MATHEMATICS

CHEN CHIN-I

ON A UNIQUENESS THEOREM FOR THE SOLUTION OF A MIXED PROBLEM FOR SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS

(Presented by Academician S. L. Sobolev on 17 XII 1956)

In the note of I. M. Gel'fand and G. E. Shilov \((^{1})\), a new approach is indicated to the problem of uniqueness of the Cauchy solution for systems of linear equations of the form

\[ \frac{\partial u}{\partial t}=P\left(t,\frac{\partial}{\partial x}\right)u,\qquad u(x,0)=u_0(x). \]

In the present note this method is applied to establishing uniqueness of the solution of the mixed problem for certain types of systems of linear equations.

Let there be given a system of linear partial differential equations of the form

\[ \frac{\partial u}{\partial t}=P\left(t,\frac{\partial^n}{\partial x^n}\right)u,\qquad n\geqslant 2\ \text{integer},\qquad x\geqslant 0,\qquad t\geqslant 0, \tag{1} \]

where \(u(x,t)=\{u_1(x,t),\ldots,u_N(x,t)\}\) is an unknown vector-function; \(P\left(t,\frac{\partial^n}{\partial x^n}\right)\) is a matrix of \(N\) rows and \(N\) columns whose elements are polynomials in the operator \(D\equiv \partial^n/\partial x^n\) with coefficients continuously depending on \(t\). Let \(nq_j\) be the highest order of the derivatives of \(u_j(x,t)\) with respect to \(x\) in system (1). To system (1) are adjoined the initial condition

\[ u(x,0)=u_0(x) \tag{2} \]

and boundary conditions of one of the following types:

\[ \left.\frac{\partial^{nl_j+m}u_j}{\partial x^{nl_j+m}}\right|_{x=0}=0,\qquad l_j=0,1,\ldots,q_j-1, \tag{3} \]

where \(m\) takes one (fixed) of the values \(0,1,\ldots,n-1\).

Theorem. Suppose that system (1) has reduced order \((^{1,2})\)

\[ 1\leqslant p_0<\frac{n}{n-2}. \tag{4} \]

If there exists a vector-function \(u(x,t)\) satisfying system (1), conditions (2), (3), and, for every \(t\geqslant 0\), the inequality

\[ |u(x,t)|\leqslant A\exp\!\left[B|x|^{p_0'-\varepsilon}\right], \qquad p_0'=\frac{p_0}{p_0-1},\qquad \varepsilon>0, \tag{5} \]

then \(u(x,t)\) is the unique solution of the mixed problem (1)—(3).

Proof. First we shall solve the adjoint problem in the basic space \({}^{(2)}\Phi\) of entire vector-functions \(\varphi(x)\) with the following properties:

\[ |x^k\varphi^{(q)}(x)| \leq C_\varphi A^k B^q k^{\alpha k} q^{\beta q}, \qquad x \geq 0, \qquad k,q=0,1,\ldots,\quad \beta<1; \tag{6} \]

\[ \varphi^{(nj)}(0)=\varphi^{(nj+1)}(0)=\ldots=\varphi^{(nj+n-m-2)}(0) =\varphi^{(nj+n-m)}=\ldots \]

\[ \ldots=\varphi^{(nj+n-1)}(0)=0,\qquad j=0,1,2,\ldots \tag{7} \]

As G. E. Shilov showed in \({}^{(3)}\), the space \(S_\alpha^\beta\) of functions \(\varphi(x)\) with properties (6) on the whole axis \((-\infty<x<\infty)\) is nonempty for \(\alpha+\beta\geq 1\) \((\alpha,\beta\ne0)\); moreover, for \(\beta>0\) the inequalities (6) on the whole axis \(x\) are equivalent to the following inequalities on the whole complex plane:

\[ \varphi(x+iy)\leq C_\varphi \exp\left[-C_1 |x|^{\frac1\alpha}+C_2 |y|^{\frac1{1-\beta}}\right]. \tag{6'} \]

Since there exist entire functions of order \(p\) \((1/2<p<1)\) which decrease exponentially with order \(p\) on the half-axis \(x\geq0\) (and increase on the half-axis \(x\leq0\)) , the space \(\pi\sigma_\alpha^\beta\) of basic functions \(\varphi(x)\) with properties (6) on the half-axis \(x\geq0\), or, as is easy to verify, equivalently, with properties (6′) in the right half-plane, is nonempty even for \(\beta>-1\) and \(\alpha+\beta=1\), where \(\beta\) may be equal to zero. In addition, as in \({}^{(2)}\), one can prove that \(\pi\sigma_\alpha^\beta\) is a space sufficiently rich in functions*.

