MATHEMATICS

V. F. ZHDANOVICH

SOLUTION BY THE FOURIER METHOD OF NON-SELF-ADJOINT MIXED PROBLEMS FOR HYPERBOLIC SYSTEMS IN THE PLANE

(Presented by Academician I. G. Petrovskii, 10 XII 1956)

We shall solve the mixed problem for a hyperbolic system in the narrow sense \({}^{1}\)

\[ \frac{\partial}{\partial t}u(x,t)=A(x)\frac{\partial}{\partial x}u(x,t)+B(x)u(x,t) \tag{1} \]

\((0\le x\le l,\ 0\le t\le T<+\infty)\) with boundary and initial conditions

\[ \begin{aligned} \text{a)}\quad &M\frac{\partial}{\partial t}u(0,t)+Nu(0,t)+P\frac{\partial}{\partial t}u(l,t)+Qu(l,t)=0;\\ \text{b)}\quad &u(x,0)=f(x), \end{aligned} \tag{2} \]

where \(u(x,t)\) is an \(n\)-dimensional vector function with complex coordinates. The matrix \(A(x)\) is twice, and \(B(x)\) once, continuously differentiable for \(x\in[0,l]\); moreover, the eigenvalues of the matrix \(A(x)\) do not vanish on the segment \([0,l]\). The matrices \(M,N,P,Q\) are complex, and

\[ \operatorname{rang}\left\|\begin{matrix} M & P\\ N & Q\end{matrix}\right\|=n+q, \qquad \operatorname{rang}\|M,P\|=q \quad (0\le q\le n). \tag{3} \]

Let \(D_2(0,l)\) be the Banach space of classes of measurable functions \(f(x)\) \((0\le x\le l)\), equivalent with respect to the norm
\[ \|f(x)\|_{D_2(0,l)} = \left(\int_0^l \|f(x)\|^2\,dx+\|Mf(0)+Pf(l)\|^2\right)^{1/2}, \]
and suppose that in condition (2b) \(f(x)\in D_2(0,l)\).

Put in the boundary-value problem (1), (2a)
\[ u(x,t)=y(x)e^{\lambda t}. \]
To find \(y(x)\) \((0\le x\le l)\) and \(\lambda\) we obtain the parametric problem

\[ \begin{aligned} \text{a)}\quad &A(x)y'(x)+B(x)y(x)=\lambda y(x);\\ \text{b)}\quad &(M\lambda+N)y(0)+(P\lambda+Q)y(l)=0. \end{aligned} \tag{4} \]

For system (4a) one can construct \({}^{2}\) a fundamental matrix \(Y(x,\lambda)\), analytic in \(\lambda\) for each \(x\in[0,l]\) in each of the regions:
1) \(\operatorname{Re}\lambda<-\gamma\), 2) \(|\operatorname{Re}\lambda|<\gamma\), 3) \(\operatorname{Re}\lambda>\gamma\), where \(\gamma\) is a sufficiently large positive number, and having the asymptotic representation

\[ Y(x,\lambda)=\left[K(x)+O\left(\frac{1}{\lambda}\right)\right] \exp\left[\lambda\int_0^x \Lambda(\xi)\,d\xi\right] \tag{5} \]

uniformly with respect to \(x\in[0,l]\) as \(\lambda\to\infty\). Here \(\Lambda(x)=[\nu_1(x),\nu_2(x),\ldots,\nu_n(x)]\) is the diagonal form of the matrix \(A^{-1}(x)\):
\[ \Lambda(x)=K^{-1}(x)A^{-1}(x)K(x), \]
and \(K(x)\) may be chosen so that the functions \(\nu_i(x)\) \((i=1,2,\ldots,n;\ 0\le x\le l)\) will be continuous and will be arranged in decreasing order—

order:
\(\nu_1(x)>\nu_2(x)>\cdots>\nu_m(x)>0>\nu_{m+1}(x)>\cdots>\nu_n(x)\). To find the eigenvalues of problem (4) we form the characteristic determinant
\(\Delta(\lambda)=\det[(M\lambda+N)Y(0,\lambda)+(P\lambda+Q)Y(0,l)]\) and expand it, using formula (5), as follows:
\[ \Delta(\lambda)=\lambda^q\left[\varphi(\lambda)+\sum_{i=1}^{2^n} b_i(\lambda)e^{\alpha_i\lambda}\right], \]
where \(b_i(\lambda)\to0\) as \(\lambda\to\infty\) \((i=1,2,\ldots,2^n)\);
\[ \varphi(\lambda)=\sum_{i=1}^{2^n} a_i e^{\alpha_i\lambda} \]
is a certain Dirichlet polynomial, and the \(\alpha_i\) \((i=1,2,\ldots,2^n)\) are the numbers arranged in decreasing order:
\[ 0,\quad \int_0^l \nu_i(\xi)\,d\xi \]
\[ (i=1,2,\ldots,2^n), \]
the sums of these numbers taken two at a time, three at a time, and so on up to \(n\); in particular,
\[ \alpha_1=\int_0^l\sum_{i=1}^m\nu_i(\xi)\,d\xi,\qquad \alpha_{2^n}=\int_0^l\sum_{i=m+1}^n\nu_i(\xi)\,d\xi . \]

Definition 1. The boundary conditions (4b) (and also (2a)) are called regular if \(a_1\ne0,\ a_{2^n}\ne0\).

