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MATHEMATICS
M. N. OLEVSKII
ON THE CAUCHY PROBLEM FOR AN ITERATED LINEAR DIFFERENTIAL-OPERATOR EQUATION
(Presented by Academician I. N. Vekua on 14 VIII 1962)
- Let \(u(x_1,\ldots,x_m,t;t_0;f)\) (briefly: \(u(x,t;t_0;f)\)) be a solution of the differential-operator equation
\[ Lu \equiv (T_t-X)u=0,\qquad T_t \equiv \frac{\partial^2}{\partial t^2}+a_1(t)\frac{\partial}{\partial t}+a_2(t) \tag{1} \]
(\(X\) is a linear operator independent of \(t\)), satisfying the initial conditions
\[ u\big|_{t=t_0}=0,\qquad \frac{\partial u}{\partial t}\bigg|_{t=t_0}=f(x). \tag{2} \]
Let \(w_n(x,t;t_0;f)\) be a solution of the iterated equation
\[ L^n w=0, \tag{3} \]
satisfying the initial conditions
\[ w_n\big|_{t=t_0}= \frac{\partial w_n}{\partial t}\bigg|_{t=t_0} =\cdots= \frac{\partial^{2n-2}w_n}{\partial t^{2n-2}}\bigg|_{t=t_0} =0,\qquad \frac{\partial^{2n-1}w_n}{\partial t^{2n-1}}\bigg|_{t=t_0} =f(x). \tag{4} \]
Below (secs. 2, 3) a formula is established that expresses the solution of the Cauchy problem (3)—(4) in terms of the solution of the Cauchy problem (1)—(2); in secs. 4—6 certain generalizations and applications of this formula are considered.
- Denote by \(\varphi_1(t,\tau)\) the solution of the equation
\[ T_t\varphi_1=T_\tau^{*}\varphi_1 \tag{5} \]
(\(T^*\) is the operator adjoint to \(T\)), satisfying the conditions
\[ \varphi_1\big|_{\tau=t}=\frac12(t-t_0), \tag{6} \]
\[ \varphi_1\big|_{\tau=t_0}=0, \tag{7} \]
and let
\[ \varphi_k(t,\tau)=\int_{\tau}^{t}\varphi_{k-1}(t,z)\varphi_1(z,\tau)\,dz,\qquad k\geqslant 2. \tag{8} \]
The following formula holds:
\[ w_n(x,t;t_0;f)=\frac{1}{\,n-1!\,}\int_{t_0}^{t}\varphi_{n-1}(t,\tau)u(x,\tau;t_0;f)\,d\tau *\qquad n\geqslant 2, \tag{9} \]
under the assumption that the operator \(X\), acting with respect to the variables \(x_1,\ldots,x_m\), commutes with integration with respect to \(\tau\) in the right-hand side of (9).
Remark. The solution of the problem (6)—(7) for the second-order differential equation (5) of hyperbolic type, under the assumption that the functions \(a_k(t)\in C^{2-k}\) \((k=1,2)\), exists and is unique \((^{1,2})\).
- Let us outline the proof of formula (9). First of all, using (8), (5), (6), (7), we establish that the function \(\varphi_k(t,\tau)\), \(k\geqslant 2\), satisfies—
\[ \* \text{Or, more generally:} \]
\[ w_k(x,t;t_0;f):=\frac{(l-1)!}{(k-1)!}\int_{t_0}^{t}\varphi_{k-l}(t,\tau)w_l(x,\tau;t_0;f)\,d\tau, \tag{9'} \]
\[ k\geqslant 2;\qquad l=1,\ldots,k-1;\qquad w_1\equiv u. \]
satisfies the equation
\[ \vartheta \varphi_k=\varphi_{k-1}, \qquad \vartheta = T_t - T_\tau^{*} \tag{10} \]
and the conditions
\[ \varphi_k\big|_{\tau=t}=0, \qquad \varphi_k\big|_{\tau=t_0}=0 . \tag{11} \]
Further, denoting the right-hand side of (9) by \(v_n\), one can show, by virtue of the assumption made in item 2 concerning the operator \(X\) and the first of conditions (2), that
\[ Lv_n=\int_{t_0}^{t} u(x,\tau;t_0;f)\,\vartheta\varphi_{n-1}(t,\tau)\,d\tau+ \]
\[ +\,2\,\frac{d\varphi_{n-1}(t,t)}{dt}\,u(x,t;t_0;f) +\varphi_{n-1}\big|_{\tau=t_0}\,\frac{\partial u}{\partial t}, \qquad n\geqslant 2. \tag{12} \]
From (12), by virtue of (10) and (11), it follows that \(Lv_n=v_{n-1}\) \((n>2)\), and consequently \(L^k v_n=v_{n-k}\), \(1\leq k\leq n-2\). From the last equality, for \(k=n-2\), we find that \(L^{n-1}v_n=u\) (since from (12), by virtue of (5), (6), (7), it follows that \(Lv_2=u\)). Thus we discover that \(v_n\) satisfies equation (3). The conditions (4) for the function \(v_n\) can be checked directly from its definition (taking into account (6), (7), (11)). Consequently, \(v_n\equiv w_n\), as was required to prove.
