Full Text
UDC 517.944
MATHEMATICS
V. M. BOROK
ON THE CAUCHY PROBLEM FOR GENERAL LINEAR EQUATIONS
(Presented by Academician I. G. Petrovskii on 20 II 1967)
The present paper is devoted to establishing necessary and sufficient conditions for uniqueness of the solution of the Cauchy problem for equations of the form
\[ P\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial t}\right) u(x,t) = \sum_{0 \leqslant k \leqslant m} P_k\left(\frac{\partial}{\partial x}\right) \frac{\partial^k u(x,t)}{\partial t^k} = 0, \tag{1} \]
\[ -\infty < x < \infty,\quad 0 \leqslant t < \infty, \]
where \(P(s,\lambda)\) is an arbitrary polynomial with constant coefficients of degree \(N\) in \(s\) and degree \(m\) in \(\lambda\), \(P_m(s) \not\equiv 0\), under the initial conditions
\[ \partial^j u(x,0) / \partial t^j = 0,\quad j = 0,\ldots,m-1. \tag{2} \]
At the same time we shall consider only solutions of normal type in \(t\), i.e., solutions of equation (1) which, together with all their derivatives entering equation (1), grow in \(t\) no faster than \(\exp\{at\}\) with some \(a>0\).
The Cauchy problem for equations (and systems) of the form (1) was first studied in a special case by S. L. Sobolev \((^1)\), and in the general formulation by S. A. Galpern \((^{2,3})\). Works \((^{4,5})\) are devoted to the same problem. In works \((^{1-4})\) the question of correct solvability of the Cauchy problem for the corresponding equations was studied, and in \((^5)\) the uniqueness of the solution of this problem. In all these works restrictions were imposed on the form of the polynomial \(P(s,\lambda)\) that were not dictated by the essence of the problem of uniqueness classes. We shall impose here no restrictions on the form of the polynomial \(P(s,\lambda)\).
Let the roots \(s_1(\lambda),\ldots,s_N(\lambda)\) (not necessarily distinct) of the polynomial \(P(s,\lambda)\) in a neighborhood of the infinitely distant point have the form \((^6)\)
\[ s_j(\lambda) = a_j \lambda^{\gamma_j}(1+o(1)),\quad j=1,\ldots,N,\quad a_j \ne 0,\quad o(1)\underset{|\lambda|\to\infty}{\longrightarrow}0. \tag{3} \]
Denote
\[ A = \min_{\{j:\, s_j(\lambda)\equiv \mathrm{const}\}} |\operatorname{Re}s_j(\lambda)|;\quad a = \min_{\{j:\,\gamma_j=0\}} |\operatorname{Re}s_j(\lambda)|. \tag{4} \]
We divide the investigation into the following 5 cases, covering all possible cases: 1) \(P_m(0)=0\); \(A>0\), or roots of the form \(s_j(\lambda)\equiv \mathrm{const}\) are absent; 2) \(P_m(0)\ne 0\); \(P_m(s)\not\equiv \mathrm{const}\); \(a=0\), \(A>0\), or roots of the form \(s_j(\lambda)\equiv \mathrm{const}\) are absent; 3) \(P_m(0)\ne 0\), \(P_m(s)\not\equiv \mathrm{const}\); \(a>0\); 4) \(A=0\); 5) \(P_m(s)\equiv \mathrm{const}\).
Theorem 1. For uniqueness of the solution of the Cauchy problem (1)—(2) in cases 1) and 2) in the class of functions satisfying the estimate
\[ \left| \partial^k u(x,t) / \partial x^k \right| \leqslant C \exp\{\beta t + |x|H(x)\}, \tag{5} \]
\[ k=0,\ldots,N-1,\quad -\infty < x < \infty,\quad 0 \leqslant t < \infty,\quad \beta>0, \]
where \(H(x)>0\) is an even continuous function, it is necessary and sufficient that
\[ \inf H(x) = 0. \tag{6} \]
Proof. Denote by \(y(x,\lambda)\) the Laplace transform (in \(t\)) of the solution \(u(x,t)\) of problem (1)—(2). Then
\[ P(\partial/\partial x,\lambda)y(x,\lambda)=0, \tag{7} \]
and we must establish that condition (6) is necessary and sufficient in order that every solution \(y(x,\lambda)\) of equation (7), analytic in some right half-plane and satisfying in this half-plane the estimate (following from (5)):
\[ \left|\partial^k y(x,\lambda)/\partial x^k\right|\leq C_1\exp\{|x|H(x)\}, \tag{8} \]
\(k=0,\ldots,N-1,\ -\infty<x<\infty\), be identically equal to zero.
