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

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7.5. The Inhomogeneous Equation

417

Since uniqueness of weak solutions of (7.5.1) reduces to uniqueness

of solutions of the homogeneous equation (7.3.1) (which has been estab-

lished in Theorem 7.3.1), the proof is complete.

The next result assumes the hypotheses used in Section 7.2 and

generalizes to the time-dependent case Example 2.5.7.

7.5.2

Theorem.

Let Assumptions (I), (II), (III), and (IV) in Section

7.2 hold. Assume uo is an arbitrary element of E and f is Holder continuous in

s < t < T, that is, for some Q > 0,

Ilf(t')-f(t)II <Clt'-tl1

(s<t,t'<T).

(7.5.7)

Then the function u(t) defined by (7.5.4) is the only genuine solution of

the inhomogeneous equation (7.5.1) in the sense of Definition (A), Section 7.2.

Proof.

Uniqueness again follows from uniqueness for the homoge-

neous equation, thus we can assume that uo = 0 in (7.5.4), which we do in

the sequel.

In view of (7.2.10), we have

S(t - s; A(r))

21ri

_---f

ret (

s)R(X; A(r)) dX, (7.5.8)

where IF is the contour used in Section 7.2. Hence

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

f Xe"('-s)(R(a; A(t))- R(A; A(s))) dX

2,ri r

(7.5.9)

in 0 < s < t < T. By virtue of Assumption (III),

IIR(X; A(t))- R(X; A(s))II < (t - s) sup IIDR(X; A(T))II

S _< T < t

<C(t-s)IXI°-

and we obtain from (7.5.9) that

IIA(t)S(t-s;A(t))-A(s)S(t-s;A(s))II<C/(t-s)P (0<s<t<T).

Combining this inequality with (7.2.45) and (7.2.59), we get

IIA(t)S(t,s)-S(t,s)A(s)II<C/(t-s)'-' (0<s<t<T)

(7.5.10)

with U as defined in Corollary 7.2.4. This will be used later.

Write

f'S(t,o)f(a)do=j'S(t,a)(f(a)-f(t))do+ jtS(t,a)f(t)do.

S s

(7.5.11)

418

The Abstract Cauchy Problem for Time-Dependent Equations

Combining (7.5.7) with (7.2.34), we see that IIA(t)S(t, a)(f(a)- f(t))II is

integrable in s < a < t, so the first integral belongs to D(A(t)) and A (t) can

be introduced under the integral sign. In relation to the second integrand,

we note that

A(t)S(t,a)f(t)

=(A(t)S(t,a)- S(t,a)A(a))f(t)+ S(t,a)A(a)f(t)

where the first term on the right is integrable in s < a < t in view of (7.5.10);

for the second we have, making use of (7.2.33):

S(t,a)A(a)f(t)do=- ft-hD,,S(t,a)f(t)do

=S(t,s)f(t)-S(t,t-h)f(t).

Hence (7.5.11) belongs to the domain of A(t) for all t > s and

A(t) ftS(t,a)f(a)do= f'A(t)S(t,a)(f(a)-f(t))do

+ f t(A(t)S(t, a)- S(t, a)A(a)) f (t) do

+S(t,s)f(t)-f(t).

(7.5.12)

An examination of this equality, (7.2.34), (7.5.7), and (7.5.10), shows that

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

is continuous in s < t < T and that

A(t) ft hS(t,a)f(a)do- A(t)u(t)=A(t) ftS(t,a)f(a)do

s s

uniformly there. Hence, if s < t'< T we have

f t A(t)u(t) dt

= lim

A(t) f t hS(t, o) f (a) dodt,

s h

0+ s+h

where

f t A(t) f t-hS(t,o)f(a) dadt= f

- h f

A(t)S(t,o)f(a) dtda

s+h

s +h

ft'- h ft'

ft-hS(t',a)f(a)do

DDS(t,a)f(a)dtdo

s h

-ft'-hS(a+h,(Y)f(o)do.

