ON THE SOLVABILITY OF THE CAUCHY PROBLEM WITH FINITE INITIAL DATA

A. A. GORIN, E. A. GORIN

Introduction. There exist extensive classes of partial differential equations for which the Cauchy problem is solvable with arbitrary finite initial data, and the solution turns out to be a sufficiently regular function. For example, the heat-conduction equation

\[ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} \]

and the wave equation

\[ \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2} \]

possess this property. However, already for the equation

\[ \frac{\partial^2 u}{\partial t^2}=\frac{\partial u}{\partial x} \]

the Cauchy problem is, generally speaking, solvable only for very smooth finite initial data, while for the equation

\[ \frac{\partial u}{\partial t}=i\frac{\partial u}{\partial x} \]

among finite initial data only the zero data are admissible.

The purpose of the present article is to describe those partial differential equations with constant (complex) coefficients for which there exists a not too rapidly growing solution of the Cauchy problem for at least some nontrivial finite initial data. Among irreducible equations (and it is precisely such equations that it is natural to consider here) these prove to be the conditionally well-posed ones, and only them. Recall that the equation

\[ P\left(\frac{\partial}{\partial t},\, i\frac{\partial}{\partial x}\right)u=0 \tag{1} \]

is called conditionally well-posed if, for real \(\sigma=(\sigma_1,\ldots,\sigma_n)\), the real parts of the roots of the polynomial \(P(\lambda,\sigma)\) grow at infinity no faster than \(|\sigma|^h\), where \(h<1\) (see [1], p. 179).

For equations of a special form this situation was investigated by F. John [2], whose result can be formulated as follows. Suppose that the degree of the polynomial \(P(\lambda,\sigma)\) in the aggregate of the variables coincides with its degree in the variable \(\lambda\) (in the terminology of [1], these are precisely the polynomials \(P(\lambda,s)\) with reduced order \(\leq 1\): for all

\(s\) for them \(|\lambda(s)| \leq a|s|+b\). If the polynomial \(P(\lambda,\sigma)\) is irreducible and the Cauchy problem for equation (1) is solvable at least for some nontrivial finite initial data, then this equation is weakly hyperbolic, i.e., the roots of the polynomial \(P_0(\lambda,\sigma)\), where \(P_0(\lambda,\sigma)\) is the principal part of the polynomial \(P(\lambda,\sigma)\), are purely imaginary.

It can be shown that, for equations with reduced order \(\leq 1\), weak hyperbolicity is equivalent to conditional correctness (see § 4). Moreover, as follows from Holmgren’s principle, the solution of the Cauchy problem for such an equation with finite initial data is itself finite, so that any restrictions on growth are automatically satisfied. Thus the theorem formulated above contains, in particular, F. John’s result. At the same time, our result turns out to be substantially more general, since F. John’s scheme does not include, for example, the heat-conduction equation.

The proof given below differs somewhat from F. John’s arguments. It is more algebraic, is based on the technique of generalized functions, and does not use information about the case of one spatial variable. Instead we apply the following result (which will be formulated more accurately later). If an entire function \(f(s)\), \(s=(s_1,\ldots,s_n)\), of first order, not growing too rapidly in the real plane, decreases exponentially on a certain algebraic curve in the plane, then it is identically equal to 0 on this curve. It follows from this that if decrease of the indicated kind occurs on a family of algebraic curves filling the set of uniqueness, then \(f(s)\equiv 0\). This result is hardly new; however, despite its simplicity, we find it difficult to give an exact reference. It is quite clear that it can be extended to a broader class of domains.

We shall now demonstrate, with the example of an equation of first order in \(t\), how this assertion will work.

