Mathematics

M. L. Rasulov

Conditions for the Well-Posedness of One-Dimensional Mixed Problems

(Presented by Academician N. I. Muskhelishvili on 9 III 1961)

In this note we give easily verifiable conditions for the well-posedness of mixed problems of the type

\[ \frac{\partial^q u^{(i)}}{\partial t^q} = \sum_{\substack{mk+l\le p\\ k\le q-1}} A_{kl}^{(i)}(x)\frac{\partial^{k+l}u^{(i)}}{\partial t^k\partial x^l} + f^{(i)}(x,t) \quad \text{for } x\in(a_i,b_i); \tag{1} \]

\[ \sum_{i=1}^{n}\sum_{l=0}^{p-1} \left\{ \alpha_{sl}^{(i)}\!\left(\frac{\partial}{\partial t}\right) \frac{\partial^l u^{(i)}}{\partial x^l}\Bigg|_{x=a_i} + \beta_{sl}^{(i)}\!\left(\frac{\partial}{\partial t}\right) \frac{\partial^l u^{(i)}}{\partial x^l}\Bigg|_{x=b_i} \right\}=0; \tag{2} \]

\[ \frac{\partial^k u^{(i)}}{\partial t^k}\Bigg|_{t=0} = \Phi^{(i)}(x) \quad \text{for } x\in(a_i,b_i)\quad (k=0,\ldots,q-1;\ i=1,\ldots,n), \tag{3} \]

where
\[ \alpha_{sl}^{(i)}(z)=\sum_{k=0}^{q}\alpha_{slk}^{(i)}z^k,\quad \beta_{sl}^{(i)}(z)=\sum_{k=0}^{q}\beta_{slk}^{(i)}z^k; \quad \alpha_{slk}^{(i)},\ \beta_{slk}^{(i)}\text{ are constants};\quad p=mq; \]
\(m\) is a natural number; \((a_i,b_i)\) are mutually non-overlapping intervals having common endpoints.

Let the following conditions be satisfied:

\(1^\circ.\) For \(x\in[a_i,b_i]\) the roots \(\varphi_s^{(i)}(x)\) of the characteristic equations

\[ \theta^p A_{0p}^{(i)}(x) + \theta^{p-m} A_{1,(q-1)m}^{(i)}(x) +\cdots+ \theta^m A_{q-1,m}^{(i)}(x)-1=0 \tag{4} \]

are distinct and different from zero; their arguments and the arguments of their differences do not depend on \(x\).

\(2^\circ.\) On the interval \([a_i,b_i]\) the functions \(A_{kl}^{(i)}(x)\) are continuously differentiable \(2-s\) times when \(mk+l=p-s\) \((s=0,1,2)\).

The straight lines determined by the equations
\[ \operatorname{Re}\lambda\varphi_k^{(i)}(x) - \operatorname{Re}\lambda\varphi_s^{(i)}(x)=0 \quad (k,s=1,\ldots,p) \]
divide the \(\lambda\)-plane into sectors \(\Sigma_j\), in each of which, under a suitable numbering of the roots of the characteristic equations (4), the inequalities

\[ \operatorname{Re}\lambda\varphi_1^{(i)}(x) \le \operatorname{Re}\lambda\varphi_2^{(i)}(x) \le \cdots \le \operatorname{Re}\lambda\varphi_p^{(i)}(x) \quad \text{for } \lambda\in\Sigma_j, \]

\[ x\in(a_i,b_i)\quad (i=1,\ldots,n) \]

hold, where \(\operatorname{Re} z\) is the real part of \(z\).

Consequently, according to a theorem of Ya. D. Tamarkin \((^1)\), under conditions \(1^\circ,2^\circ\) the homogeneous equations corresponding to equations (9) have a fundamental system of particular solutions
\[ y_k^{(i)}(x,\lambda)\quad (k=1,\ldots,p;\ i=1,\ldots,n), \]
which, in the sector \(\Sigma_j\) for large \(\lambda\), admit asymptotic

representations

\[ \frac{d^\nu y_k^{(i)}(x,y)}{dx^\nu} = \lambda^\nu \left\{ \sum_{s=0}^{1}\lambda^{-s}\eta_{k\nu s}^{(i)}(x) + \frac{E_{k\nu}^{(i)}(x,\lambda)}{\lambda^2} \right\} \exp\left(\lambda\int_{a_i}^{x}\varphi_k^{(i)}(\xi)\,d\xi\right), \tag{5} \]

where the functions \(\eta_{k\nu s}^{(i)}(x)\) on the interval \([a_i,b_i]\) have continuous derivatives up to order \(2-s\) inclusive \((s=0,1)\).

