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

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7.3. Abstract Parabolic Equations: Weak Solutions 407

where Sh is the operator defined by (7.2.35) and (7.2.36), so that (7.3.6) is

justified by Lemma 7.2.6(i). On the other hand,

<Sh(t,s)u,u*'(t))dt=-(Sh(s+k,s)u,u*(s+k))

-fT

<D,Sh(t,S)u-A(t)Sh(t,s)u,a*(t))dt

-fT

(Sh(t,s)u,A(t)*u*(t))dt (ash-O+)

s+k

- (S(s + k, s)u, u*(s + k))

_fT

(S(t,s)u,A(t)*u*(t))dt (ask-0)

+ k

- (u, u*(s)) - f T(u(t), A(t)*u*(t)) dt, (7.3.7)

which shows that (7.3.3) holds for u(-); note that in the first limit we have

used (7.2.38). We must now prove uniqueness of weak solutions and to this

end we bring into play the operator Sh defined in (7.2.51). Let u(-) be a

weak solution of (7.3.1) in s < t < T, and let s < t'< T,

a continuously

differentiable function with values in E* and support in (s, t'). Since

Sh(t, s)A(s) is a bounded operator for s + h < t (Lemma 7.2.11 (ii)) we see

(Section 4) that

Sh(t, s)*E* c D(A(s)*)

and

A(s)*Sh(t, s)* = (Sh(t, s)A(s))* = (Sh(t, s)A(s)

is continuous in the norm of E* in the same region. On the other hand, also

by Lemma 7.2.11 (ii), DS Sh (t, s) exists and is continuous in the norm of (E)

in s + h < t, thus DsSh(t, s)* exists and is continuous in the norm of (E*)

for these values of s and t. It follows that if h (is sufficiently small,

uh(t)-Sh(t',t)*I(t)

will satisfy the requirements in Definition (B). Accordingly, we have

ft,

(S(t', t)u(t),''(t)) d t = lim

f ` (Sh(t', t)u(t),V(t)) dt

h0+ s

= lim

` (u(t), Sht', t)*41'(t)) dt

h0+

1lim

ft (u (t), uh'(t)+A(t)*uh(t)) dt

-O+ S

- lim

f " ((DSh(t', t) + Sh(t', t)A(t) )u(t),1(t))dt.

h0+ s

(7.3.8)

408

The Abstract Cauchy Problem for the Time Dependent Equations

The first term on the right-hand side vanishes since u is a weak solution of

(7.3.1) and uh(s) = 0; the second limit is seen to be zero taking (7.2.53) into

account (note that 4, vanishes near t'). Taking now 'Y(s) = q>(s)u* in

(7.3.8), u* an arbitrary element of E* and p a smooth scalar function with

support in (s, t'), we obtain

jt <S(t', t)u(t), u*)cp'(t) dt = 0, (7.3.9)

thus (S(t', t)u(t), u*) (a fortiori S(t', t)u(t)) must be constant. Hence

S(t', s)u(s) = S(t', t')u(t') = u(t'), and we obtain essentially as in the end

of Section 7.2, that weak solutions are unique and that S = S. The proof of

Theorem 7.3.1 is then complete.

It is natural to ask whether the assumptions in this or in the previous

section imply that the intersection (7.1.16) is dense in E. This is in fact not

so, as the following example shows.

7.3.2 Example. There exists an operator-valued function t - A(t) de-

fined in 0 < t < 1 in E = L2(0,1) such that hypotheses (I) to (IV) in Section

7.2 are satisfied, but

D(A(t)) = (0). (7.3.10)

06t<t

The operators are defined by

A 1 7 3 11(t)u(x) u(x), ( . . )

x)z

-(t

D(A(t)) consisting of all u such that (7.3.11) belongs to

L2.

Clearly each

A(t) is self-adjoint and (A(t)u, u) < -I1u112 for u E D(A(t)) so that a(A(t))

is contained in (- oo, - 1] for all t; (I) is satisfied for any T E (0, v/2). If

X (-00,-1],wehave

(t -

x)z

R X A 7 12

( ;

(t))u(x) =

u(x) ( . )

and

A(t-x)z+l

2(t - x)

D R X A 7 13

D ( ; (t))u(x)= u(x).

