ON A CERTAIN METHOD FOR SOLVING THE MIXED PROBLEM FOR A HYPERBOLIC EQUATION WITH DISCONTINUOUS COEFFICIENTS

Z. O. MELNIK

The mixed problem for hyperbolic equations with discontinuous coefficients is at present very poorly studied, even in the case of two independent variables.

In the present note a method is indicated for investigating the mixed problem for a general two-dimensional hyperbolic equation of the second order in the case when the coefficients, the free term of the equation, and the initial data of the problem are discontinuous along certain lines (for simplicity, the paper assumes the presence of one line of discontinuity). The proposed method also makes it possible, in fact, to construct the classical solution of the problem with arbitrarily high accuracy.

In most of the constructions carried out in the paper, the results of [1] are used essentially.

§ 1. STATEMENT OF THE PROBLEM

Denote by \(\Pi\) the open rectangle in the \((x,t)\)-plane bounded by the straight lines \(x=0\), \(x=l\), \(t=0\), and \(t=T\). Let this rectangle be divided by the straight line \(x=x_0\) \((0<x_0<l)\) into two open parts \(\Pi^{-}\) and \(\Pi^{+}\) (the sign “\(-\)” corresponds to values of \(x\) smaller than \(x_0\)).

In \(\Pi_1=\Pi^{-}\cup\Pi^{+}\) consider the hyperbolic equation

\[ au \equiv \sum_{i=0}^{2}\sum_{j=0}^{i} a_{ij}(x,t)\, \frac{\partial^i u}{\partial t^{\,i-j}\partial x^j} = f(x,t)\qquad (a_{20}(x,t)\equiv 1). \tag{1.1} \]

Hyperbolicity of the equation is understood in the sense that in the expansion

\[ \sum_{j=0}^{2} a_{2j}(x,t)\lambda^{2-j}\xi^{j} = \prod_{i=1}^{2}(\lambda-\lambda_i(x,t)\xi) \]

the functions \(\lambda_1(x,t)\) and \(\lambda_2(x,t)\) are real and distinct for all \((x,t)\in\overline{\Pi}\). Let, for example, \(\lambda_1(x,t)<0<\lambda_2(x,t)\). (The case of other combinations of signs of the functions \(\lambda_i(x,t)\) will be considered in § 3.)

It is assumed that the coefficients and the free term of equation (1.1) have discontinuities of the first kind when crossing the straight line \(x=x_0\), i.e.

\[ a_{ij}(x,t)= \begin{cases} a_{ij}^{-}(x,t), & (x,t)\in\overline{\Pi}^{-},\\ a_{ij}^{+}(x,t), & (x,t)\in\overline{\Pi}^{+}, \end{cases} \qquad (0\le i\le 2;\quad 0\le j\le i), \]

\[ f(x,t)= \begin{cases} f^{-}(x,t), & (x,t)\in \overline{\Pi}^{-},\\ f^{+}(x,t), & (x,t)\in \overline{\Pi}^{+}. \end{cases} \]

Then, obviously,

\[ \lambda_i(x,t)= \begin{cases} \lambda_i^{-}(x,t), & (x,t)\in \overline{\Pi}^{-},\\ \lambda_i^{+}(x,t), & (x,t)\in \overline{\Pi}^{+} \end{cases} \qquad (i=1,2), \]

\[ u(x,t)= \begin{cases} u^{-}(x,t), & (x,t)\in \overline{\Pi}^{-},\\ u^{+}(x,t), & (x,t)\in \overline{\Pi}^{+}. \end{cases} \]

The functions \(a_{2j}(x,t)\) \((j=1,2)\) are assumed to be three times, and the functions \(a_{ij}(x,t)\) \((0\leq i<j\leq 1)\) and \(f(x,t)\) twice, uniformly continuously differentiable in \(\Pi_1\).

In \(\Pi_1\) it is required to find a twice continuously differentiable solution of equation (1.1), satisfying the initial conditions

\[ u\big|_{t=0}=g_0(x), \qquad \left.\frac{\partial u}{\partial t}\right|_{t=0}=g_1(x) \qquad (0\leq x\leq l), \tag{1.2} \]

the boundary conditions

\[ \gamma_1^{-}(t)\frac{\partial u^{-}(0,t)}{\partial t} +\gamma_2^{-}(t)\frac{\partial u^{-}(0,t)}{\partial x} +\gamma_3^{-}(t)u^{-}(0,t)=h^{-}(t), \]

\[ \gamma_1^{+}(t)\frac{\partial u^{+}(l,t)}{\partial t} +\gamma_2^{+}(t)\frac{\partial u^{+}(l,t)}{\partial x} +\gamma_3^{+}(t)u^{+}(l,t)=h^{+}(t) \tag{1.3} \]

\[ (0\leq t\leq T) \]

and the conjugation conditions on the line of discontinuity

\[ u^{-}(x_0,t)=u^{+}(x_0,t), \]

\[ k_1(t)\frac{\partial u^{-}(x_0,t)}{\partial x} = k_2(t)\frac{\partial u^{+}(x_0,t)}{\partial x} \qquad (0\leq t\leq T). \tag{1.4} \]

Here and below \(F^{-}(x_0)\) \((F^{+}(x_0))\) denotes the boundary value of the function \(F(x)\) as \(x\) tends to \(x_0\) from the left (right).

