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UDC 513.88+517.946
MATHEMATICS
S. G. KREIN
WELL-POSEDNESS OF THE CAUCHY PROBLEM AND ANALYTICITY OF SOLUTIONS OF AN EVOLUTION EQUATION
(Presented by Academician I. G. Petrovskii, 16 II 1966)
- In a Banach space \(E\) one considers the differential equation
\[ dx/dt=Ax, \tag{1} \]
where \(A\) is a closed unbounded operator with domain of definition \(D(A)\) dense in \(E\). By a solution of equation (1) on the interval \([0,T]\) is meant a continuous function \(x(t)\) on \([0,T]\), with values in \(D(A)\), which has, for all \(t\in[0,T]\), a derivative satisfying equation (1).
The Cauchy problem is the problem of finding a solution for a prescribed initial condition
\[ x(0)=x_0\in D(A). \tag{2} \]
The Cauchy problem is called well-posed if the solution of (1)—(2) exists for every \(x_0\in D(A)\); the solution is unique and depends continuously on the initial data in the sense that from \(x_n(0)\to x(0)\) it follows that \(x_n(t)\to x(t)\) for all \(t\in[0,T]\). In view of the time independence of the operator \(A\), from the well-posedness of the Cauchy problem on the interval \([0,T]\) it follows that it is well posed on any finite interval \([0,T_1]\) \((T_1>0)\).
An operator \(U(t)\) is introduced such that, for the solution,
\[ x(t)=U(t)x_0 \quad (x_0\in D(A)). \tag{3} \]
It turns out that \(U(t)\) is a strongly continuous, for \(t>0\), semigroup of bounded operators.
If \(x_0\in \overline{D(A)}\), then the function \(U(t)x_0\) is called a generalized solution of equation (1).
It is known that \(\|U(t)\|\) grows at infinity no faster than an exponential \(({}^{1},\) p. 260).
Suppose now that the operator \(A\) has at least one regular point \(\lambda_0\), and at it a resolvent \(R(\lambda_0)\). Then the operator \(A\) has the resolvent \(R(\lambda)=(A-\lambda I)^{-1}\) for all \(\lambda\) with \(\operatorname{Re}\lambda>\omega\), and for \(x_0\in D(A)\)
\[ R(\lambda)x_0=-\int_0^\infty e^{-\lambda t}U(t)x_0\,dt; \tag{4} \]
if, however, \(x\) is any element of \(E\), then one may apply formula (4) to the element \(x_0=R(\lambda_0)x\) and obtain the formula
\[ R(\lambda)x=R(\lambda_0)x-(\lambda-\lambda_0)\int_0^\infty e^{-\lambda t}U(t)R(\lambda_0)x\,dt. \tag{5} \]
The function \(U(t)R(\lambda_0)x\) is continuous on \([0,T]\) for any \(x\in E\); therefore the operators \(U(t)R(\lambda_0)\) are uniformly bounded on \([0,T]\). Taking into account the behavior of \(U(t)\) at infinity, we may write that
\[ \|U(t)R(\lambda_0)\|\le M_\varepsilon e^{(\omega+\varepsilon)t}\quad (t\ge 0). \tag{6} \]
From this estimate and (5) one obtains the inequality
\[ \|R(\lambda)\|\leq C(1+|\lambda|), \tag{7} \]
valid in the half-plane \(\operatorname{Re}\lambda\geq \omega_2\). This estimate shows that the requirement of well-posedness of the Cauchy problem imposes rather stringent restrictions on the resolvent of the operator \(A\). We note that from formula (4) it also follows that, for \(x_0\in D(A)\),
\[ \|R(\lambda)x_0\|\to 0, \tag{8} \]
when \(\operatorname{Re}\lambda\to\infty\).
- If for equation (1) the Cauchy problem is well posed, then it will also be well posed for the equation \(x'=\mu Ax\) for any \(\mu>0\). If the corresponding semigroup is denoted by \(U_\mu(t)\), then obviously
\[ U_\mu(t)=U(\mu t). \tag{9} \]
We now consider the equation
\[ dx/dt=\xi Ax, \tag{10} \]
where \(\xi\) is a complex number.
Definition 1. The totality of all \(\xi\) for which the Cauchy problem for equation (10) is well posed will be called the well-posedness set \(K_A\) of the operator \(A\).