Taking limits, we obtain

f t A(t)u(t) dt = u(t') -f t f (t) dt,

7.6. Parabolic Equations with Time-Dependent Coefficients

419

which shows that u(.) is indeed a solution of (7.5.1). This completes the

proof of Theorem 7.5.2.

7.6. PARABOLIC EQUATIONS WITH

TIME-DEPENDENT COEFFICIENTS

We examine here time-dependent versions of the elliptic operators in

Chapter 4. Since the methods are clearly visible in space dimension one and

most of the technical complications are absent, we look mostly at this case.

In the following preparatory results, /30 (resp. /31) is a boundary condition at

0 (resp. at 1 > 0) and Ap(/3o, /3,) is the operator (written in variational form)

AP(/3o,/3,)u(x) = (a(x)u'(x))'+b(x)u'(x)+c(x)u(x). (7.6.1)

The coefficients a, b are assumed to be continuously differentiable in

0 < x < 1 while c is continuous there. As in Section 4.3, AP (/3o, /3,) is thought

of as an operator in LP(0, 1) with domain consisting of all u E W2 , P(0,1)

that satisfy both boundary conditions. Use will also be made of the operator

AP, defined in a similar way, but without including boundary conditions in

the definition of the domain (that is, D(AP)=W2'p(0,1)). The admissible

range of p will be 1 < p < oo.

The first observation concerns solutions of the equation

(XI-AP)u=f (7.6.2)

for f E LP with nonhomogeneous boundary conditions; to fix ideas, assume

both boundary conditions are of type (I), so that one wishes to find

u E W2'(0, 1) satisfying (7.6.2) and

u'(0) = you(0)+ do, u'(1) = y,u(1)+ d,. (7.6.3)

To solve this problem we use our information on the operator AP(/30, /3,).

Let p be a real-valued twice continuously differentiable function satisfying

the same boundary conditions as u:

P'(0)=yop(0)+do, P'(1)=y,P(1)+dl.

Then v = u - p belongs to D(Ap(/30, #I)) and satisfies

(XI-A(Ro,ar))v= f -(XI-AP)P.

(7.6.4)

Observe next that

IIAPPII < C'11pl1w2' < C"(Idol + I d,I ),

(7.6.5)

where the constant C" does not depend on

I do 1,

I d I I if p is adequately

chosen. We deduce then that

if A E p(AP(f3o, /3,)),

the solution of

(7.6.2), (7.6.3) given by

u=R(A; AP(/3o,Q,))(f -(XI-AP)P)+p

(7.6.6)

420

The Abstract Cauchy Problem for Time-Dependent Equations

can be estimated in terms of I I f 11, Idol, Id, 1. It will be important in what

follows to obtain estimates that are "uniform with respect to Ap(/30, /3,)" in

a sense to be precised below. We consider a class !1 of operators Ap(13o, #I)

whose coefficients satisfy the inequalities:

Ia(x)I, la'(x)I, I b(x)I , I b'(x)I ,

I c(x)I , IYol, IY,I < C,

a(x) >,K > 0,

(7.6.7)

uniformly for 0 < x < I and Ap(/3o, /3I) E W.

7.6.1 Lemma. Let q)p be the angle in Theorem 7.3.1,

2 1/2

q)p=arctg(/

((pp-2) -11

Then if 0 < g < q)p, there exists w such that the sector 2 = E(q) + 77/2) =

(A; I arg A I< (p + ¶r/2) belongs to the resolvent set of every A( /3o, /3,)- wI,

Ap(13o, /3,) E W, and the solution u of (7.6.2),(7.6.3) for AP - wI satisfies

d + d

7 6 8

I ,l),Ilull<

111lfll

. . )

(

..v < C I1f II+ IAI'/2(Id l + IdHull l))

(

(7.6.9)

lx+ III/,

IlullW2,P

<C"(Ilfhl+ IAI(Idol + Id,l))

(7.6.10)

for A E I and A(/30, /3I) E W, where the constants C, C', C" are independent

off, do, d1, A,A,(Qo,#I)EW.