Sufficiency. Let \(y_1(x,\lambda),\ldots,y_N(x,\lambda)\) be a fundamental system of solutions of equation (7), and let \(y(x,\lambda)\) be a solution of (7) satisfying condition (8). Representing \(y(x,\lambda)\) in the form
\[ y(x,\lambda)\equiv \sum_{1\leq k\leq N} C_k(\lambda)y_k(x,\lambda), \tag{9} \]
we obtain
\[ C_k(\lambda)=w^{-1}(x,\lambda)w_k(x,\lambda),\qquad k=1,\ldots,N, \tag{10} \]
where \(w(x,\lambda)\) is the Wronskian of the system of functions \(y_1(x,\lambda),\ldots,y_N(x,\lambda)\), and \(w_k(x,\lambda)\) is the determinant obtained from \(w(x,\lambda)\) by replacing the \(k\)-th column by the column of functions \(y(x,\lambda),\ldots,y^{(N-1)}(x,\lambda)\). Taking into account that
\[ w(x,\lambda)=w(0,\lambda)\exp\left\{\sum_{1\leq j\leq N}s_j(\lambda)x\right\} \]
and the estimate (8), from (10) we obtain
\[ |C_k(\lambda)|\leq C_2(1+|x|)^{M_1}|\lambda|^{M_2} \exp\{-\operatorname{Re}s_k(\lambda)x+|x|H(x)\}, \tag{11} \]
\(-\infty<x<\infty,\ \operatorname{Re}\lambda\geq\sigma_0>0;\ M_1,M_2>0\).
From (11), taking (6) into account, we conclude that \(C_k(\lambda)=0\) for all \(\lambda,\ \operatorname{Re}\lambda\geq\sigma_0\), with the exception of at most a finite number of rays. Indeed, if \(\gamma_k>0\), or \(\gamma_k<0\), or \(\gamma_k=0\) and \(\operatorname{Re}a_k=0\), then \(\operatorname{Re}s_k(\lambda)\ne0\) for any \(\lambda\) in some right half-plane, except, possibly, for a finite number of rays where \(\operatorname{Re}s_k(\lambda)=0\). For \(\gamma_k=0\) and \(\operatorname{Re}a_k\ne0\), the inequality \(\operatorname{Re}s_k(\lambda)\ne0\) is valid for all \(\lambda\) from some right half-plane. Fixing \(\lambda\) so that \(\operatorname{Re}s_k(\lambda)\ne0\), consider values of \(x\) such that \(\operatorname{sign}x=\operatorname{sign}\operatorname{Re}s_k(\lambda)\), and choose from them a sequence \(x_n\), \(|x_n|\to\infty\), such that \(H(x_n)\to0\). Then as \(n\to\infty\) the right-hand side of expression (11) tends to zero, and consequently \(C_k(\lambda)=0\). Thus the function (9) is equal to zero for \(\operatorname{Re}\lambda\geq\sigma_0\), with the exception of at most a finite number of rays; consequently, \(y(x,\lambda)\equiv0\).
Necessity. In case 1), from the condition \(P_m(0)=0\) and the Newton diagram we conclude that there is a root \(s_j(\lambda)\) with \(\gamma_j<0\). Let \(\inf H(x)=C>0\). Then the function \(y(x,\lambda)=\exp\{s_j(\lambda)x\}\), not being identically zero, satisfies condition (8).
Analogously, in case 2) an example is provided by the function \(\exp\{s_j(\lambda)x\}\), where
\[ s_j(\lambda)=a_j+a'_j\lambda^{\gamma_j}(1+o(1)),\quad \operatorname{Re}a_j=0,\quad a'_j\ne0,\quad \gamma_j<0. \]
Theorem 2. In case 3), the solution \(u(x,t)\) of the Cauchy problem (1)—(2), satisfying the estimate
\[ \left|\partial^k u(x,t)/\partial x^k\right|\leq C\exp\{\beta t+\alpha|x|\}, \tag{12} \]
\(k=0,\ldots,N-1,\ -\infty<x<\infty,\ 0\leq t<\infty,\ \beta>0\), is identically equal to zero if \(\alpha<a\). If \(\alpha>a\), then there exists a nontrivial solution of problem (1)—(2) satisfying estimate (12).