Suppose that the Cauchy problem for the equation

\[ \frac{\partial u}{\partial t}=P\left(i\frac{\partial}{\partial x}\right)u \]

is solvable for some nontrivial finite initial datum \(u(0,x)=\varphi(x)\), and moreover \(u(t,x)\in S'\) for every \(t\geq 0\). Here \(S'\) is the space of slowly increasing generalized functions (see [3]). Passing to the Fourier transform, we obtain

\[ v(t,\sigma)=e^{tP(\sigma)}\psi(\sigma), \]

where \(v\) and \(\psi\) are the Fourier transforms of the functions \(u\) and \(\varphi\). The function \(\psi(s)\), \(s=\psi(\sigma+i\tau)\), is entire and satisfies the inequality

\[ |\psi(s)|\leq C_k(1+|\sigma|^k)e^{a|\tau|}. \]

Since \(u(t,x)\in S'\), it follows that \(v(t,\sigma)\in S'\). This means that \(|v(t,\sigma)|\) grows as \(|\sigma|\to\infty\) no faster than a power. Suppose now that our equation is not conditionally correct. Then, as one can verify using the Seidenberg–Tarski theorem, on a certain family of (unbounded) algebraic curves filling an entire domain, the inequality

\[ \operatorname{Re} P(\sigma)\geq c|\sigma|^l \]

will hold. On all these curves the function \(\psi(\sigma)\) must decrease exponentially

and, by what was said above, tends to 0. But then \(\psi(s)\equiv 0\), and we arrive at a contradiction.

1. Entire functions. The following proposition is well known.

Lemma 1. Let \(f(\zeta)\) be holomorphic for \(\operatorname{Im}\zeta \geqslant 0\) and satisfy there the inequality

\[ |f(\zeta)| \leqslant Ce^{a|\zeta|}. \]

If on the positive semiaxis

\[ |f(\xi)| \leqslant C_1 e^{-b_1\xi}, \quad b_1>0, \]

and on the negative semiaxis

\[ |f(\xi)| \leqslant C_2 e^{b_2|\xi|}, \]

where \(b_2<b_1\), then \(f(\zeta)\equiv 0\).

Proof. Let \(0<\varepsilon<\dfrac12(b_1-b_2)\) and

\[ f_\varepsilon(\zeta)=f(\zeta)e^{(b_2+\varepsilon)\zeta}. \]

Then

\[ |f_\varepsilon(\zeta)| \leqslant Ce^{2h|\zeta|}, \]

and for real \(\xi\)

\[ |f_\varepsilon(\xi)| \leqslant \max(C_1,C_2)e^{-\varepsilon|\xi|}. \]

By Carlson’s theorem (see [4], Ch. 5) \(f_\varepsilon(\zeta)\equiv 0\), and therefore \(f(\zeta)\equiv 0\).

Let us note that in Lemma 1 the upper half-plane can be replaced by a half-plane with a semicircle removed, \(|\zeta|\leqslant r\), \(\operatorname{Im}\zeta\geqslant 0\).

Let \(g_k(\zeta)\), \(1\leqslant k\leqslant n\), be functions (of one variable), holomorphic in the domain \(|\zeta|\geqslant r\), \(\operatorname{Im}\zeta\geqslant 0\). Suppose that \(g_k(\xi)\) are real for \(\zeta=\xi\geqslant r\) and have the representation

\[ g_k(\zeta)=a_k\zeta+h_k(\zeta), \]

where \(\sum_{k=1}^n |a_k|>0\), and \(|h_k(\zeta)|=O(|\zeta|^\rho)\), \(\rho<1\). Consider in the space \(\mathbb{C}^n\) the curve

\[ \Gamma=\{s\in\mathbb{C}^n:\ s_k=g_k(\zeta),\ 1\leqslant k\leqslant n\}. \]

Let us note that \(\Gamma\) has an unbounded intersection with the real subspace \(R^n=\{s\in\mathbb{C}^n:\operatorname{Im}s=0\}\). Put, for \(s=\sigma+i\tau\in\mathbb{C}^n\),

\[ |s|=\sum_{k=1}^n |s_k|,\quad |\sigma|=\sum_{k=1}^n |\sigma_k| \quad \text{and} \quad |\tau|=\sum_{k=1}^n |\tau_k|. \]

Lemma 2. Let \(f(s)\), \(s\in\mathbb{C}^n\), be an entire function, and suppose

\[ |f(s)| \leqslant Ce^{\alpha|\sigma|+\beta|\tau|}. \tag{2} \]

Suppose that on the real part of the curve \(\Gamma\), more precisely on the set \(\{\sigma\in R^n:\sigma_k=g_k(\xi),\xi\geqslant r\}\), the function \(f(\sigma)\) decreases exponentially,

\[ |f(\sigma)| \leqslant C_1 e^{-\gamma|\sigma|},\quad \gamma>0. \tag{3} \]

If \(\alpha\) is a sufficiently small (positive) number, then \(f(s)=0\) on the curve \(\Gamma\).