The functions \(g_0(x)\) and \(g_1(x)\) are continuously differentiable, respectively three and two times, on \([0,l]\setminus\{x=x_0\}\); the functions \(\gamma_i^{\pm}(t)\) \((i=1,2,3)\), \(h^{\pm}(t)\), \(k_1(t)\), and \(k_2(t)\) are continuously differentiable for all \(t\in[0,T]\) twice. It is natural to assume that

\[ g_i(x)= \begin{cases} g_i^{-}(x), & 0\leq x<x_0,\\ g_i^{+}(x), & x_0\leq x\leq l \end{cases} \qquad (i=0,1), \]

\[ g_i^{-}(x_0)=g_i^{+}(x_0), \qquad k_1(0)\frac{dg_0^{-}(x_0)}{dx} = k_2(0)\frac{dg_0^{+}(x_0)}{dx}. \]

Let \(x=\varphi_i(t;\xi,\tau)\) \((i=1,2)\) be the solutions of the characteristic equations for equation (1.1), passing through the point \((\xi,\tau)\in \overline{\Pi}\). Obviously,

\[ \varphi_i(t;\xi,\tau)= \begin{cases} \varphi_i^{-}(t;\xi,\tau), & (\xi,\tau)\in \overline{\Pi}^{-},\\ \varphi_i^{+}(t;\xi,\tau), & (\xi,\tau)\in \overline{\Pi}^{+} \end{cases} \qquad (i=1,2). \]

It is assumed that nowhere in \(\Pi^{-}\) do the characteristics \(x=\varphi_1^{-}(t;0,0)\) and \(x=\varphi_2^{-}(t;x_0,0)\) intersect, and nowhere in \(\Pi^{+}\) do the characteristics \(x=\varphi_1^{+}(t;x_0,0)\) and \(x=\varphi_2^{+}(t;l,0)\) intersect. Otherwise the problem is reduced to the successive solution of several identical problems [1].

In addition, it is assumed that the coefficients of the boundary conditions (1.3) and of the conjugation conditions (1.4) satisfy the relations

\[ \gamma_2^{-}(t)+\lambda_1^{-}(0,t)\gamma_1^{-}(t)\ne 0,\qquad \gamma_2^{+}(t)+\lambda_2^{+}(l,t)\gamma_1^{+}(t)\ne 0, \]

\[ \frac{k_1(t)}{k_2(t)}\ne \frac{\lambda_2^{-}(x_0,t)}{\lambda_1^{+}(x_0,t)} . \tag{1.5} \]

We shall call these conditions the conditions for the unique solvability of the problem posed. It will be shown below that if at least one of these conditions is not satisfied, then the problem posed may have an infinite set of solutions (or may have no solution at all).

Finally, it is assumed that at the corner points \((0,0)\), \((x_0,0)\), and \((l,0)\) the initial conditions, the coefficients and the free term of the equation, and the coefficients of the boundary conditions and conjugation conditions satisfy compatibility conditions ensuring twice continuous differentiability of the solution upon crossing the characteristics issuing from the indicated points. We shall not write out these conditions because of their unwieldiness.

§ 2. REDUCTION OF THE PROBLEM TO A SYSTEM OF VOLTERRA INTEGRAL EQUATIONS

Introduce new unknown functions \(u_1,u_2,u_3\) by the substitution
\(u_1=\dfrac{\partial u}{\partial t}\), \(u_2=\dfrac{\partial u}{\partial x}\), \(u_3=u\). Then problem (1.1)—(1.4) is reduced to an equivalent mixed problem for a first-order hyperbolic system [2]. Reducing the resulting system to canonical form [2] and introducing new unknown functions \(z_1,z_2,z_3\) by the substitution

\[ u_1=-\frac{\lambda_1}{\lambda_2-\lambda_1}\,z_1 +\frac{\lambda_2}{\lambda_2-\lambda_1}\,z_2;\qquad u_2=-\frac{1}{\lambda_2-\lambda_1}\,z_1+ \]

\[ +\frac{1}{\lambda_2-\lambda_1}\,z_2;\qquad u_3=z_3, \]

we reduce the latter problem to the following equivalent problem:

\[ \frac{\partial z_i}{\partial t} -\lambda_i(x,t)\frac{\partial z_i}{\partial x} = \sum_{j=1}^{3} p_{ij}(x,t)z_j+f_i(x,t) \quad (i=1,2,3); \tag{2.1} \]

\[ z_1(x,0)=g_1(x)-\lambda_2(x,0)g_0'(x);\quad z_2(x,0)= \]
\[ =g_1(x)-\lambda_1(x,0)g_0'(x);\quad z_3(x,0)=g_0(x); \tag{2.2} \]