After these remarks one can show that the space \(\Phi\) of entire functions with properties (6) and (7) is nonempty and sufficiently rich in functions. Indeed, first take the space \(\pi\sigma_{\alpha'}^{\beta'}\) \((\beta'>-1,\ \alpha'+\beta'=1)\) with elements \(\psi(x)\). With the aid of the easily verified inequalities \({}^{(3)}\)

\[ A_1|\xi|^h-A_2|\eta|^h \leq |\xi-\eta|^h \leq \max(2^{h-1},1)\bigl(|\xi|^h+|\eta|^h\bigr),\qquad h\geq0, \]

and the known Young inequality

\[ |\xi\eta|\leq \frac{\varepsilon^r|\xi|^r}{r} +\frac{|\eta|^{r'}}{\varepsilon^{r'}r'},\qquad \varepsilon>0,\qquad r>1,\qquad \frac1r+\frac1{r'}=1, \]

noting that \(\dfrac1{\alpha'}=\dfrac1{1-\beta'}\), one can prove that for any integer \(n\), for any \(\psi(x)\in\pi\sigma_{\alpha'}^{\beta'}\), we have

\[ |\varphi(z)|=|\psi(z^n)| \leq C_\psi \exp\left[-A_1|\operatorname{Re} z^n|^{\frac1{\alpha'}} +A_2|\operatorname{Im} z^n|^{\frac1{1-\beta'}}\right]\leq \]

\[ \leq C_\varphi \exp\left[-B_1|x|^{\frac1\alpha} +B_2|x|^{\frac1{1-\beta}}\right], \]

where

\[ \frac1\alpha=\frac{n}{\alpha'},\qquad \frac1{1-\beta}=\frac{n}{1-\beta'}\quad(\beta'>-1). \]

Hence

\[ \alpha=\frac{\alpha'}{n},\qquad \beta=\frac{\beta'}{n}+\frac{n-1}{n}. \tag{8} \]

* The existence of such entire functions is easily established by applying V. L. Bernstein’s theorem to the trigonometrically convex periodic function with period \(2\pi\),
\[ h(\theta)=A\cos p(\pi-\theta), \]
as an indicator, where \(A\) is any positive number \({}^{(4)}\).

** That is, if for some function \(f(x)\) and every \(\varphi(x)\in\pi\sigma_\alpha^\beta\)
\[ \int_0^\infty f(x)\varphi(x)\,dx=0, \]
then \(f(x)\equiv0\) almost everywhere on \((0,\infty)\).

Let us note that, since \(n \geqslant 2\) and \(\beta' > -1\), the inequality

\[ \beta > \frac{n-2}{n} \geqslant 0. \tag{9} \]

holds.

Thus, if \(\psi(x) \in \pi\sigma_{\alpha'}^{\beta'}\), then \(\psi(x^n) \in \pi\sigma_{\alpha}^{\beta}\); therefore, the space \(\pi\sigma_{\alpha}^{\beta}\) contains all functions of the form

\[ \varphi(x)=x^{n-m-1}\varphi(x^n), \qquad \psi \in \pi\sigma_{\alpha'}^{\beta'}. \]

Since \(\psi(x)\) are entire functions, we may expand them in series

\[ \psi(x)=\sum_{k=0}^{\infty} a_k x_k . \]

It is easy to verify that all functions of the form

\[ \varphi(x)=x^{n-m-1}\psi(x^n) =x^{n-m-1}\sum_{k=0}^{\infty} a_k x^{nk} \]

possess the properties (7). Hence the space \(\Phi\) is nonempty.

Let us show that the space \(\Phi\) is sufficiently rich in functions. Indeed, if for all \(\varphi(x)\in\Phi\)

\[ \int_0^\infty f(x)\varphi(x)\,dx=0, \]

where \(f(x)\) is some function satisfying the inequality

\[ |f(x)|\leqslant C_1 \exp\left[C_2 |x|^{\frac1\alpha-\varepsilon}\right], \]

so that the integrals \(\int_0^\infty f(x)\varphi(x)\,dx\) make sense for all \(\varphi(x)\in\Phi\), then for any \(\psi\in\pi\sigma_{\alpha'}^{\beta'}\) we have

\[ \int_0^\infty f(x)x^{n-m-1}\psi(x^n)\,dx = \int_0^\infty \frac{f(\sqrt[n]{y})}{n\sqrt[n]{y^m}}\psi(y)\,dy =0,\qquad y=x^n . \]

Since \(m/n<1\), \(f(\sqrt[n]{y})/\sqrt[n]{y^m}\) is locally integrable in a neighborhood of the point \(x=0\). Since the space \(\pi\sigma_{\alpha'}^{\beta'}\) is sufficiently rich in functions, it follows that

\[ f(\sqrt[n]{y})/\sqrt[n]{y^m}=0, \]

whence \(f(x)\equiv 0\) almost everywhere on \((0,\infty)\).