( .

. )

(t-x)2+1)

A simple estimation shows that 2 a l Xaz + 11 -z < C I X I -' /z in any sector

2((p +17/2) for 0 <a<1, so that

IID1R(X;A(t))II

<C1x1-1/2

and (III) holds. As a particular case of (7.3.12), we see that

A(t)-'u(x) = -(t - x)zu(x)

so that A(t)-' _ -(tI - M)2, M the (bounded) operator of multiplication

7.4. Abstract Parabolic Equations: The Analytic Case 409

by x and (II) and (IV) are amply satisfied. Let u E L2 be an element of

n D(A(t )). Then for every t, 0 < t < 1, we have

/'t+hIu(x)I

1-h

1+h lu(x)I42

dx1I/ll

lfr+h(t-x)4dx11/2-0

h (t - x)\ 4h

t-h

as h -+ 0, thus u(x) = 0 at every Lebesgue point. It follows that u(-)

vanishes almost everywhere.

We note that, in order to prove that the Cauchy problem for (7.3.1)

is properly posed for the family A(-) in this example, it is not necessary to

use the awesome machinery of Theorem 7.2.5: in a suitable sense, (7.3.1) is

in this case equivalent to the ordinary differential equation

u (t>_ s), (7.3.14)

(t - x)2

and S(t, s) is given by the formula

S(t,s)u(x)=

s-x

)u(x)

exp

(

t-x

(x<sorx>t)

(7.3.15)

0 (s<x<t).

The example above may be used to illustrate why the obvious

generalization of the definition of weak solution used in Section 2.4 for

time-independent equations would be inadequate in the present case: in

fact, the generalization of (2.4.8) would read as follows:

(B') For every u* E n D(A(t)*) and every test function q, with support

in (- oo, T], we have

fT((u(t), a*)q)'(t)+(u(t), A(t)*u*)(p (t)) dt

= - (u0, u*)qv(s). (7.3.16)

However, in the case considered in Example 7.3.2 (where E* = E and

A(t)* = A(t)), we have rl D(A(t)*) = (0). Hence, u* = 0 is the only candi-

date to appear in (7.3.16).

7.4.

ABSTRACT PARABOLIC EQUATIONS:

THE ANALYTIC CASE

We prove in this section a complex variable version of Theorem 7.2.5.

7.4.1 Theorem.

Let 0 be an open convex neighborhood of the inter-

val [0, T ] in the complex plane, and let (A(z); z E 0) be a family of operators

410

The Abstract Cauchy Problem for the Time Dependent Equations

satisfying:

(I') For each z E A, A(z) is densely defined, R(X; A(z)) exists in a

sector 2 = 2((p + 77/2) for some qp independent of z, 0 < q, < it/2;

if A' is a bounded neighborhood of [0, T ] with 0' c A,

IIR(X;A(z))II<C/IXI

(XE2,zEo'),

(7.4.1)

where C may depend only on A'.

(II') The function

z - A(z) -t E (E)

is analytic in A.

Then the Cauchy problem for

(7.4.2)

u'(t) = A(t)u(t) (7.4.3)

is properly posed (with D = E) in 0 < t < T with respect to solutions defined in

(A), Section 7.2. The propagator S(t, s) is strongly continuous in 0 < s < t and

admits an analytic extension

z) (as a function of the two variables) to

(7.4.4)

Moreover, for these values of t, z,

z)E c D(A(s)) (so that

z) is

everywhere defined and bounded),

z) A(z) is bounded in D(A(z)) and the

equalities

z) = z), z) = z)A(z) (7.4.5)

hold. Finally, for any A' as described above,

(7.4.6)

(the constant C possibly depending on A') for , z in

z); , z e A',

z, Iarg( - z)I < q p).