\[ [\gamma_2^-(t)+\lambda_1^-(0,t)\gamma_1^-(t)]z_1^-(0,t)-[\gamma_2^-(t)+\lambda_2^-(0,t)\gamma_1^-(t)]z_2^-(0,t)- \]
\[ -\gamma_3^-(t)[\lambda_2^-(0,t)-\lambda_1^-(0,t)]z_3^-(0,t) =-[\lambda_2^-(0,t)-\lambda_1^-(0,t)]h^-(t); \tag{2.3} \]

\[ [\gamma_2^+(l,t)+\lambda_1^+(l,t)\gamma_1^+(t)]z_1^+(l,t)- \]
\[ -[\gamma_2^+(t)+\lambda_2^+(l,t)\gamma_1^+(t)]z_2^+(l,t)-\gamma_3^+(t)[\lambda_2^+(l,t)- \]
\[ -\lambda_1^+(l,t)]z_3^+(l,t)=-[\lambda_2^+(l,t)-\lambda_1^+(l,t)]h^+(t); \]

\[ \frac{\lambda_1^-(x_0,t)}{\lambda_2^-(x_0,t)-\lambda_1^-(x_0,t)}\,z_1^-(x_0,t) -\frac{\lambda_2^-(x_0,t)}{\lambda_2^-(x_0,t)-\lambda_1^-(x_0,t)}\,z_2^-(x_0,t)= \]
\[ =\frac{\lambda_1^+(x_0,t)}{\lambda_2^+(x_0,t)-\lambda_1^+(x_0,t)}\,z_1^+(x_0,t) -\frac{\lambda_2^+(x_0,t)}{\lambda_2^+(x_0,t)-\lambda_1^+(x_0,t)}\,z_2^+(x_0,t); \tag{2.4} \]

\[ k_1(t)\left[ \frac{z_1^-(x_0,t)}{\lambda_2^-(x_0,t)-\lambda_1^-(x_0,t)} -\frac{z_2^-(x_0,t)}{\lambda_2^-(x_0,t)-\lambda_1^-(x_0,t)} \right]= \]
\[ =k_2(t)\left[ \frac{z_1^+(x_0,t)}{\lambda_2^+(x_0,t)-\lambda_1^+(x_0,t)} -\frac{z_2^+(x_0,t)}{\lambda_2^+(x_0,t)-\lambda_1^+(x_0,t)} \right], \]

where \(\lambda_3(x,t)\equiv 0\), \(f_1(x,t)\equiv f_2(x,t)=f(x,t)\), \(f_3(x,t)=0\), and the coefficients \(p_{ij}(x,t)\) are expressed in an obvious way through the coefficients of equation (1.1). Moreover, as is not difficult to see, \(p_{ij}(x,t)\) \((i,j=1,2,3)\), together with their second derivatives, are uniformly continuous in \(\Pi_1\).

We introduce two more unknown functions \(H_0(t)\) and \(H_1(t)\) by means of the equalities

\[ H_0(t)=u^-(x_0,t)=u^+(x_0,t), \]

\[ H_1(t)=k_1(t)\frac{\partial u^-(x_0,t)}{\partial x} =k_2(t)\frac{\partial u^+(x_0,t)}{\partial x}. \]

Then from (2.4) we obtain

\[ z_2^-(x_0,t)=H_0'(t)-\frac{\lambda_1^-(x_0,t)}{k_1(t)}\,H_1(t), \]
\[ z_1^+(x_0,t)=H_0'(t)-\frac{\lambda_2^+(x_0,t)}{k_2(t)}\,H_1(t). \tag{2.5} \]

We divide the rectangle \(\Pi\) into six parts:
\(\overline{\Pi_0^-}\{0\le t\le T;\ \varphi_1^-(t;0,0)\le x\le \varphi_2^-(t;x_0,0)\}\);
\(\overline{\Pi_1^-}\{0\le t\le T;\ 0\le x\le \varphi_1^-(t;0,0)\}\);
\(\overline{\Pi_2^-}\{0\le t\le T;\ \varphi_2^-(t;x_0,0)\le x\le x_0\}\);
\(\overline{\Pi_0^+}\{0\le t\le T;\ \varphi_1^+(t;x_0,0)\le x\le \varphi_2^+(t;l,0)\}\);
\(\overline{\Pi_1^+}\{0\le t\le T;\ \varphi_1^+(t;l,0)\le x\le l\}\);
\(\overline{\Pi_2^+}\{0\le t\le T;\ x_0\le x\le \varphi_1^+(t;x_0,0)\}\).
By integration along the characteristics [1] in the domains \(\Pi_i^k\) \((k=0,1)\), for the unknowns \(z_i^\pm(x,t)\) we obtain a system of integral equations

Volterra, solvable by the method of successive approximations. Let the solution of the system of equations obtained be the functions

\[ z_i^{\pm}(x,t)=Z_{ik}^{\pm}(x,t)\quad \text{for }(x,t)\in \Pi_k^{\pm}\quad (k=0,1). \]

In this connection the first two conditions (1.5) are used in an essential way.

