MATHEMATICS

Yu. F. KOROBEINIK

ON SOME PROBLEMS IN THE ANALYTIC THEORY OF PARTIAL DIFFERENTIAL EQUATIONS

(Presented by Academician I. G. Petrovskii, March 21, 1960)

§ 1. In the present work we study analytic solutions of one class of partial differential equations:

\[ \frac{\partial^{r}u}{\partial x^{r}}=ay^{m}\frac{\partial^{m+1}}{\partial y^{m+1}}, \tag{1} \]

where \(r\) and \(m\) are natural numbers, and \(a\) is a constant number. The class (1) includes the heat equation \((r=2,\ m=0)\), and also the equation of mixed type

\[ \frac{\partial^{2}u}{\partial x^{2}}=ay\frac{\partial^{2}u}{\partial y^{2}}. \tag{2} \]

Throughout the whole work we shall consider solutions of equation (1) that are entire in \((x,y)\), i.e., functions \(u(x,y)\) analytic in any bicylinder \(|x|\le R,\ |y|\le R_1\). From equation (1) it is not difficult to obtain the general form of such an expansion:

\[ u(x,y)= \sum_{l=0}^{m-1} P_l(x)y^l+f_m(x)y^l+ \sum_{k=1}^{\infty} y^{m+k} \frac{f_m^{(kr)}(x)(k-1)!(k-2)!\ldots 2!} {a^k(m+k)!(m+k-1)!\ldots(m+1)!}. \tag{3} \]

Here the functions \(P_l(x)\), \(l=0,1,\ldots,m-1\), must be polynomials of degree not exceeding \(r-1\), while the function \(f_m(x)\) is an entire function whose growth is restricted by the requirement of uniform convergence of the series (3) in any bicylinder \(|x|\le R,\ |y|\le R_1\).

For this condition to be satisfied, it is necessary and sufficient that the function \(f_m(x)\) be an entire function of order

\[ \frac{1}{1-(m+1)/r} \]

and of zero type, if \(m+1<r\), and an arbitrary entire function, if \(m+1\ge r\). We shall denote the class of such functions by \(H_{m,r}\). Every class \(H_{m,r}\) contains all entire functions of exponential type. The function \(\partial^m u/\partial y^m \big|_{y=0}\) will be called the growth function of the given solution.

Theorem 1. In order that the function \(u(x,y)\), defined by the series (3), be an entire in \((x,y)\) solution of equation (1), it is necessary and sufficient that its growth function belong to the class \(H_{m,r}\).

The Cauchy problem with respect to the variable \(y\) for equation (1) in the class of solutions entire in \(x,y\) can be formulated as follows:

Find a solution of equation (1) satisfying the conditions:

1) \(\partial^s u/\partial y^s \big|_{y=0}=P_s(x)\), \(s=0,1,\ldots,m-1\); \(P_s(x)\) are polynomials of degree \(\le r-1\);

2) \(\partial^m u/\partial y^m \big|_{y=0}=f(x)\), where \(f(x)\in H_{m,r}\).

The Cauchy problem has a unique solution in the class of solutions entire in \((x,y)\) for any prescribed polynomials \(P_s(x)\) \((s=0,1,\ldots,m-1)\), of degree \(\le r-1\), and for any entire function \(f(x)\) from \(H_{m,r}\).

§ 2. For analytic solutions of equation (1) one can also pose certain new problems. We shall consider here one such problem (we shall call it problem A).

Problem A. Let \(y=\mu(x)\equiv \sum_{i=0}^{l} A_i x^i\) be the equation of a parabola of order \(l<r\). It is required to find an entire solution in \((x,y)\) of equation (1) such that:

a) \(\left.\partial^s u/\partial y^s\right|_{y=0}=P_s(x),\quad s=0,1,\ldots,m-1,\) where \(P_s(x)\) are given polynomials of degree not exceeding \(r-1\);

b) \(u(x,\mu(x))=\lambda(x)\), where \(\lambda(x)\) is a given entire function.

The function \(\lambda(x)\) cannot be prescribed completely arbitrarily. From representation (3), putting \(y=\mu(x)\), we obtain that the function

\[ \nu(x)= \frac{\lambda(x)-\sum_{l=0}^{m-1}[\mu(x)]^l P_l(x)} {[\mu(x)]^m} \tag{4} \]

must be entire. The solution of problem A is reduced to the determination of the function \(f_m(x)\) from the condition

\[ \nu(x)=f_m(x)+\sum_{k=1}^{\infty}[\mu(x)]^k \frac{2!\,3!\ldots(k-1)!}{(m+1)!(m+2)!\ldots(m+k)!}\, \frac{f_m^{(kr)}(x)}{a^k} \tag{5} \]

or

\[ \nu(x)=Z(x)+\sum_{k=1}^{\infty} Q_k(x) Z^{(k)}(x), \]

where \(Z(x)=f_m(x)\), and \(Q_k(x)\) is a polynomial of degree \(\leq k-1\), i.e., to the solution of a differential equation of infinite order of a definite type considered in \((^1)\).

