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

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7.10. Miscellaneous Comments

457

results are already in [1962: 1]. The applications in Section 7.6 are rather

particular cases of results sketched in Kato-Tanabe [1967:

1]; see also

Tanabe [1979: 11. The material in Sections 7.7, 7.8, and 7.9 is due to Kato

[1970: 11; some of the results in Section 7.9 were discovered earlier by Kato

[1953: 1], [1956: 1].

It is of interest to trace the evolution of the results in evolution

equations leading to the present exposition. The first theorem on the

equation (7.1.1), where the A (t) are permitted to depend on t, is due to Kato

[1953: 1]; we note in passing a result of Phillips [1953: 1] corresponding to

the case A(t) = A + P(t) with A E (2+ and P (t) strongly continuous and

taking values in (E): the propagator is constructed by means of a perturba-

tion series of the type of (7.1.42). Kato's result, on the other hand, is a

forerunner of Theorems 7.7.5 and 7.8.3 in the sense that the propagator

S(t, s) is obtained in the same "product integral" form. However, the

hypotheses on are different; the domain D(A(t )) is assumed to be

independent of t, each A(t) belongs to (2+(1, 0) and smoothness hypotheses

are placed on the ((E)-valued) function B(t, s) = (1- A(t))(I - A(s))-1.

The assumptions in Kato [1956: 1] are of the same type, but more general.

Other early works are due to Ladyzenskaya [1955: 11, [1956: 11, [1958: 1],

Visik [1956: 1], Vi§ik-Ladizenskaya [1956: 1], although these refer to special

kinds of operators A(t) such as self-adjoint operators and perturbations

thereof and will be commented on in (b). The same observation applies to

the works of Sobolevskii [1957: 1], [1957: 2], and [1958: 2]. In this last paper

the operators A(t) are self-adjoint and an effort is made to relax the

hypothesis, prevalent until then, that D(A(t)) must be independent of t.

Sobolevskii assumes instead that D(A( t )h) is constant for certain h > 0 and

makes use of the integral equations (7.2.4) and (7.2.6) in the construction of

the propagator. This method was taken up by Kato [1961: 1] for general

A(t) in C?; as shown there, the hypothesis of constancy of D(A(t)h) is

(somewhat surprisingly) verified in some important cases where D(A(t))

varies with time. These results of Kato generalize earlier theorems of Tanabe

[1959: 1], [1960: 1], [1960: 2], [1961: 1], where D(A(t)) is assumed constant.

Tanabe introduced the construction of the propagator through the integral

equations (7.2.8) and (7.2.48). The same method was then carried a big step

forward by Kato and Tanabe in [1962: 1], where essentially no relations are

postulated between the different D(A(t)), as Example 7.3.2 puts in evidence

in a rather extreme way. We note that independence on t of the domain of

A(t) is truly essential in treating initial-boundary value problems where the

boundary conditions depend on t, or even operators like d (t) in Section 7.8,

where D(d(t)) is not easily identified (and may be suspected of changing

with time).

As pointed out before, some of the ideas leading to the results in

Section 7.7 and following sections can be traced back to Kato [1953: 1].

Other basic ingredients in the proof of Theorems 7.7.5 and 7.8.3 such as the

458

The Abstract Cauchy Problem for Time-Dependent Equations

introduction of the space F, and the requirement of (F; E)-smooth depen-

dence can be found in Heyn [ 1962: 1 ] and Kisynski [ 1966: 11; the use of E-

and F-stable families appears for the first time in Heyn's paper, while

several results in the style of Example 7.7.3 were given by Kisynski. A

similar approach to the equation (7.7.1), where many of the basic ideas in

Kato [1970: 1] are already in evidence, is Goldstein [1969: 4]. We point out

finally that the operators K(t) make their apparition in Kato [1956: 11 and

that the proof of Theorem 7.8.3 follows Yosida [1965/66: 1] (see also [1978:

1]). Certain simplifications to the various arguments have been proposed by

Dorroh [1975: 1]. We note that in the work of Heyn and (partly) in that of

Kisyiiski mentioned above, the approach to the equation (7.7.1) is through

the Yosida approximation (Example 2.3.13), that is, the solution u(.) is

constructed as lim u,(-), where un

is the solution of

u;, (t) = nA(t)R(n;

A similar approach was used by Elliott [1962:

1] and Yosida [1963:

1],

[1965/66: 11 the latter working in locally convex spaces.

