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

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7.2. Abstract Parabolic Equations

thus

t-T P

IIJ2II

<C(t-s) f,

(t-r)p(r-T)p

=C/(t-T)'-

(t-s)'

which implies (7.2.29) for J2. By virtue of Lemma 7.2.3,

IIJII <C(t-T)

(t - r)(T - r)p(r - s)p

397

(7.2.30)

+C(t-T)a fT

s (t - r)(r - s)p

I+I2. (7.2.31)

We estimate these integrals as follows. Making use of the inequality a''b' -''

< a + b valid whenever a, b >, 0, 0 < y < 1, we obtain

(t-T)'-

(7.2.32)

t - r

(T -r)'

Using this for y = K and noting that p + K < 1,

<C(t-T)KfT(T-r)p+K(r-s)P

<(tT)'

<C//

(t - T)K

(T-S)2p+K-I

(T - S)P

The estimation of I2 runs along similar lines; we use now (7.2.32) for

y =1- a + K, obtaining

<C(t-T)K

(T-r)'-a+K(r-s)P

(t - T)"

(t T)'

< C

(T-S)P-a+K

(TS)'

We have already gathered the information necessary to handle the

Cauchy problem for (7.2.1). The relevant notion of solution is:

(A)

is a solution of (7.2.1) in s <I< T if and only if is

continuous in s < t < T, continuously differentiable in s < t < T,

u(t) E D(A(t)), and (7.2.1) is satisfied in s < t < T.

7.2.5 Theorem. Let (A(t); 0 < t <T) be a family of operators in E

satisfying Assumptions (I), (II), (III), and (IV). Then the Cauchy problem for

398

The Abstract Cauchy Problem for the Time Dependent Equations

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

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

continuously differentiable (in the norm of (E)) with respect to s and t in

0 < s < t < T. Moreover, S(t, s)E c D(A(t)), A(t)S(t, s), and S(t, s)A(s)

are bounded,

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

DSS(t, s) = - S(t, s)A(s),

(7.2.33)

and

IID:S(t,s)II,<C/(t-s),

IIDSS(t,s)II <C/(t-s)

(7.2.34)

in0<s<t<T.

The proof of Theorem 7.2.5 depends on the following auxiliary

result, which will find further use in the next section.

let

7.2.6 Lemma.

Let Assumptions (I), (II), and (III) be satisfied and

Sh(t, s) = S(t - s; A(t))+I

hS(t

- T; A(t))R(T, s) dT

(7.2.35)

for 0 < s < t - h < T- h,

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

(7.2.36)

for t - h < s < t. Then4 (i)

Sh(t,s)-*S(t,s) (7.2.37)

in the norm of (E) as h -* 0, uniformly in 0 < s < t < T. (ii) Dt Sh (t, s) exists

and is continuous in the norm of (E), Sh(t, s)E c D(A(t)), and A(t)Sh(t, s)

is continuous in (E)for 0<s<t-h<T-h.(iii)For each uEEand k>0,

D,Sh(t,s)u-A(t)Sh(t,s)u-*0

(7.2.38)

as h -* 0, uniformly in 0 < s < t - k < T - k. (iv) S(t, s) is (E)-continuous in

0 < s < t < T, strongly continuous in 0 < s < t < T, and

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

(7.2.39)

Proof. (i) follows immediately from the fact that, if we set tB,

h =

max(s, t - h), then

S(t,s)-Sh(t,s)=f

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

ts, h

everywhere in the triangle 0 < s < t < T, and from Lemma 7.2.2. We have

already observed (see (7.2.17)) that S(t - s; A(t)) is continuously differen-

tiable as a (E )-valued function with respect to s and tin 0 < s < t < T. This

'The function S(t, s) is defined by (7.2.7); existence of the integral is a consequence of

the estimate (7.2.21).