The proof follows from careful observation of the material in Section

4.3. Note first that under the present uniform bounds on the coefficients of

Ap(/30, /3,), the estimate (4.3.15) can be obtained with w independent of

Ap(/30, /3,); it suffices to increase that w to w + 1 and (7.6.8) results from

(7.6.6). To obtain (7.6.10) we note that the norm of W2'° is equivalent to,

say, the graph norm of Ap(/3o, /3,)- wI. Finally, to show (7.6.9), we use the

fact (Adams [1975: 1, p. 70]) that for every u E W 2' p (0, 1) we have

Ilullw.

<KSIIullw2.P+KS-'IIuIILP

(7.6.11)

for 0 < S < 1, where K only depends on p and 1. Setting S = I A + 1 l -1/2

and using (7.6.8) and (7.6.10), the desired inequality is obtained.

In the following result we consider time-dependent operators and

boundary conditions: here /3o(t), $,(t) are, for each t, boundary conditions

of type (I),

u'(0)-y0(t)u(0),

u'(t) = Y,(t)u(1), (7.6.12)

and

Ap(t,/3o(t),/3,(t))= (a(x,t)u'(x))'+b(x,t)u'(x)+c(x,t)u(x).

(7.6.13)

7.6. Parabolic Equations with Time-Dependent Coefficients

421

We assume that yo and y,, are defined and continuously differentiable in

O< t < T and that D1a, a', D1a', b', Dtb, Dc exist and are continuous in

0 < x < 1, 0 < t < T together with a, b, c. Moreover,

a(x,t)>0 (0<x<l,0<t<T).

(7.6.14)

In what follows, co is still the parameter in Lemma 7.6.1 corresponding to

the family t = (A, (t; /30 (t ), /31(t)); 0 < t < T).

7.6.2 Theorem.

Under the hypotheses above the family of operators

(Ap(/30(t),/31(t))-cwI;0<t<T)

(7.6.15)

in LP(0,1) satisfies the assumptions of Theorem 7.3.1; in particular, the

Cauchy problem for the equation

u'(t)=Ap(t;/3o(t),/31(t))u(t)

(0<t<T) (7.6.16)

is properly posed (with respect to the weak solutions defined in (B), Section

7.3).

Proof We only have to verify assumptions (I), (II), and (III) in

Section 7.2. Since the family (7.6.15) fits into Lemma 7.6.1, (1) follows from

(7.6.8) with do=d1=0 and

The second estimate implies that

IIu&, t)II w,o <CIIIII

(7.6.17)

uniformly for 0 < t < T, A E 2. This shows uniform boundedness of ux(O, t),

u,(1, t) since it follows from (4.3.40) that

IUJO,t)I,1ua(l,t)1<Cllua(-,011w"

(7.6.18)

for 0 < t < T and A E 2 fixed (how C depends on A is unimportant for the

moment). The next step is to observe that the function

wa(x, t) = u,\ (x, t + h)- ux(x, t)

satisfies the nonhomogeneous boundary value problem

(XI-(Ap(t)-coI))wx(x,t)=(A,(t+h)-A,(t))ua(x,t+h),

(7.6.19)

w((0,t)=yo(t)w;, (0,t)+(yo(t+h)-yo(t))ux(0,t+h),

w'(1,t)=y1(t)w,,(l,t)+(y,(t+h)-y!(t))ux(l,t+h). (7.6.20)

A look at (7.6.10) and (7.6.18) shows that, for A E 2 fixed, the function

t u&'t)EW"P (7.6.21)

is continuous in the W 2' norm in 0 < t < T uniformly for f in 11f 11 < 1. We

422

The Abstract Cauchy Problem for Time-Dependent Equations

consider next the problem

(XI-(Ap(t)-wI))v,(x, t)

= (Da(x,t)u'(x,t))'

+ D,b(x,t)u'(x,t)+ D1c(x,t)u(x,t)

