cdots \int_0^{t_p}\alpha_i^*(t_{p+1})\,dt_{p+1}\cdots dt_1, \tag{11} \]

then problem (5)—(6) has a unique solution \(u\), possessing continuous derivatives in \(\overline D\)

\[ \frac{\partial^k}{\partial x^{\,k-i}\partial y^i}u \quad (i=0,1,2;\ i\le k\le p+2). \]

This solution is stable in the following sense: let the functions \(f_i,\mu_i,\nu_i\) \((i=1,2)\) correspond to the solutions \(u_i\). Then for every \(\varepsilon>0\) there exists a \(\delta=\delta(\varepsilon)>0\) such that from the estimate

\[ \sum_{k=0}^{2p+3}\max_{\overline D} \left|\frac{\partial^k}{\partial x^k}(f_1-f_2)\right| + \sum_{k=0}^{2p+4}\max_x \left|\frac{\partial^k}{\partial x^k}(\nu_1-\nu_2)\right| + \sum_{k=0}^{2p+5}\max_x \left|\frac{\partial^k}{\partial x^k}(\mu_1-\mu_2)\right| <\delta \tag{12} \]

it follows that

\[ \sum_{i=0}^{2}\sum_{k=i}^{p+2}\max_D \left| \frac{\partial^k(u_1-u_2)}{\partial x^{\,k-i}\partial y^i} \right| <\varepsilon. \tag{13} \]

2°. Fix some domain \(\Omega\) of the complex plane \(z=x+it\), containing the segment \((t=0,\ 0\le x\le 1)\). A continuous function \(\varphi(x,y)\) in \(\overline D\) will be assigned to the class \(H(D,\Omega)\) if, for each fixed \(y\), it admits an analytic continuation in \(x\) to the domain \(\Omega\).

Theorem 2. If the functions \(A,B,\Delta,a,b,c,f,\mu,\nu\) belong to the class \(H(D,\Omega)\), then problem (5)—(6) has a unique solution \(u(x,y)\in H(D,\Omega_1)\), where the domain \(\Omega_1\) contains the segment \((t=0,\ 0\le x\le 1)\) and depends only on \(\Omega\).

This solution is stable in the following sense: let the functions \(f_i,\mu_i,\nu_i\) \((i=1,2)\) correspond to the solutions \(u_i\). Then for every \(\varepsilon>0\) there exists a \(\delta=\delta(\varepsilon)>0\) such that if

\[ |f_1(z,y)-f_2(z,y)|+|\mu_1(z)-\mu_2(z)|+|\nu_1(z)-\nu_2(z)|<\delta, \qquad z\in\Omega,\ y\ge 0, \tag{14} \]

then

$$ |u_1(z,y)-u_2(z,y)|<\varepsilon,\qquad z\in \Omega,\ y\geqslant 0. \tag{15} $$

The example of I. S. Berezin mentioned above \((^9)\) shows that, from estimate (14), when \(\operatorname{Im} z=0\), in general estimate (15) for \(\operatorname{Im} z=0\) does not follow. At the same time, from this example (analogous to Hadamard’s well-known example for the Laplace equation) and from Theorem 2 it follows that problem (5)—(6) may possess properties inherent in the Cauchy problem for an elliptic equation.

\(3^\circ\). Theorem 1 strengthens all the above-mentioned criteria for the correctness of problem (1)—(2). Indeed, let

$$ \varlimsup_{y\to +0}\frac{|a(x,y)|}{\Delta'(y)}\leq q<+\infty \qquad (0\leq x\leq 1). \tag{16} $$

Then estimate (10) is certainly satisfied if

$$ \int_0^y t^{p+2}[\Delta(t)]^{1+p-q}\Delta'(t)\,dt<+\infty \qquad (p\geqslant 0). \tag{17} $$

Thus, problem (1)—(2) is correct in the sense of Theorem 1 if \(q<p+2\). For \(\Delta(y)=y^\alpha\) \((\alpha>0)\), it suffices to have the estimate \(a_0<(\alpha+1)\times(p+2)\) (in the notation of (4)). Protter’s criterion \((^4)\), also, evidently, strengthened by Theorem 1, turns out to be the more stringent the faster \(\Delta(y)\) tends to zero. For example, for \(\Delta(y)=\exp\{-y^{-\beta}\}\) \((\beta>0)\), according to criterion (17), it suffices to have the estimate

$$ \varlimsup_{y\to +0}\frac{y^{\beta+1}|a(x,y)|}{\Delta(y)}<\beta(p+2) \qquad (0\leq x\leq 1). \tag{18} $$

We also note that if the function \(\{\min_x \Delta\}^{-1}\) is integrable for \(y\geqslant 0\), then the coefficients of equation (5) are free from restrictions, and even its right-hand side may be taken to be a nonlinear function satisfying the usual Lipschitz conditions in such cases.