Theorem 2. Let \(u(x,y)\) be an entire solution in \((x,y)\) of equation (1). Then:

1) the functions \(\left.\partial^s u/\partial y^s\right|_{y=0}=P_s(x),\quad s=0,1,\ldots,m-1,\) must be polynomials of degree not exceeding \(r-1\);

2) if \(\lambda(x)\) is the value of the solution \(u(x,y)\) on the parabola \(y=\mu(x)\equiv \sum_{k=0}^{l} A_k x^k,\ l<r,\) then the function \(\nu(x)\), defined by equality (4), must be entire.

Conversely, for any parabola \(y=\mu(x)\) of order \(<r\) one can indicate a certain subclass \(G\) of sufficiently slowly growing entire functions \((G\subset H_{m,r})\) such that, whatever the prescribed polynomials \(P_0,P_1,\ldots,P_{m-1}\) of degree \(\leq r-1\) and whatever the prescribed entire function \(\lambda(x)\), provided only that \(\nu(x)\in G\), the solution of problem A always exists. This solution is unique in the class of entire solutions in \((x,y)\) of equation (1) of sufficiently slow growth (namely, in the class of functions \(u(x,y)\) such that their growth functions \(\left.\partial^m u/\partial y^m\right|_{y=0}\) belong to \(G\)).

Using the results of paper \((^1)\), one can give a method for the approximate solution of problem A (with an estimate of the error), and also show the correct dependence of the solution (in a certain metric) on the data of the problem.

The class \(G\) is defined by specifying the parabola \(y=\mu(x)\) and, generally speaking, changes in passing from one parabola of order \(<r\) to another. However, it always contains the class \(y_0\) of entire functions \(f(x)\) of zero order such that

\[ \sum_{m=0}^{\infty} |f^{(m)}(0)|\left(\prod_{k=0}^{m} (k!)\right)<\infty \]

(in this case, as a rule, the class \(y\) is substantially wider than the class \(G_0\)).

We shall give some concrete results for special types of parabolas.

1. \(\mu(x)\equiv Cx^l,\ 0\leq l<r\). Introduce the auxiliary entire function

\[ F_m(x)=\sum_{s=1}^{\infty}\frac{x^s m!(m-1)!\ldots 2!}{s!(s+1)!\ldots(s+m)!} \]

and denote by \(d_0\) the unique positive root of the equation \(F_m(x)=1\). Further, by \(G_C\) denote the class of entire functions of order \(1-\frac{l}{r}\) and type

\[ <\frac{r}{r-l}\left(\frac{d_0}{C}\right)^{1/r}. \]

Theorem 3. Let \(u(x,y)\) be a solution of problem A for the parabola \(y=Cx^l,\ l<r\). Then:

1) the functions \(\left.\partial^s u/\partial y^s\right|_{y=0},\ s=0,1,\ldots,m-1,\) will be polynomials of degree not exceeding \(r-1\);

2) the function \(\lambda(x)\) has the form

\[ \lambda(x)=\sum_{k=0}^{m-1} C^k x^{lk} P_k(x)+\varphi(x)x^{lm}, \]

where \(\varphi(x)\) is an entire function.

Conversely, if the function \(\lambda(x)\) satisfies condition 2) and belongs to \(G_C\), then the solution of problem A exists and is unique in the class of entire-in-\((x,y)\) solutions whose functions of growth belong to \(G_C\).

1. \(\mu(x)\equiv C\). Let \(G_{m,r}\) be the class of entire functions of the first order of finite type, for which none of the Borel-associated singularities of the functions coincides with zeros of the function

\[ 1+\sum_{k=1}^{\infty}\frac{C^k 2!3!\ldots(k-1)!\,x^{kr}}{(m+1)!(m+2)!\ldots(m+k)!}. \]

Using (2), we arrive at the following result:

Theorem 4. The problem of determining a solution \(u(x,y)\) of equation (1) under the conditions:

1) \(\left.\partial^s u/\partial y^s\right|_{y=0}=P_s,\ s=0,1,\ldots,m-1;\ P_s\) are polynomials of degree \(\leq r-1\);

2) \(u(x,c)=\lambda(x),\ \lambda(x)\in G_{m,r},\)

is solvable and, moreover, uniquely so in the class of entire-in-\((x,y)\) solutions \(v(x,y)\) whose functions of growth belong to \(G_{m,r}\).

Theorems 2–4 ensure the existence and uniqueness of the solution of problem A in the class of entire-in-\((x,y)\) solutions of equations (1) of sufficiently slow growth. It is necessary to note that in the whole class of entire-in-\((x,y)\) solutions (defined by the condition that the functions of growth belong to \(G_{m,r}\)) uniqueness, generally speaking, will no longer hold.

If the solution of equation (1) is determined by prescribing it on a parabola of order \(\geq r\), then the study of problem A becomes considerably more complicated, and the nature of the results is different. For the solvability of problem A it is necessary that the function \(\lambda(x)=u(x,\mu(x))\) satisfy, in addition to the natural conditions of type (4) (for analytic solutions), further conditions of an algebraic character. Uniqueness of the solution of problem A no longer holds in any class of entire functions \(v(x,y)\) of arbitrarily slow growth that contains all polynomials.