(7.6.22)

with boundary conditions

v'(0,t) = yo(t)va(0,t)+ DtYo(t)uJ0,t),

va(l, t) = yi(t)vx(1, t)+DDy,(t)uJ1, t). (7.6.23)

A similar argument,

this time applied to vx(x, t + h) - vx(x, t),

shows that

t-->

t) EW2°p

(7.6.24)

is continuous for 0 < t < Tin the W 2, P norm uniformly in I I f 11 < 1; here we

make use of the continuity of (7.6.21) and of the fact, again consequence of

(4.3.40), that ux(O, t), ua(l, t) are continuous in 0 < t < T uniformly in

I If II < 1. It is obvious that (II) will be amply satisfied if we can show that

va(.,t)=D,R(X;A(t;Qo(t),ar(t))-wI)f()

(7.6.25)

(which is intuitively obvious since the problem (7.6.22), (7.6.23) is obtained

differentiating formally with respect to t the boundary value problem

satisfied by ux(x, t)). To establish (7.6.25) rigorously, we consider

zjx, t) = (h-'(ua(x, t+h)-ua(x, t))- va(x, t)).

An amalgamation of (7.6.19) and (7.6.22) shows that za satisfies

(XI -(AP(t)-wI)) z,(x, t)

_ ((h-'(a(x,t+h)-a(x,t))-Da(x,t))u'(x,t))'

+(h-'(b(x, t+h)-b(x, t))-Db(x, t))u' (x, t)

+(h-'(c(x,t+h)-c(x,t))-D1c(x,t))ux(x,t)

+ h-'(Ap(t + h)- AP(t))(ua(x, t + h)- ux(x, t)),

za(O,t)=yo

(t)zx(O,t)+(h-'(Yo(t+h)-Yo(t))-DrYo(t))ua(O,t)

+h-'(yo(t+h)-yo(t))(uJ0,t+h)-u,,(0,t)),

z'(1,t)=Y(t)z,(1,t)+(h-'(Yr(t+h)-Yr(t))-Dyjt))uJI,t)

+ h -'(y,(t + h)-Y1(t))(ua(l, t + h)- ux(1, t)).

thus za - 0 as h -* 0.

It remains to show that Assumption (III) holds. In possession of

(7.6.25) we go back to the boundary value problem defining v. and obtain,

7.6. Parabolic Equations with Time-Dependent Coefficients

using (7.6.8), that

423

Ilvx(

t)II <

IX+l l

Ilullwz,o+C(l ua(O, 01 + l ua(l, t)I ).

We use next (7.6.9) and (7.6.10) for u, keeping in mind that u satisfies a

homogeneous boundary value problem. In view of (7.6.18), we obtain

l l vx (' , 011 <

I X+ l l l/z

(X E 2, 0< t< T),

(7.6.26)

which completes the verification of (III) and thus the proof of Theorem

7.6.2.

7.6.3 Theorem.

Let the hypotheses in Theorem 7.6.2 be satisfied.

Assume in addition that Dra, D1a', Drb, D,c, are Holder continuous in t

uniformly with respect to x and that D,yo, Dty, are Holder continuous. Then the

family (7.6.15) satisfies the assumptions of Theorem 7.2.5; in particular, the

Cauchy problem for (7.6.16) is properly posed with respect to the strong

solutions defined in (V), Section 7.2).

Proof. We only have to verify Assumption (IV) in Section 7.2. We

can in fact show that if a is the Holder exponent of D1 a, D, a', Deb, D, c,

Dryo, Dry,, then

(0<t<t'<T),

where yk(x, t, t') = (t'- t) -°`(va(x, t')- vk(x, t)). This is done by examina-

tion of the boundary value problem satisfied by yk, which is easily deduced

from (7.6.22). Details are omitted.

The case where one of the boundary conditions is of type (II) is

handled in a similar way. When both boundary conditions are of type (II)

the argument becomes in fact much simpler, since all functions employed

(uk, v k, ...) satisfy the homogeneous boundary conditions and no results on

the nonhomogeneous problem are necessary. Of course the same is true if

/30, /3 are of type (I) but independent of t.