We shall now solve in the basic space \(\Phi\) the mixed problem adjoint to (1)—(3):

\[ \frac{\partial\varphi}{\partial t} = -\,P^*(t,D^*)\varphi,\qquad D^*=(-1)^nD; \tag{1′} \]

\[ \varphi(x,t_0)=\varphi_0(x),\qquad t_0>0\ \text{arbitrary},\qquad \varphi_0\in\Phi; \tag{2′} \]

\[ \varphi(0)=\varphi'(0)=\cdots=\varphi^{(n-m-2)}(0) =\varphi^{(n-m)}(0)=\cdots=\varphi^{(n-1)}(0)=0, \tag{3′} \]

where \(P^*\) is the matrix transposed to the matrix \(P\).

Since, by the hypothesis of the theorem, system (1) has reduced order \(p_0<\dfrac{n}{n-2}\), and since system (1′) has the same reduced order \(p_0\), the operator \(Q^*(D^*,t_0,t)\) \((^1,^2)\) is bounded in the space \(\Phi\).

with \(\beta=\dfrac{1}{p_0}>\dfrac{n-2}{n}\) (1), so that, according to (9) and (8), we have \(\beta'>-1\), and therefore \(\pi_{\alpha'}^{\beta'}\), and hence \(\Phi\), is nonempty.

As is easy to verify, the solution of the mixed problem \((1')-(3')\) is given by the formula

\[ \varphi(x,t)=Q^*(D^*,t_0,t)\varphi_0(x). \tag{10} \]

Indeed, we have \((2)\)

\[ \frac{\partial \varphi}{\partial t}=-P^*Q^*\varphi_0=-P^*\varphi,\qquad \varphi(x,t_0)=\varphi_0(x). \]

Finally, since \(Q^*\) is an entire function of the operator \(\partial^n/\partial x^n\) and \(\varphi_0(x)\) possesses the properties (7), it follows that \(\varphi(x,t)\) also satisfies the conditions \((3')\).

Thus, the application of the operator \(Q^*\) to each \(\varphi_0(x)\) from \(\Phi\) gives a solution of the mixed problem \((1')-(3')\).

Now suppose that the mixed problem (1)—(3) has a solution \(u(x,t)\) satisfying condition (5). We shall prove that it is the unique solution of the given problem. Indeed, by virtue of (5), \(u(x,t)\) is a functional in the space \(\Phi\), and, under conditions (3) and properties (7), for all \(\varphi_0\in\Phi\) we have

\[ (Pu,\varphi_0)=(u,P^*\varphi_0),\qquad (Pu,Q^*\varphi_0)=(u,P^*Q^*\varphi_0). \]

Therefore, for any \(\varphi_0\in\Phi\),

\[ \begin{aligned} \frac{\partial}{\partial t}(u,Q^*\varphi_0) &=(u_t,Q^*\varphi_0)+\left(u,\frac{\partial Q^*}{\partial t}\varphi_0\right) =(Pu,Q^*\varphi_0)+(u,-P^*Q^*\varphi_0)= \\ &=(u,P^*Q^*\varphi_0)-(u,P^*Q^*\varphi_0)=0. \end{aligned} \]

If for \(t=0\) \(u(x,0)=0\), then for all \(\varphi_0\in\Phi\), for every \(t\geq 0\),

\[ (u,Q^*\varphi_0)=0. \]

In particular, for \(t=t_0\) we have, for all \(\varphi_0\in\Phi\),

\[ (u,\varphi_0)=0. \]

Hence, by virtue of the sufficient supply of functions in the space \(\Phi\) and the arbitrariness of \(t_0>0\), we obtain

\[ u(x,t)\equiv 0, \]

as was required to prove.

Clearly, the theorem is also valid in the multidimensional case \(x=(x_1,\ldots,x_L)\) in domains where all or some of the \(x_j\geq 0\).

In conclusion I take the opportunity to express my deep gratitude to G. E. Shilov for valuable advice in carrying out this work.

Received
11 XII 1956

References

\(^{1}\) I. M. Gel'fand, G. E. Shilov, Dokl. Akad. Nauk SSSR, 102, 6 (1955).
\(^{2}\) I. M. Gel'fand, G. E. Shilov, Uspekhi Mat. Nauk, 8, 6 (1953).
\(^{3}\) G. E. Shilov, Dokl. Akad. Nauk SSSR, 102, 5 (1955).
\(^{4}\) B. Ya. Levin, Distribution of Zeros of Entire Functions, Moscow, 1956, p. 124.