- A more general Cauchy problem related to equation (3), namely, the problem of determining a solution \(w(x,t;t_0;f_0,f_1,\ldots,f_{2n-1})\) of equation (3) satisfying the conditions
\[ \left.\frac{\partial^k w}{\partial t^k}\right|_{t=t_0} =f_k(x) \qquad (k=0,1,\ldots,2n-1) \]
can be reduced to the problem (3)—(4).
Indeed, if by \(w_n^{(l)}(x,t;t_0;f)\) \((l=1,\ldots,np)\) we denote a solution of the equation\(^*\)
\[ (T_p-X)^n w=0,\qquad T_t \equiv \frac{\partial^p}{\partial t^p}+a_1(t)\frac{\partial^{p-1}}{\partial t^{p-1}}+\cdots+a_p(t), \tag{13} \]
satisfying the conditions
\[ \left.\frac{\partial^q w_n^{(l)}}{\partial t^q}\right|_{t=t_0} = \begin{cases} 0, & q\ne l-1,\\ f(x), & q=l-1, \end{cases} \qquad q=0,1,\ldots,np-1, \tag{14} \]
then, by virtue of the linearity of equation (13), it is sufficient to express \(w_n^{(l)}\), \(l=1,\ldots,\ldots,np-1\), in terms of \(w_n^{(np)}\). To this end, putting
\[
w_n^{(l)}=\frac{(t-t_0)^{l-1}}{(l-1)!}\,f(x)+\widetilde w,
\]
from equation (13) we obtain for \(\widetilde w\) the nonhomogeneous equation
\[ (T_p-X)^n\widetilde w=\Phi(x,t;t_0);\qquad \Phi(x,t;t_0)=-(T_t{}_p-X)^n\frac{(t-t_0)^{l-1}}{(l-1)!}\,f(x), \]
and, by virtue of (14), the initial conditions
\[ \left.\frac{\partial^q\widetilde w}{\partial t^q}\right|_{t=t_0}=0, \qquad q=0,1,\ldots,np-1. \]
Applying Duhamel’s principle (10) to determine \(\widetilde w\), we find the following formula:
\[ w_n^{(l)}(x,t;t_0;f) =\frac{(t-t_0)^{l-1}}{(l-1)!}\,f(x) +\int_{t_0}^{t} w_n^{(np)}(x,t;\xi;\varphi)\,d\xi; \]
\[ \varphi=\Phi(x,\xi;t_0), \qquad (l=1,\ldots,np-1). \tag{15} \]
\[ \text{* Bearing in mind item 6, we do not restrict ourselves here to the case } p=2;\ w_n \text{ from items 1–2, in the notation of item 4, is } w_n^{(2n)}. \]
With the proper smoothness of \(w_n^{(np)}\) (see below formula (16)), the solution \(w_n^{(l)}\) \((l=1,\ldots,np-1)\) can also be expressed in another form. Denote by \(L_{np}\) the operator \((T_p-X)^n\) in expanded form,
\[ {}_{t}L_{np}\equiv ({}_{t}T_p-X)^n =\frac{\partial^{np}}{\partial t^{np}} +b_1(t,X)\frac{\partial^{np-1}}{\partial t^{np-1}} +\cdots+b_{np}(t,X), \]
and by \({}_{t}L_q^*\) the operator adjoint with respect to \(t\) to the operator \({}_{t}L_q\), i.e.
