Fattorini H.O., Kerber A. The Cauchy Problem

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6.6. Miscellaneous Comments

377

f(A)g(A). An instance of an operational calculus for a fairly arbitrary

unbounded operator A was sketched in Example 3.12: as seen in Section

6.4, this calculus is essentially sufficient to define fractional powers of

certain operators (but not of those satisfying only (6.3.1)). For information

on operational calculi, see Hille-Phillips [1957: 1], Dunford-Schwartz [1958:

1], Balakrishnan [1959:

11, Kantorovitz [1970: 1], Faraut [1970:

1], and

Hirsch [1972: 2]. In the Hilbert space case, advantage can be taken of the

"reduction to self-adjoint operators" outlined in Section 3.8(b)): see Sz.-

Nagy-Foias [1969: 1].

In view of some of the applications, it is important to compute the

fractional powers of ordinary and partial differential operators, or at least

to identify their domains. This is no easy task, and the results are sometimes

less than intuitively obvious.

6.6.2 Example (Yosida [1978: 1]). Let E = L2(- oo, oo), Au = u" with

domain consisting of all u E E such that u" (taken in the sense of distri-

butions) belongs to E. Then A satisfies (6.3.1) (in fact,

it belongs to

(2+ (1, 0)). The operator - (- A )1/2 is the singular integral operator

-(-A

)1/2U(X)=

lim

1 r- u(x - )-u(x)

d (6.6.4)

h-O+

?TJ_0

t2+h2

with domain D((- A)'/2) consisting of all u E E such that the limit exists in

the L2 sense (and not the first derivative operator Bu = iu' as one may

expect). However, we do have

D((- A)1/2) = D(B).

To prove these statements we do not need the general theory of fractional

powers since A is a self-adjoint operator. Through the Fourier-Plancherel

transform, A is equivalent to the multiplication operator Au(a) a2u(a)

in LQ, thus (- j)1/2U(0) _ I a I u(a) and formula (6.6.4) is an immediate

consequence of the fact that

etax

dx=he'

oox2+h2

We note that

B2=-A

and the "discrepancy" between (- A)'/2 and B is in fact a consequence of

the fact that if a is a real number, a 2 equals a and not a. In fact, we

have (in the sense of the functional calculus for self-adjoint operators)

A)',2

= IBI

For results on the identification of fractional powers of differential opera-

tors see the treatise of Triebel [1978: 1] and bibliography therein, Evzerov

378 Some Improperly Posed Cauchy Problems

[1977:

11,

Evzerov-Sobolevskii [1973:

1],

[1976:

1],

Krasnosel'skii-

Pustyl'nik-Zabreiko [ 1965: 11, [1965: 2], Segal-Goodman [ 1965: 1 ], Masuda

[1972: 2], Murata [1973: 11, Sobolevskii [1972: 11, [1977: 11, and Koledov

[1976: 1].

Generalizations to the higher-order equation

u(n)(t) = Au(t) (6.6.5)

were carried out by Radnitz [ 1970: 11 and the author and Radnitz [ 1971: 2].

The problem here is that of existence and uniqueness of solutions of (6.6.5)

that satisfy the estimate

IIu(t)II=O(e')

ast - oo

(6.6.6)

and the incomplete initial conditions

u(k)(0)=ukEE

(kEa), (6.6.7)

where a is a predetermined subset of the set (0, 1, ... , n - 1); also, the

dependence of the solution on the initial data (6.6.7) is examined. The

author's paper [1973: 1] proceeds along similar lines, but the selection of

the solution of (6.6.5) satisfying the incomplete initial conditions (6.6.7) is

made not necessarily on the basis of the growth condition (6.6.6) but in an

unspecified way producing existence, uniqueness, and linear and continuous

dependence on the initial data. In a sense, these investigations are related to

Hille's results on the higher-order Cauchy problem (see 2.5(c)) and with

results for particular higher-order equations due to Lions [1961: 1].

A problem of a somewhat similar sort is the boundary value problem

for a higher order equation

u(n)(t)+A,(t)u(n-')(t)+ .

+An-,(t)u'(t)+An(t)u(t) = f(t).