Theorem 1. If the boundary conditions are regular, then problem (4) has a countable set of eigenvalues, and all of them are located in the strip
\[ -\gamma<\operatorname{Re}\lambda<\gamma<+\infty . \]

Let \(\{\lambda_s\}\) \((s=0,\pm1,\ldots)\) be the zeros of the function \(\Delta(\lambda)\), renumbered in increasing order of their imaginary parts; let \(\{k_s\}\) be their multiplicities; and let \(\{p_s\}\) be the multiplicities of the eigenvalues \(\lambda_s\); then, as is known \((^3)\), \(p_s\le k_s\). If \(p_s<k_s\) for at least one \(s\), then in the boundary-value problem (1), (2) we shall take the unknown function \(u(x,t)\) to be a rectangular matrix, and set
\[ u(x,t)=\mathscr{Y}_{nk}(x)e^{J_k(\lambda)t}\qquad (k=1,2,\ldots), \tag{6} \]
where \(\mathscr{Y}_{nk}(x)\) \((0\le x\le l)\) is an unknown rectangular matrix with \(n\) rows and \(k\) columns, and \(J_k(\lambda)\) is a \(k\)-dimensional Jordan cell. To find \(\mathscr{Y}_{nk}(x)\) \((k=1,2,\ldots)\) we obtain the parametric problems
\[ \begin{aligned} \text{a)}\quad & A(x)\mathscr{Y}'_{nk}(x)+B(x)\mathscr{Y}_{nk}(x)=\mathscr{Y}_{nk}(x)J_k(\lambda);\\ \text{b)}\quad & M\mathscr{Y}_{nk}(0)J_k(\lambda)+N\mathscr{Y}_{nk}(0)J_k(\lambda) +P\mathscr{Y}_{nk}(l)J_k(\lambda)+Q\mathscr{Y}_{nk}(l)=0. \end{aligned} \tag{7} \]

Using the theory of eigenfunctions and associated functions \((^{3,4})\), we establish the following properties of these problems: 1) all problems (7), for \(k=1,2,\ldots\), have common eigenvalues, namely the numbers \(\lambda_s\) \((s=0,\pm1,\ldots)\); 2) the number of linearly independent solutions of problem (7), as \(k\) increases, does not decrease and for \(k\ge k_s\) is equal to \(k_s\); 3) these \(k_s\) solutions can be chosen so that they consist of \(p_s\) groups of matrices:
\[ \|0,\ldots,0,y_1^{(i)}(x),y_2^{(i)}(x),\ldots,y_{m_i}^{(i)}(x)\|, \]
\[ \|0,\ldots,0,0,y_1^{(i)}(x),\ldots,y_{m_i-1}^{(i)}(x)\|,\ldots, \|0,\ldots,0,0,0,\ldots,0,y_1^{(i)}(x)\| \]
\((i=1,2,\ldots,p_s)\), where \(y_j^{(i)}(x)\) \((0\le x\le l;\ i=1,2,\ldots,p_s;\ j=1,2,\ldots,m_i)\) are \(n\)-dimensional columns. From the matrices
\(\mathscr{Y}_{nm_i}^{(s)}(x)=\|y_1^{(i)}(x),y_2^{(i)}(x),\ldots,y_{m_i}^{(i)}(x)\|\)
\((i=1,2,\ldots,p_s)\), by formula (6) we construct the matrices
\[ U_{nm_i}^{(s)}(x,t)=\mathscr{Y}_{nm_i}^{(s)}(x)\exp J_{m_i}(\lambda_s)t \quad (s=0,\pm1,\ldots;\ i=1,2,\ldots,p_s), \]
and these latter, for each \(s\) \((s=0,\pm1,\ldots)\), are combined into one
\[ U_{nk_s}^{(s)}(x,t)=\mathscr{Y}_{nk_s}^{(s)}(x)e^{I_s t}, \tag{8} \]
where
\[ I_s=[J_{m_1}(\lambda_s),\ J_{m_2}(\lambda_s),\ \ldots,\ J_{m_{p_s}}(\lambda_s)] \]
is a quasidiagonal matrix;
\[ \mathscr{Y}_{nk_s}^{(s)}(x)=\|\mathscr{Y}_{nm_1}^{(s)}(x),\ \mathscr{Y}_{nm_2}^{(s)}(x),\ \ldots,\ \mathscr{Y}_{nm_{p_s}}^{(s)}(x)\|. \tag{9} \]