\(4^\circ\). In the work of Chi Min-yu \((^{10})\) the general solution of the equation

$$ -y^2u_{xx}+u_{yy}=au_x\qquad (y>0) \tag{19} $$

for \(a=\mathrm{const}\) was obtained in explicit form. In the special case when \(a=4n+1\) (\(n\geqslant 0\) an integer), the Cauchy problem with initial data (2) for \(\nu\equiv 0\) has the unique solution

$$ u=\sum_{k=0}^{n} c_k y^{2k}\mu^{(k)}\!\left(x+\frac{y^2}{2}\right) $$

(\(c_k\) are constants). Hence it follows that, for large \(a\), problem (19)—(2) with \(\nu\equiv 0\) has only a generalized solution if the function \(\mu\) is not infinitely differentiable. At the same time, this formula gives an explicit dependence of the character of stability of the solution on the magnitude of \(a\).

A direct application to this problem of the criterion indicated in the preceding paragraph gives somewhat more restrictive conditions of correctness; however, this same example shows that Theorem 1 cannot be substantially improved.

\(5^\circ\). The results obtained carry over to the following Cauchy problem for the system

$$ u_{iy}+A_i u_{1x}+B_i u_{2x}=a_i u_1+b_i u_2+f_i, $$

$$ u_i(x,+0)=\mu_i(x)\qquad (i=1,2;\ y>0;\ 0\leq x\leq 1), \tag{20} $$

where

$$ \Delta^2(x,y)=(A_1-B_2)^2+4A_2B_1>0\quad (y>0),\qquad \Delta(x,+0)\geqslant 0. \tag{21} $$

A special case of the problem was studied by S. A. Tersenov \((^7)\).

Obviously, one may assume that \(A_1 \geqslant B_2\). In the case \(A_2 \equiv B_1 \equiv 0\) correctness is obvious. Let us additionally suppose that for \(y>0\), \(A_2>0\). Denote

\[ c=A_1-B_2+\Delta,\quad 2d=A_1+B_2,\quad 2A_2e=c, \]

\[ \alpha_1=b_1+e_y+de_x+e(a_1-ea_2-b_2), \tag{22} \]

\[ \omega A_2=\Delta,\quad \alpha_2=\alpha_1+\omega_y+d\omega_x,\quad \omega\alpha=|\alpha_1|+|\alpha_2|. \]

Theorem 1 remains valid, with the corresponding changes, if the functions \(A_i, B_i, f_i\) and \(c\) are continuous in \(\overline D\) together with their derivatives with respect to \(x\) up to order \((2p+3)\) (\(p\geqslant0\)), \(\mu_i\) up to order \((2p+4)\), \(A_i, B_i\) are continuously differentiable in \(D\) with respect to \(y\), the functions \(\left(\dfrac{\partial^k}{\partial x^k}a_i\right)^*\) and \(\left(\dfrac{\partial^k}{\partial x^k}b_i\right)^*\) \((i=1,2;\ k=0,1,\ldots,2p+2)\) (see notation (8)) are integrable for \(y\geqslant0\), and for some \(i\) \((=1,2)\)

\[ \int_0^y t\omega^*(t)\,d\left\{f_p^i(t)\exp\left(\int_t^y\alpha^*(\tau)\,d\tau\right)\right\}<+\infty \quad (y>0), \tag{23} \]

where

\[ f_p^i(y)=\int_0^y A_2^*(t_1)\int_0^{t_1}\alpha_i^*(t_2)\cdots \int_0^{t_{2p}}A_2^*(t_{2p+1}) \int_0^{t_{2p+1}}\alpha_i^*(t_{2p+2})\,dt_{2p+2}\cdots dt_1. \tag{24} \]

As with problem (5)—(6), problem (20) is correct with nonlinear right-hand sides if the function \(\{\min_x\omega\}^{-1}\) is integrable for \(y\geqslant0\).

Finally, we note that Theorem 2 remains valid also for problem (20), if the functions \(A_i, B_i, a_i, b_i, f_i\) \((i=1,2)\), \(c, d, e\), and \(\omega\) belong to the class \(H(D,\Omega)\).

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
2 VI 1965

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