MATHEMATICS

A. I. VAGABOV

CONDITIONS FOR THE WELL-POSEDNESS OF ONE-DIMENSIONAL MIXED PROBLEMS FOR HYPERBOLIC SYSTEMS

(Presented by Academician I. N. Vekua, January 6, 1964)

Consider the mixed problem for a Kovalevskaya system:

\[ \frac{\partial^p u}{\partial t^p} = \sum_{\substack{k_0 \le p-1\\ k_0+k_1\le p}} A^{(k_0k_1)}(x)\, \frac{\partial^{k_0+k_1}u}{\partial t^{k_0}\partial x^{k_1}} + f(x,t); \tag{1} \]

\[ \sum_{\substack{k_1\le p-1\\ k_0+k_1\le p}} \left\{ \left. \alpha^{(k_0k_1)} \frac{\partial^{k_0+k_1}u}{\partial t^{k_0}\partial x^{k_1}} \right|_{x=a} + \left. \beta^{(k_0k_1)} \frac{\partial^{k_0+k_1}u}{\partial t^{k_0}\partial x^{k_1}} \right|_{x=b} \right\}=0; \tag{2} \]

\[ \left. \frac{\partial^k u}{\partial t^k} \right|_{t=0} = \psi_k(x) \qquad (k=0,\ldots,p-1), \tag{3} \]

where \(A^{(k_0k_1)}(x)\) are \(n\)-dimensional matrices of functions; \(f(x,t)\), \(u(x,t)\), \(\psi_k(x)\) are \(n\)-dimensional columns of functions; \(\alpha^{(k_0k_1)}\), \(\beta^{(k_0k_1)}\) are constant rectangular matrices of size \(np\times n\).

Introduce the following matrix notation:

\[ A(x)= \left\| \begin{array}{ccccc} A^{(p-1,1)}(x) & A^{(p-2,2)}(x) & \cdot & \cdot & A^{(0,p)}(x)\\ E & 0 & \cdot & \cdot & 0\\ 0 & E & \cdot & \cdot & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & E & 0 \end{array} \right\|, \]

where \(E\) is the identity matrix of order \(n\); further:

\[ \alpha^{(k)} = \left\| \alpha^{(k,0)},\ \alpha^{(k-1,1)},\ldots,\alpha^{(0,k)},\ 0,\ldots,0 \right\|, \tag{4} \]

\[ \beta^{(k)} = \left\| \beta^{(k,0)},\ \beta^{(k-1,1)},\ldots,\beta^{(0,k)},\ 0,\ldots,0 \right\| \]

\[ (k=0,\ldots,p); \]

zeros denote zero matrices of size \(np\times n\). Their number is chosen so that the matrices \(\alpha^{(k)}\) and \(\beta^{(k)}\) are square. For \(k=p\), as an exception, we assume that the matrices (4) end respectively with the terms \(\alpha^{(1,p-1)}\) and \(\beta^{(1,p-1)}\).

In what follows we shall assume that:

1) The elements of the matrices \(A^{(k_0k_1)}(x)\), for which \(k_0+k_1=k\), are continuously differentiable on \([a,b]\) \(k+1\) times; \(\psi_k(x)\) \((k=0,\ldots,p-1)\) are continuously differentiable \(2p-k\) times and vanish at the endpoints \(a\) and \(b\), together with all derivatives up to order \(2p-k-1\); \(f(x,t)\) has a second derivative with respect to \(t\) and a \((p-1)\)-st derivative with respect to \(x\), continuous on \([a,b]\times[0,T]\), and \(f(a,t)\equiv f(b,t)\equiv 0\).

2) \(\theta\) are the roots of the equation

\[ \det\|A(x)-\theta E\|=0 \tag{5} \]

are real, everywhere distinct, and nonzero functions. It is easy to show that this requirement means hyperbolicity of system (1) in the narrow sense of I. G. Petrovskii.

We agree to regard the roots of equation (5) as numbered so that they satisfy the inequalities:

\[ \theta_1(x)<\cdots<\theta_\tau(x)<0<\theta_{\tau+1}(x)<\cdots<\theta_{np}(x). \]

From assertion 2) there follows the existence of matrices \(m(x)\) with the property

\[ m^{-1}(x)A(x)m(x)= \begin{pmatrix} \theta_1(x) & 0\\ & \ddots\\ 0 & & \theta_{np}(x) \end{pmatrix}. \]

The set of such matrices forms the class \(\mathfrak M_A\).

Passing to the consideration of the boundary data (3), we first construct, with the aid of the matrices (4), a special matrix \(\|\alpha,\beta\|\) of size \(np\times 2np\). First of all, one may assume that in the matrix \(\|\alpha^{(p)},\beta^{(p)}\|\) the first \(q_p\) \((0\le q_p\le np)\) rows are linearly independent, while the remaining ones are zero. This can be achieved by a linear combination of the conditions (3) and a corresponding change in the numbering of the rows. These \(q_p\) rows will constitute the first \(q_p\) rows of the matrix \(\|\alpha,\beta\|\). Just as above, one may assume that in the matrix \(\|\alpha^{(p-1)},\beta^{(p-1)}\|\) the rows going after number \(q_p\) up to number \(q_{p-1}\) \((0\le q_{p-1}\le np)\) are linearly independent (they will constitute the next rows of the matrix \(\|\alpha,\beta\|\), of course, if \(q_p<q_{p-1}\)).