(6.6.8)

Here, solutions are sought in a finite interval (say, 0 < t < T) and boundary

conditions are (given at each endpoint:

u(k)(0) = Uk 0

(k E a0),

u(k'(T) = uk T

(k E aT), (6.6.9)

where a0, aT are fixed subsets of (0,1,...,n - 1). (We may also have "mixed"

boundary conditions involving both extremes.) We seek conditions on the

Aj, a0, aT, which make the problem properly posed in the sense that ex-

istence, uniqueness, and continuous dependence on the boundary data and

on f hold. To give an idea of the difficulties involved, consider the following

6.6.3

Example. Let E = L2(0,77), Au = u" with domain D(A) consisting

of all u E E with u" E E and u(0) = u(7r) = 0. Any solution u(t)(x) = u(x, t)

of the abstract differential equation

u"(t) = Au(t) (6.6.10)

6.6. Miscellaneous Comments

379

can be written in the form

u(x,t)= E (ancosnt+bnsinnt)sinnx.

n=1

Assume we impose the initial and final conditions

u(x,0) = 0, u(x, T) = u(x) _ E cnsin nx E E. (6.6.11)

n=1

Then an = 0 and

bn=cn/sinnT (n>-l). (6.6.12)

If sin nT = 0 for some n (that is, if T/Ir is rational), condition (6.6.12) forces

cn to be zero, thus the subspace D of final data for which the problem has

solutions is not dense in E. On the other hand, if T/Ir is irrational, D

includes all u with finite Fourier series and is thus dense in E. However, in

order to obtain a "continuous dependence" estimate of the type

Ilu(t)II<cIIu(T)II

(0<t<T),

(6.6.13)

we must have I sin nT I >, e > 0 for all n >, 1, in other words

(nT) >, S>0 (n> 1),

(6.6.14)

where (nT) indicates the distance from nT to the nearest point of the form

kv. But it is an immediate consequence of Kronecker's theorem in number

theory (Hardy-Wright [1960: 1, p. 376]) that (6.6.14) never happens when

T/Ir is irrational, thus a bound of the type of (6.6.13) has no chance to exist.

Since the bound does hold for u E D when T/17 is rational, existence of

solutions to the boundary value problem for "sufficiently many" boundary

data and continuous dependence clash in an obvious sense. A sort of way

out can be attempted by replacing the inequality (6.6.13) by the weaker

version

Ilu(t)II<cII(Dx)ku(T)II

(0, t, T).

(6.6.15)

To obtain this, we only need to show that I sin nTI >_ en-k for all n >, 1 for

some e > 0, or, equivalently,

(nT) >, Sn-k

(n>_ 1)

(6.6.16)

for some S > 0. According to a theorem of Liouville (Hardy-Wright

[1960: 1, p. 161]), this is the case if T/ir is an algebraic number of degree

k + 1. The preceding considerations, which are particular cases of those in

Bourgin-Duffin [1939: 1] (and which can be easily adapted to the case

u(x,0) - 0) provide an answer to the problem of choosing T in order to

make the boundary value problem (6.6.10), (6.6.11) properly posed (al-

though the answer would probably not be appreciated very much by a

physicist or engineer, dealing with the very practical problem of "interpolat-

380

Some Improperly Posed Cauchy Problems

ing" the movement of a string, its position known at two times t = 0 and

t = T).

Although no necessary and sufficient conditions on the Aj(t ), T and

the sets a0, aT in order that the boundary value problem be well posed are

apparently known, even in the particular case where the Ai do not depend

on time, numerous results in both directions exist. Necessary conditions

regarding the number of boundary conditions and their distribution be-

tween the two endpoints were found by the author [1974: 1] for the equation

u(")(t) = Au(t)

(6.6.17)

using a modification of an idea of Radnitz [1970: 1] to obtain the resolvent

of A in terms of the solutions of (6.6.17), (6.6.9). The following references

deal mostly with sufficient conditions in order that the boundary value

problem (6.6.8), (6.6.9) be properly posed in various senses; most of the

material is on second-order equations with one boundary condition at each

endpoint, or with mixed boundary conditions involving both endpoints.

See the treatise of Krein [1967: 1] and Krein-Laptev [1962: 1], [1966: 1],

[1966: 2], Favini [1975:

21, Gorbacuk-Koeubei [1971:

1],

Gorbacuk-

Gorbacuk [1976: 1], Honig [1973: 11, Jakubov [1973: 1], Jurcuc [1974: 1],

[ 1976: 11, [1976: 2], Kislov [ 1972: 1], [1975: 1 ], Kutovoi [ 1976: 1 ], McNabb

[1972: 11, McNabb-Schumitzky [1973:

11, [1974:

11, Mihailec [1974: 11,

[ 1975: 11, Misnaevskii [ 1976: 11, Orudzev [ 1976: 11, Pavec [ 1971: 11, Romanko

[1976:

1], [1977:

1], Venni [1975: 11, Laptev [1966:

11, [1968:

11, J. M.