We proceed next to the domains \(\Pi_2^{-}\) and \(\Pi_2^{+}\). We write the integral equations corresponding to the solution of system (2.1) in these domains. In view of (2.5) we obtain:

a) for \((x,t)\in \Pi_2^{-}\)

\[ z_1^{-}(x,t)=G_1^{-}(x,t)+ \int_{t_1^{-}(x,t)}^{t} \sum_{j=1}^{3} p_{1j}^{-}(\varphi_1^{-}(\tau;x,t),\tau)\, z_j^{-}(\varphi_1^{-}(\tau;x,t),\tau)\,d\tau, \]

\[ \begin{aligned} z_2^{-}(x,t)={}&G_2^{-}(x,t)+H_0'\bigl(t_2^{-}(x,t)\bigr) -\frac{\lambda_1^{-}\bigl(x_0,t_2^{-}(x,t)\bigr)} {k_1\bigl(t_2^{-}(x,t)\bigr)} H_1\bigl(t_2^{-}(x,t)\bigr) \\ &+\int_{t_2^{-}(x,t)}^{t} \sum_{j=1}^{3} p_{2j}^{-}(\varphi_2^{-}(\tau;x,t),\tau)\, z_j^{-}(\varphi_2^{-}(\tau;x,t),\tau)\,d\tau, \end{aligned} \tag{2.6} \]

\[ z_3^{-}(x,t)=G_3^{-}(x,t)+ \int_{t_3^{-}(x)}^{t} \sum_{j=1}^{3} p_{3j}^{-}(x,\tau)\,z_j^{-}(x,\tau)\,d\tau; \]

b) for \((x,t)\in \Pi_2^{+}\)

\[ \begin{aligned} z_1^{+}(x,t)={}&G_1^{+}(x,t)+H_0'\bigl(t_1^{+}(x,t)\bigr) -\frac{\lambda_2^{+}\bigl(x_0,t_1^{+}(x,t)\bigr)} {k_2\bigl(t_1^{+}(x,t)\bigr)} H_1\bigl(t_1^{+}(x,t)\bigr) \\ &+\int_{t_1^{+}(x,t)}^{t} \sum_{j=1}^{3} p_{1j}^{+}(\varphi_1^{+}(\tau;x,t),\tau)\, z_j^{+}(\varphi_1^{+}(\tau;x,t),\tau)\,d\tau, \end{aligned} \]

\[ z_2^{+}(x,t)=G_2^{+}(x,t)+ \int_{t_2^{+}(x,t)}^{t} \sum_{j=1}^{3} p_{2j}^{+}(\varphi_2^{+}(\tau;x,t),\tau)\, z_j^{+}(\varphi_2^{+}(\tau;x,t),\tau)\,d\tau, \tag{2.7} \]

\[ z_3^{+}(x,t)=G_3^{+}(x,t)+ \int_{t_3^{+}(x)}^{t} \sum_{j=1}^{3} p_{3j}^{+}(x,\tau)\,z_j^{+}(x,\tau)\,d\tau, \]

where the functions \(t_k^{\pm}(x,t)\) \((k=1,2,3)\) satisfy the identities

\[ \varphi_2^{-}\bigl(t_3^{-}(x);x_0,0\bigr)\equiv x;\qquad \varphi_1^{+}\bigl(t_3^{+}(x);x_0,0\bigr)\equiv x; \]

\[ \varphi_2^{-}\bigl(t_2^{-}(x,t);x,t\bigr)=x_0;\qquad \varphi_1^{+}\bigl(t_1^{+}(x,t);x,t\bigr)=x_0; \]

\[ \varphi_1^{-}\bigl(t_1^{-}(x,t);x,t\bigr) =\varphi_2^{-}\bigl(t_1^{-}(x,t);x_0,0\bigr); \]

\[ \varphi_1^{+}\bigl(t_2^{+}(x,t);x_0,0\bigr) =\varphi_2^{+}\bigl(t_2^{+}(x,t);x,t\bigr) \]

and \(G_k^{\pm}(x,t)\) \((k=1,2,3)\) are known functions, expressed in an obvious way in terms of the data of the original problem and in terms of the solutions \(Z_{ik}^{\pm}(x,t)\). Substituting \(z_i^{-}(x,t)\) and \(z_i^{+}(x,t)\) \((i=1,2)\) from formulas (2.6) and (2.7) into the equalities (2.4), for the functions \(H_0(t)\) and \(H_1(t)\) we obtain the system of equations

\[ \left[ \frac{\lambda_2^{-}(x_0,t)} {\lambda_2^{-}(x_0,t)-\lambda_1^{-}(x_0,t)} + \frac{\lambda_1^{+}(x_0,t)} {\lambda_2^{+}(x_0,t)-\lambda_1^{+}(x_0,t)} \right] H_0'(t) - \]

\[ - \left[ \frac{1}{k_1(t)} \frac{\lambda_1^{-}(x_0,t)\lambda_2^{-}(x_0,t)} {\lambda_2^{-}(x_0,t)-\lambda_1^{-}(x_0,t)} + \frac{1}{k_2(t)} \frac{\lambda_1^{+}(x_0,t)\lambda_2^{+}(x_0,t)} {\lambda_2^{+}(x_0,t)-\lambda_1^{+}(x_0,t)} \right]H_1(t) = \]