(7.4.7)

Proof. We begin by verifying that the assumptions in Theorem 7.2.5

are satisfied for the restriction (A(t); 0 < t <T) of to [0, T]. This is

immediately obvious for (I), (II), and (IV), but not quite so for (III). To

establish this assumption we note that (I') and (II') imply that R(X; A(z)) is

analytic in A, z for A E 2, z E A; to see this we only have to write the

complex variable analog of (7.2.12),

R(A; A(z))

=A(z)-'(AA(z)-t_ I) -i

and note that inverses and products of operator-valued analytic functions

are analytic. Making use of analyticity of R(A; A(z)) with respect to z, we

can write

DZR(X; A(z)) = 1

R(X; A(z')) dz (7.4.8)

2771

Iz'- zI=a

(Z_ Z)2

7.4. Abstract Parabolic Equations: The Analytic Case

411

for z' E A' (A' as in (7.4.1)), where a is so small that all the circles

(z; I z'- z I < a) for z in A' are contained in another bounded neighborhood

A" of [0, T] with A" C A. We note next that, using techniques very similar to

those in Lemma 4.2.3 combined with inequality (7.4.1), we can show that

R(X; A(z)) actually exists in a sector E((p"+ it/2) with q) < q," < rr/2 for

every z (=- A' (q)" may of course depend on A') and (7.4.1) holds as well in

I (q)" + 7r/2) X A'. We can then estimate the integral (7.4.8) obtaining: if

v, < v

IIDDR(X; A(z))II < C/IXI

(7.4.9)

where C may depend only on qq' and A'. It is clear that this implies (7.2.14)

for p >, 0. (The substitution of q) by p' was not necessary in this argument,

but will be essential in the proof of Lemma 7.4.2.)

The preceding considerations show that we can apply Theorem 7.2.5

and, in view of the conclusions of that theorem, we only have to establish

the statements in Theorem 7.4.1 regarding analytic extension of S(t, s). To

do this we reexamine the construction of S in Section 2 and extend all the

necessary results to the complex plane. We denote below by A' a subset of A

like the one described in the statement of Theorem 7.4.1; the symbol A'

indicates the set in (7.4.7).

7.4.2

Lemma.

S(t; A(s)) admits an analytic extension

A(z))

to 7, (p)X A such that

I (7.4.10)

for

z) E 2(gq)X A'. Moreover, R1Q, z) = -(Dr + z; A(z))

satisfies

IIR1Q,z)II<C/It-zIP

(zEA',

(7.4.11)

for p > 0 arbitrary (with C depending on p and A').

Proof.

As noted in the comments preceding (7.4.9), for every A'

there exists a qq'> qp such that R(X; A(z)) exists in E(T'+ it/2)X A' and

satisfies an estimate of the form (7.4.1) there. The formula

A(z)) =

21ri

(z)) A,

(7.4.12)

where t' is the boundary of 1:(q)'+ it/2) (see (7.2.10)), then provides an

analytic extension of S(t; A(z)) to 2(q))X A'. Using the estimate for

R(X; A(z)) and proceeding much in the same way as in the second part of

Theorem 4.2.1, we obtain the required inequality. The estimate (7.4.11) is

obtained working with the analytic continuation of (7.2.19), again using the

techniques in the second part of Theorem 4.2.1 and inequality (7.4.9).

We examine next the integral equation (7.2.8) giving birth to R(t, s).

In view of Lemma 7.4.2, each member of the sequence s)) defined by

412

The Abstract Cauchy Problem for the Time Dependent Equations

(7.2.22) admits an analytic continuation to 0., given by

z) = f R1(J, T)Rn-1(T, z) dT,

(see Figure 7.4.1) where the path of integration is, say, the segment I joining

z and (note that if (z, ) E Aro and T (=- I, then both T) and (T, z) belong

to A as well, and a similar statement holds for z ). It is easy to deduce

inductively from (7.4.11) that if p > 0, then

C"F(l

I)(1

P) P

z)II <

F(n(1- p))

(7.4.13)

on each A'; thus the series ER, z) is uniformly convergent in A' and it

provides an analytic extension of R(t, s) to AV which satisfies an estimate

of the form (7.4.11) in each A'. Making then again use of Lemma 7.4.2 and

of (7.2.7), we see that the desired analytic extension of S is given by

(7.4.14)

It only remains to prove equalities (7.4.5), and the statements preceding

them, as well as inequalities (7.4.6). To do this, we write the first equality

(7.4.5) for s, t real in the form

A(t)-1D,S(t,s)=S(t,s)

(0<s<t<T).