A far-reaching generalization of the ideas in Sections 7.7 and 7.8 is

due to Kato [1973: 1] (see Examples 7.9.5 and 7.9.6) roughly along the lines

of [1970: 11, but relaxing in an essential way many of the hypotheses.

Numerous other approaches have been used in relation to the

equation (7.7.1), most of them starting with the assumption that each

A(t) belongs to (2+ or to a subclass thereof but making different require-

ments on t-dependence of A(t); the following is a list of references. See

Baumgartel [ 1964: 11, [1964: 2], Bragg [ 1974: 11, Domslak [ 1961: 1], [1962:

11, Dyment-Sobolevskii [ 1970: 11, Enikeeva [ 1972: 1 ], Fitzgibbon [ 1976: 1 ],

Foias-Gussi-Poenaru [1957: 1], Fujie-Tanabe [1973: 1], Gajewski [1971: 1],

Gibson-Clark [ 1977: 1 ], Globevnik [ 1971: 11, Goldstein [ 1972: 11, [1974: 11,

[1975/76: 11, Hackman [1968: 1], Inoue [1972: 1], [1974: 1], V. N. Ivanov

[ 1966: 11, Jakubov [ 1964: 11, [1964: 2], [1966: 11, [1966: 2], [1966: 3], [1966:

4], [1966: 5], [1967: 4], [1968: 11, [1970: 1], [1970: 2], [1973: 11, Jakubov-

Samedova [1975: 1],

Kartsatos-Zigler [1976:

1], Krasnosel'skii-Krein-

Sobolevskii [ 1956: 1], [1957: 1 ], Kohnen [ 1970: 11 [1971: 1], [1972: 11, Kree

[1970: 11, Krein [1957:

11, [1966:

11, [1967:

1], Krein-Daleckii [1970:

1],

Krein-Laptev [1969:

11, Krein-Sav&nko [1972:

1], Lopatinskii [1966:

1],

Lovelady [1975: 2], Mamedov [1964:

11, [1964: 2], [1965:

11, [1966:

1],

Massey [1972: 1], Masuda [1967: 11, [1967: 2], [1968: 1], Mlak [1959: 11,

[ 1960: 11, [1960/6 1:

11, Murakami [ 1966: 11, Nemyckii-Vainberg-Gusarova

[1964:

1], Oja [1974: 1], Poulsen [1965:

1],

Raskin-Sobolevskii [1967:

1], [1968:

1], [1968: 2], Rota [1963:

11, Sadarly [1969:

1], Sarymsakov-

Murtazaev [1970:

1], Sohr [1973:

1],

[1973:

2], [1973:

3],

Solomjak

[1960: 1], Strygina-Zabreiko [1971:

1], Suryanarayana [1965: 1], Tanabe-

Watanabe [1966: 1], Tsokos-Rama Mohana Rao [1969:

1], Watanabe

[1967: 11, [1968: 1], Plju§cev [1973: 1], [1975: 11, Yagi [1976: 1], Da Prato-

7.10. Miscellaneous Comments

459

lannelli [1976: 1], Datko [1972: 1], Derguzov [1963:

11, [1964:

2], [1964: 3], [1973: 1], Derguzov-Jakubovic [1963:

11,

[1969:

11,

Derguzov-Melanina-Timcenko [1975:

1], and Djacenko [1976: 1]. For a

thorough treatment of the subject and additional references, see Tanabe

[1979: 1].

(a) Propagators.