\[ =R_1(t) + \int_{t_1^{-}(x_0,t)}^{t} \sum_{j=1}^{3} A_j^{-}(t,\tau)\, z_j^{-}\bigl(\varphi_1^{-}(\tau;x_0,t),\tau\bigr)\,d\tau + \]

\[ + \int_{t_2^{+}(x_0,t)}^{t} \sum_{j=1}^{3} A_j^{+}(t,\tau)\, z_j^{+}\bigl(\varphi_2^{+}(\tau;x_0,t),\tau\bigr)\,d\tau; \tag{2.8} \]

\[ \left[ \frac{k_1(t)} {\lambda_2^{-}(x_0,t)-\lambda_1^{-}(x_0,t)} + \frac{k_2(t)} {\lambda_2^{+}(x_0,t)-\lambda_1^{+}(x_0,t)} \right] H_0'(t) - \]

\[ - \left[ \frac{\lambda_1^{-}(x_0,t)} {\lambda_2^{-}(x_0,t)-\lambda_1^{-}(x_0,t)} + \frac{\lambda_2^{+}(x_0,t)} {\lambda_2^{+}(x_0,t)-\lambda_1^{+}(x_0,t)} \right]H_1(t) =R_2(t)+ \]

\[ + \int_{t_1^{-}(x_0,t)}^{t} \sum_{j=1}^{3} B_j^{-}(t,\tau)\, z_j^{-}\bigl(\varphi_1^{-}(\tau;x_0,t),\tau\bigr)\,d\tau + \]

\[ + \int_{t_2^{+}(x_0,t)}^{t} \sum_{j=1}^{3} B_j^{+}(t,\tau)\, z_j^{+}\bigl(\varphi_2^{+}(\tau;x,t),\tau\bigr)\,d\tau, \]

where \(R_i(t)\), \(A_j^{\pm}(t,\tau)\), \(B_j^{\pm}(t,\tau)\) are known functions expressed in terms of the data of problem (2.1)—(2.4).

By virtue of the third of conditions (1.5), from (2.8) we obtain

\[ H_0'(t)=R_3(t) + \int_{t_1^{-}(x_0,t)}^{t} \sum_{j=1}^{3} C_j^{-}(t,\tau)\, z_j^{-}\bigl(\varphi_1^{-}(\tau;x_0,t),\tau\bigr)\,d\tau + \]

\[ + \int_{t_2^{+}(x_0,t)}^{t} \sum_{j=1}^{3} C_j^{+}(t,\tau)\, z_j^{+}\bigl(\varphi_2^{+}(\tau;x_0,t),\tau\bigr)\,d\tau; \]

\[ H_1(t)=R_4(t)+ \int_{t_1^-(x_0,t)}^{t}\sum_{j=1}^{3}D_j^-(t,\tau)z_j^-\bigl(\varphi_1^-(\tau;x_0,t),\tau\bigr)\,d\tau+ \]

\[ +\int_{t_2^+(x_0,t)}^{t}\sum_{j=1}^{3}D_j^+(t,\tau)z_j^+\bigl(\varphi_2^+(\tau;x_0,t),\tau\bigr)\,d\tau . \]

Here \(R_3(t)\), \(R_4(t)\), \(C_{ij}^{\pm}(t,\tau)\), \(D_j^{\pm}(t,\tau)\) are completely determined functions expressed in terms of the data of the original problem.

Substituting the found values \(H_0'(t)\) and \(H_1(t)\) into (2.6) and (2.7), we then obtain, for the unknown functions in the domains \(\Pi_i^{\pm}\), a system of integral equations of the form

\[ z_1^-(x,t)=G_1^-(x,t)+ \int_{t_1^-(x,t)}^{t}\sum_{j=1}^{3}p_{1j}^-\bigl(\varphi_1^-(\tau;x,t),\tau\bigr) z_j^-\bigl(\varphi_1^-(\tau;x,t),\tau\bigr)\,d\tau, \]

\[ z_1^+(x,t)=\widetilde G_1^+(x,t)+ \int_{t_1^+(x,t)}^{t}\sum_{j=1}^{3}p_{1j}^+\bigl(\varphi_1^+(\tau;x,t),\tau\bigr) z_j^+\bigl(\varphi_1^+(\tau;x,t),\tau\bigr)\,d\tau+ \]

\[ +\int_{T_1^-(x,t)}^{t_1^+(x,t)} \sum_{j=1}^{3}E_j^-(x;t,\tau)z_j^-\bigl(\psi_1^-(\tau;x,t),\tau\bigr)\,d\tau+ \]

\[ +\int_{T_1^+(x,t)}^{t_1^+(x,t)} \sum_{j=1}^{3}E_j^+(x;t,\tau)z_j^+\bigl(\psi_1^+(\tau;x,t),\tau\bigr)\,d\tau; \tag{2.9} \]