Introduce the notation:

\[ w_k^{(i)}=\int_{a_i}^{b_i}\varphi_k^{(i)}(x)\,dx,\qquad \alpha_k=\arg w_k^{(i)}, \]

\[ A_{ks}^{(i)}(\lambda)= \sum_{l=0}^{p-1}\alpha_{kl}^{(i)}(\lambda)\lambda^l \bigl(\varphi_s^{(i)}(a_i)\bigr)^l,\qquad B_{ks}^{(i)}(\lambda)= \sum_{l=0}^{p-1}\beta_{kl}^{(i)}(\lambda)\lambda^l \bigl(\varphi_s^{(i)}(b_i)\bigr)^l. \]

According to condition \(1^\circ\), the equations

\[ \operatorname{Re}\lambda w_k^{(i)}=0,\qquad (k=1,\ldots,p) \]

determine \(2\mu\) distinct rays \(d_j\) \((j=1,\ldots,2\mu\leq 2p)\) issuing from the origin of the \(\lambda\)-plane. Obviously, the argument of the ray \(d_j\) is equal to \(\pi/2-\alpha_j\).

Next take the set of rays \(d'_j\), distinct from the rays \(d_j\) and arranged in the sequence

\[ d'_1,\ d_1,\ d'_2,\ d_2,\ldots,\ d'_{2\mu},\ d_{2\mu},\ d'_1. \tag{6} \]

By the boundaries of the sectors \(\Sigma_j\) and the rays (6), the plane is divided into sectors \((T_j)\), in each of which, under a suitable numbering of the roots of the characteristic equation (3), the inequalities

\[ \operatorname{Re}\lambda w_1^{(i)}\leq \operatorname{Re}\lambda w_2^{(i)}\leq\cdots\leq \operatorname{Re}\lambda w_p^{(i)}. \tag{7} \]

hold.

Denote by

\[ w_{1j}^{(i)},\ w_{2j}^{(i)},\ldots,\ w_{\nu_j j}^{(i)} \]

those of the numbers \(w_k^{(i)}\) that lie on the straight line making an angle \(\alpha_j\) with the positive direction of the real axis of the \(\lambda\)-plane. According to condition \(1^\circ\), the arguments of the numbers \(w_k^{(i)}\) do not depend on the index \(i\).

For the sector \(T_j\) put

\[ w_{kj}^{(i)}=\mu_{kj}^{(i)}e^{\sqrt{-1}\alpha_j}\qquad (k=1,\ldots,\nu_j), \]

where \(\mu_{kj}^{(i)}\) are real numbers numbered in increasing order:

\[ \mu_{1j}^{(i)}<\mu_{2j}^{(i)}<\cdots<\mu_{s_jj}^{(i)}<0< \mu_{s_j+1,j}^{(i)}<\cdots<\mu_{\nu_jj}^{(i)}. \]

If from the set of numbers \(w_1^{(i)},\ldots,w_p^{(i)}\) all the numbers \(w_{kj}^{(i)}\) are excluded, then the remaining numbers \(w_k^{(i)}\) can be divided into two groups \((w_k^{(i,1)})\), \((w_k^{(i,2)})\). To the first group assign those of the numbers \(w_k^{(i)}\) for which in the sector \((T_j)\) we have \(\operatorname{Re}\lambda w_k^{(i)}\to-\infty\) as \(|\lambda|\to\infty\). To the second group assign those for which we have \(\operatorname{Re}\lambda w_k^{(i)}\to+\infty\) as \(|\lambda|\to\infty\).

For the sector \((T_j)\) we shall regard the numbers \(w_k^{(i)}\) as numbered in the following sequence:

\[ w_1^{(i,1)},\ldots,w_{\chi_j}^{(i,1)},\ w_{1j}^{(i)},\ldots,w_{s_jj}^{(i)},\ w_{s_j+1,j}^{(i)},\ldots,w_{\nu_jj}^{(i)},\ w_{\chi_j+\nu_j+1}^{(i,2)},\ldots,w_p^{(i,2)}, \]

where, for \(\lambda\in T_j\), the inequalities hold

\[ \operatorname{Re}\lambda w^{(i,1)}_1 \ll \cdots \ll \operatorname{Re}\lambda w^{(i,1)}_{\chi_j} \ll \operatorname{Re}\lambda w^{(i)}_{l_j} \ll \cdots \ll \operatorname{Re} w^{(i)}_{s_j} \ll 0 \ll \]

\[ \ll \operatorname{Re}\lambda w^{(i)}_{s_j+1,j} \ll \cdots \ll \operatorname{Re}\lambda w^{(i)}_{\nu_j j} \ll \operatorname{Re}\lambda w^{(i,2)}_{\chi_j+\nu_j+1} \ll \cdots \ll \operatorname{Re}\lambda w^{(i,2)}_p . \]