Proof. As in the preceding theorem, it is enough to determine the existence of a nontrivial solution \(y(x,\lambda)\) of equation (7), analytic in the right half-plane \(\operatorname{Re}\lambda\geq\sigma_0>0\) and satisfying there the estimate
\[ \left|\partial^k y(x,\lambda)/\partial x^k\right|\leq C_1\exp\{a|x|\},\qquad k=0,\ldots,N-1,\quad -\infty<x<\infty. \tag{13} \]
Using (13), we arrive, analogously to (11), at the estimate \((k+1,\ldots,N)\)
\[ |C_k(\lambda)|\leq C_2(1+|x|)^{M_1}|\lambda|^{M_2} \exp\{-\operatorname{Re}s_k(\lambda)x+\alpha|x|\} \tag{14} \]
for the coefficients of the expansion (9). Let \(\alpha<a\). From the condition \(P_m(0)\ne 0\) it follows that \(\gamma_k\geq 0,\ k=1,\ldots,N\). Then from (3) and (4), for any \(\varepsilon>0\), for sufficiently large values of \(|\lambda|\) we have \(|\operatorname{Re}s_k(\lambda)|\geq a-\varepsilon,\ k=1,\ldots,N\). For \(a-\varepsilon>\alpha\) the left-hand side of (14) tends to zero as \(|x|\to\infty\) and \(\operatorname{sign}x=\operatorname{sign}\operatorname{Re}s_k(\lambda)\). Consequently, \(C_k(\lambda)\equiv 0\).
If \(\alpha>a\), then the function \(y(x,\lambda)=\exp\{s_k(\lambda)x\}\), where \(|\operatorname{Re}a_k|=a,\ \gamma_k=0\), is analytic for sufficiently large values of \(|\lambda|\) and satisfies the estimate (13).
Remark. The class of functions satisfying the estimate (12) with \(\alpha=a\) may turn out in case 3) to be both a uniqueness class for the solution of the problem (1)—(2) and a nonuniqueness class. This is indicated by the following examples: 1) \(\partial^2u/\partial x\partial t-\partial u/\partial t=u(x,t)\); 2) \(\partial^2u/\partial x\partial t-\partial u/\partial t=-u(x,t)\).
Theorem 3. In case 4), for uniqueness of the solution of the Cauchy problem (1)—(2) in the class of functions satisfying the estimate
\[ |\partial^ku(x,t)/\partial x^k|\leq \alpha(x)\exp\{\beta t\}, \tag{15} \]
\[ k=0,\ldots,N-1,\quad -\infty<x<\infty,\quad 0\leq t<\infty,\quad \beta>0, \]
\(\alpha(x)>0\) an even function monotone for \(x>0\), it is necessary and sufficient that
\[ \lim_{x\to\infty}\alpha(x)=0. \tag{16} \]
Proof. Suppose condition (16) is fulfilled. Analogously to (11) and (17), in the case under consideration, using (15), we obtain \((k=1,\ldots,N)\):
\[ |C_k(\lambda)|\leq C_1(1+|x|)^{M_1}|\lambda|^{M_2}\alpha(x) \exp\{-\operatorname{Re}s_k(\lambda)x\} \tag{17} \]
\[ -\infty<x<\infty,\quad M_1,M_2>0. \]
If \(\operatorname{Re}s_k(\lambda)\ne 0\), then from (17) it is obvious that \(C_k(\lambda)\equiv 0\). But \(\operatorname{Re}s_k(\lambda)\ne 0\) for all \(\lambda\) in some right half-plane, with the exception of a finite number of rays, only if \(s_k(\lambda)\not\equiv\mathrm{const}\) or \(s_k(\lambda)\equiv\mathrm{const}\), but \(\operatorname{Re}s_k(\lambda)\ne 0\). Thus the function \(y(x,\lambda)\) is a linear combination of functions of the form \(\exp\{s_k(\lambda)x\}\) and of products of such functions by powers of \(x\), where \(s_k(\lambda)\equiv\mathrm{const},\ \operatorname{Re}s_k=0\). But then from (16) and the estimate \(|y(x,\lambda)|\leq C\alpha(x)\), \(-\infty<x<\infty\), it follows that \(y(x,\lambda)\equiv 0\).
If condition (16) is not fulfilled, then equation (1) has a nontrivial solution of the form \(u(x,t)=Ct^m\exp\{s_kx\}\), \(\operatorname{Re}s_k=0\), satisfying condition (15) and the initial condition (2).
In case 5), equation (1) is an equation of Kovalevskaya type. For equations of this type with reduced order \(p_0>1\), in (7) a necessary and sufficient condition for uniqueness of the problem (1)—(2) was established, but in a form inconvenient for verification. We shall give here a more convenient criterion.