7.2. Abstract Parabolic Equations

399

s<t-hors+h<t

Sh(t,s)

S(t, S)

(S, 0

S(t,S)

S(IS)

T-h T

Lines indicate domains of integration in the definition of

S, Sh, S. Sh

FIGURE 7.2.1

justifies the following calculation in the interval 0 < s < t - h:

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

+ f t hDtS(t - T; A(t))R(T, s) dT

_ (D, + Ds)S(t - s; A(t))- DsS(t - s; A(t))

+ S(h; A(t))R(t - h, s)

+ f t

h(Dt

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

-ft- hDS(t

- T; A(t))R(T, s) dT

=A(t)S(t-s;A(t))-R,(t,s)- ft-hR,(t,T)R(T,s)dT

ft hA(1)S(t

- T; A(t))R(T, s) dT

+ S(h; A(t))R(t - h, s). (7.2.40)

400

The Abstract Cauchy Problem for the Time Dependent Equations

This equality can be rewritten in the form

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

h+ft Ri(t,T)R(T,s)dT, (7.2.41)

where we have used the integral equation (7.2.8). It follows immediately

from these equalities that (ii) holds. To see that (iii) is verified, we write

(7.2.41) in the form

D(Sh(t, s)- A(t)Sh(t, s) = S(h; A(t))R(t - h; s)- R(t, s)

h+ft Rl(t,T)R(T,s)dT.

It is obvious that the combination of the first two terms tends to zero

strongly as h -* 0 in s < t - k. The integral tends to zero uniformly when

h -+ 0 under the same conditions; this follows easily from the estimates

(7.2.16) and (7.2.21) and yields (iii) immediately. The proof of (iv) follows

from (i) and from the fact that Sh(t, s) is (E)-continuous in 0 < s < t < T

and strongly continuous in 0 < s < t < T; (7.2.39) is evident.

Availing ourselves now of hypothesis (IV), we continue with the

proof of Theorem 7.2.5. We have

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

+ft hA(t)S(t-T;A(t))(R(T,s)-R(t,s))dT

-ft-hD,S(t-T;

A(t))R(t,s)dT

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

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

+S(t-s;A(t))R(t,s)-S(h;A(t))R(t,s).

(7.2.42)

Observe that, due to the estimate for R (t, s) - R (T, s) obtained in Corollary

7.2.4, the integral on the right-hand side of (7.2.42) (which integral is

obviously a (E)-continuous function of s, t in 0 < s < t - h < T - h) con-

verges uniformly on triangles 0 < s < t - k < T - k, k > 0 to

ftA(t)S(t-T;A(t))(R(T,s)-R(t,s))dT,

(7.2.43)

which must therefore be itself (E)-continuous in 0 < s < t < T. Since A(t) is

closed, we deduce from (7.2.42) that S(t, s)E c D(A(t)) and

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

+ f'A(t)S(t - T; A(t))(R(T, s)- R(t, s)) dT

+S(t-s;A(t))R(t,s)-R(t,s)

(7.2.44)

7.2. Abstract Parabolic Equations

401

for 0 < s < t < T, which shows that A(t)S(t, s) is continuous there in view

of the comments surrounding (7.2.43). We estimate now the integral on the

right-hand side using (7.2.11) and (7.2.29):

ftA(t)S(t - T; A(t))(R(T, s)- R(t, s)) dT

C ft

t-S ,

(Ts)P+ts(t-

)t_"

<C/

(t - S)P +

(t-S)1-K

C//

(t-S)1-K

This (and again (7.2.11)) shows that

IIA(t)S(t,s)II<C/(t-s) (0<s<t<T),

which inequality will yield the first estimate (7.2.34) as soon as we prove

that D1S(t, s) = A(t)S(t, s). Note that, as a consequence of the preceding

arguments, we have obtained

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

(7.2.45)

an inequality that will find application in Section 7.5.