It was proved in Chapter 2 that there is a com-

plete equivalence between propagators of time invariant equations and

strongly continuous semigroups, that is, strongly continuous operator func-

tions that satisfy the two equations (2.3.1). The equivalence breaks down in

the time-varying case: although the propagator of the time-dependent

equation (7.1.1) is strongly continuous and satisfies (7.1.7), (7.1.8), not every

strongly continuous (E )-valued function satisfying these equations is the

propagator of a differential equation. For a trivial counterexample (with

dim E = 1), it suffices to consider

S(t, s) = a(t)/a(s),

where a is a continuous, positive, nowhere differentiable function. The same

counterexample shows that nothing is gained by requiring (E)-continuity of

S. Necessary and sufficient conditions on a (E)-valued function S satisfying

(7.1.7) and (7.1.8) in order that S be the propagator of a properly posed

Cauchy problem are apparently not known; likewise, the class of operator-

valued functions A(.) giving rise to a properly posed Cauchy problem for

(7.1.1) has never been fully characterized. For results in these and related

areas see Masuda [1972: 11 (Example 7.4.4), Y. Komura [1970: 1], Lovelady

[ 1975: 2], Pljul6ev [ 1975: 11, Herod [ 1971: 2], and Lezanski [ 1970: 1], [1974:

1]. See also Carroll [1975: 1] and references therein for very general relations

between the equation (7.1.1) and the variation-of-constants formula (7.1.34).

(b) Operational Differential Equations. The equation (7.1.1) has

been extensively investigated using tools on the whole different from

semigroup theory. To this category belong some of the early works by

Vilik-Ladyzenskaya, Ladyzenskaya, Sobolevskii, and others, quoted at the

beginning of this section. A systematic treatment was undertaken by Lions

starting with [1958: 1], [1958: 2], [1959: 1] and culminating in this treatise

[1961: 1]. Some salient features of Lions' theory can be roughly described as

follows. Let V, H be Hilbert spaces with V C H, V dense in H and the

injection i : V -* H continuous. The operators A(t) are defined by time-

dependent bounded sesquilinear forms a(t; u, v) in V X H, that is,

(A(t)u, v) = a(t; u, v).

In more general situations V may depend on t. The solutions under

consideration range from strong or genuine solutions in the sense of Section

7.1 to weak solutions as those in Section 7.5 to solutions in the sense of

vector-valued distributions. The methods used in the construction of the

460

The Abstract Cauchy Problem for Time-Dependent Equations

solutions are, on the one hand, duality arguments combined with a priori

estimates (such as versions of the Lax-Milgram lemma) and, on the other

hand, approximation methods (for instance, based on finite difference

approximations of (7.1.1)) also used in combination with a priori estimates.

A notable way to construct approximate solutions of (7.1.1) is the so-called

Faedo-Galerkin method, an abstract adaptation of the classical Rayleigh-Ritz

approximation procedure. It can be roughly exemplified as follows. Choose

a linearly independent sequence w1, w2, ...

in V such that finite linear

combinations of the (wk) are dense in V; take uo = u(O) in V and select

coefficients rylnk such that

U. = k 1jnkWk' u0

k=1

Solve then the ordinary differential system

n n

E nn'k(t)(Wj,wk)= L 11nk(t)a(t,wj,wk)

(lam j <n)

k=1 k=1

with initial conditions

'ljk(0) = rIjk

and take

un(t)

= 'qnk(t)Wk

k=1

as an approximation of u(t).

For an extensive bibliography of early work in this direction,

see Lions [1961: 1]; for other applications and additional references, see

Lions [1969: 1], Lions-Magenes [1968: 11, [1968: 2], [1970: 1], and Carroll

[1969: 2].

Chapter 8

The Cauchy Problem in the Sense of

Vector-Valued Distributions

Although the Cauchy problem for certain differential equations and

systems is not well posed in the sense of Chapter 1, it fits into the

mold of mildly well posed problems (Section 8.6) where the solution of

the abstract differential equation u'(t) = Au(t) depends continuously

on u(O) if the initial condition is measured in the graph norm of a

suitable power of A. This concept is closely related to that of properly

posed problem in the sense of Hadamard and turns out to be

equivalent to a formulation of the Cauchy problem in the sense of

distributions for the nonhomogeneous equation u'( t) = Au(t)+ f(t)

studied in Sections 8.4 and 8.5. The fact that the equation under

consideration is a differential equation plays no role here, thus the

theory refers to convolution equations.