Proof. The function

\[ F(\zeta)=f\bigl(g_1(\zeta),\ldots,g_n(\zeta)\bigr) \]

is holomorphic on the set \(\operatorname{Im}\zeta \geqslant 0,\ |\zeta|\geqslant r\). It is clear that, for all \(\zeta\),

\[ |F(\zeta)|\leqslant C_2 e^{a|\zeta|}. \]

Since the \(a_k\) are real, and \(|h_k(\zeta)|=O(|\zeta|^\rho)\), where \(\rho<1\), it is easy to derive from (2) that, for real \(\xi\),

\[ |F(\xi)|\leqslant C_3 e^{\alpha b|\xi|}, \]

where \(b\) depends only on the functions \(g_k(\zeta)\) and on the number \(\beta\). From (3) it follows that, for \(\xi\geqslant r\),

\[ |F(\xi)|\leqslant C_4 e^{-\gamma_1 \xi},\qquad \gamma_1>0. \]

Comparing the last three inequalities and using Lemma 1, we see that \(F(\zeta)\equiv 0\), provided only that \(\alpha\) is sufficiently small.

Corollary. Let

\[ R(w,s)=f_m(s)w^m+\cdots+f_1(s)w+f_0(s). \]

Suppose that the functions \(f_k(s)\) satisfy condition (2), and that the equation \(R(w,s)=0\) has a solution \(w=w(s)\) for which, on the real part of the curve \(\Gamma\) occurring in Lemma 2, condition (3) is satisfied, i.e.

\[ |w(\sigma)|\leqslant C_1 e^{-\gamma|\sigma|}. \]

Then \(f_0(s)=0\) on \(\Gamma\).

Indeed, if \(R(w,s)=0\), then

\[ f_0(s)=-\sum_{k\geqslant 1} f_k(s)w^k, \]

and therefore it satisfies all the conditions of Lemma 2.

In what follows we shall have to deal with entire functions that are Fourier transforms of finite functionals on the space \(S\) of infinitely differentiable functions decreasing at infinity, together with all their derivatives, in a power-like manner. Finite functionals on \(S\) have finite order, and their Fourier transforms satisfy estimate (2) for every \(\alpha>0\) (see [3]).

2. The character of the growth of algebraic functions. We list here several facts without dwelling on detailed proofs. The proofs are obtained by the usual scheme, by applying the Seidenberg–Tarski theorem (see, for example, [5]).

Let \(P(\sigma)\) be a real polynomial. Put

\[ \mu(r)=\max_{|\sigma|=r} P(\sigma). \]

The function \(\mu(r)\) is piecewise algebraic. Since \(\mu(r)\) is continuous, for large \(r\) it may be regarded as algebraic. In particular, \(\mu(r)\) can be expanded in a Puiseux series in rational powers of \(r\), convergent in a neighborhood of infinity (see [6]). Hence it follows that \(\mu(r)\) has the form

\[ \mu(r)=ar^h(1+o(1)). \]

Further, if \(\mu(r)\) has this form, then one can indicate such an algebraic curve \(\sigma_k=g_k(\xi)\), \(1\le k\le n\), on which

\[ P(\sigma)=\mu(|\sigma|)=a|\sigma|^h(1+o(1)). \]

Let us note that the parametrization of the curve \(\sigma_k=g_k(\xi)\) can be chosen with such a calculation that the functions \(g_k(\xi)\) satisfy the conditions listed on p. 1642. In what follows this is always assumed. Finally, if \(P(\sigma)\ge a'|\sigma|^h\) on the given algebraic curve and if \(a''<a'\), then \(P(\sigma)\ge a''|\sigma|^h\) on all sufficiently close algebraic curves.

Everything said in this paragraph applies not only to polynomials, but also to arbitrary (continuous) algebraic functions.

