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UDC 517.944
ON THE CAUCHY PROBLEM WITH DATA ON A DEGENERACY LINE OF TYPE FOR A HYPERBOLIC EQUATION
S. A. TERSENOV
Consider the differential equation
\[ A u_{xx}+2B u_{xy}+C u_{yy}+a u_x+b u_y+c u=f \tag{1} \]
of hyperbolic type for \(y>0\), which may degenerate parabolically on the straight line \(y=0\), i.e., for \(y>0\), \(AC-B^2<0\), while for \(y=0\), \(AC-B^2\leq 0\); moreover, for \(y=0\) not all the coefficients \(A\), \(B\), and \(C\) can simultaneously vanish.
It is known (see [1–3]) that the Cauchy problem in its usual formulation with data on the degeneracy line is, generally speaking, not well posed.
In works [1–15], for various cases of equation (1), both the Cauchy problem in the usual formulation and in the modified formulation proposed by A. V. Bitsadze in [1] have been investigated.
In the present paper, under certain smoothness conditions imposed on the coefficients of equation (1) and on the initial data, the Cauchy problem in the modified formulation is studied. We note that if, for an equation, the Cauchy problem in the usual formulation is well posed, then the problem in the modified formulation coincides with the usual one.
§ 1. Let the function \(f(x,y)\), given in the rectangle \(R:\alpha\leq x\leq \beta\), \(0\leq y\leq \delta\), satisfy the following conditions: 1) it is continuous in \(R\) together with its derivatives with respect to \(x,y\) up to order \(p\) inclusive; 2) there exists a function \(F(y)\) such that
\[ \int_0^\delta \left|\frac{\partial^k f(x,y)}{\partial x^k}\right|\,dy \leq \int_0^\delta F(y)\,dy<\infty,\quad k=0,1,\ldots,p . \tag{1.1} \]
Then, as is known, for any \(\xi\), \(\alpha<\xi<\beta\), there exists a unique solution \(x=\varphi(y;\xi)\) of the equation
\[ \frac{dx}{dy}=f(x,y), \]
satisfying the condition \(\varphi(0,\xi)=\xi\). The function \(\varphi(y,\xi)\) has continuous derivatives with respect to \(\xi\) up to order \(p\). Moreover, the equation \(x-\varphi(y,\xi)=0\) is uniquely solvable with respect to \(\xi\). The function \(\xi=\xi(x,y)\) has continuous partial derivatives with respect to \(x\) and \(y\) (\(y>0\)) up to order \(p\), and \(\xi(x,0)=x\). We shall use this observation below.
Let us now consider equation (1). Let the coefficients and the right-hand side be defined in the strip \(R_0: -\infty<x<\infty,\ 0\leq y\leq \delta\). We shall assume,
that \(C(x,y)>0\) for \(y>0\) and \(C(x,0)\geq 0\). The characteristic directions of (1) satisfy the equations
\[ \frac{dx}{dy}=\frac{B+\sqrt{B^2-AC}}{C}=K_1,\qquad \frac{dx}{dy}=\frac{B-\sqrt{B^2-AC}}{C}=K_2(x,y). \]
Assume that the functions \(K_1\) and \(K_2\) are such that every characteristic line issuing from an interior point \(R_0\) reaches the straight line \(y=0\). Obviously, for this it is necessary and sufficient that
\[ \int_0^\delta K_i(x,y)\,dy<\infty,\quad i=1,2. \]
Let the domain \(S\) be the characteristic triangle formed by two characteristics issuing from the point \((x_0,y_0)\) and by the segment \((\alpha,\beta)\) of the \(x\)-axis; \(\alpha\) and \(\beta\) are the points of intersection of the characteristic lines with the \(x\)-axis.
Assume that the functions \(K_1\) and \(K_2\), for \(y>0\), have continuous derivatives with respect to \(x\) up to second order and with respect to \(y\) of first order. In addition, there exists a function \(K(y)\) such that
\[ \int_0^y \left|\frac{\partial^n K_i}{\partial x^n}\right|\,dy \leq \int_0^y K(y)\,dy<\infty . \tag{1.2} \]
The final conditions imposed on the coefficients and on the right-hand side of the equation will be formulated below.