The rest of the chapter covers preparatory material on vector-

valued distributions.

8.1.

VECTOR-VALUED DISTRIBUTIONS. SUPPORTS,

CONVERGENCE, STRUCTURE RESULTS

Let 0 be an open interval in R, 6 (St) the space of test functions in Q. A

sequence

is convergent to zero in 6 (Q) if and only if:

(a) There exists a compact set K C SZ such that

9 K for all n.

(8.1.1)

(b) q),(,m)(x) -> 0 uniformly on R for all m > 0.

461

462

The Cauchy Problem in the Sense of Vector-Valued Distributions

More generally, (p,) converges to 99 E 6 (S2) if and only if qp - T - 0

according to the previous definition.

Given a Banach space E the space 6'(12; E) of E-valued distributions

defined in 0 consists of all linear operators U, V... from 6 (S2) into E that

are continuous in the following fashion:

if %, - 0

in 6 (S2), then 0

in E, (8.1.2)

where is a sequence in 6D(0).' Clearly 6 '(S2; E) is a linear space. Every

piecewise continuous (or, more generally, locally integrable) function

f defined in 0 with values in E can be identified with a distribution

U E 6 '(S2, E) through the formula

U(p)= f(p(t)f(t)dt

(9pE6l(S2)) (8.1.3)

and two functions f, g give rise to the same distribution if and only if they

coincide almost everywhere (if f and g are piecewise continuous, this means

f = g except perhaps at discontinuity points).

We introduce some notational conventions. When SZ = R, we write

6D(0) = 6D,

as agreed in Section 7. We also write 6'(68; E) = 6D'(E),

6i '(S2,C) = 6'(12), 6 '(C) = 6D,. If f : Sl -* E, we indicate the function f (or

the distribution in 6 '(S2; E) it defines) by f(i); f(t) denotes instead the

value off at t. Distributions will be often written in "functional" notation;

for instance, the Dirac measure 8 E 6D' will be denoted by S(t) while, say,

8(t - 3) indicates the Dirac measure centered at t = 3, that is, the distribu-

tion 99 - qp(3) in 6D'. Likewise we shall abuse sometimes the notation writing

U(q)) as Jq)(t) U(t) dt, where U E 6 '(S2; E) and p e @(U). Finally, given

U E 6'(12) and u e E, we indicate by U®u the distribution in 6D '(U; E)

defined by

(U®u)(p)=U(p)u. (8.1.4)

As in the scalar case, the derivative of a distribution U E 6 '(S2; E) is

defined by

U'(p) = - U(p')

(8.1.5)

'The definition of the topology of 6D is not complete, since it cannot be entirely

characterized by sequences; there are nets ((pp) converging to zero in 6i which do not satisfy

(8.1.1). The correct topology of 61. is the inductive limit topology of (say) the sequence 6D,,,

where 6i is the (Frechet) space of all q' E 6D with support in I t I < n endowed with the topology

of uniform convergence of all derivatives; the inductive limit topology is the finest (that is, the

largest) one making all injections i : 6D -> continuous. A fundamental property of this

inductive limit topology is that an operator U: 6 - E is continuous if and only if all its

restrictions U. to 6i are continuous. It follows then that our definition of continuity (8.1.2) is

the correct definition of continuity with respect to the topology of 6D; moreover, since each I1

is a Frechet (thus a metric) space, we may well use a sequence ((p,,) in (8.1.1) and (8.1.2). The

same observations apply to arbitrary Q.

8. I. Vector-Valued Distributions. Supports, Convergence, Structure Results

463

and behaves in an entirely similar way. When f is a continuous, piecewise

continuously differentiable function with values in E, its ordinary derivative

defines the same distribution as the one obtained differentiating f as a

distribution.