Theorem 4. Let \(H(x)>0\) be an even function increasing for \(x>0\). If the reduced order of equation (1) is \(p_0>1\), then for uniqueness of the solution of the problem (1)—(2) in the class of functions satisfying the estimate
\[ |\partial^ku(x,t)/\partial x^k|\leq C\exp\left\{\beta t+\left|\int_0^x H(t)\,dt\right|\right\}, \tag{18} \]
\[ k=0,\ldots,N-1,\quad -\infty<x<\infty,\quad 0\leq t<\infty,\quad \beta>0, \]
it is necessary and sufficient that
\[ \int^\infty [H(x)]^{1-p_0}\,dx=\infty. \tag{19} \]
The proof follows the same scheme as above. Similarly to (11), (14), and (17), we obtain \((k=1,\ldots,N)\):
\[ |C_k(\lambda)| \leq C_1(1+|x|)^{M_1}|\lambda|^{M_2} \exp\left\{-\operatorname{Re}s_k(\lambda)x+\left|\int_0^x H(t)\,dt\right|\right\},\qquad M_1,\ M_2>0. \tag{20} \]
The functions \(C_k(\lambda)\), \(k=1,\ldots,N\), are analytic for \(\operatorname{Re}\lambda\geq\sigma_0>0\) (\(\sigma_0\) sufficiently large). Under the assumption (19) we shall show the strong decrease of each of the functions \(C_k(\lambda)\) along one of the half-lines \(\lambda=\sigma_1+i\tau\), \(\sigma_1>\sigma_0\), \(0<\tau<\infty\), or \(-\infty<\tau<0\), or \(\lambda=\sigma(1+Ai)\), \(0<\sigma<\infty\), from which we shall be able to conclude that \(C_k(\lambda)\equiv0\), \(\operatorname{Re}\lambda\geq\sigma_0\). Since \(\min\gamma_k=p_0^{-1}\) (7), then for \(\gamma_k=p_0^{-1}<1\) on one of the half-lines \(\lambda=\sigma_1+i\tau\), \(\tau\geq\tau_0\) or \(\tau\leq-\tau_0\), \(|\operatorname{Res}_k(\lambda)|\geq C_k|\tau|^{p_0^{-1}}\). Substituting this estimate into (20), choosing the sign of \(x\) in the required way, and putting \(|x|=x(|\tau|)\), where \(H(x(|\tau|))\equiv \frac12 C_k|\tau|^{p_0^{-1}}\), we obtain from (20)
\[ |C_k(\sigma_1+i\tau)|\leq C_2\exp\{-C'|\tau|^{p_0^{-1}}x(|\tau|)\}, \]
whence, using the definition of \(x(|\tau|)\) and condition (19), we conclude that
\[ \int^\infty |\tau|^{-2}\ln|C_k(\sigma_1+i\tau)|\,d|\tau|=-\infty. \]
It follows from this, by (8), that \(C_k(\lambda)\equiv0\). For \(\gamma_k>p_0^{-1}\) an analogous conclusion is obtained by considering \(C_k(\lambda)\) for \(\lambda=\sigma(1+Ai)\), \(\sigma>0\), \(A\) sufficiently large.
If condition (19) is not fulfilled, then an example of a nontrivial solution \(u(x,t)\) of problem (1)—(2), satisfying condition (18), is given by the inverse Laplace transform of the function \(y(x,\lambda)=C(\lambda)\exp\{s_j(\lambda)x\}\), \(\gamma_j=p_0^{-1}\), where \(C(\lambda)\) is a function analytic in the right half-plane such that
\[ |C(\lambda)|\leq C_1\exp\{-C_2|\lambda|^{p_0^{-1}}x(|\lambda|)\}, \]
with \(H(x(|\lambda|))\equiv C_3|\lambda|^{p_0^{-1}}\), \(C_3<C_2\). The existence of such a \(C(\lambda)\ne0\) follows from the Carleman–Ostrovsky criterion, since
\[ \int^\infty |\lambda|^{p_0^{-1}-2}x(|\lambda|)\,d|\lambda|<\infty. \]
Remark. If \(p_0\leq1\), then problem (1)—(2) has a unique solution in the class of functions (18) for any increasing function \(H(x)>0\), since condition (19) will always be fulfilled; the proof is analogous to the first part of the proof of Theorem 4.
Kharkov State University
named after A. M. Gorky
Received
17 II 1967
References
- S. L. Sobolev, Izv. AN SSSR, ser. matem., 18, No. 1 (1954).
- S. A. Gal'pern, UMN, 8, 5 (57) (1953).
- S. A. Gal'pern, DAN, 119, No. 4 (1958).
- A. G. Kostyuchenko, G. I. Eskin, Tr. Mosk. matem. obshch., 10, 273 (1961).
- G. I. Eskin, Tr. Mosk. matem. obshch., 10, 285 (1961).
- N. G. Chebotarev, Theory of Algebraic Functions, Moscow–Leningrad, 1948.
- G. N. Zolotarev, Uch. zap. Ivanovsk. ped. inst., 34, 34 (1963).
- S. Mandelbrojt, Adhering Series. Regularization of Sequences. Applications, IL, 1955.