Finally, we comment briefly on the extension of the results to partial

differential operators. Using again the divergence form, we write

A(t;/3(t))=E F, DJ(ask(x,t)Dku)+Ebj (x,t)Diu+c(x,t)u,

where the coefficients ask, bj, c are defined in SZ X [0, T], SZ a bounded

domain in Rm, and /3(t) is a time-dependent boundary condition on the

boundary IF. As in dimension one, the case where /3(t) is the Dirichlet

boundary condition is much simpler, and we indicate how to adapt the

arguments above. The assumptions are as follows. For each t, the coeffi-

cients of the operator and the domain satisfy the smoothness assumptions in

Theorem 4.9.3; moreover, ask, Djajk, bj and c are continuously differentia-

424

The Abstract Cauchy Problem for Time-Dependent Equations

ble with respect to t in F2 ttx [0, T]

and

Eajk(x,t)SjSkiKI512 (5ER

, (x, t)ES2X[0,T])

for some IC > 0. The one-dimensional reasoning adapts without changes, the

only estimates needed being those for the homogeneous boundary value

problem (we note incidentally that the "intermediate" estimate (7.6.9) is

unnecessary). Under these conditions the family

(A (t; #(t)); 0 < t <T) (7.6.27)

satisfies the conclusion of Theorem 7.6.3. If one wishes to fit (7.6.27) into

Theorem 7.6.4, it is enough to require that Dtajk, DlDjajk, Dtbj, and Dc be

Holder continuous in t uniformly with respect to x E 9.

The case where the boundary condition is of type (I) must be

handled with the help of estimates on the nonhomogeneous boundary value

problem (see Example 7.6.6 below).

7.6.4

Example.

State and prove analogues of Theorems 7.6.3 and 7.6.4 in

the case m > 1 for Dirichlet boundary conditions using the suggestions

above and the results on partial differential operators in Chapter 4.

*7.6.5

Example. A multidimensional analogue of (7.6.5). Let 9 be a

bounded domain of class C(2),

a function defined and continuously

differentiable on the boundary F. Consider the space W',P(2), 1 < p < cc.

If d(.) is a function in W"(), then there exists a function p(-) in

W2^P(SZ) such that

D"p(x) = y(x)p(x)+d(x)

(x E I')

in the sense of (4.7.37). Moreover, p can be chosen in such a way that there

exists a constant C independent of d(.) such that

1IPIIw2' (O)<CIIdIlw'o( )

(7.6.28)

The outline of the proof is as follows. By means of boundary patches

and their associated maps, we can reduce locally the problem to the case

SZ = R m, IF = R and then reason as in Example 4.8.19 via an approxima-

tion argument.

7.6.6 Example. Using Example 7.6.5, establish a multi-dimensional ana-

logue of Lemma 7.6.1.

7.6.7

Example.

State and prove extensions of Theorem 7.6.2 and Theo-

rem 7.6.3 in the case m > 1 for boundary conditions of type (I).

7.7. THE GENERAL CASE

We study in this section the abstract differential equation

u'(t) = A(t)u(t) (7.7.1)

under hypotheses that can be roughly described as follows: each A(t) is

7.7. The General Case

425

assumed to belong to (2+ and relations among the operators A (t), A(t'),...

are imposed in order to make possible the construction of S(t, s) under the

"product integral" form sketched in Example 7.1.8 in the bounded case.

To simplify future statements, we introduce several ad hoc notations

and definitions, several of which were already used in Section 5.6. If E, F

are two Banach spaces, we write F - E to indicate that F is a dense

subspace of E and that the identity map from F to E is bounded.