Let \(\varphi_1(y,\xi)\) and \(\varphi_2(y,\eta)\) be solutions of the equations
\[ \frac{dx}{dy}=K_1,\qquad \frac{dx}{dy}=K_2 \tag{1.3} \]
respectively, satisfying the conditions:
\[ \varphi_1(0,\xi)=\xi,\quad \varphi_2(0,\eta)=\eta,\quad \alpha\leq \xi\leq\beta,\quad \alpha\leq\eta\leq\beta. \]
Such solutions exist, are unique, and satisfy the integral equations
\[ x=\xi+\int_0^y K_1(x,t)\,dt,\qquad x=\eta+\int_0^y K_2(x,t)\,dt . \]
Denote by \(\xi(x,y)\) and \(\eta(x,y)\) the solutions of the equations
\[ x-\varphi_1(y,\xi)=0,\qquad x-\varphi_2(y,\eta)=0 \]
respectively. Obviously, these equations are uniquely solvable with respect to \(\xi\) and \(\eta\), respectively, since
\[ \frac{\partial\varphi_1}{\partial\xi}\neq 0,\qquad \frac{\partial\varphi_2}{\partial\eta}\neq 0. \]
Observe that \(\eta(x,y)-\xi(x,y)>0\) for \(y>0\), and \(\eta(x,0)=\xi(x,0)=x\). Introduce new independent variables
\[ t=\frac{(\eta-\xi)^2}{16},\qquad \tau=\frac{\eta+\xi}{2}. \tag{1.4} \]
Then equation (1) takes the form
\[ t u_{tt}-u_{\tau\tau}+a_1 u_t+b_1 u_\tau+c_1 u=f_1, \tag{1.5} \]
where
\[ a_1=-\frac{1}{2}+\sqrt{t}\,\overline{A}_1-\sqrt{t}\,\overline{B}_1,\quad b_1=\overline{A}_1+\overline{B}_1, \]
\[ c_1=\frac{4c}{\xi_x\eta_x C(K_1-K_2)^2},\quad f_1=\frac{4f}{\xi_x\eta_x C(K_1-K_2)^2}, \]
\[ A_1=\frac{2}{\eta_x C(K_1-K_2)^2} \left(CK_{1y}-b+aK_1-2B_xK_1+C_xK_1^2+A_x\right), \]
\[ B_1=\frac{2}{\xi_x C(K_1-K_2)^2} \left(CK_{2y}-b+aK_2-2B_xK_2+C_xK_2^2+A_x\right). \]
Obviously, under the substitution (1.4) the straight line \(y=0\) is transformed into the straight line \(t=0\), and the characteristic triangle \(S\) is transformed into a certain characteristic triangle \(S_0\) of equation (1.5), resting on the segment \((\alpha,\beta)\) of the straight line \(t=0\). Let the function \(a_1(\tau,t)\) in the rectangle \(D\) (\(S_0\subset D\)): \(\alpha_1\le \tau\le \beta_1,\ 0\le t\le t_0\) be twice continuously differentiable with respect to \(\tau\). Consider the function
\[ \omega(\tau,t)=\int_t^{t_0}\exp\left\{\int_{t_1}^{t_0}a_1(\tau,t_2)t_2^{-1}\,dt_2\right\}dt, \]
which, obviously, satisfies the equation
\[ t\omega_{tt}+a_1\omega_t=0. \tag{1.6} \]
The function \(\omega\) is continuous in \(D\) and at all points of the segment \((\alpha_1,\beta_1)\), where \(a_1(\tau,0)<1\). It becomes infinite where \(a_1(\tau,0)>1\). Introduce a new unknown function
\[ u=\omega v. \tag{1.7} \]
Then, if \(u\) satisfies (1.5), \(v\) must satisfy the following equation:
\[ t v_{tt}-v_{\tau\tau}+a_2 v_t+b_2 v_\tau+c_2 v=f_2, \tag{1.8} \]
where
\[ a_2=a_1+2t\frac{\omega_t}{\omega},\quad b_2=b_1+2\frac{\omega_\tau}{\omega}, \]
\[ c_2=-\frac{\omega_{\tau\tau}}{\omega} +b_1\frac{\omega_\tau}{\omega}+c_1,\quad f_2=\frac{f_1}{\omega}. \]
It is easy to see that, by virtue of the conditions imposed on \(a_1(\tau,t)\), the function \(a_2(\tau,0)\le 1\) on the segment \((\alpha,\beta)\).
