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

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7.1. The Abstract Cauchy Problem for Time-Dependent Equations

387

on continuity of t -A (t) (and modifying accordingly the definition of

solution employed). One variant in the direction of greater generality is:

7.1.5

Example. Let A(t) be everywhere defined and bounded for almost

all t, 0 < t < T. Assume that for each u the function t - A(t) u is strongly

measurable and that

IIA(t)II <a(t) (0<t<T),

(7.1.26)

where $ is locally integrable. Then the Cauchy problem for (7.1.1) is

properly posed forward and backwards if we weaken the definition of

solution thus: a continuous function is said to be a solution of (7.1.1) if and

only if the integral equation (7.1.9) is satisfied, the integral understood in

the sense of Bochner. (Note, incidentally, that A(T)u(T, s) is strongly

measurable in T if u is continuous; this can be seen approximating u by step

functions.) Integrability follows from boundedness of u, since IIA(T)u(T, s)II

< C$(T). The proof of Theorem 7.1.1 can be readily adapted to the present

setting; we outline the main steps. The iteration scheme to solve (7.1.9) is

the same; inequality (7.1.12) becomes

Ilun(t,s)-un-1(t,s)II <

(f t$(T)dT)nIIu01I

Inequality (7.1.15) is now

I1u011+ f ds,

(n 1,2,...)-

(7.1.27)

which implies, via Lemma 7.1.2, that

IIS(t, s)II <

exp(f ta(T) dT).

(7.1.28)

Continuity of S(t, s) is proved with two inequalities similar to (7.1.19) and

(7.1.20). Examples 7.1.3 and 7.1.4 have obvious generalizations to the

present situation.

The following result, on the other hand, strengthens the conclusion

of Theorem 7.1.1 under stronger assumptions.

7.1.6

Example. If t - A(t) is continuous (in the norm of (E)), Theorem

7.1.1 holds with the following refinements: the propagator S(t, s) is con-

tinuously differentiable (in the norm of (E)) with respect to both variables

in the square 0 < s, t < T and

D,S(t,s)=A(t)S(t,s), DS(t,s)=-S(t,s)A(s) (7.1.29)

there. In fact, S(t, s) is differentiable in the norm of (E) in 0 < s, t < T also

under the hypotheses in Theorem 7.1.1 and both equalities (7.1.29) hold (the

derivatives of course are not in general continuous in (E)).

388

The Abstract Cauchy Problem for the Time Dependent Equations

7.1.7

Example.

Let A(-),

be two operator-valued functions satisfy-

ing the assumptions in Example 7.1.5 and let

IIA(t)-Ao(t)II<e (0<t<T').

If S is the propagator of (7.1.1), So the propagator of u'(t) = A0(t)u(t), we

have

IIS(t,s)-So(t,s)II<ET'exp(2 fT113(T)dT).

(7.1.30)

The result follows from subtracting the integral equation (7.1.9) satisfied by

S and the corresponding one for So; we obtain

(S(t,s)-S0(t,S))u= f

t(A(T)-AO(T))S(T,s)udT

+ f tAo(T)(S(T,s)-S0(T,s))udT

so that, if q(t) = II(S(t, s)- S0(t, s))uII, we have, making use of (7.1.28),

rl(t)<eT'(exp f T Q(T)dT)IIuII+ f IQ(T)rl(T)dT

from which (7.1.30) follows using Lemma 7.1.2.

7.1.8 Example.

Under the hypotheses in Example 7.1.6, let 0 < s < t < T

and choose for each n =1, 2.... a partition s = tn,o < tn,1 <

< tn,

m(n) = t

in such a way that 8n = sups (t

n i - to 1 _ 1) -+ 0 as n - oo. Then

m(n)

S(t, s) = lira

11 exp((tn,l - tn, j_i)A(tn,j)) (7.1.31)

n-00 f=1

in the norm of (E), where the product is ordered backwards:

m(n)

eXp((tn,j-tn,l-1)A(tn,j))

r=1

exp((tn

m(n)-tn,m(n)

I)A(tn,m(n)))...eXp((tn,1-tn,0)A(tn,1))

(7.1.32)

The limit is uniform on compacts in the following sense: if 0 < s, t < ' < T,

I I S(t, s) - Sn (t, s )II can be made small with Sn, independently of s, t or the

particular partition chosen. To see this we proceed as follows: having fixed

s, t and the partition tn,0+... I tn,m(n), we define An(-) thus:

A(T) (0<T<s)

An(T)

A(Tn,l)

(tn, j-1 <T <tn,J, J=1,2,...,m(n))

A(T) (t<T<T').