\[ z_2^-(x,t)=\widetilde G_1^-(x,t)+ \int_{t_2^-(x,t)}^{t}\sum_{j=1}^{3}p_{2j}^-\bigl(\varphi_2^-(\tau;x,t),\tau\bigr) z_j^-\bigl(\varphi_2^-(\tau;x,t),\tau\bigr)\,d\tau+ \]

\[ +\int_{T_2^-(x,t)}^{t_2^-(x,t)} \sum_{j=1}^{3}M_j^-(x;t,\tau)z_j^-\bigl(\psi_2^-(\tau;x,t),\tau\bigr)\,d\tau+ \]

\[ +\int_{T_2^+(x,t)}^{t_2^-(x,t)} \sum_{j=1}^{3}M_j^+(x;t,\tau)z_j^+\bigl(\psi_1^+(\tau;x,t),\tau\bigr)\,d\tau, \]

\[ z_2^+(x,t)=G_2^+(x,t)+ \int_{t_2^+(x,t)}^{t}\sum_{j=1}^{3}p_{2j}^+\bigl(\varphi_2^+(\tau;x,t),\tau\bigr) z_j^+\bigl(\varphi_2^+(\tau;x,t),\tau\bigr)\,d\tau; \]

\[ z_3^{-}(x,t)=G_3^{-}(x,t)+ \int_{t_3^{-}(x)}^{t}\sum_{j=1}^{3}p_{3j}^{-}(x,\tau)z_j^{-}(x,\tau)\,d\tau, \]

\[ z_3^{+}(x,t)=G_3^{+}(x,t)+ \int_{t_3^{+}(x)}^{t}\sum_{j=1}^{3}p_{3j}^{+}(x,\tau)z_j^{+}(x,\tau)\,d\tau, \]

where

\[ T_1^{-}(x,t)=t_1^{-}(x_0,t_1^{-}(x,t)), \qquad T_1^{+}(x,t)=t_2^{+}(x_0,t_1^{+}(x,t)), \]

\[ T_2^{-}(x,t)=t_1^{-}(x_0,t_2^{-}(x,t)), \qquad T_2^{+}(x,t)=t_2^{+}(x_0,t_2^{+}(x,t)), \]

\[ \psi_1^{-}(\tau;x,t)=\varphi_1^{-}(\tau;x_0,t_1^{-}(x,t)), \qquad \psi_1^{+}(\tau;x,t)=\varphi_2^{+}(\tau;x_0,t_2^{-}(x,t)), \]

\[ \psi_2^{-}(\tau;x,t)=\varphi_1^{-}(\tau;x_0,t_2^{-}(x,t)), \qquad \psi_2^{+}(\tau;x,t)=\varphi_2^{+}(\tau;x_0,t_1^{+}(x,t)), \]

and \(\hat G_i^{\pm}(x,t)\), \(E_j^{\pm}(x;t,\tau)\), \(M_j^{\pm}(x;t,\tau)\) are known functions which, together with their second derivatives with respect to all arguments, are uniformly continuous in their domains of definition.

Solving the system of equations (2.9) and the corresponding system obtained from it by differentiating each equation with respect to \(x\) and \(t\), by the method of successive approximations, we are readily convinced of the validity of the theorem (taking into account the solutions \(Z_{ik}(x,t)\) obtained in the domains \(\Pi_k^{\pm}\) \((k=0,1)\) and the compatibility conditions).

Theorem. If the coefficients of equation (1.1), of the boundary conditions (1.3), and of the conjugation conditions (1.4), the free term \(f(x,t)\), the initial functions \(g_0(x)\) and \(g_1(x)\), and the boundary functions \(h^{-}(t)\) and \(h^{+}(t)\) satisfy all the requirements formulated in § 1, then in the domain \(\Pi_1=\Pi^{-}\cup\Pi^{+}\) there exists a unique twice continuously differentiable solution of equation (1.1), satisfying the initial conditions (1.2), the boundary conditions (1.3), and the conjugation conditions (1.4).

The existence of derivatives up to the second order of the solution follows from the fact that, as is easy to see, the free terms and kernels of the obtained system of integral equations are twice continuously differentiable in all their arguments. The fact that the solution satisfies conditions (1.2)—(1.4) follows immediately from the method of reducing the original problem to integral equations.

§ 3. ON OTHER COMBINATIONS OF SIGNS OF THE ROOTS OF THE CHARACTERISTIC EQUATION

In addition to the case considered in the preceding sections, equation (1.1) may have characteristic roots whose signs coincide with one of the following combinations:

\[ \begin{aligned} \text{a)}\;& \lambda_1^{-}<\lambda_2^{-}<0;\quad \lambda_1^{+}<\lambda_2^{+}<0;\\ \text{b)}\;& 0<\lambda_1^{-}<\lambda_2^{-};\quad 0<\lambda_1^{+}<\lambda_2^{+};\\ \text{c)}\;& \lambda_1^{-}<0;\quad \lambda_2^{-}>0;\quad \lambda_1^{+}<\lambda_2^{+}<0;\\ \text{d)}\;& \lambda_1^{-}<\lambda_2^{-}<0;\quad \lambda_1^{+}<0;\quad \lambda_2^{+}>0;\\ \text{e)}\;& 0<\lambda_1^{-}<\lambda_2^{-};\quad \lambda_1^{+}<\lambda_2^{+}<0; \end{aligned} \tag{3.1} \]

\[ \begin{aligned} &\text{e)}\quad \lambda_1^-<0;\quad \lambda_2^->0;\quad 0<\lambda_1^+<\lambda_2^+;\\ &\text{zh)}\quad 0<\lambda_1^-<\lambda_2^-;\quad \lambda_1^+<0;\quad \lambda_2^+>0;\\ &\text{z)}\quad \lambda_1^-<\lambda_2^-<0;\quad 0<\lambda_1^+<\lambda_2^+ . \end{aligned} \]