Let us denote by \(M^{(i)}_{1j}(\lambda)\), \(M^{(i)}_{\sigma_j j}(\lambda)\), respectively, the following matrices:

\[ \left( \begin{array}{cccccc} A^{(i)}_{1,1}(\lambda) & \ldots & A^{(i)}_{1,\chi_j}(\lambda) & B^{(i)}_{1,\chi_j+1}(\lambda) & \ldots & B^{(i)}_{1,\chi_j+s_j}(\lambda)\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ A^{(i)}_{np,1}(\lambda) & \ldots & A^{(i)}_{np,\chi_j}(\lambda) & B^{(i)}_{np,\chi_j+1}(\lambda) & \ldots & B^{(i)}_{np,\chi_j+s_j}(\lambda) \end{array} \right. \]

\[ \left. \begin{array}{cccccc} A^{(i)}_{1,\chi_j+s_j+1}(\lambda) & \ldots & A^{(i)}_{1,\chi_j+\nu_j}(\lambda) & B^{(i)}_{1,\chi_j+\nu_j+1}(\lambda) & \ldots & B^{(i)}_{1,p}(\lambda)\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ A^{(i)}_{np,\chi_j+s_j+1}(\lambda) & \ldots & A^{(i)}_{np,\chi_j+s_j+1}(\lambda) & B^{(i)}_{np,\chi_j+\nu_j+1}(\lambda) & \ldots & B^{(i)}_{np,p}(\lambda) \end{array} \right), \]

\[ \left( \begin{array}{cccccc} A^{(i)}_{1,1}(\lambda) & \ldots & A^{(i)}_{1,\chi_j+s_j}(\lambda) & B^{(i)}_{1,\chi_j+s_j+1}(\lambda) & \ldots & B^{(i)}_{1,p}(\lambda)\\ A^{(i)}_{np,1}(\lambda) & \ldots & A^{(i)}_{np,\chi_j+s_j}(\lambda) & B^{(i)}_{np,\chi_j+s_j+1}(\lambda) & \ldots & B^{(i)}_{np,p}(\lambda) \end{array} \right). \]

Next, let \(M_{1j}(\lambda)\), \(M_{\sigma_j j}(\lambda)\) be, respectively, the matrices consisting of the matrix cells \(M^{(i)}_{1j}(\lambda)\), \(M^{(i)}_{\sigma_j j}(\lambda)\) \((i=1,\ldots,n)\):

\[ M_{1j}(\lambda)=\bigl(M^{(1)}_{1j}(\lambda),\,M^{(2)}_{1j}(\lambda),\,\ldots,\,M^{(n)}_{1j}(\lambda)\bigr), \]

\[ M_{\sigma_j j}(\lambda)=\bigl(M^{(1)}_{\sigma_j j}(\lambda),\,M^{(2)}_{\sigma_j j}(\lambda),\,\ldots,\,M^{(n)}_{\sigma_j j}(\lambda)\bigr). \]

Assume that, in addition to conditions \(1^\circ\), \(2^\circ\), the following condition is also satisfied:

\(3^\circ\). The determinants of the matrices \(M_{1j}(\lambda)\), \(M_{\sigma_j j}(\lambda)\), for all \(j=1,\ldots,2\mu\), are polynomials in \(\lambda\) of the same degree \(d\), distinct from the identically zero polynomial. All determinants of orders \(pn\), \(pn-1\), formed from all possible combinations of columns of the matrices \(M_{1j}(\lambda)\), \(M_{\sigma_j j}(\lambda)\), are polynomials of degrees not exceeding \(d\).