8.1.1 Lemma. Let U E 6 '(SZ; E) satisfy

U(m) = 0

for some m >, 0. Then U coincides with a polynomial

m-1

P(t) = Y_ tiuj (t E 0), (8.1.6)

i=o

where uo,...,um_ I E E.

Proof We consider first the case m = 1. Since U' = 0, it follows that

U((p') = 0 for all q? E 6D(2) or, equivalently, that U(q)) = 0 if Jqq dt = 0. Let

quo be any function in 6D (0) with nonzero integral 71= Jq)o dt. If J E 61 (2),

we write qq = - X %o, A = (1/'q)fJ dt. Then Jw dt = 0 and U(O) = XU(g)o)

= (1/'q)U(q)o)fO dt, showing that U= u = (1/,q)U(4po). If m > 1, we obtain

using the previous result that U(m-') = u for some u E E from which (8.1.6)

follows by repeated integration.

We define convergence in the space 6D'(2; E). In order to do this, we

introduce first an auxiliary notion. A set Yc 6D(2) is bounded if for every

sequence

c `

and every sequence of real numbers with e - 0 we

have

0 in 6D (0). (8.1.7)

It is immediate to verify that a set `Y is bounded if and only if:

(a) There exists a compact set K C St such that

supp(cp) c K (qp E `y).

(8.1.8)

(b) For every m >, 0 there exists a constant Cm < oo with

IvW(m'(t)I <Cm (q)E=- X,tEK).

(8.1.9)

A sequence (or generalized sequence) {Ua} of distributions in 6 '(SZ; E)

is said to converge to a U E 6 '(SZ; E) if and only if

U«(T)-U((P)

uniformly on bounded subsets of 6D.

It is easy to see that if { fa} is a generalized sequence of piecewise

continuous (or, more generally, locally integrable) functions such that fa - f

pointwise and I I fa(t )I I < g(t), where g is locally integrable, then fa - fin the

sense of distributions.

A distribution U E G1'(S2; E) is said to vanish in an open subset 0' of

SZ (in symbols, U = 0 in St') if

UkO = 0 (8.1.10)

464

The Cauchy Problem in the Sense of Vector-Valued Distributions

whenever supp(q)) c 2'. Denote by SZ(U) the union of all the open subsets

of 2 where U vanishes. We define the support of U (in symbols supp(U)) as

follows:

supp(U) = 9\0(U).

It is clear that supp(U) is closed in Q. We may easily verify that the present

definition of support coincides with the usual one when U is a smooth

function. The following result shows that the property that U = 0 can be

"pieced together."

8.1.2 Lemma U=0 in 9 \supp(U).

Proof.

A local compactness argument easily shows that there exist

two countable families of open intervals such that (i) I c J,,. (ii)

U I = 2 \supp(U). (iii) J n J n J,,, = fd if n < m - 1. (v)

U= 0 in J,,. Let %, be a nonnegative function in 6D (0) with support in J

which equals I in I. The sum I(t) = Eggn(t) (where all the terms, except for

at most two, vanish for each t) is positive in Q. Then, if 4, = T,,/I, it is

clear that ,, E 6D(52), c J,,, and E (t) =1 (t E S2(U)). Hence if

qp E 6D (2) and supp T c 9\supp(U),

U(T)=U(wY n)=Y. U(T J=0

(8.1.11)

since supp(p ) c J and U vanishes in J,,. Note that the introduction of U

into the series is justified by the fact that the vanish identically except

for a finite number.

J _ I J

FIGURE 7.1.1

In the case SZ = R, we shall make use of several subspaces of

6D'(2; E) = 6 '(E ). We denote by P(E) (resp. 6D (E )) the set of all ele-

ments of 6 D'(E) with compact support (resp. with support in t > 0). We shall

also write too(E) = F'(E)n 6 o(E) and denote by 6D'(E) the subspace of all

U E 6D'(E) with supp(U) bounded below. Use will be made of 6D0 '(R; E)

(distributions in 6D'(2; E) with support in t > 0) and of the similarly defined

&'(2; E).