Expressions like A E (2+(w; E), A E C+(C, w, E) indicate that A belongs to

the indicated classes in the space E. Given an operator A E (2,(E), we say

that F is A-admissible if and only if

S(t;A)FcF (t>,O)

(7.7.2)

and t - S(t; A) is strongly continuous in F for t > 0. We denote by AF the

largest restriction of A to F with range in F, that is, the restriction of A with

domain (u E D(A) n F; Au E F). The reader is referred to Section 5.6 for

examples and relations between these definitions, especially Lemma 5.6.2,

which will be useful in the sequel.

A family (A(t ); 0 < t < T) of densely defined operators in the

Banach space E is said to be stable (in E) with stability constants C, w if and

only if n p(A(t)) Q (w, cc) and

fl R(X; A(tj)) <C(X- w)

(X> w)

(7.7.3)

j=1

for every finite set (tj), 0 < t, < t2 <

< t,, < T. Here and in what follows,

products like the one in (7.7.3) are arranged in descending order of indices,

that is,

flBJ=BnBn-t...B,.

(7.7.4)

j=1

It is plain (as we see taking, = t2 = = tn) that each A(t) in a stable

family belongs to (2+(C, w). Two equivalent forms of the definition are

proved below.

7.7.1 Lemma. (a) Inequality (7.7.3) is equivalent to

F1 R(Aj; A(tj))

j=l

<CF1

(Xi-w)1

j=1

(7.7.5)

for all (tj),(Xj)such that 0<tI<t2< <tn<Tand X1,A2,...,Xn>w.(b)

A family A(.) of operators in e+(w) is stable with stability constants C, co if

and only if

11 S(sj; A(tj)) <Ce"(s'(7.7.6)

j=l

for all (tj) as above, s,, s21 ...,sn > 0, S(sj; A(tj)) = exp(sjA(tj)).

Proof.

Assume that A(.) is stable, and let the (sj) (s s2,

....

sn ? 0)

be rational, sj = pj/qj (pj, qj integers). Let n be a positive integer. Consider

426

The Abstract Cauchy Problem for Time-Dependent Equations

the operator

B, ,

{-n'R(-n';A(ti))} , (7.7.7)

j=1

sj Si

where nj = nqt

gnsj = nqt '

' q; - i p;qj+ i '

' qn. Since nj /sj is indepen-

dent of j, we can use (7.7.3) for A = n j /sj and the family

(tt.... ,t1, t2,...,t2,...4n,...401

each tj repeated nj times, obtaining

fS_J

IIBnII1< C1

(1- (7.7.8)

J=I ni

Observe next that, in view of (2.1.27) each of the n factors on the right-hand

side of (7.7.7) converges strongly to S(sj; A(tj)), while the product obtained

deleting any number of factors from (7.7.7) is uniformly bounded, just as

the total product, by virtue of the stability assumption. This is easily seen to

imply that

Inn

Bn - 1 1 S(sj; A(ti))

/_1

strongly, thus we obtain (7.7.6) for sj rational from (7.7.8). The general case

is proved exploiting the strong continuity of each S(s; A(te)). Inequality

(7.7.5) is then obtained noting that

fl R(Aj; A(ti)) u = f

e-X(s,+...+sn)

11 S(sj; A(tj))udsi ... dsn.

l=1

J=t

(7.7.9)

Finally, it is obvious that (7.7.5) implies (7.7.3) if we take all the X j equal to

X. This completes the proof of Lemma 7.7.1.

7.7.2 Remark. To verify the stability condition (7.7.3) (in any of its

equivalent forms) is usually no easy task, except of course in the trivial case

where all the A(t) are m-dissipative, or, more generally, when all the

A(t)- wI E e (1,0) for some fixed w; in fact, in this case, if A > w and

0<t,< <tn<T,

FIR (X; A(ti))II < (X - w)

(7.7.10)

The following result generalizes this observation to the case where

the above condition is satisfied after a certain (time-dependent) renorming

of the space.

7.7.3

Example (Kato [ 1970; 1 ]).

For each t, 0 < t < T, let I I' I I, be a norm

in E equivalent to the original one. Assume that there exists a constant c > 0