UDC 517.948:513.88

MATHEMATICS

S. G. KREIN, G. I. LAPTEV, G. A. TSVETKOVA

ON THE HADAMARD WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR AN EVOLUTION EQUATION

(Presented by Academician I. N. Vekua on 27 X 1969)

The uniformly well-posed Cauchy problem for an evolution equation in a Banach space has been well studied (see \((^1)\)). If one considers systems of partial differential equations with constant coefficients of the form \(\partial u/\partial t = A(D)u\), then necessary and sufficient conditions for the uniform well-posedness of the Cauchy problem for them in the space \(\mathcal L_2(R^n)\) were obtained by H. O. Kreiss \((^2)\) and are complicated and difficult to verify. On the other hand, if a system is Petrovskii correct, i.e., if all eigenvalues of its characteristic matrix lie in some fixed half-plane, then, as I. G. Petrovskii showed \((^3)\), the Cauchy problem for it is Hadamard well-posed. Thus, Hadamard well-posed Cauchy problems form a considerably broader class than uniformly well-posed ones.

In the present paper a definition is given of Hadamard well-posedness of the abstract Cauchy problem for an evolution equation in a Banach space, and necessary and sufficient conditions for Hadamard well-posedness are found.

In a Banach space \(E\) with norm \(\|\cdot\|\), consider the differential equation

\[ dx/dt = Ax(t) \qquad (0 \le t < \infty) \tag{1} \]

with a linear closed operator \(A\), having an everywhere dense domain of definition \(D(A)\) in \(E\).

A solution of equation (1) on the half-axis \([0,\infty)\) is a function \(x(t)\) with values in \(D(A)\), strongly continuously differentiable on \([0,\infty)\) and satisfying equation (1) for all \(t \in [0,\infty)\). By the Cauchy problem is meant the problem of finding a solution on \([0,\infty)\) satisfying the initial condition

\[ x(0)=x_0 \in D(A). \tag{2} \]

Let in the space \(E\) there be given a linear set \(E_1\), contained in \(D(A)\), on which another norm \(\|\cdot\|_1\) and a seminorm \(\|\cdot\|_2\) are defined.

Definition 1. The Cauchy problem (1), (2) is Hadamard well-posed with type \(\omega\) on the set \(E_1\) if there exist constants \(C_1\) and \(C_2\) such that for any \(x_0 \in E_1\) there exists a solution \(x(t)\) satisfying the conditions

\[ \|x(t)\| \le C_1 e^{\omega t}\|x_0\|_1,\qquad \|Ax(t)\| \le C_2 e^{\omega t}\|x_0\|_2, \]

and this solution is unique.

If the Cauchy problem is Hadamard well-posed, then by a change of the unknown function it is reduced to the same problem with type \(\omega=0\). In what follows we shall consider Hadamard well-posed problems for which

\[ \|x(t)\| \le C_1\|x_0\|_1,\qquad \|Ax(t)\| \le C_2\|x_0\|_2 \quad (x_0 \in E_1). \tag{3} \]

We make the main assumption.

I. The positive half-axis does not belong to the point spectrum of the operator \(A\), i.e., for \(\lambda > 0\) there exists the operator \(J_\lambda=(A-\lambda I)^{-1}\) (generally speaking, unbounded).

Lemma 1. Under assumption I, for a given \(x_0\in D(A)\) there cannot exist more than one solution of problem (1), (2), bounded on the half-axis.

Corollary. Under assumption I, the uniqueness requirement in the definition of Hadamard well-posedness may be discarded.

Definition 2. The maximal domain of Hadamard well-posedness is the linear set \(D\) of all elements \(x_0\) for which there exists a solution of problem (1), (2), bounded together with its derivative on the half-axis \([0,\infty)\).

This definition is justified by the following consideration: by virtue of Lemma 1, on \(D\) the norm and seminorm are uniquely determined:
\[ \|x_0\|_{\mathrm I}=\sup_{0\leq t<\infty}\|x(t)\|,\qquad \|x_0\|_{\mathrm {II}}=\sup_{0\leq t<\infty}\|Ax(t)\|, \]
where \(x(t)\) is the solution referred to in Definition 2. It is obvious that, in the norms \(\|\cdot\|_{\mathrm I}\) and \(\|\cdot\|_{\mathrm {II}}\), problem (1), (2) is Hadamard well-posed on \(D\). Furthermore, if problem (1), (2) is Hadamard well-posed with type \(\omega=0\) on some set \(E_1\), then \(E_1\subset D\) and \(\|x\|_{\mathrm I}\leq C_1\|x\|_1\) and \(\|x\|_{\mathrm {II}}\leq C_2\|x\|_2\).

Under assumption I, let us study the properties of the operators \(J_\nu(\nu>0)\).

\(1^\circ\). The operator \(J_\nu(\nu>0)\) is defined on \(D\), and the formula
\[ J_\nu x_0=-\int_0^\infty e^{-\nu\tau}x(\tau)\,d\tau \qquad (x_0\in D,\ \nu>0), \tag{4} \]
holds, where \(x(t)\) is the corresponding solution of problem (1), (2).