(The case in which one of the characteristic roots is equal to zero is not considered.)

Since in the cases indicated the reasoning almost completely coincides with the arguments given in § 2, here we shall restrict ourselves only to general remarks.

Depending on the signs of the characteristic roots, the mixed problem is posed as follows: in \(\Pi_1\) it is required to find a twice continuously differentiable solution of equation (1.1), satisfying the initial conditions (1.2), the conjugation conditions (1.4), and boundary conditions which are prescribed according to the following rule: at \(x=0\) two linear combinations of the function \(u\) and its first derivatives are prescribed in cases a), g), z), one such linear combination in cases v), e), and nothing is prescribed in cases b), d), zh); at \(x=l\) two linear combinations are prescribed in cases b), e), z), one linear combination in cases g), zh), and nothing is prescribed in cases a), v), d).

Let us examine case a). We prescribe the boundary conditions in the form

\[ \gamma_1^-(t)\frac{\partial u^-(0,t)}{\partial t} +\gamma_2^-(t)\frac{\partial u^-(0,t)}{\partial x} +\gamma_3^-(t)u^-(0,t)=h_1^-(t), \]

\[ \delta_1^-(t)\frac{\partial u^-(0,t)}{\partial t} +\delta_2^-(t)\frac{\partial u^-(0,t)}{\partial x} +\delta_3^-(t)u^-(0,t)=h_2^-(t) \tag{3.2} \]

(nothing is prescribed on the side \(x=l\)). It is easy to see that if
\(\gamma_1^-(t)\delta_2^-(t)-\gamma_2^-(t)\delta_1^-(t)\ne0\), then the mixed problem posed, in the case under consideration, for arbitrary nonzero \(h_1(t)\) and \(h_2(t)\), always has a unique solution. Indeed, the initial conditions (1.2) and the boundary conditions (3.2) uniquely determine the solution in the rectangle \(\overline{\Pi^-}\) [3] and, consequently, uniquely determine \(u^+(x_0,t)\) and

\[ \frac{\partial u^+(x_0,t)}{\partial x}. \]

These latter quantities uniquely determine the solution in \(\Pi^+\).

In exactly the same way one verifies that the corresponding mixed problem in case b), for arbitrary \(h_1(t)\) and \(h_2(t)\), also always has a unique solution, provided only that the coefficients of the corresponding boundary conditions satisfy a certain natural condition.

Carrying out similar reasoning, we easily see that in cases v)—zh) the corresponding mixed problems are never uniquely solvable. Depending on the coefficients of the equation and the boundary conditions, on the free term of the equation and on the initial functions, these problems either have an infinite set of solutions or have no solution at all. Since these cases are practically of no interest, we shall not analyze them.

Remark. The last assertion on nonunique solvability of the problems refers only to the case under consideration, i.e., to the case when two conjugation conditions are prescribed on the line of discontinuity. It is possible

it would also be possible in cases c)—3) to pose the question of the existence of a unique solution, by specifying a different number of conjugation conditions at \(x=x_0\). A. D. Myshkis drew my attention to this circumstance.

§ 4. SOME REMARKS

1. It is easy to see that all the arguments of the preceding sections carry over without essential changes to the case where the equation, the boundary conditions, and the conjugation conditions have a more general form. (In this section we shall speak only about the case \(\lambda_1<0,\ \lambda_2>0\). Everything said will apply equally to the cases developed in § 3.) One could consider the equation

\[ a u+\int_0^t \sum_{i=0}^{1}\sum_{j=0}^{i} a_{ij}(x;t,\tau)\, \frac{\partial^{i}u(x,\tau)}{\partial\tau^{\,i-j}\partial x^{\,j}}\,d\tau =f(x,t) \]

with the initial conditions (1.2), boundary conditions

\[ \gamma_1^{-}(t)\frac{\partial u^{-}(0,t)}{\partial t} +\gamma_2^{-}(t)\frac{\partial u^{-}(0,t)}{\partial x} +\gamma_3^{-}(t)u^{-}(0,t)+ \]

\[ +\int_0^t\left[ \gamma_1^{-}(t,\tau)\frac{\partial u^{-}(0,\tau)}{\partial\tau} +\gamma_2^{-}(t,\tau)\frac{\partial u^{-}(0,\tau)}{\partial x} + \right. \]

\[ \left. +\gamma_3^{-}(t,\tau)u^{-}(0,\tau) \right]\,d\tau =h^{-}(t) \]