§ 3. Consider some particular cases of equation (1).

1. For equation (2), problem A consists in determining the solution from its values on the straight lines \(y=0\) and \(y=Cx+B\): \(u(x,0)=\lambda_1(x)\), \(u(x,Cx+B)=\lambda(x)\). For the solvability of this problem, by Theorem 2, it is necessary that the function \(\lambda_1(x)\) be linear: \(\lambda_1(x)=ax+b\), and that the function \(\lambda(x)\) be entire, with \(\lambda(-B/C)=-aB/C+b\). The last condition is the natural compatibility condition (coincidence of the values of the functions \(\lambda_1(x)\) and \(\lambda(x)\) at \(x=-B/C\)). Let \(d_0\) be the positive root of the equation \(J_1(2i\sqrt{x})=2i\sqrt{x}\). From Theorem 3 it follows:

Theorem 5. The problem of determining an entire-in-\((x,y)\) solution of equation (2) from the data \(u(x,0)=ax+b;\ u(x,Cx+B)=\lambda(x)\) is always solvable if the function \(\lambda(x)\) is an entire function of order \(1/2\) and type

\(< 2\sqrt{d_0/|Ca|}\), with \(\lambda(-B/C)=-aB/C+b\). The solution is unique in the class of functions \(v(x,y)\) for which \(\partial v/\partial y|_{y=0}\) is an entire function of order \(1/2\) and type \(<2\sqrt{d_0/|Ca|}\).

If \(C=0\), we arrive at the following problem: to find a solution of equation (2) such that \(u(x,0)=ax+b,\ u(x,d)=\lambda(x)\). Here the line \(y=d\) may lie both in the hyperbolic and in the elliptic part of the plane (depending on the sign of \(d\)). By Theorem 4 the problem is solvable if \(\lambda(x)\) is an entire function of exponential type for which no singularity of the Borel-associated function coincides with the zeros of the entire function \(1+\sum_{k=1}^{\infty}\frac{d^k}{a^k}\frac{x^{2k}}{k!(k+1)!}\). The solution is unique in the class of entire functions \(v(x,y)\) in \((x,y)\) such that \(\partial v/\partial y|_{y=0}\in G_{1,2}\).

1. Let us also consider the heat equation \(\partial^2 u/\partial x^2=\partial u/\partial y\). Problem A consists in determining a solution from its values on the line \(y=Cx+B\). By the linear change \(x=x_1-B/C\) we reduce this problem to the following: to find a solution of the heat equation such that \(u(x,Cx)=\lambda(x)\), where \(\lambda(x)\) is a given entire function. Denoting by \(G_C\) the class of entire functions of order \(1/2\) and type \(<2\sqrt{\ln 2/|C|}\), and applying Theorem 3, we obtain:

Theorem 6. Problem A, under the condition that \(\lambda(x)\in G_C\), is uniquely solvable in the class of solutions entire in \((x,y)\) whose growth functions belong to \(G_C\).

In connection with Theorem 6 it is appropriate to note one result of Holmgren \((^3)\), who showed that a regular (i.e., continuous together with first-order derivatives) integral \(v(x,y)\) of the heat equation in the whole plane is uniquely determined by prescribing on the line \(y=Cx,\ C\ne0\), the values \(v(x,Cx)\) and \(\partial v(x,Cx)/dx\). Therefore there exists an infinite set of regular integrals \(v(x,y)\) taking a prescribed value \(\lambda(x)\) on the line \(y=Cx\). At the same time, as follows from Theorem 6, if \(\lambda(x)\) is a sufficiently smooth entire function \((\lambda(x)\in G_C)\), then in the class of entire integrals in \((x,y)\) of sufficiently slow growth the specification of the function \(\lambda(x)=u(x,Cx)\) uniquely determines the corresponding solution.

Problem A for the heat equation admits a clear physical interpretation. As is known, the heat equation characterizes the propagation of temperature in an infinite thin thermally insulated rod. Suppose that along the rod, with constant velocity \(\alpha=dx/dt\), there moves an instrument which at each instant of time \(t\) determines the temperature of the rod at the corresponding point \(x=\alpha t\) (it is assumed, for definiteness, that at \(t=0\) the instrument is at the point \(x=0\)). It is required, from the readings of the instrument—the function \(\varphi(x)\)—to determine the law of propagation of temperature in the rod. It follows from Theorem 6 that this problem is solvable if \(\varphi(x)\) is an entire function of order \(1/2\) and type \(<2\sqrt{|\lambda|\ln 2}\). The solution is unique in the class of sufficiently slowly growing entire functions in \((x,t)\) (namely, such that \(v(x,0)\) is an entire function of order \(1/2\) and type \(<2\sqrt{|\lambda|\ln 2}\)).

Rostov-on-Don State University

Received
17 III 1960

CITED LITERATURE

1. Yu. F. Korobeinik, DAN, 122, No. 3 (1958).
2. A. O. Gelfond, Calculus of Finite Differences, 1952, pp. 439–455.
3. É. Goursat, A Course of Mathematical Analysis, 3, part 1, 1933, p. 266.