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

389

Then

IIA(T)-An(T)II <E (O<T<T')

uniformly in 0 < T < T' if S is sufficiently small, independently of the

particular partition chosen. It follows then from Example 7.1.7 that

IIS(t, s)-SIP, s)II <Ce,

where S is the propagator of u'(t) = A,, (t) u(t ). But it is easy to see that

s) is the operator in (7.1.32), thus the result follows.

7.1.9 Example. Under the hypotheses of Example 7.1.6, assume that A(t )

and JstA(T) dT commute for each s, t. Then

S(t, s) = exp(f tA(T) dT). (7.1.33)

This formula is also valid with the hypotheses in Theorem 7.1.1 or Example

7.1.5, the integral JA(T) dT interpreted "elementwise."

7.1.10

Example. Let ft-) be continuous in t >_ s, and assume the premises

of Theorem 7.1.1 hold. Then, if uo E E,

u(t,s)=S(t,s)uo+ ftS(t,a)f(a)da,

(7.1.34)

is a genuine solution of the inhomogeneous equation

u'(t)=A(t)u(t)+f(t) (s<t<T).

(7.1.35)

satisfying the initial condition (7.1.2) (genuine solutions are defined in the

same way as for the homogeneous equation). It is the only such solution

satisfying (7.1.2). Under the hypotheses of Example 7.1.5, if f is locally

summable in t >, s, formula (7.1.34) furnishes the only solution (in the sense

of Example 7.1.5) of (7.1.35), which equals uo for t = s.

7.1.11

Example. We assume A(.) continuous in the norm of (E) as in

Example 7.1.6. Let t -* F(t) a continuous (E)-valued function, Uo E (E).

Then

U(t,s)=S(t,s)U0+ ftS(t,a)F(a)da

(7.1.36)

is the only solution of the initial-value problem

DtU(t,s)=A(t)U(t,s)+F(t), U(t,t)=Uo,

(7.1.37)

whereas

V(t,s)=VOS(t,s)- ftF(T)S(T,s)dT

(7.1.38)

is the only solution of

DSV(t, s) = - V(t, s)A(s)+ F(s),

V(t, t) = Vo. (7.1.39)

390

The Abstract Cauchy Problem for the Time Dependent Equations

The derivatives here are understood in the norm of (E) and continuous in

the same norm.

7.1.12 Example.

Let A(-),

be strongly continuous in 0 < t < T, as in

Theorem 7.1.1, and let S be the solution operator of (7.1.1), S the solution

operator of

u'(t) _ (A(t)+P(t))u(t). (7.1.40)

Then S can be obtained as the sum of a perturbation series motivated by

formula (7.1.34) (see Theorem 5.1.1) as follows. Define So = S,

S,,(t,s)u= ftS(t,(Y)P((Y)Sn_1(a,s)udo

(0<s,t<T,n>,1).

(7.1.41)

Then

S(t,s)= E Sn(t,s)

(7.1.42)

n=0

in the same range of s, t. The convergence is uniform in 0 S s, t s T' for any

T'< T. To see this we use (7.1.17) and denote by M a bound for I I P(t )I I in

0 S t S T'; a simple inductive argument shows that

It-sln

aCat-l

(n a1)

(7.1.43)

IISn(t,s)II<

in 0 < s, t 5 T', C the constant in (7.1.17).