Since the two holomorphic functions

z),

FIGURE 7.4.1

7.4. Abstract Parabolic Equations: The Analytic Case

413

coincide for , z real, they must coincide wherever both are defined, in

particular in AV The second equality (7.4.5) is equivalent to

z)A(z)-I = - z)

and this is established in an entirely similar way. To prove inequalities

(7.4.6), we need to extend the estimate (7.2.29) to complex values of t and s.

This is done as follows: we have already observed that for any p > 0,

z)II

zIP (7.4.15)

for z) E 0' and it follows from (7.4.13) that

(7.4.16)

there as well. The relevant estimates in Section 7.2 can be extended to the

complex plane (and at the same time simplified, due to the fact that a = 1):

(7.2.27) becomes

((c,z)Eo').

(7.4.17)

With the help of this inequality, (7.2.24) is extended to

valid for any p > 0 (with C depending on p): here T is any complex number

in the segment joining z and . The proof of Corollary 7.2.4 can then be

extended to the complex plane: the result is the estimate

zIP+IIT-ZIKJ

(q,z)E0

(7.4.19)

which holds for any p, K > 0 with p + K < 1. Making use of this inequality

combined with (7.4.16) in (7.4.14), the first estimate (7.4.6) follows. The

second can be obtained working in the same way with the integral equation

defining S(t, s). Details are left to the reader.

7.4.3

Example.

The "real analytic" version of Theorem 7.4.1 is false, as

can be seen with the function t - A(t) produced in Example 7.3.2; although

the real counterparts of the hypotheses of Theorem 7.4.1 are satisfied, the

propagator S(t, s) is not real analytic in the neighborhood of any point

(s, t), 0 < s < t < T. To see this

it suffices to show that the function

u(t, s, x) = S(t, s)u(x) provided by (7.3.15) (say, for u(x) =1) is merely

C(OO) (but not real analytic) as a L2-valued function. Details are omitted; we

only point out that the verification is related to the classical counterexample

of an infinitely differentiable nonanalytic function

e '/E ( > 0),

AO = 0 ( < 0).

414

The Abstract Cauchy Problem for the Time Dependent Equations

Since Assumption (II') is the same in its real or complex form, the

total failure of Theorem 7.4.1 to function in the real domain must be traced

to (I'). In fact, even though the family (A (t)) can be extended analytically to

the entire complex plane by the obvious formula

A(z)u(x) _ - I u(x)

(z - x)2

(note incidentally that A(z) is bounded off the real axis), we have a(A(z))

= (_(Z - X)-2; 0 < x < 1); Assumption (I') is not satisfied in any complex

neighborhood of [0,1] (a(A(a + se'P)) contains the point - see-2i', which

surrounds the origin as qq moves from 0 to 7r).

We note a few additional results that can be obtained by careful

examination of the proof of Theorem 7.4.1. In the first place, it can be

proved that z) is strongly continuous in the closure of A' (we only have

proved strong continuity up to = z for , z real); then, equalities (7.1.7)

and (7.1.8) can be extended to the complex plane:

S(Z,z)=I,

(7.4.20)

for (h', T), (T, Z) E 0,, (note that z) E OT). On the basis of these observa-

tions, the following result shows that the conditions in Theorem 7.4.1 are

essentially necessary.

7.4.4 Example (Masuda [1972: 1]). Let A, p be as in the statement of

Theorem 7.4.1. Assume that z); z) E A,,} is a family of operators in

(E) such that:

(a) (7.4.20) is satisfied for

T), (T, Z) in AV

(b) S(., ) can be extended to a strongly continuous, uniformly

bounded function in the closure of AT in such a way that

S(z, Z) = I

(z e A). (7.4.21)

For each z e A, define an operator A (z) by

A(z)u= lim h-'(S(z+h,z)-I)u, (7.4.22)

h -. 0 +

the domain of A(z) consisting precisely of those u for which (7.4.22) exists.

Then the family (A(z)) has the following properties:

(d) A (z) is densely defined for all z E A.

(e) For any compact subset K of A and any s > 0 there exists

constants C, M such that

2=2(q)+1r/2-s)n(X; IXI>,M)

belongs to p(A(z)) for z E K,

IIR(X;A(z))II<C/IXI (XE2,zEK)

7.5. The Inhomogeneous Equation

415

and for each X E 2, R(X; A(z)) is holomorphic in z in some

neighborhood of K.