\(2^\circ\). The set \(D\) is invariant with respect to the operators \(J_\nu(\nu>0)\), and the solution \(y(t)\) of equation (1) with the initial condition \(y(0)=J_\nu x_0\) is determined by the formula
\[ y(t)=-\int_0^\infty e^{-\nu\tau}x(t+\tau)\,d\tau . \tag{5} \]
Moreover,
\[ \|y(t)\|\leq \frac{1}{\nu}\|x_0\|_{\mathrm I},\qquad \|Ay(t)\|\leq \frac{1}{\nu}\|x_0\|_{\mathrm {II}}. \]

\(3^\circ\). Hilbert’s identity holds:
\[ J_\nu J_\lambda x_0=\frac{1}{\lambda-\nu}(J_\lambda-J_\nu)x_0 \qquad (x_0\in D;\ \lambda,\nu>0), \]
as do the formulas for derivatives
\[ J_\lambda^n x_0=\frac{1}{(n-1)!}\frac{d^{\,n-1}J_\lambda x_0}{d\lambda^{\,n-1}} \qquad (x_0\in D,\ \lambda>0,\ n\geq 1). \tag{6} \]

\(4^\circ\). For \(x_0\in D\), \(\lambda>0\), and \(n\geq 0\), the inequalities
\[ \|J_\lambda^n x_0\|\leq \lambda^{-n}\|x_0\|_{\mathrm I},\qquad \|J_\lambda^n A x_0\|\leq \lambda^{-n}\|x_0\|_{\mathrm {II}} \tag{7} \]
hold.

\(5^\circ\). The equalities (cf. (5))
\[ \|x_0\|_{\mathrm I}=\sup_{n\geq 0}\sup_{\lambda>0}\|\lambda^n J_\lambda^n x_0\|, \tag{8} \]
\[ \|x_0\|_{\mathrm {II}}=\sup_{n\geq 0}\sup_{\lambda>0}\|\lambda^n J_\lambda^n A x_0\| \tag{9} \]
hold.

From formula (5) it follows that the solution \(z(t)\) of equation (1) with initial value \(z(0)=\nu J_\nu x_0+x_0\), for \(x_0\in D(A)\) and \(\nu>0\), is representable in the form
\[ z(t)=-\int_0^\infty e^{-\nu\tau}Ax(t+\tau)\,d\tau . \]

Hence, from (7) it follows

\(6^\circ\). The inequality holds
\[ \|J_\lambda^n(\nu J_\nu x_0+x_0)\|\leq \frac{C_2}{\nu\lambda^n}\|x_0\|_1,\qquad x_0\in D. \]

\(7^\circ\). For \(x_0\in D\) the relation
\[ \lim_{\nu\to\infty}\|\lambda^n J_\lambda^n(\nu J_\nu A x_0+A x_0)\|=0 \tag{10} \]
holds uniformly with respect to \(\lambda>0\) and \(n\geq 0\).

Let us introduce the set \(G\), consisting of all elements \(y\in E\) for which
\[ \|y\|_G=\sup_{n\geq 0}\sup_{\lambda>0}\|\lambda^n J_\lambda^n y\|<\infty . \tag{11} \]

From the closedness of the operators \(J_\lambda\) it follows that \(G\), with respect to \(\|\cdot\|_G\), is a Banach space. From \(1^\circ\), \(2^\circ\), \(4^\circ\), and \(5^\circ\) it follows that \(D\subset G\) and \(\|x\|_1=\|x\|_G\) for \(x\in D\). For every \(y\in G\) the function \(J_\nu y\) can be analytically continued by means of the expansion
\[ J_\nu y=\sum_{k=0}^{\infty}(\nu-\lambda)^k J_\lambda^{k+1}y\qquad(\lambda>0), \tag{12} \]
which converges in the disk \(|\nu-\lambda|<\lambda\). Varying \(\lambda\), we obtain an analytic continuation of \(J_\nu x\) into the right half-plane.

The following assertion is essential:

Lemma 2. The operator \(J_\mu\) \((\mu>0)\) is a bounded operator on \(G\) in the norm (11), and, moreover, for it the relation
\[ \|J_\mu x\|_G\leq \frac{1}{\mu}\|x\|_G \tag{13} \]
holds.

For the proof, the quantity \(\|\lambda^n J_\lambda^n(\mu J_\mu x)\|\) for \(\mu<\lambda\) can be estimated with the aid of the expansion (12):
\[ \|\lambda^n J_\lambda^n(\mu J_\mu x)\|\leq \frac{\mu}{\lambda}\sum_{k=0}^{\infty}\left(1-\frac{\mu}{\lambda}\right)^k \|\lambda^{n+k+1}J_\lambda^{n+k+1}x\|\leq \|x\|_G . \]

For \(\lambda<\nu\) the same estimate can be obtained by expanding the function \(\lambda^n J_\lambda^n x\) in a series at the point \(\lambda=\mu\). From these estimates the inequality (13) follows.