§ 2. Let the coefficients of equation (1.8) satisfy the following conditions:
1) the functions \(b_2,c_2,f_2\) in the domain \(S_0\) have the form
\[ b_2=\frac{\delta(\tau,t)}{\sqrt{t}},\quad c_2=\frac{\gamma(\tau,t)}{t^r},\quad f_2=\frac{F(\tau,t)}{t^r}, \tag{2.1} \]
where \(r<1\), and \(\delta,\gamma\) and \(F\) are functions continuous in \(\overline{S}_0\). Let \(p\) be such an integer that
\[ \max |1-2a_2-2\delta|+\max |1-2a_2+2\delta|\le \]
\[ \le k\,[2p(1-r)-2q(1-r)-2r+3-2a_2(\tau,0)]. \tag{2.2} \]
where
\[ 0<k<1,\quad 1+\max(1-a_2(\tau,0))>q(1-r)\geq \max(1-a_2(\tau,0)); \]
2) the functions \(\delta,\gamma,F\) have, in the domain \(\overline S_0\), continuous derivatives with respect to \(\tau\) up to order \(2(p+1)\). In addition, \(\delta\) has a continuous derivative with respect to \(t\) for \(t>0\);
3) the function \(a_2\) in \(\overline S_0\) has a continuous first derivative with respect to \(t\) and continuous derivatives with respect to \(\tau\) up to order \(2(p+1)\). Moreover,
\[ a_2(\tau,t)=a_2(\tau,0)+O(t^s),\quad (s<1)\quad \text{in } \overline S_0; \tag{2.3} \]
\(O\) is Landau’s symbol.
We shall seek solutions of equation (1.8) in the class of functions having in \(S_0\) continuous derivatives up to second order with respect to \(t\) and continuous derivatives up to order \(2(p+2)\) with respect to \(\tau\), up to the segment \((\alpha,\beta)\) of the axis \(t=0\). We denote this class of functions by \(\Omega_p\). Let \(\sigma(\tau),\nu(\tau)\) be functions prescribed on the segment \((\alpha,\beta)\), having continuous derivatives up to order \(2(p+2)\).
Introduce the notation:
\[ L_1(v)\equiv v_{\tau\tau}-b_2v_\tau-c_2v, \]
\[ M_1(v)\equiv -H\int_t^l t_1^{-1}H_{t_1}^{-1}v\,dt_1-\int_0^t t_1^{-1}H_{t_1}^{-1}Hv\,dt_1, \]
\[ M_2(v)\equiv \int_0^t t_1^{-1}H_{t_1}^{-1}(\tau,t_1)\,[H(\tau,t)-H(\tau,t_1)]\,v\,dt_1, \]
where
\[ H(\tau,t)=\int_0^t \exp\left\{\int_{t_1}^l a_2(\tau,t_2)t_2^{-1}\,dt_2\right\}dt_1\quad (l<t_0). \]
Obviously, the function \(H\) satisfies the equation
\[ tH_{tt}+a_2H_t=0. \]
Consider the functions
\[ \Phi(\tau,t)=\sum_{n=1}^{p}\Phi_n(\tau,t),\quad \Phi_n=M_1(\psi_n),\quad 1\leq n\leq q, \]
\[ \Phi_n=M_2(\psi_n),\quad n\geq q+1, \]
where
\[ \psi_1=L_1(\tau)+f_2,\quad \psi_n=L_1(\Phi_{n-1}),\quad 2\leq n\leq q, \]
\[ \psi_{q+1}=L_1(\Phi_q)+L_1(H\nu),\quad \psi_n=L_1(\Phi_{n-1}),\quad n\geq q+2. \]
It is easy to see that the functions \(\Phi_n\) are solutions of the recurrent system of equations
\[ t\Phi_{n tt}+a_2\Phi_{nt}=\psi_n,\quad \Phi_n(\tau,0)=0. \]
By virtue of the conditions imposed on the coefficients, and of the fact that
\[ \frac{\partial^j H}{\partial \tau^j}=O\bigl(\psi|\log t|^j\bigr), \]
where
\[ \psi= \begin{cases} \dfrac{(1-a_2(\tau,0))\,t^{1-a_2(\tau,0)}}{1-t^{1-a_2(\tau,0)}}, & \text{if } a_2(\tau,0)<1,\\[1.2em] -(\log t)^{-1}, & \text{if } a_2(\tau,0)=1, \end{cases} \]
it is not hard to show that
\[ L_1(\Phi_p)=O\left(t^{1-a_2(\tau,0)+r_0}|\log t|^m\right), \tag{2.4} \]
where \(r_0=(p-q)(1-r)-r\), and \(m\) is some integer.