7.2. ABSTRACT PARABOLIC EQUATIONS

Many of the results in this chapter on the equation

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

(7.2.1)

will be obtained, roughly speaking, on the basis of hypotheses of the

following type: (1) the Cauchy problem for the "equation with frozen

coefficients" u'(t) = A(s)u(t) (s fixed) is properly posed (that is, A(s) E (2+,

or a subclass thereof for all s) and (2) the operator function t - A(t)

depends smoothly on t in one way or other (the assumptions in the previous

section are clearly of this

type).2

The methods used in this section and the next are based on the

following considerations, which are for the moment purely formal. Let

S(t, s) be the propagator of (7.2.1), and let S(t; A(s)) be the propagator of

the time-invariant equation.

u'(t) = A(s)u(t). (7.2.2)

2Since A(t) is in general unbounded, "smooth dependence" in this section may mean

smoothness of the function t - A(t)- 1 or of other related functions.

7.2. Abstract Parabolic Equations

391

Then

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

= A(t)S(t - s; A(s))+(A(s)-A(t))S(t - s; A(s)) (t >-s)

(7.2.3)

with S(s - s; A(s)) = I so that, in view of (7.1.36), we may expect an

equality of the type

S(t - s; A(s)) = S(t, s) - f tS(t, a)(A(a)-A(s))S(a - s; A(s)) do.

(7.2.4)

On the other hand,

DDS(t - s; A(t)) = - S(t - s; A(t))A(t)

_ - S(t - s; A(t))A(s)+S(t - s; A(t))(A(s)- A(t))

(s < t)

(7.2.5)

with S(t - t, A(t)) = I, hence (7.1.38) suggests that

S(t - s; A(t)) = S(t, s) - f tS(t - T; A(t))(A(T)- A(t))S(T, s) dT,

(7.2.6)

thus we can hope to fabricate S(t, s) as the solution of one of the Volterra

integral equations (7.2.4) or (7.2.6). We shall in fact use both, begin-

ning with the last. To solve (7.2.6) directly, however, would need rather

strong conditions on S(t - T; A(t))(A(T)- A(t)), thus we shall follow

a more roundabout way. Ignoring the dependence on t of the function

(A(T)- A(t))S(T, s) in (7.2.6), we modify the equation thus:

S(t,s)=S(t-s;A(t))+ ftS(t-T;A(t))R(T,s)dT,

(7.2.7)

and seek to obtain an integral equation for R(T, S). In order to do this, we

note that (again formally) (7.2.7) implies

D,S(t, s) = D1S(t - s; A(t))+R(t, s) + f tDS(t - T; A(t))R(T, s) dT.

Also, applying A(t) on the left to both sides of (7.2.7),

A(t)S(t, s) DS(t - s; A(t))- f tDTS(t - T; A(t))R(T, s) dT.

Combining the last two equalities, we obtain

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

= (D, + DD)S(t - s; A(t))+R(t, s)

+ f t(D1 + D,T)S(t - T; A(t))R(T, s) dT,

392

The Abstract Cauchy Problem for the Time Dependent Equations

so that R(t, s) satisfies the integral equation

R(t,s)- f`R,(t,T)R(T,s)dT=R,(t,s), (7.2.8)

where we have set RI(t, s) = -(D1 + Ds) S(t - s; A(t)). We impose now

conditions on the function t --> A(t) that will allow us to justify the formal

calculations above. As in Section 4.2, if 0 < q) < it/2, we denote by 2(q)) the

sector I arg X I < T. The assumptions are (I), (II), (III), and (IV) below, where

0<T<oo.

(I) For each t, 0 < t <T, A(t) is densely defined, R(X; A(t)) exists in

a sector I = I (T + Ir/2) for some (p, 0 < q) < ?T12 and

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

(X(=-2,O<t<T),

(7.2.9)

where neither rp nor C depend on t.

We recall that, by virtue of Theorem 4.2.1, assumption (I) implies

that each A(t) belongs to @ (qq -). If I' is the boundary of the sector 2

oriented clockwise with respect to it, we have

S(s; A(t))

2 --

fee?sR(X; A(t)) dX (s > 0) (7.2.10)

,r i

(see (4.2.8)) and we deduce from this formula and (7.2.9) that

IIA(t)S(s;A(t))II <C/s (s>0),

(7.2.11)

where C is independent of t (that an estimate of this type exists for each t is

of course a consequence of the fact that A (=- d; see Section 4.2).