The results in the remainder of this section will show that, at least

locally, a E-valued distribution U is nothing more than the derivative (in the

sense of distributions) of a continuous function, which can be chosen to

8.1. Vector-Valued Distributions. Supports, Convergence, Structure Results

465

vanish in t < a if U does. Moreover, this correspondence between functions

and distributions can be established in such a way that boundedness in

6D'(2; E) translates into local boundedness of the corresponding sequence

of continuous functions.

8.1.3

Lemma. Let U E 6 '(SZ; E), K a compact subset of 2. Then

there exists an integer p , 0 and a constant C > 0 such that

IIU(ro)II < C11THP

(8.1.12)

for every *p E 6D with support in K, where

IIPIIp =

sup sup

Imv1(t)1.

(8.1.13)

0_<j _<p -oo<t<oo

Proof

Assume (8.1.12) does not hold for any p. Then there exists a

sequence (q)n) c 6D such that

supp(Tn) c K (8.1.14)

and

IIU(mn)II > nII9'nlln

(n ,1).

(8.1.15)

Let n = %, /n II9'n I I

Clearly

I I to I I

n =

n -' - 0 as n , oo, which, together

with (8.1.14), implies that

` ,,

- 0 in 6D. But it follows from (8.1.15) that

1, which is a contradiction. This ends the proof.

Let K be, as before, a compact set. We denote by 6D p(K) the space of

all p times continuously differentiable functions defined in R with support

in K endowed with the norm (8.1.13). It is clear that 6 p(K) is a Banach

space.

8.1.4 Lemma.

Let K be a compact subset of Q. Then, given T E

6D(K) and 8 > 0, there exists t/ E 6D(2) such that

11T-PIIp<8,

where II'Iip indicates the norm in 6 p(K6), K8 = (t; dist (t, K) < 8).

The proof is a standard application of mollifiers (Section 8) and is

thus omitted.

8.1.5 Structure Theorem.

Let U E 6D'(2; E), 9' an open bounded

subset of SZ with D'(: St. Then there exists a continuous function f : R -* E, and

an integer m , 0 such that

U= f (n') in St' (8.1.16)

(the derivative understood in the sense of distributions).

Proof.

Let K = S2' and let e > 0 be such that KE = (t; dist(t, K) < E)

is contained in 2. Applying Lemma 8.1.4 to KF (see (8.1.12)), we deduce that

U can be extended by continuity to all elements of 6 p(KE) and that the

466

The Cauchy Problem in the Sense of Vector-Valued Distributions

extension (which we denote by the same symbol) satisfies

IIU(T)II < CllTllp

(T (=-ODP(KE)),

(8.1.17)

where 11-11p indicates the norm of 6D (KE). Loosely speaking, this partial

result states that "a distribution can be applied to a sufficiently differentia-

ble function." Define now

1(t)=

tp+'/(p+1)!

(t%0)

(t < 0).

(8.1.18)

Clearly n is p times continuously differentiable. Then if X E 6D and supp X c

K, the function X(9)71(t - s) belongs to 6D( K£) for each t and, moreover,

the map t - X(s)rl(t - s) from R into 6 p(KF) is continuous (as can be

easily verified by repeated differentiation). Hence, the function

f(t)=U(X(s)q(t-s))

(8.1.19)

from O into E is as well continuous. We observe next that if qq E 6D,

where tji is the element of 6D defined by

(s) = X(s) f n(t - s)tp(t) dt

=X(s) fn(t)q)(s+t)dt

(this can be easily proved approximating the integrals involved by Riemann

sums). Take now X with X(t) =1 for t e K and cp E GD with support in 2'.

Let m = p + 2. Then

f (m) (T) _

(-1) m fwI"')

(t)f(t) dt

where

m(S)=X(S) f ' (t-S)T(m)(t)dt

_ (-1)map(s)

since qp(s) = 0 outside of K. This ends the proof.

As the classical example

(8.1.20)

U= E S(") (t - n)

n=0