Theorem 1. If the coefficients \(a_2, b_2, c_2, f_2\) and the functions \(\sigma(\tau)\), \(\nu(\tau)\) satisfy the requirements formulated above, then there exists a unique solution \(\vartheta\in\Omega_p\) in the domain \(S_0\) of equation (1.8), satisfying the conditions
\[ \vartheta(\tau,0)=\sigma(\tau),\quad \lim_{t\to 0} H_t^{-1}(\vartheta_t-\Phi_t)=\nu(\tau). \tag{2.5} \]
It is not hard to calculate that the conditions (2.5), for \(C=1\), in the variables \(x,y\), are equivalent to the ordinary Cauchy conditions. In this case an analogous theorem, under somewhat less restrictive conditions imposed on the coefficients, was proved by A. B. Nersesyan1. Following [12], we shall seek the solution \(\vartheta(\tau,t)\) in the form \(\vartheta=\sigma+H\nu+\Phi-w\).
Taking into account the properties of the functions \(H\) and \(\Phi\), Theorem 1 reduces to the proof of existence and uniqueness of a solution \(w\) of the equation
\[ tw_{tt}-w_{\tau\tau}+a_2w_t+b_2w_\tau+c_2w=L_1(\Phi_p)=\varphi \tag{2.6} \]
under the initial conditions
\[ w(\tau,0)=0,\quad \lim_{t\to 0} H_t^{-1}w_t=0. \tag{2.7} \]
Moreover, by virtue of (2.4) the function \(\varphi\) admits the estimate
\[ \varphi=O\left(t^{1-a_2(\tau,0)+r_0-\varepsilon}\right). \tag{2.8} \]
As in [12], problem (2.6), (2.7), by introducing new unknown functions
\[ u_1=w,\quad u_2=\sqrt{t}\,w_t+w_\tau,\quad u_3=\sqrt{t}\,w_t-w_\tau, \]
is reduced to a system of Volterra-type integral equations:
\[ u_1=\frac{1}{2}\int_0^t t_1^{-1/2}(u_2+u_3)\,dt_1, \]
\[ u_2=\int_0^{s_2}\left\{ t_1^{-1/2}\varphi+ \frac{1-2a_2-2\delta}{4}\,t_1^{-1}u_2+ \frac{1-2a_2+2\delta}{4}\,t_1^{-1}u_3 -\frac{1}{2}\gamma t_1^{-1/2-r}u_1 \right\}\frac{dt_1}{ds_2}\,ds_2, \]
\[ u_3=\int_0^{s_3}\left\{ t_1^{-1/2}\varphi+ \frac{1-2a_2-2\delta}{4}\,t_1^{-1}u_2+ \frac{1-2a_2+2\delta}{4}\,t_1^{-1}u_3 -\frac{1}{2}\gamma t_1^{-1/2-r}u_1 \right\}\frac{dt_1}{ds_3}\,ds_3, \]
to which, by virtue of conditions (2.2) and (2.8), the method of successive approximations is applicable.
- Differential equations
The solution \(u\) of equation (1) is connected with the solution \(v\) of equation (1.8) by the formula \(u=\omega v\). Then, by virtue of the theorem proved, there exists a unique solution \(u\) of equation (1) in the domain \(S\), satisfying the conditions
\[ \lim_{y\to 0}\frac{u}{\omega}=\sigma(x),\quad \lim_{y\to 0}\left[\left(\frac{u}{\omega}-\Phi\right)_x H_0+ \left(\frac{u}{\omega}-\Phi\right)_y H_1\right]=\nu(x), \]
where
\[ H_0=H_t^{-1}\frac{\partial x}{\partial t},\quad H_1=H_t^{-1}\frac{\partial y}{\partial t}. \]
From this it is not difficult to draw the following conclusions.
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If \(\omega\) is bounded everywhere in \(\overline S_0\), then in conditions (2.9) one may put \(\omega=1\), i.e., for equation (1) Theorem 1 also holds.
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There exists a unique solution \(u\), bounded in \(\overline S_0\), of equation (1), satisfying the conditions
\[ u(x,0)=\sigma(x), \]
\[ \lim_{y\to 0}\left[\left(\frac{u}{\omega}-\Phi\right)_x H_0+ \left(\frac{u}{\omega}-\Phi\right)_y H_1\right]=\nu(x) \quad \text{on } \overline{(\alpha,\beta)}-G, \]
where \(G\) is the set of points of the segment \((\alpha,\beta)\) where \(\lim \omega=\infty\).
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Received by the editors
30 September 1965
Novosibirsk State University
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This result was reported in March 1965 at a seminar at the Institute of Mathematics of the Siberian Branch of the Academy of Sciences of the USSR. ↩