Note also that, since R(X; A(t)) is assumed to exist for A = 0, A(t)

exists for all t. The next three postulates bear on its dependence on t.

(II) The function t - A(t)-' is continuously differentiable (in the sense

of the norm of (E)) in 0 < t < T.

We note that (II) implies that t -* R(X; A(t)) is as well continuously

differentiable in the norm of (E). To see this, write

R(A;A(t))=A(t)-'(AA(t)-'-I)-'

(7.2.12)

and notice that products and inverses of continuously differentiable func-

tions are likewise continuously differentiable. Moreover, making use of the

formulas D(FG) = (DF)G + F(DG) and D(F-') = - F-'(DF)F-', we

obtain from (7.2.12) (after some rearrangement of terms) the equality

D,R(A; A(t))

A(t)R(A; A(t))DA(t)-'A(t)R(A; A(t)).

(7.2.13)

(III) There exist constants C and p, C > 0, 0 < p < 1, such that for

every A E 2 and every t, 0 < t < T, we have

IID,R(A;A(t))II

<C/IAI'-P.

(7.2.14)

7.2. Abstract Parabolic Equations

393

(IV) The function DA(t)-1 is Holder continuous in (E), that is, there

exist constants C, a > 0 such that

IIDA(t)-'-DA(s)-'II

<CIt-sI" (0<s,t<T).

(7.2.15)

We proceed next to show that (under Assumptions (I), (II), and (III)

alone) the integral equation (7.2.8) can be solved. The first result is an

auxiliary estimate for R I (t, s). Here and in other inequalities, C, C',...

denote constants, not necessarily the same in different expressions.

7.2.1 Lemma. R , (t, s) is continuous in 0 < s <t < T in the norm of

(E) and satisfies there the inequality

IIRI(t,s)II<C/(t-s)P

(7.2.16)

Proof. Let F be as in the comments following Assumption (I). By

virtue of (7.2.10), we have

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

2Iri

f Xe"(`s)R(X;A(t))dX

1 f

A(t)) dX,

(7.2.17)

2Iri F

f Xe"(`-s)R(X; A(t)) dX

(7.2.18)

DSS(t - s; A(t)) = -

217i F

(the differentiations under the integral sign can be easily justified) so that

R,(t, s) = -(D,+ Ds)S(t - s; A(t))

1 fe"('-s)DtR(X;

A(t)) A.

(7.2.19)

21ri r

Hence continuity of R I (t, s) in 0 < s < t < T follows. Making use of (III) we

obtain

IIRI(t,s)II<C

freReX(t-s)IXIP-'Id1vI=C'(t-s)-P

(7.2.20)

thus completing the proof.

Inequality (7.2.16) suffices to show that (7.2.8) has a solution. In fact,

we have:

7.2.2

Lemma. There exists a solution R(t, s) of (7.2.8), which is

continuous in 0 < s < t < Tin the norm of (E) and satisfies there the inequality

IIR(t, s)II <C'/(t - s)". (7.2.21)

Proof. Starting with R i (t, s), we define a sequence of (E)-valued

functions recursively by

Rn(t,s)= f tR1(t,T)Rn-1(T,s)dT

(n=2,3,...). (7.2.22)

394

The Abstract Cauchy Problem for the Time Dependent Equations

A simple inductive argument based on (7.2.16)3 shows that each Rn(t, s)

exists and is continuous in 0 < s < t < T and

Cnr(I-p)n(t-s)(n-1)(I-P)-P

IIR.(t, S)II <

I'(n(I - p))

there, where C is the same constant in (7.2.16). It follows that

R(t, s) = E Rn(t, s) (7.2.23)

n=I

converges in the norm of (E) in 0 < s < t < T and defines a continuous

(E )-valued function satisfying (7.2.21), where

C'=E(CnI'(l-

p)n/I'(n(I-p)))T(n-1)(1-P),

Since the partial sums of (7.2.23) satisfy the same estimate, an application of

the Lebesgue dominated convergence theorem shows, adding up both sides

of (7.2.22) from 2 to oo, that R(t, s) is a solution of the integral equation

(7.2.8) as claimed.