With the aid of the asymptotic representations (5), by the method of contour integration published in the notes \(({}^2\text{–}{}^4)\), one proves:

Theorem 1. Under conditions \(1^\circ\)–\(3^\circ\), if under the mapping \(\lambda^m=z\) all rays \(d_k\) \((k=1,\ldots,2\mu)\) pass into rays lying in the left half-plane \(z^*\), and if on the interval \([a_i,b_i]\) the functions \(\Phi^{(i)}_{s-1}(x)\) \((s=1,\ldots,p)\) are continuously differentiable \(p-ms\) times, while the functions \(\partial^k f^{(i)}(x,t)/\partial t^k\) \((k=0,\ldots,q)\) are once differentiable for \(t\in[0,T]\), then problem (1)–(3) has a unique solution \(u^{(i)}(x,t)\), represented by the formula

\[ u^{(i)}(x,t)= \frac{-1}{2\pi\sqrt{-1}}\sum_{\nu}\int_{c_\nu} \lambda^{m-1} \left\{ \sum_{j=1}^{n}\int_{a_j}^{b_j} G^{(i,j)}(x,\xi,\lambda) \left(F^{(j)}(\xi,\Phi,\lambda^m)+ \right. \right. \]

\[ \left. \left. +\int_{0}^{t} f^{(j)}(\xi,\tau)\exp(-\lambda^m\tau)\,d\tau\right) \,d\xi \right\} \exp(\lambda^m t)\,d\lambda, \tag{8} \]

where \(c_\nu\) is a simple closed contour surrounding only one pole \(\lambda_\nu\) of the integrand; \(G^{(i,j)}(x,\xi,\lambda)\) is the Green’s function of the spectral-

\[ \text{* The imaginary axis is not included in the half-planes.} \]

problem:

\[ \sum_{\substack{mk+l\le p\\ k\le q-1}} \lambda^{mk} A_{kl}^{(i)}(x)\frac{d^l v^{(i)}}{dx^l} -\lambda^p v^{(i)} = F^{(i)}(x,\Phi,\lambda^m); \tag{9} \]

\[ \sum_{i=1}^n \sum_{l=0}^{p-1} \left\{ \alpha_{sl}^{(i)}(\lambda^m) \left.\frac{d^l v^{(i)}}{dx^l}\right|_{x=a_i} + \beta_{sl}^{(i)}(\lambda^m) \left.\frac{d^l v^{(i)}}{dx^l}\right|_{x=b_i} \right\}=0: \tag{10} \]

\[ F^{(i)}(x,\Phi,\lambda^m) = \sum_{j=0}^{q-1}\lambda^{m(q-1-j)}\Phi_j^{(i)}(x) - \]

\[ - \sum_{\substack{1\le k\le q-1\\ mk+l\le p}} A_{kl}^{(i)}(x) \bigl(\lambda^{m(k-1)}\Phi_0^{(i)}(x)+\cdots+\Phi_{k-1}^{(i)}(x)\bigr); \]

the sum over \(\nu\) in (8) is extended over all poles of the integrand. In this case \(u^{(i)}(x,t)\) depends continuously on the initial data \(\Phi^{(i)}(x)\) and on the free terms \(f^{(i)}(x,t)\) of equations (1).

From Theorem 2 of note (5) it follows

Theorem 2. Under the conditions of Theorem 1, if under the mapping \(\lambda^m=z\) one of the rays \(d_k\) \((k=1,\ldots,2\mu)\) passes into a ray lying in the right \(z\)-half-plane, then problem (1)—(3) does not have a sufficiently smooth solution*.

For \(p=q=2\) (then, obviously, \(m=1\)) the following stronger assertion is proved:

Theorem 3. Under the conditions of Theorem 1, if \(p=q=2\), then for the correctness of problem (1)—(3) (moreover, for the existence of a sufficiently smooth solution of this problem) the coincidence of the rays \(d_k\) \((k=1,2)\) with the imaginary half-axes of the \(\lambda\)-plane is necessary and sufficient.

In other words, under the conditions of Theorem 1, if \(p=q=2\), then for the existence of a sufficiently smooth solution of problem (1)—(3), the reality of the roots of the characteristic equations

\[ \theta^2 A_{02}^{(i)}(x)+\theta A_{11}^{(i)}(x)-1=0 \qquad (i=1,\ldots,n) \]

is necessary and sufficient.

Azerbaijan State University
named after S. M. Kirov

Received
18 II 1961

CITED LITERATURE

1. Ya. D. Tamarkin, On certain general problems of the theory of ordinary linear differential equations and on the expansion of arbitrary functions in series, Petrograd, 1917.
2. M. L. Rasulov, DAN, 125, No. 1 (1959).
3. M. L. Rasulov, DAN, 125, No. 2 (1959).
4. M. L. Rasulov, DAN, 131, No. 1 (1960).
5. M. L. Rasulov, DAN, 120, No. 1 (1958).

* It is assumed that not all \(\Phi^i(x)\), \(f^i(x,t)\) are identically equal to zero. Sufficient smoothness is understood in the sense of note (5) and is a weaker requirement than the continuity of the derivatives occurring in the problem.