3. The main theorem. Let \(P(\lambda,\sigma)\) be a polynomial with constant (complex) coefficients, and suppose that the coefficient of \(\lambda\) in the highest degree \(m\) is equal to 1 (does not depend on \(\sigma\)). We shall consider the Cauchy problem for the equation

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

The initial data are assumed to be finite functionals of the class \(S'\), and the solutions (and their derivatives with respect to \(t\)) are arbitrary functionals (with respect to the spatial variables) of the same class. The solutions are considered in the strip \(0\le t\le T\). Let us note that, under our assumptions, the solution of the Cauchy problem is unique (see [1]).

Theorem. Suppose that the polynomial \(P(\lambda,\sigma)\) is irreducible. Then the Cauchy problem for the equation

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

is solvable for at least one set of nontrivial finite initial data \(\varphi_0,\varphi_1,\ldots,\varphi_{m-1}\) if and only if the equation is conditionally correct.

Proof. The main point in this theorem is to prove necessity.

Assume that the Cauchy problem is solvable in the sense indicated above, and let \(v(t,\sigma)\) be the Fourier transform of the solution, \(\psi=(\psi_0,\psi_1,\ldots,\psi_{m-1})\) the Fourier transforms of the initial data. Let

\[ P(\mu,\sigma)-P(\lambda,\sigma)=(\mu-\lambda)\sum_{k=0}^{m-1}Q_k(\lambda,\sigma)\mu^k \]

and

\[ w(\lambda,t,\sigma)=\sum_{k=0}^{m-1}Q_k(\lambda,\sigma)\frac{d^k v}{dt^k}. \tag{4} \]

If \(\lambda=\lambda(\sigma)\) is any root of the equation \(P(\lambda,\sigma)=0\), then, as is easy to see,

\[ \frac{d}{dt}w(\lambda(\sigma),t,\sigma)=\lambda(\sigma)w(\lambda(\sigma),t,\sigma), \]

so that

\[ w(\lambda(\sigma),t,\sigma)=e^{t\lambda(\sigma)}w(\lambda(\sigma),0,\sigma), \]

i.e.,

\[ w(\lambda(\sigma), t, \sigma)=e^{t\lambda(\sigma)} \sum_{k=0}^{m-1} Q_k(\lambda(\sigma),\sigma)\psi_k(\sigma). \tag{5} \]

Let us note that, as follows from (4), \(w(\lambda(\sigma),t,\sigma)\in S'\). Put now

\[ H(z,\lambda,s)=z-\sum_{k=0}^{m-1} Q_k(\lambda,s)\psi_k(s) \]

and let \(R(z,s)\) be the resultant with respect to \(\lambda\) of the polynomials \(P(\lambda,s)\) and \(H(z,\lambda,s)\). Assuming that the equation is not conditionally well posed, we shall show that \(R(0,s)\equiv 0\). Since \(R(0,s)\) is (up to sign) the resultant with respect to \(\lambda\) of the polynomials \(P(\lambda,s)\) and

\[ \sum_{k=0}^{m-1} Q_k(\lambda,s)\psi_k(s), \]

and the degree of the latter with respect to \(\lambda\) is strictly less than the degree with respect to \(\lambda\) of the polynomial \(P(\lambda,s)\), this will mean that \(P(\lambda,s)\) is reducible in the field of meromorphic functions of \(s\) (see [7], Ch. 1). But then \(P(\lambda,s)\) is reducible in the ordinary sense as well, contrary to the original assumption.

Thus, it remains only for us to prove that \(R(0,s)\equiv 0\), if \(P(\lambda,\sigma)\) is not conditionally well posed.

Let \(\lambda=\lambda(\sigma)\) be a root at which conditional well-posedness is violated. Taking into account (4), (5), and using the considerations of § 2, we can indicate a family of algebraic curves on which

\[ \sum_{k=0}^{m-1} Q_k(\lambda(\sigma),\sigma)\psi_k(\sigma) \]

will decrease exponentially. Since

\[ H\left(\sum_{k=0}^{m-1} Q_k(\lambda(\sigma),\sigma)\psi_k(\sigma),\lambda(\sigma),\sigma\right)\equiv 0 \]

(and \(P(\lambda(\sigma),\sigma)\equiv 0\)), it follows that

\[ R\left(\sum_{k=0}^{m-1} Q_k(\lambda(\sigma),\sigma)\psi_k(\sigma),\sigma\right)\equiv 0. \]

But the last equality, with what was said above taken into account, means that \(R(z,s)\) satisfies the condition of the corollary to Lemma 2. Therefore \(R(0,s)\equiv 0\), as was required. Necessity is established.