We have not used up to now hypothesis (IV). With its help we prove

two additional results.

7.2.3 Lemma. The function R 1(t, s) satisfies

IIR1(t,s)-R1(T,s)II<C(

t-T

P+(t

t_ )a) (7.2.24)

l (t-S)(T-S) J

for 0,<S< T<t,<T.

Proof. We make use of formula (7.2.19):

R1(t,s)-R1(T,s)

2Iri

frea('-s)(DR(X;A(t))-DTR(X;A(T)))dX

1 f

2Tri

= I1 + I2. (7.2.25)

3We/use here the well known equality involving the beta and gamma functions,

J t(t - T).- I (T - S)S-1 dT = (t -

S)a+fl-I

B(a,p)

_ (t -

s)'+[3-'r(a)r($)/r(a+

fi)

valid for a, $ > 0. This identity will be utilized many times in the future without explicit

mention.

7.2. Abstract Parabolic Equations

395

By virtue of (7.2.13) we have

D,R(X; A(t))- DR (A; A (T))

_ -{A(t)R(X; A(t))-A(T)R(A; A(T)))D1A(t)-'A(t)R(X; A(t))

- A(T)R(X; A(T))(D,A(t)-'

DTA(T)-')A(t)R(X; A(t))

- A(T)R(X; A(T))DTA(T) '{A(t)R(X; A(t))-A(T)R(X; A(T))).

(7.2.26)

But

A(t)R(X; A(t))- A(T)R(X; A(T)) = X(R(X; A(t))- R(X; A(T)))

= X

f'D,.R(X; A(r)) dr

so that, in view of (III),

IIA(t)R(X; A(t))-A(T)R(X; A(T))II < IXI f

ICIXIP-'dr

=C(t-T)IXIP.

Note that, by virtue of Assumption (I), A(t)R(X; A(t)) = XR(X; A(t))- I

is uniformly bounded in norm in IF x [0, T]. On the other hand, Assumption

(II) implies that DDA(t)-' is bounded in [0, T]. Using these two facts and

the preceding inequality, we estimate the first and third terms on the

right-hand side of (7.2.26); for the second term we use Assumption (IV). We

obtain

IID:R(X;A(t))-DTR(X;A(T))II

SC((t-T)IXIP+(t-T)«)

(7.2.27)

and we proceed now to estimate I. We have

111111<1 C

freRea(t-s)((t-T)IXIP+(t-T)°`)IdXl

C {

t - T +

(t-S)P+1

t - S

C,{

S)(

- S)P

+ (tt-

(7.2.28)

As for I2, we have

- f

t-sDr{2_1 fre"DTR(x;A(T))dX}dr

ft-s{ fxexrDR(x;

A(T)) dX l dr.

2;i T - S

396

The Abstract Cauchy Problem for the Time Dependent Equations

Making use of (III) we see that

XexDTR(X; A(T)) dX

hence

= C,,

(t-S)(T-S)P

Combining this inequality with (7.2.28), Lemma 7.2.3 follows.

7.2.4

Corollary.

Let K < min(1- p, a). Then the function R(t, s)

satisfies

IIR(t,s)-R(T,s)II

(t -

)( T

+ (t-T)K

(7.2.29)

111

T - S)

t - S

for 0 < s < T < t < T.

Proof. We obtain from (7.2.8) that

R(t, s)- R(T, s) = R1(t, s)- R1(T, s)+ f tR,(t, r)R(r, s) dr

+ fT(R1(t, r)-RI(T, r))R(r, s) dr

=J1+J2+J3.

It is obvious from Lemma 7.2.3 that J, satisfies the estimate (7.2.29). For J2

we use Lemma 7.2.1 and Lemma 7.2.2:

IIJ2II <C

f`(t-r)d(r-s)P

If T < r < t, we have

<C f e(ReX)rlXIPI dAI <C'Y-(P+

t - S

111211<C"I

Jr T

-(P+ 1) dr

_ 1

(TS)'

(tS)'

Crr

P { 1 -

(

t -

s y'I

(T - S)

C,, T - S

(T-S)"

t - S

t- T

r- T t- T

r -s t - s