Cooper [ 1971: 1 ], and Karasik [ 1976: 1 ].

Chapter 7

The Abstract Cauchy Problem for

Time-Dependent Equations

Many deep and useful results for the time-dependent equation u'(t) =

A(t)u(t) can be obtained by extensions and modifications of the

theory pertaining to the equation u'(t) =A u(t). There are two main

avenues of approach. In the first (Sections 7.1 to 7.6) one assumes that

each A(t) belongs to the class d introduced in Chapter 4, with a

uniform bound on the norm of R(X; A(t)). The Cauchy problem is

then shown to be properly posed, the definition of solution depending

on the smoothness of the function t - A(t)-'. There are applications

to the time-dependent version of the parabolic equations considered

in Chapter 4.

In the second approach we only assume that each A(t)

belongs to C'+; the hypotheses on the function t -> A(t) make possible

the constructions of the solutions in "product integral" form, corre-

sponding to approximation of A(.) by a piecewise constant operator-

valued function. The natural application here is to time-dependent

symmetric hyperbolic equations; many of the auxiliary tools needed

were already developed in Section 5.6.

7.1. THE ABSTRACT CAUCHY PROBLEM FOR

TIME-DEPENDENT EQUATIONS

Let 0 < T < oo and (A(t ); 0 5 t < T) a family of operators in an arbitrary

Banach space E. Strong or genuine solutions of the equation

u'(t)=A(t)u(t)

(7.1.1)

381

382

The Abstract Cauchy Problem for the Time Dependent Equations

in any interval I contained in 0 < t < T are defined, as in the time-indepen-

dent case, to be continuously differentiable functions

such that u(t) E

D(A) and (7.1.1) is satisfied everywhere in I.

We shall say that the Cauchy problem for (7.1.1) is well posed (or

properly posed) in 0 < t < T if the following two assumptions are satisfied:

(a) (Existence of solutions for sufficiently many initial data) There

exists a dense subspace D of E such that for every s, 0 < s < T and

every u0 E D there exists a solution t -* u(t, s) of (7.1.1) in

s<t<Twith

u(s,s) = u0. (7.1.2)

(b) (Continuous dependence of solutions on their initial data) There

exists a strongly continuous (E)-valued function S(t, s) defined in

the triangle 0 < s < t < T such that if u(t, - ) is a solution of (7.1.1)

in s < t < T, we have

u(t, s) = S(t, s)u(s, s).

(7.1.3)

The operator S will be called the propagator (or evolution operator or solution

operator) of (7.1.1). Because of the uniform boundedness theorem, (b)

implies

IIS(t,s)II<C (0<s<t<T')

(7.1.4)

for any T' < T, so that the label given to (b) is justified. Note that solutions

depend continuously not only on their initial data, but also on the time s at

which these data are imposed. It is clear that when A(t) = A and T = oo, our

definition is equivalent to that in Section 1.2 for the time-independent

Cauchy problem. Just as in there, if 0 < s < T and u0 E E, we call the

function

u(t) = S(t, s)u0 (7.1.5)

a generalized solution of (7.1.1). We shall discuss later the relation among

generalized solutions of (7.1.1) and weak solutions (that is, solutions in the

sense of distributions).

Note that (b) implies that D is contained in D(A(s)) for any s in

[0, T), thus the set

n D(A(t))

(7.1.6)

O<t<T

must be dense in E. This imposition can be avoided if need be by requiring

the solution u( , s) in (a) to satisfy (7.1.1) only in 0 < s < T (but, naturally,

retaining continuity in 0 < s < T to give sense to the initial condition) or by

allowing D to depend on s, in which case D(s) c D(A(s)) and denseness of

each D(s) only implies denseness of the domain of each A(s), hardly an

extravagant assumption. Unlike in the time-independent case, the definition

7.1. The Abstract Cauchy Problem for Time-Dependent Equations

383

of solution of (7.1.1) is critical, and several different notions will have to be

used below.

The following two equalities are obtained in the same way as (2.1.2):

S(s,s)=I (0<s<T),

(7.1.7)

S(t,r)S(r,s)=S(t,s) (0<s<r<t<T).