From what was said above it is clear that this set consists of a totality of rays issuing from the point \(\xi=0\). Denote by \(U_\xi(t)\) the semigroup generated by problem (10)—(2) for \(\xi\in K_A\), and put \(U(\xi)=U_\xi(1)\). By virtue of (9),
\[ U(\mu\xi)=U_{\mu\xi}(1)=U_\xi(\mu)\qquad(\mu>0). \tag{11} \]
Lemma 1. If the operator \(A\) has at least one regular point, then the operators \(U(\xi)\) form a semigroup in the sense that, for \(\xi_1,\xi_2\) and \(\xi_1+\xi_2\) belonging to \(K_A\),
\[ U(\xi_1+\xi_2)=U(\xi_1)U(\xi_2). \tag{12} \]
The proof consists in showing, using the well-posedness of the Cauchy problem, that for \(x_0\in D(A^3)\) the functions \(U((\xi_1+\xi_2)t)x_0\) and \(U(\xi_1 t)U(\xi_2 t)x_0\) give a solution of one and the same Cauchy problem for the equation \(x'=(\xi_1+\xi_2)Ax\). By uniqueness of the solution they coincide, whence (12) follows.
Theorem 1. Under the conditions of Lemma 1, the operator-valued function \(U(\xi)\) is analytic at every interior point of the set \(K_A\).
If \(\xi_0\) is an interior point of \(K_A\), then it belongs to an open sector from \(K_A\). Using this and the semigroup property (12), one can show that, for \(x_0\in D(A)\), the function \(U(\xi)x_0\) has at the point \(\xi_0\) one and the same derivative in two non-collinear directions, and from this derive its analyticity. The analyticity of the function \(U(\xi)x\) for arbitrary \(x\in E\) is proved by means of its uniform approximation by analytic functions \(U(\xi)x_n\), with \(x_n\in D(A)\).
From the proof of Theorem 1 and known considerations it follows that
Theorem 2. If the operator \(A\) has at least one regular point, then for every \(x_0\in D(A)\) there exists a solution \(x(\xi)\) of the equation
\[ dx/d\xi=Ax(\xi)\qquad (x(0)=x_0), \]
defined and analytic in the open kernel of the well-posedness set \(K_A\) of the operator \(A\). The exponential types of all solutions along rays belonging to a closed sector of the open kernel of the set \(K_A\) are uniformly bounded.
It is not difficult to give examples of operators \(A\) for which the well-posedness set consists only of zero. Such, for example, is an operator whose spectrum is located on the real and imaginary axes and goes to infinity in the directions \(\pm\infty\) and \(\pm i\infty\). If the spectrum of the operator \(A\)
lies on the imaginary axis and goes to infinity at both of its ends, then the set of correctness can consist only of points of the real axis. For a bounded operator the set of correctness coincides with the whole plane.
Theorem 3. If the operator \(A\) is unbounded and has at least one regular point, then its set of correctness lies in some closed half-plane.
If the spectrum of the operator \(A\) has a sequence of points \(\lambda_n\) tending to infinity, then one may assume that \(\arg \lambda_n \to \alpha\). Then the set of correctness lies in the half-plane
\[
\pi/2+\alpha \leq \arg \zeta \leq 3\pi/2+\alpha .
\]
If, on the other hand, the spectrum of the operator \(A\) is bounded, then the space can be decomposed into a direct sum \(E=E_1+E_2\) of invariant subspaces in such a way that in \(E_1\) the operator \(A\) is bounded, while in \(E_2\) its spectrum is empty. If \(x_0\in E_2\) and \(f\in E^*\), then the function \(f(R(\lambda)x_0)\) is analytic in the entire plane. If we assume that the set of correctness contains three rays not lying in one half-plane, then for the resolvent \(R(\lambda)\) estimate (7) will be valid outside some triangle. Consequently, \(f(R(\lambda)x_0)\) is linear. From (8) it follows that it is equal to zero, i.e. \(x_0=0\). Thus \(E_2\) is empty and the operator is bounded, which contradicts the hypothesis. The theorem is proved.
- The Cauchy problem for equation (1) is called uniformly correct on \([0,T]\) if it is correct and, in addition, from \(x_n(0)\to x(0)\) it follows that \(x_n(t)\to x(t)\) uniformly on \([0,T]\).
In order that the Cauchy problem be uniformly correct, it is necessary and sufficient that the operator \(A\) be the infinitesimal generator of a strongly continuous semigroup satisfying the \(C_0\)-condition (see \((2\text{–}4)\)). For this, in turn, it is necessary and sufficient that the following inequalities hold for the resolvent of the operator (see (1), p. 373):
\[
\|R^n(\lambda)\|\leq M/(\operatorname{Re}\lambda-\omega)^n
\quad
(\operatorname{Re}\lambda>\omega,\ n=1,2,\ldots).
\tag{13}
\]
Definition 2. The totality of all \(\zeta\) for which the Cauchy problem for equation (10) is uniformly correct will be called the set \(K_A^c\) of uniform correctness of the operator \(A\).