\[ {}_{t}L_q^*v\equiv \sum_{s=0}^{q}(-1)^{q-s} \frac{\partial^{q-s}\bigl(b_s(t,X)v\bigr)}{\partial t^{q-s}}, \qquad q\le np; \]
then the formula* holds
\[ w_n^{(l)}(x,t;t_0;f) ={}_{t_0}L_{np-l}^*w_n^{(np)}(x,t;t_0;f) \qquad (l=1,\ldots,np-1). \tag{16} \]
If the coefficients of the operator \(T_p\) are constant, then \(\partial w_n^{(np)}/\partial t=-\partial w_n^{(np)}/\partial t_0\), and, consequently, the operator \({}_{t_0}L_{np-l}^*\) in formula (16) may be replaced by the operator \({}_{t}L_{np-l}\).
- Let us note the following application of formula (9).
5.1. Put \(a_1(t)=a_2(t)=0\) in equation (1). In this case the function \(\varphi_1(t,\tau)=\frac12(\tau-t_0)\), and thus, for the solution of the Cauchy problem (4) corresponding to the equation
\[ \left(\frac{\partial^2}{\partial t^2}-X\right)^n w=0, \tag{17} \]
we obtain the formula
\[ w_n(x,t;t_0;f)= \frac{1}{2^{2n-3}}\frac{n-1}{(n-1)!^2} \int_{t_0}^{t}(\tau-t_0)\bigl[(t-t_0)^2-(\tau-t_0)^2\bigr]^{\,n-2} u(x,\tau;t_0;f)\,d\tau. \tag{18} \]
5.2. Denote by \(v^{\{a\}}(x,t;f)\) the solution of the following Cauchy problem for the generalized Euler–Poisson–Darboux equation:
\[ \frac{\partial^2 v}{\partial t^2} +\frac{a}{t}\frac{\partial v}{\partial t}=Xv, \tag{19} \]
\[ v\big|_{t=0}=f(x),\qquad \frac{\partial v}{\partial t}\bigg|_{t=0}=0. \tag{20} \]
In (³) a relation was established between \(v^{\{a_1\}}(x,t;f)\) and \(v^{\{a_2\}}(x,t;f)\), which, as applied to the case of interest to us, can be written as follows:
\[ v^{\{a\}}(x,t;f)= \frac{2\Gamma\left(\frac{a+1}{2}\right)} {\Gamma\left(\frac12\right)\Gamma\left(\frac a2\right)} \,t^{1-a} \int_{0}^{t}(t^2-\tau^2)^{\frac{a-2}{2}} v^{\{0\}}(x,\tau;f)\,d\tau, \qquad a>0. \tag{21} \]
From (18) and (21) we find the following relation between the solution of the Cauchy problem (4) for equation (17) and the solution of the singular Cauchy problem (20) for the generalized Euler–Poisson–Darboux equation (19):
\[ w_n(x,t;t_0;f)= \frac{1}{2n-1!}(t-t_0)^{2n-1} v^{\{2n\}}(x,t-t_0;f). \tag{22} \]
5.3. Formula (22), in conjunction with the results of § 6 of work (¹²), makes it possible to establish that the Huygens principle for the generalized polywave equation
\[ \left[ \frac{\partial^2}{\partial t^2} -\sum_{i=1}^{m} \left( \frac{\partial^2}{\partial x_i^2} +\frac{\lambda_i}{x_i}\frac{\partial}{\partial x_i} \right) \right]^n w=0 \tag{23} \]
(\(\lambda_i\) are constants, \(|x_i|>|t|\ge 0\) for all \(i=1,\ldots,m\), if at least
* For \(n=1\) it was established in (⁴).
one of the numbers \(\lambda_i\) is different from zero) takes place only in the case when:
\[ \begin{gathered} 1)\quad m-2n=2p+1\quad (p=0,1,\ldots);\\ 2)\quad \lambda_i=\alpha_i(1-\alpha_i)\quad (\alpha_i=1,2,\ldots,p+1);\\ 3)\quad \alpha_1+\ldots+\alpha_m\leqslant m+p. \end{gathered} \tag{24} \]
In particular, for the polywave equation* \((\lambda_i=0;\ i=1,\ldots,m)\) the conditions (24) reduce to one: \(m-2n=2p+1,\ p=0,1,\ldots\) (see \((^{6,8,11})\)).