Each column of the matrix (8) is a solution of the boundary-value problem (1), (2a), and therefore, if \(f(x)\in \overline{D}_2(0,l)\) is represented in the form of the sum of the series

\[ f(x)=\sum_{s=-\infty}^{+\infty} y_{nk_s}^{(s)}(x)a_s, \tag{10} \]

then the formal solution of problem (1), (2) will have the form

\[ \sum_{s=-\infty}^{+\infty} y_{nk_s}^{(s)}(x)e^{\lambda_s t}a_s. \tag{11} \]

Definition 2. The boundary conditions

\[ s^{*}v(t)=-R\frac{\partial}{\partial t}v(0,t)+Sv(0,t)-V\frac{\partial}{\partial t}v(l,t)+Wv(l,t)=0 \tag{12} \]

are called adjoint to the boundary conditions (2a) if there are matrices \(\|H_1,H_2\|\) and \(\|G_1,G_2\|\) such that, for arbitrary \(u(x,t)\in C^{(1)}(\Omega)\), \(v(x,t)\in C^{(1)}(\Omega)\) \((\Omega=[0,l]\times[0,T])\),

\[ v^{*}(x,t)A(x)u(x,t)\bigg|_{x=0}^{x=l} = \frac{\partial}{\partial t}[Rv(0,t)+Vv(l,t)]^{*}[Mu(0,t)+Pu(l,t)]+ \]

\[ +[H_1v(0,t)+H_2v(l,t)]^{*}su(t)+[s^{*}v(t)]^{*}[G_1u(0,t)+G_2u(l,t)], \tag{13} \]

where \(su(t)\) is the left-hand side of the boundary condition (2a)\(^*\).

The matrices \(R,S,V,W\) are found by comparing the coefficients in the identity (13). Without restricting the generality of the results, in the condition (2a) one may assume

\[ \|M,P,N,Q\|= \left\| \begin{array}{cccc} M_{qn} & P_{qn} & N_{qn} & Q_{qn}\\ 0_{n-qn} & 0_{n-qn} & N_{n-qn} & Q_{n-qn} \end{array} \right\|, \]

where \(0_{n-qn}\) is a matrix of zeros. Then it is not difficult to prove that

\[ \|R,V,S,W\|= \left\| \begin{array}{cccc} R_{qn} & V_{qn} & S_{qn} & W_{qn}\\ 0_{n-qn} & 0_{n-qn} & S_{n-qn} & W_{n-qn} \end{array} \right\|. \]

Definition 3. The boundary-value problem for the system

\[ -\frac{\partial}{\partial t}v(x,t) = -\frac{\partial}{\partial x}\bigl[A^{*}(x)v(x,t)\bigr]+B^{*}(x)v(x,t) \tag{14} \]

with the boundary conditions (12) is called adjoint to problem (1), (2a).

Carrying out separation of variables in problem (14), (12) by the formula \(v(x,t)=z(x)e^{-\mu t}\), we obtain the parametric problem:

\[ \begin{aligned} \text{a)}\quad &-\frac{d}{dx}\bigl[A^{*}(x)z(x)\bigr]+B^{*}(x)z(x)=\mu z(x);\\ \text{b)}\quad &(R\mu+S)z(0)+(V\mu+W)z(l)=0. \end{aligned} \tag{15} \]

Theorem 2. If \(\lambda_s\) is a zero of the function \(\Delta(\lambda)\) of multiplicity \(k_s\) and an eigenvalue of problem (4) of multiplicity \(p_s\), then \(\mu_s=\overline{\lambda}_s\) will be a zero of the characteristic determinant \(\Delta_1(\mu)\) of problem (15) of the same multiplicity \(k_s\) and an eigenvalue of problem (15) of the same multiplicity \(p_s\).

Let \(F_{nk}(x)\) and \(G_{nm}(x)\) \((k,m=1,2,\ldots;\ 0\leq x\leq l)\) be two rectangular matrices. Introduce the notation

\[ [F_{nk}(x),G_{nm}(x)]= \]

\[ =\int_0^l G_{nm}^{*}(x)F_{nk}(x)\,dx +[RG_{nm}(0)+VG_{nm}(l)]^{*}[MF_{nk}(0)+PF_{nk}(l)]. \]

\[ \underline{\qquad} \]

\(^*\) \(v^{*}u\) here and below denotes the scalar product of the vectors \(u\) and \(v\).