Let \(F\) now denote the collection of all elements of \(G\) for which
\[ \|\mu J_\mu y+y\|_G\to 0\quad \text{as }\mu\to\infty . \tag{14} \]

By virtue of the uniform boundedness of the operators \(\mu J_\mu\), \(F\) is a closed subspace of \(G\). From the commutativity of the operators \(J_\nu\) and \(J_\mu\), and the boundedness of \(J_\nu\) in \(G\), it follows that \(J_\nu F\subset F\). The operators \(J_\nu\) bounded in \(F\) form a pseudo-resolvent, and since they vanish only at zero, they are the resolvent of some operator \(\widehat A\). By virtue of (14), the domain of definition of this operator is dense in \(F\). It is not difficult to verify that the operator \(\widehat A\) is the restriction of the operator \(A\) to the space \(F\), i.e., it is defined on all elements \(x_0\in F\cap D(A)\) for which \(Ax_0\in F\), and on them coincides with \(A\).

By what has been said and by Lemma 2, for the operator \(\widehat A\) the conditions of the Hille–Yosida theorem are satisfied, and consequently it is the infinitesimal generator of a contraction semigroup \(U(t)\) of operators in \(F\) (see \((1,4)\)). For every \(x_0\in D(\widehat A)\) there exists a solution \(x(t)=U(t)x_0\) of the equation
\[ x'=\widehat A x \]
in \(F\), satisfying the condition \(x(0)=x_0\). This solution will also be

is a solution of problem (1), (2), since the norm in \(F\) is stronger than the norm in \(E\). Further:

\[ \sup_{0\leq t<\infty}\|x(t)\|\leq \sup_{0\leq t<\infty}\|U(t)x_0\|_G=\|x_0\|_G<\infty, \]

\[ \sup_{0\leq t<\infty}\|Ax(t)\|\leq \sup_{0\leq t<\infty}\|U(t)\hat A x_0\|_G=\|Ax_0\|_G<\infty, \]

i.e. \(x_0\in D\).

Thus, \(D(\hat A)\subset D\). On the other hand, by properties \(1^0\)—\(7^0\), \(D\subset D(\hat A)\), whence \(D=D(\hat A)\).

From all that has been said the following main results follow.

Theorem 1. Under assumption I the maximal domain of Hadamard correctness consists of all elements \(x_0\) from \(D(A)\) possessing the properties:

1) \(x_0\in G\) and \(\|\mu J_\mu x_0+x_0\|_G\to 0\) as \(\mu\to\infty\);

2) \(Ax_0\in G\) and \(\|\mu J_\mu Ax_0+Ax_0\|_G\to 0\) as \(\mu\to\infty\).

Theorem 2. Let I be satisfied. In order that the Cauchy problem (1), (2) be Hadamard correct on a set \(E_1\) with type \(\omega=0\), the following conditions are necessary and sufficient:

1) for each \(x_0\in E_1\) all operators \(J_\lambda\) \((\lambda>0)\) and all their possible products are defined;

2) the inequalities are valid \((x_0\in E_1)\):

\[ \|J_\lambda^n x_0\|\leq C_1\lambda^{-n}\|x_0\|,\qquad \|J_\lambda^n Ax_0\|\leq C_2\lambda^{-n}\|x_0\|_2; \]

3) uniformly with respect to \(n\geq 0\) and \(\lambda>0\) the relations hold

\[ \lim_{\mu\to\infty}\|\lambda^n J_\lambda^n(\mu J_\mu x_0+x_0)\|=0, \]

\[ \lim_{\mu\to\infty}\|\lambda^n J_\lambda^n(\mu J_\mu Ax_0+Ax_0)\|=0. \]

Theorem 3. Let I be satisfied and let the Cauchy problem be Hadamard correct with type \(\omega=0\) on the set \(E_1\). Then there exists a space \(F\), algebraically and topologically embedded in the space \(E\) and containing the set \(E_1\), such that for the equation

\[ dx/dt=\hat A x, \]

where \(\hat A\) is the restriction of \(A\) in the space \(F\), the Cauchy problem is uniformly correct and has type \(\omega=0\).

Voronezh State University
named after the Lenin Komsomol

Latvian State University
named after P. Stuchka
Riga

Received
13 X 1969

CITED LITERATURE

\(^{1}\) S. G. Krein, Linear Differential Equations in Banach Space, “Nauka,” 1967.
\(^{2}\) H. O. Kreiss, Sborn. per. Matematika, 7, No. 2, 39 (1963).
\(^{3}\) I. G. Petrovskii, Bull. Moscow State Univ., Ser. Math. and Mech., 1, 7, 1 (1938).
\(^{4}\) E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups, IL, 1962.
\(^{5}\) W. Feller, Ann. Math., 58, 1, 166 (1953).