Next we pass to the matrix \(\|\alpha^{(p-2)},\beta^{(p-2)}\|\). At each step we shall, generally speaking, adjoin successive rows to the matrix being constructed. Obviously, the construction of the matrix will be completed when \(np\) rows have been obtained.

In doing so we assume that:

3) The boundary conditions are regular in the sense that the determinants \(m(\alpha,a,\beta,b)\) and \(m(\beta,b,\alpha,a)\) are nonzero, where

\[ m(\alpha,a,\beta,b)= \]

\[ = \left| \begin{array}{cccccc} \displaystyle\sum_{j=1}^{np}\alpha_{1,j}m_{j,1}(a) & \cdots & \displaystyle\sum_{j=1}^{np}\alpha_{1,j}m_{j,\tau}(a) & \displaystyle\sum_{j=1}^{np}\beta_{1,j}m_{j,\tau+1}(b) & \cdots & \displaystyle\sum_{j=1}^{np}\beta_{1,j}m_{j,np}(b)\\ \cdot & & \cdot & \cdot & & \cdot\\ \cdot & & \cdot & \cdot & & \cdot\\ \displaystyle\sum_{j=1}^{np}\alpha_{np,j}m_{j,1}(a) & \cdots & \displaystyle\sum_{j=1}^{np}\alpha_{np,j}m_{j,\tau}(a) & \displaystyle\sum_{j=1}^{np}\beta_{np,j}m_{j,\tau+1}(b) & \cdots & \displaystyle\sum_{j=1}^{np}\beta_{np,j}m_{j,np}(b) \end{array} \right|, \]

and \(\alpha_{ij}, \beta_{ij}, m_{ij}\) denote the elements of the matrices designated by the corresponding letters, \(m(x)\in\mathfrak M_A\) is an arbitrary matrix.

With the aid of the residue method and the method of the contour integral, set forth in the works \((^{1-3})\), it is proved that

Theorem. Under conditions 1), 2), 3), problem (1)—(3) has a unique solution, continuously dependent on the given functions and representable by the formula:

\[ u(x,t)=-\frac{1}{2\pi\sqrt{-1}}\sum_\nu \int_{C_\nu}\lambda^{1-p}e^{\lambda t}\,d\lambda \int_a^b G(x,\xi,\lambda)\bigl(A^{(0p)}(\xi)\bigr)^{-1} \left(F(\xi,\psi,\lambda)+ \int_0^t e^{-\lambda\tau}f(\xi,\tau)\,d\tau\right)\,d\xi, \tag{6} \]

where \(C_\nu\) is a simple closed contour enclosing only one pole \(\lambda_\nu\) of the integrand; \(G(x,\xi,\lambda)\) is the Green’s function of the spectral problem:

\[ \sum_{\substack{k_0 \le p-1\\ k_0+k_1 \le p}} \lambda^{k_0} A^{(k_0 k_1)}(x)\frac{d^{k_1}v}{dx^{k_1}} -\lambda^p v = F(x,\psi,\lambda); \tag{7} \]

\[ \sum_{\substack{k_1 \le p-1\\ k_0+k_1 < p}} \lambda^{k_0} \left\{ \alpha^{(k_0 k_1)} \left.\frac{d^{k_1}v}{dx^{k_1}}\right|_{x=a} + \beta^{(k_0 k_1)} \left.\frac{d^{k_1}v}{dx^{k_1}}\right|_{x=b} \right\} =0; \tag{8} \]

\[ F(x,\psi,\lambda) = \sum_{k_0=0}^{p-1} \lambda^{p-1-k_0}\psi_{k_0}(x) - \]

\[ - \sum_{\substack{1\le k_0\le p-1\\ k_0+k_1\le p}} A^{(k_0 k_1)}(x)\frac{d^{k_1}}{dx^{k_1}} \{\lambda^{k_0-1}\psi_0(x)+\cdots+\psi_{k_0-1}(x)\}. \]

It is also proved that, in the case of simple poles of the spectral problem (7)—(8), the hyperbolicity condition is necessary for the existence of a solution of the mixed problem (1)—(3).

In particular, for \(p=1\) the corresponding results were obtained earlier in \((^4)\).

Azerbaijan State University
named after S. M. Kirov

Received
20 XII 1963

CITED LITERATURE

\(^1\) M. L. Rasulov, Matem. sborn., 30 (72), no. 3, 509 (1952).
\(^2\) M. L. Rasulov, Matem. sborn., 48 (90), 3 (1959).
\(^3\) M. L. Rasulov, Doctoral dissertation, V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 1960.
\(^4\) A. I. Vagabov, Uch. zap. Azerb. gos. univ. im. S. M. Kirova, 3, 4 (1963).