(7.1.8)

We note finally that the notion of properly posed Cauchy problem

can be formulated equally well in intervals other than [0, T).

The case where the operators A(t) are everywhere defined and

depend continuously of t in one way or another is of scarce interest to us for

the reasons already pointed out in the time-independent case in Section 1.2.

However, the results are considerably less trivial when A(t) is not a constant

function and we include one of them, which is an infinite-dimensional

extension of a classical theorem on existence and uniqueness of solutions of

systems of ordinary differential equations.

7.1.1 Theorem.

Let A(t) be bounded and everywhere defined in

0 < t < T; moreover, let t - A(t) E (E) be strongly continuous in 0 < t < T.

Then the Cauchy problem for (7.1.1) is properly posed in 0 < t < T and the

function (s, t) -* S(t, s) E (E) is continuous in 0 < s < t <T.

Proof.

The proof is a standard application of the successive ap-

proximation method. We begin by observing that if u0 E E, the differential

equation (7.1.1) and the initial condition (7.1.2) are equivalent to the single

integral equation[

u(t, S) = uo + f 'A(T)u(T, s) dT. (7.1.9)

We take u0(t) ° u0 as initial approximation and define the following ones

recursively by

un(t,s)=u0+ f'A(T)un-t(T,s)dT,

(n=1,2,...). (7.1.10)

By the uniform boundedness theorem, if T' < T

IIA(t)II<C

(0<t<T'),

(7.1.11)

which yields, via a simple inductive argument applied to (7.1.10), the

If v(-) is a continuous E-valued function, A v

is continuous. This is a simple

consequence of the strong continuity of A(.) and of the bound (7.1.1 1). This will be used many

times in the sequel.

384

The Abstract Cauchy Problem for the Time Dependent Equations

sequence of inequalities

IIun(t s)- un-1(i s)II <

Cn(n, S)nIluoll

(n=1,2,...) (7.1.12)

in s < t < T'. Accordingly, the series Ell un (t, s) - un _ 1(t, s )I I converges (and

thus u(t, s) = lim un(t, s) exists) uniformly on 0 < s < t < T'. Taking limits

in (7.1.10), we see that u(

, s)

is a solution of (7.1.9) and then a solution of

the initial-value problem (7.1.1), (7.1.2).

To prove uniqueness of u(t, s) and continuity of the solution opera-

tor, we make use of the following result (Gronwall's lemma), which we state

in a version more general than necessary now, with a view to future use.

7.1.2 Lemma.

Let /3 be locally integrable and nonnegative in t >-a.

Assume the nonnegative continuous function q(t) satisfies

i(t)<a+ f`fO(s)rt(s)ds

(7.1.13)

in t >_ a, where a >, 0. Then

in tea.

fi(t)<aexp(f'/3(s)ds) (7.1.14)

Proof. We assume first that a > 0. It follows from (7.1.13) that

$(t)rl (t)(a+ f r$(s)n(s)

ds)

<$(t)

(t >,a).

Integrating from a to t gives

log(a+ f'#(s),q(s) ds)-loga< f 1$(s)

ds,

whence (7.1.14) follows using again (7.1.13). The case a = 0 is dealt with

taking a -* 0+.

We go back to the uniqueness problem. Let u(t, s) be a solution of

(7.1.9) in s < t < T'. If we set ra(t) = II u(t, s)II, we have

(t)<Iluoll+C

f'n(s)ds,

(7.1.15)

thus it follows from Lemma 7.1.2 that

Il u(t, s)II < ec('-s)lluoll

(s < t <T'),

(7.1.16)

which shows that u(t, s) = 0 if uo = 0. Since T' < T is arbitrary, existence

and uniqueness of the solution is established in s < t < T for any s < T. We

note, incidentally, that the arguments leading to existence and uniqueness of

the solutions of (7.1.1) work just as well for t < s as for t >_ s so that

solutions actually exist in 0 < t < T' (i.e., the Cauchy problem can be solved

7. I. The Abstract Cauchy Problem for Time-Dependent Equations

385

not only forward but also backwards). Hence the propagator S(t, s) can be

defined in the square 0 < s, t < T rather than only in the triangle 0 < s < t < T

and satisfies in each square 0 < s < t < T' an estimate of the form

IIS(t,s)II

<eCIt-SI.