(f)

z)E c

z)A(z) is bounded in D(A(z)) and

z) = z), z) = - SQ, z)A(z) .

7.5. THE INHOMOGENEOUS EQUATION

We prove in this section two results on the equation

u'(t) = A(t)u(t)+ f(t),

(7.5.1)

which are based, respectively, on the hypotheses in Section 7.3 and 7.2. The

first one refers to weak solutions of (7.5.1). If f is, say, locally summable in

t > 0, these solutions are defined by generalizing in an obvious way defini-

tion (B) in Section 7.3; a function u(.), continuous in s < t < T, is a weak

solution of (7.5.1) with initial condition

u(s) = uo (7.5.2)

for f locally summable in t >, 0 if and only if

fT(u(t), u*'(t)+A*(t)u*(t)) dt = -(uo, u*(s))-

fT(f(t), u*(t)) dt

(7.5.3)

for all functions u*(.) satisfying the assumptions in Definition (B).

As noted in Section 7.1, the analogue of the Lagrange formula (2.4.3)

in the time-dependent case is

u(t)=S(t,s)uo+ f'S(t,a)f(a)do.

(7.5.4)

It does not seem to be known in general whether every weak solution of

(7.5.1) must be representable in the form (7.5.4), at least if we only assume

that the Cauchy problem for (7.5.1) is properly posed as defined in Section

7.1; note that the corresponding result for the time-independent case (see

Theorem 2.4.6) was proved using the Phillips adjoint of S, whose theory has

not been fully extended to the time-dependent case. However, the result

holds in the abstract parabolic setting; this will be seen as a consequence of

the following result.

7.5.1 Theorem. Let Assumptions (I), (II), and (III) in Section 7.2 be

satisfied. Assume uo is an arbitrary element of E and f is continuous in

s < t < T (or, more generally, is locally summable there). Then the function

u(t) defined by (7.5.4) is the only weak solution of (7.5.1), (7.5.2) in s < t < T.

Proof.

The first term on the right-hand side of (7.5.4) has already

been shown to be a weak solution of the homogeneous equation (7.3.1) in

416 The Abstract Cauchy Problem for Time-Dependent Equations

Theorem 7.3.1, thus we may assume that uo = 0. We begin by showing that

u(t) = f tS(t, a)f (a) do

is continuous in s < t < T (under the sole hypothesis of strong continuity of

S). In fact, if s < t < t'< T, we have

IIu(t')-u(t)II < f tIIS(t',a)IIIIf(o)Ildo

+ f

tll(S(t'-s)-S(t-s))f(s)II ds

and continuity follows as in Example 2.4.5. We prove now that u(.) satisfies

the inhomogeneous equation in the weak sense. We have

fT(u(t),u*'(t))dt= fT( f'S(t'o)f(a)do,u*'(t))dt

s s s

fT(S(t,a)f(a),u*'(t))dtdo

lm lim

fT-k fT (Sh(t,a)f(a),u*'(t))dtdo.

k h-O+

But

fT-k /'+k

(Sh(t,a)f(a),u*'(t))dtdo

fT k(Sh(o+k,a)f(o),u*(o+k))do

T ((D'S,h(t,a)-A(t)S(t,o))f(a),u*(t))dtdo

1+k

T (Sh(t,o)f(a),A(t)*u*(t))dtdo. (7.5.5)

-°s 'o+k

Making use of the properties of Sh(t, s) established in Lemma 7.2.6, we see

that (7.5.5) tends (as h - 0+) to

_1T -k(S(o+k,o)f(o),u*(o+k))do

T k

(S(t,o)f(a),A(t)*u*(t))dtdo

(7.5.6)

-°s °a+k

On the other hand, this expression tends to

-fT(f(t),u*(t))dt- fT fT(S(t,o)f(a),A(t)*u*(t))dtdo

s s

fT(f (t), u*(t)) dt -f T(u(t), A(t)*u*(t)) dt

when k - 0+, which proves (7.5.3) taking into account that u(s) = uo = 0.