If, for all roots of the equation \(P(\lambda,\sigma)=0\), the condition

\[ \operatorname{Re}\lambda(\sigma)\leq a|\sigma|^h+b, \]

where \(h<1\), is fulfilled, then (see [1], p. 179) the equation \(P\left(\dfrac{\partial}{\partial t}, i\dfrac{\partial}{\partial x}\right)u=0\) is solvable with initial data from the space \(S^\alpha\) under the condition that \(1<\alpha<1/h\). This space contains finite functions. Further, their Fourier transforms belong to the space \(S_\alpha\), and therefore for real \(\sigma\) they decrease as \(e^{-|\sigma|^{1/\alpha}}\). Hence it follows that the corresponding solution will even be a decreasing function.

4. The case of F. John. The following lemma establishes the connection between the theorem proved above and the result of F. John. In what follows, \(P_0(\lambda,\sigma)\) denotes the principal part of the polynomial \(P(\lambda,\sigma)\).

Lemma 3. In order that a polynomial \(P(\lambda,\sigma)\) with reduced order \(\leq 1\) be conditionally well posed, it is necessary and sufficient that it be weakly hyperbolic.

Proof. By hypothesis, \(P(\lambda,\sigma)\) admits the representation

\[ P(\lambda,\sigma)=\sum_{k=0}^{m} a_k(\sigma)\lambda^k, \]

where \(\deg a_k \leq m-k\). Consider the following four equations:

\[ P(\lambda,\sigma)=0,\quad P(\mu|\sigma|,\sigma)=0,\quad P_0(\lambda_0,\sigma)=0,\quad P_0(\mu_0|\sigma|,\sigma)=0. \]

It is clear that \(\lambda=\mu|\sigma|\) and \(\lambda_0=\mu_0|\sigma|\). Moreover, by the theorem on the continuous dependence of the roots of an algebraic equation on its coefficients, \(|\mu-\mu_0|\to 0\) as \(|\sigma|\to\infty\). Therefore, by the Seidenberg–Tarski theorem there exists a constant \(\rho>0\) such that for all sufficiently large \(|\sigma|\)

\[ |\mu-\mu_0|<|\sigma|^{-\rho}. \]

Hence

\[ |\lambda-\lambda_0|<|\sigma|^{1-\rho} \]

and, all the more,

\[ |\operatorname{Re}\lambda-\operatorname{Re}\lambda_0|<|\sigma|^{1-\rho}. \]

Consequently, if \(\operatorname{Re}\lambda_0\equiv 0\), then \(\operatorname{Re}\lambda \leq |\operatorname{Re}\lambda|<|\sigma|^{1-\rho}\).

Conversely, suppose that for sufficiently large \(|\sigma|\) the inequality \(\operatorname{Re}\lambda \leq |\sigma|^h\) holds with \(h<1\). Then

\[ \operatorname{Re}\lambda_0 \leq |\sigma|^h+|\sigma|^{1-\rho}<|\sigma|^{h_1},\quad h_1<1. \]

But \(\lambda_0\) is a solution of the homogeneous equation; hence \(\operatorname{Re}\lambda_0\equiv 0\). The lemma is proved.

References

1. Gelfand I. M. and Shilov G. E. Some questions in the theory of differential equations. Generalized Functions, issue 3. Fizmatgiz, 1958.
2. John F. Comm. pure and appl. Math., 10, No. 3, 391–398, 1957.
3. Gelfand I. M. and Shilov G. E. Spaces of basic and generalized functions. Generalized Functions, issue 2. Fizmatgiz, 1958.
4. Titchmarsh E. Theory of Functions. Gostekhizdat, 1951.
5. Gorin E. A. UMN, vol. XVI, 1 (97), 91–118, 1961.
6. Chebotarev N. G. Theory of Algebraic Functions. Gostekhizdat, 1948.
7. Zariski O., Samuel P. Commutative Algebra, 1. IL, 1963.

Received by the editors
February 17, 1965

Institute of Precision Mechanics
and Computer Engineering, Academy of Sciences of the USSR,
Moscow State University
named after M. V. Lomonosov