The solution (18) in this case may be written (if one uses the explicit expression for \(u\) \((^{10})\)) in the form
\[ w_n(x,t;t_0;f)=t^{2n-1}\sum_{l=0}^{p} a_l t^l \frac{\partial^l M(x,t;f)}{\partial t^l} \tag{25} \]
(\(a_l\) are constants), directly revealing its Huygens character; here \(M(x,t;f)\) is the mean value of the function \(f\) on the sphere of radius \(t\) with center at the point \(x\).
6.1. Formula (9) can be generalized to the case when the operator \(T\) in equation (3) has order greater than two (i.e., has the form indicated in 13)). Similarly to Sec. 3, one can show that the kernel \(\varphi_1(t,\tau)\) in formula (9) in this case must be a solution of the equation
\[ \left(T_t^p-T_\tau^{p*}\right)\varphi_1=0, \tag{26} \]
satisfying the conditions:
\[ \varphi_1\big|_{\tau=t_0}=0,\qquad \left.\frac{\partial^k\varphi_1}{\partial t^k}\right|_{\tau=t} = \begin{cases} 0, & k=1,\ldots,p-3,\\ \dfrac{1}{p}(t-t_0), & k=p-2. \end{cases} \tag{27} \]
The existence of a solution of the problem (26)—(27) can be proved under the assumption of sufficiently high smoothness of the functions \(a_i(t)\) \((i=1,\ldots,p)\), for example, in the case when they are analytic.
In particular, if the operator \(T_p\) reduces to \(\dfrac{\partial^p}{\partial t^p}\), the function
\[ \varphi_1(t,\tau)=\frac{p-1}{p!}(\tau-t_0)(t-\tau)^{p-2}. \]
6.2. In the case when the order of the differential operator \(T_p\) is equal to 1, and consequently the corresponding equation (3) can be written in the form
\[ \left(\frac{\partial}{\partial t}-X\right)^n w=0, \tag{28} \]
it is easy to show that
\[ w_n(x,t;t_0;f)=\frac{(t-t_0)^{n-1}}{(n-1)!}\,u(x,t;t_0;f); \tag{29} \]
for the case of the polycaloric equation, i.e., when \(X\) in (28) is the Laplace operator, cf. \((^{13})\).
Received
8 VIII 1962
REFERENCES
- E. Picard, Leçons sur quelques types simples d’equations aux dérivées partielles avec des applications a la physique math., Paris, 1927, pp. 152—156.
- Б. М. Левитан, Операторы обобщенного сдвига и некоторые их применения, 1962, pp. 154—155.
- М. Н. Олевский, DAN, 101, No. 1 (1955).
- М. Н. Олевский, Сборн. научн. работ ТСХА, 14, 231 (1961).
- J. P. Kormes, Bull. Am. Math. Soc., 50, 842 (1944).
- A. Weinstein, Ann. Math. pura ed Appl., 39, 245 (1955).
- L. Asgeirsson, Techn. Rep., No. 6, Dept. of Math. Univ. of California (1956), cited in \((^{8})\).
- D. Krahn, Proc. Konin. Nederl. Akad. Weten., Ser. A, 60, 492 (1957).
- E. P. Miles, Jr., Portug. Math., 18, 111 (1959).
- Р. Курант, Д. Гильберт, Методы математической физики, 2, 1945, pp. 187—188, 470.
- L. Gårding, Ann. of Math., 48, 788 (1947).
- D. Fox, J. Math. and Mech., 8, 197 (1959).
- M. Nicolescu, Studii şi cercetări matematice, 5, 243 (1954).
* The Cauchy problem for the polywave equation for \(n=2\) was considered in \((^{5,6})\); for \(n\geqslant 2\) with \(m\) odd, greater than \(2n-1\), in \((^{7})\), and in the general case in \((^{8})\), where the solution procedure indicated by the author was carried through to the end for \(m=1,\ n=3\) and \(m=3,\ n=2\); in the case when \(f(x)\) is an analytic function (under certain additional restrictions), a solution in the form of a series is indicated in \((^{9})\). The author knows of the note \((^{7})\) only from the paper \((^{8})\).