Theorem 3. If \(\mathscr{Y}_{n k_s}^{(s)}(x)\) \((s=0,\pm 1,\ldots)\) is the system of matrices constructed from the eigenfunctions and adjoint functions of problem (4), and \(Z_{n k_r}^{(r)}(x)\) \((r=0,\pm 1,\ldots)\) is the same system for problem (15), then for \(s\ne r\)
\[ [\mathscr{Y}_{n k_s}^{(s)}(x), Z_{n k_r}^{(r)}(x)] = 0 . \]

Theorem 4. If \(f(x)\in D_2(0,l)\) is represented in the form of the sum of the series (10), then
\[ a_s=[f(x), B_s^{*-1} Z_{n k_s}^{(s)}(x)], \]
where
\[ B_s=[\mathscr{Y}_{n k_s}^{(s)}(x), Z_{n k_s}^{(s)}(x)] \quad (s=0,\pm 1,\ldots). \]

To justify the scheme presented, we shall use the problems generated by (4) by the linear differential operator
\[ \mathscr{L}y=A(x)y'(x)+B(x)y(x), \]
where \(y(x)\in C^{(1)}(0,l)\) and
\[ MA(0)y'(0)+[MB(0)+N]y(0)+PA(l)y'(l)+[PB(l)+Q]y(l)=0 . \]

Theorem 5. If the boundary conditions are regular, \(f(x)\) \((0\le x\le l)\) belongs to the domain of definition of the operator \(\mathscr{L}y\), \(f_1(x)=\mathscr{L}f(x)\) is continuous and satisfies the condition
\[ N_{n-q n} f_1(0)+Q_{n-q n} f_1(l)=0, \]
and \(f'_1(x)\in \mathscr{L}_2(0,l)\), then the series (11), under a certain grouping of terms independent of the choice of the function \(f(x)\), converges uniformly on \(\Omega\) to a continuously differentiable function \(u(x,t)\) satisfying equation (1) and conditions (2).

Introduce the Banach space \(M_2(\Omega)\) of measurable functions \(f(x,t)\), \((x,t)\in\Omega\), such that for each \(t\in[0,T]\), \(f(x,t)\) belongs to \(D_2(0,l)\) and
\[ \|f(x,t)\|_{D_2(0,l)}<m_0<+\infty \]
uniformly with respect to \(t\);
\[ \|f(x,t)\|_{M_2\Omega}=\sup_{0\le t\le T}\|f(x,t)\|_{D_2(0,l)}. \]
\(M_2(\Omega)\) is a complete space.

Definition 4. A function \(u(x,t)\in M_2(\Omega)\) is called a generalized solution\({}^{5}\) of problem (1), (2), if, for any \(t\in[0,T]\) and any function \(v(x,t)\in C^{(1)}(\Omega)\) for which
\[ S_{n-q n}v(0,t)+W_{n-q n}v(l,t)=0, \]
it satisfies the integral equation
\[ \iint_{\Omega(t)} [\Phi^* v(x,\tau)]^* u(x,\tau)\,dx\,d\tau -[u(x,t),v(x,t)]+[f(x),v(x,0)] +\int_0^t [S^*v(\tau)]^*[Mu(0,\tau)+Pu(l,\tau)]\,d\tau=0, \]
where
\[ \Phi^*v(x,\tau)=\frac{\partial}{\partial \tau}v(x,\tau) -\frac{\partial}{\partial x}\,[A^*(x)v(x,\tau)] +B^*(x)v(x,\tau),\quad (x,\tau)\in\Omega(t)=[0,l]\times[0,t], \]
\[ [u(x,t),v(x,t)] =\int_0^l v^*(x,t)u(x,t)\,dx +[Rv(0,t)+Vv(l,t)]^*[Mu(0,t)+Pu(l,t)]. \]

Theorem 6. If the boundary conditions are regular and if \(f(x)\in D_2(0,l)\), then the series (11) converges, under a certain grouping of its terms, in the norm of the space \(M_2(\Omega)\) to a certain function \(u(x,t)\), \((x,t)\in\Omega\), which is the unique generalized solution of problem (1), (2).

I express my gratitude to A. D. Myshkis for a number of valuable suggestions concerning the question discussed.

Belorussian State University
named after V. I. Lenin

Received
6 XI 1956

CITED LITERATURE

1. I. G. Petrovskii, Lectures on Partial Differential Equations, 1950.
2. V. S. Pugachev, Matem. sborn., 15(57), 1, 24 (1944).
3. M. A. Naimark, Linear Differential Operators, 1954.
4. M. V. Keldysh, DAN, 72, No. 1, 11 (1951).
5. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.