(7.1.17)

We prove finally that S is continuous in the sense of the norm of (E). To

this end, we note that

IIS(t', s') - S(t, s)II < IIS(t', s')- S(t, s')II+IIS(t, s') - S(t, s)II

(7.1.18)

and proceed to estimate the two terms on the right for 0 < s < s'< T' and

0 < t < t'< T', where T' < T. In view of (7.1.11) and (7.1.17),

II(S(t',s')-S(t,s'))uII < f tIIA(T)IIIIS(T,s')uIIdT

<C(t'-t)eCT IIuII

(7.1.19)

for any u E E. The second term on the right-hand side of (7.1.18) is also

estimated by means of (7.1.11) and (7.1.17). If u E E, we have

II(S(t,s')-S(t,s))uII <

IIA(T)II1IS('r,s)ullds

f'IIA(T)IIII(S(T,S')-S(T,s))uIIdT

<C(s'-s)eCT CTf tII(S(T,S')-S(T,S))uII dS.

Applying Lemma 7.1.2 to rl(t) = II(S(t, s')- S(t, s))uII, we obtain

II(S(t,s')-S(t,s))uII <C(s'-s)e2CT'IIuII

(7.1.20)

for any u E E, which ends the proof of Theorem 7.1.1.

Motivated by some of the results above (see the observations follow-

ing (7.1.16)), we introduce a definition that will also find application in

Section 7.7 and following sections. We say that the Cauchy problem for

(7.1.1) is well posed (or properly posed) forward and backwards, or in both

senses of time if the solutions required in (a) actually exist in 0 < t < T,

rather than only in s < t < T and if the operator S(t, s) exists and is strongly

continuous not only in the triangle 0 < s < t < T but in the square

0 < s, t < T. This notion can be formulated equally well for an operator-val-

ued function A(.) defined in an arbitrary interval I, not necessarily of the

form [0, T) (say [0, T], (- T, T), etc.). It is clear that this definition

generalizes to the time-dependent case that of well-posed Cauchy problem

in (- oo, oo) (see Section 1.5).

When the Cauchy problem is well posed in both senses of time in

some interval I, equality (7.1.8) holds in the square s, t E I and it follows

386

The Abstract Cauchy Problem for the Time Dependent Equations

from it and from (7.1.7) that

S(t, s)S(s, t) = S(t, t) = I = S(s, S) _

S(s, t)S(t, s), or

S(s,t)-t=S(t,s) (s,tEl). (7.1.21)

Each operator S(t, s) is invertible.

7.1.3 Example. Assume the hypotheses on A(.) are those in Theorem

7.1.1. We look now for a (E)-valued function 9(t, s), solution of the

integral equation.

S(t,s)u=u- ftS(t,a)A(a)udr (uEE) (7.1.22)

in the square 0 < s, t < T. Existence and uniqueness of S(t, s) are proved by

means of an iteration scheme similar to the one used in (7.1.9). The function

S(t, s)u is continuously differentiable in s for all u,

D,S(t,s)u=-S(t,s)A(s)u (0<s<T),

S(t,t)u=u,

(7.1.23)

and S satisfies an estimate of the type of (7.1.17). Take u E E and define

q9 (a) = S(t, a)S(a, S) U.

Then

DgS(t, a)S(a, s)u + S(t, a)DQS(a, s)u

S(t, a)A(a)S(a, s) u + S(t, a)A(a)S(a, s) u = 0

in 0 < a < T, so that p is constant; hence S(t, s)u = q9 (s) = q) (t) = S(t, s)u,

which shows that

S(t, S) = S(t, s). (7.1.24)

We obtain in this way nontrivial information on S as a function of s.

Similar tricks will work in other situations (see Section 7.2).

7.1.4

Example.

It was observed in Section 1.5 that in the time-indepen-

dent case A(t) = A E (E), we have

IIS(t,s)-III =1IS(t-s)-III ->0

as t - s - 0. A similar result holds if we place ourselves under the hypothe-

ses of Theorem 7.1.1. In fact, it follows immediately from (7.1.19) that

IIS(t,s)-III'< Cec'°It

- sl (7.1.25)

for 0 < s, t < T. It does not seem to be known whether S(t - s) -* I in (E)

as t - s - 0 must necessarily imply that the A(t) are bounded in the general

case; when A does not depend on time, the corresponding result is Lemma

2.3.5.

Many variants of Theorem 7.1.1 can be obtained under the basic

assumption of boundedness of A(t) by simply modifying the requirements