From conditions (13) it follows immediately that, in order that the ray \(\arg \zeta=\varphi\) belong to the set of uniform correctness of the operator \(A\), it is necessary that the inequalities
\[
\|R^n(z+\eta)\|\leq M/|z-z_0|^n
\quad (n=1,2,\ldots),
\tag{14}
\]
hold, where \(z_0\) is a point on the ray \(\arg z=-\varphi\), \(z\) is any point on this ray with \(|z|>|z_0|\), and \(\eta\) is an arbitrary point on the line perpendicular to this ray. It is sufficient that conditions (14) hold for \(\eta=0\).
Lemma 2. If two rays \(\arg \zeta=\varphi_1\) and \(\arg \zeta=\varphi_2\) \((0<\varphi_2-\varphi_1<\pi)\) belong to the set of uniform correctness of the operator \(A\), then every ray \(\arg \zeta=\varphi\) for which \(\varphi_1\leq \varphi\leq \varphi_2\) also belongs to this set.
From the conditions of the lemma there follows the validity of the estimates
\[
\|R^n(z+\eta_1)\|\leq M_1/|z-z_1|^n,\qquad
\|R^n(z+\eta_2)\|\leq M_2/|z-z_2|^n,
\]
when \(z,\eta_1\) and \(\eta_2\) are situated in the manner described above. Draw at the points \(z_1\) and \(z_2\) perpendiculars to the rays \(\arg z=-\varphi_1\) and \(\arg z=-\varphi_2\). Denote their point of intersection by \(\xi\) and introduce the operator \(A_1=A-\xi I\). A simple calculation shows that for the resolvent of the operator \(A_1\) the inequalities
\[
\|z^n R_{A_1}^{\,n}(z)\|\leq M_1,\qquad
\|z^n R_{A_1}^{\,n}(z)\|\leq M_2,
\]
are valid, where \(z\) ranges respectively over the rays \(\arg z=-\varphi_1\) and \(\arg z=-\varphi_2\).
By the Phragmén—Lindelöf principle, then
\[ \|z^n R_{A,n}(z)\| \le M=\max(M_1,M_2) \]
also inside the angle \(-\varphi_2 \le \arg z \le -\varphi_1\). Hence the assertion of the lemma follows easily.
From Lemma 2 and the preceding one there follows
Theorem 4. In order that the semigroup \(U(\zeta)\) corresponding to the Cauchy problem be analytic in the sector \(\varphi_1<\arg\zeta<\varphi_2\) \((0<\varphi_2-\varphi_1-\pi)\) and satisfy in it the condition
\[ \|U(\zeta)\| \le Me^{\omega|\zeta|} \]
it is necessary and sufficient that the rays \(\arg\zeta=\varphi_1\) and \(\arg\zeta=\varphi_2\) belong to the set of uniform well-posedness.
Theorem 5. The set of uniform well-posedness of the operator \(A\) can be only one of the following: 1) the point \(\zeta=0\); 2) a ray; 3) a straight line; 4) an open, half-open, or closed sector with opening angle \(\psi\): \(0<\psi\le\pi\).
- In conclusion we give two theorems concerning equation (1) in a Hilbert space \(H\).
Theorem 6. In order that, for the uniformly well-posed Cauchy problem (1)—(2) in a Hilbert space \(H\), all generalized solutions admit analytic continuation into the sector \(-\varphi_0<\arg\zeta<\varphi_0\) with the estimate
\[ \|x(\zeta)\| \le Me^{\omega|\zeta|}\|x_0\| \qquad (|\arg\zeta|<\varphi_0), \tag{15} \]
where \(M\) and \(\omega\) do not depend on the solution, it is necessary that the operator \(A\) admit the representation
\[ A=\omega I+QB, \tag{16} \]
where \(Q\) is a positive definite self-adjoint operator, and \(B\) is a dissipative operator (see (⁴), p. 121), for which
\[ \tg \varphi_0|\operatorname{Im}(Bx,x)| \le |\operatorname{Re}(Bx,x)|. \]
Moreover, the operator \(Q^{1/2}B\) has a bounded inverse defined on all of \(H\).
Theorem 7. If the closed operator \(A\) has the representation (16), then the space \(H\) can be continuously and densely embedded in a Hilbert space \(\widetilde H\), in which all generalized solutions of equation (1) are analytic in the sector \(|\arg\zeta|<\varphi_0\), and the estimate (15) is valid (in the norm of the space \(\widetilde H\)).
Voronezh State University
Received
11 II 1966
REFERENCES
¹ E. Hille, R. Phillips, Functional Analysis and Semi-Groups, IL, 1962.
² S. G. Krein, Dokl. Akad. Nauk SSSR, 118, No. 2 (1958).
³ R. S. Phillips, Proc. Nat. Acad. Sci., 40, 244 (1954).
⁴ Functional Analysis, “Nauka,” 1964.