(a condition of the same form at \(x=l\)) and conjugation conditions

\[ k_{1i}^{-}(t)\frac{\partial u^{-}(x_0,t)}{\partial t} +k_{2i}^{-}(t)\frac{\partial u^{-}(x_0,t)}{\partial x} +k_{3i}^{-}(t)u^{-}(x_0,t)+ \]

\[ +\int_0^t\left[ k_{1i}^{-}(t,\tau)\frac{\partial u^{-}(x_0,\tau)}{\partial\tau} +k_{2i}^{-}(t,\tau)\frac{\partial u^{-}(x_0,\tau)}{\partial x} + \right. \]

\[ \left. +k_{3i}^{-}(t,\tau)u^{-}(x_0,\tau) \right]\,d\tau = k_{1i}^{+}(t)\frac{\partial u^{+}(x_0,t)}{\partial t} +k_{2i}^{+}(t)\frac{\partial u^{+}(x_0,t)}{\partial x} + \]

\[ +k_{3i}^{+}(t)u^{+}(x_0,t) +\int_0^t\left[ k_{1i}^{+}(t,\tau)\frac{\partial u^{+}(x_0,\tau)}{\partial x} + \right. \]

\[ \left. +k_{2i}^{+}(t,\tau)\frac{\partial u^{+}(x_0,\tau)}{\partial x} +k_{3i}^{+}(t,\tau)u^{+}(x_0,\tau) \right]\,d\tau+h_i(t) \]

\[ (i=1,2). \]

1. The case where the boundary conditions and the conjugation conditions contain derivatives of the unknown function with respect to \(x\) and \(t\) of arbitrary order presents nothing essentially new. By elimination (using the equation) and integration with respect to \(t\), the mixed problem in this case is reduced to the form indicated in the remark.

1. Finally, let us give an example of a mixed problem for which only the third condition (1.5) is not satisfied and which has an infinite set of solutions.

Let, in the rectangle \(\overline{\Pi}\{0 \le t \le T;\ 0 \le x \le l\}\), it be required to find a solution of the equation

\[ \frac{\partial^2 u}{\partial t^2}-a^2\frac{\partial^2 u}{\partial x^2}=0,\qquad a= \begin{cases} a^-, & 0\le x\le x_0,\\ a^+, & x_0\le x\le l \end{cases} \qquad (0<x_0<l), \]

satisfying the initial conditions
\[ u\big|_{t=0}=\frac{\partial u}{\partial t}\bigg|_{t=0}=0, \]
the boundary conditions
\[ \frac{\partial u^-}{\partial x}\bigg|_{x=0} = \frac{\partial u^+}{\partial x}\bigg|_{x=l} =0 \]
and the conjugation conditions
\[ u^-(x_0,t)=u^+(x_0,t); \qquad a^-\frac{\partial u^-(x_0,t)}{\partial x} = -a^+\frac{\partial u^+(x_0,t)}{\partial x}. \]
Here \(\lambda_2^- = a^-\), \(\lambda_1^+ = -a^+\), \(k_1=a^-\), \(k_2=-a^+\), and
\[ \frac{k_1}{k_2}=\frac{\lambda_2^-}{\lambda_1^+}. \]
By direct verification it is easy to see that a solution of the problem posed will be the function

\[ u(x,t)= \begin{cases} 0, & \text{for } 0\le x\le x_0-a^-t \text{ and for } x_0+a^+t\le x\le l,\\[6pt] H_0\!\left(t+\dfrac{x-x_0}{a^-}\right) +\displaystyle\int_0^{\,t+\frac{x-x_0}{a^-}} H_1(\tau)\,d\tau, & \text{for } x_0-a^-t\le x\le x_0,\\[10pt] H_0\!\left(t-\dfrac{x-x_0}{a^+}\right) +\displaystyle\int_0^{\,t-\frac{x-x_0}{a^+}} H_1(\tau)\,d\tau, & \text{for } x_0\le x\le x_0+a^+t, \end{cases} \]

where \(H_0(t)\) and \(H_1(t)\) are arbitrary twice continuously differentiable functions satisfying the conditions
\[ H_0(0)=H_0'(0)=H_0''(0)=H_1(0)=H_1'(0)=0 \]
and
\[ T=\min\left\{\frac{x_0}{a^-},\frac{l-x_0}{a^+}\right\}. \]
The solution can be continued, with preservation of all its properties, for any \(t>T\). Analogous examples can also be given for the cases when at least one of the first two conditions (1.5) is not satisfied. Hence it follows that conditions (1.5) are essential for the unique solvability of the problem under consideration.

References

1. Abolinya V. E., Myshkis A. D. Uch. zap. Latv. un-ta, vol. XX, issue 3, 1958, pp. 87–104.
2. Petrovskii I. G. Lectures on Partial Differential Equations. Moscow, 1961.
3. Melnik Z. O. DAN SSSR, 157, 5, 1039–1042, 1964.

Received by the editors
April 12, 1965

Lviv State University