It remains to be shown that S(t, s)u is a solution of (7.2.1) for all

u E E. In order to do this, we take k > 0 with s + k < t; in view of (7.2.38),

(7.2.43), and preceding comments,

kA(T)S(T,s)udT = urn A(T)Sh(T,s)udT

+ k

lira

DSh(T,s)udT

h-/0 s+k

=S(t,s)u-S(s+k,s)u,

thus our claim on S(t, s) u is justified. Actually, much more is true: since

A(T)S(T, s) is (E)-continuous in s + k < T < t, we can write the last in-

equality thus:

A(T)Sh(T,s)dT=S(t,s)-S(s+k,s),

s+k

hence DS(t, s) exists in the norm of (E) and

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

We have proved then part (a) in the definition of a properly posed Cauchy

problem, since the strong continuity of S has already been shown in Lemma

7.2.6.

402

The Abstract Cauchy Problem for the Time Dependent Equations

Clearly, part (b) will be fulfilled as well if we can prove that any

solution (as defined in (A)) of (7.2.1) must perforce be of the form S(t, s) u.

This will be achieved by essentially the same means as in the time-invariant

case (see the end of the proof of Theorem 2.1.1). In the present situation,

this would involve differentiation of the function s - S(t, s)u(s) for an

arbitrary solution of (7.2.1). However, we lack information on the behavior

of S as a function of s. Taking a hint from Example 7.1.3, we may expect

that

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

(7.2.46)

for each t. To prove this directly would not be simple, thus we use a trick

similar to that in Example 7.1.3; namely, we construct an operator-valued

solution S of equation (7.2.46) and show that S = S. The construction of S

is based on the integral equation (7.2.4), instead of (7.2.6), which was the

starting point for the construction of S. As we did in relation to (7.2.6), we

seek an S of the form

S(t, s) = S(t - s; A(s))+ f tf (t, a)S(a - s; A(s)) do. (7.2.47)

Formally, we have

DDS(t, s) = D5S(t - s; A(s))-A(t, s)

+ f tA(t, a) DsS(o - s; A(s)) do,

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

+ f tk(t,a)D0S(u -s; A(s)) do.

Therefore, if (7.2.46) is to be satisfied,

0 = DsS(t, s)+S(t, s)A(s)

= (D1 + DD)S(t - s; A(s))- A(t, s)

+ f tA(t, o)(DQ + DS)S(o - s; A(s)) do.

so that A must be a solution of the integral equation

A(t,s)- f'A(t,a)A, (a,s)do=AI(t,s)

(7.2.48)

with A t (t, s) = (Dt + DD)S(t - s; A(s)). The construction of S now proceeds

in a way exactly analogous to that of S, thus we limit ourselves to state the

main steps and leave the details to the reader.

7.2.7 Lemma.

A, (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.

(7.2.49)

7.2. Abstract Parabolic Equations

403

7.2.8 Lemma. There exists a solution R(t, s) of (7.2.48), which is

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

IIR(t,s)II<C'/(t-s)P. (7.2.50)

The function R(t, s) is obtained as the sum of the series

the R defined recursively by

An (t, s) = f 1(t, a)R1(a, s) da.

s),

We note that the last two results are independent of hypothesis (IV).

Using this assumption we obtain:

7.2.9 Lemma.

The function R t (t, s) satisfies

(

IIRi(t,a)-R1(t,s)IIC

a-s

(a-s)'

(t-s)(t-a)P

t - s

)

in 0<s<a<t<T.

7.2.10

Corollary.

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

satisfies

IIR(t,a)-R(t,s)II <C(

a - s

+ (a-s)K}

l (t-s)(t- a)°

t -s

for 0<s<a<t<T.

The analogue of Lemma 7.2.6 is

7.2.11

Lemma. Let Assumptions (I), (II), and (III) be satisfied, and

let

Sh(t,s)=S(t-s;A(s))+ ft R(t,a)S(a-s;A(s))da (7.2.51)

s+h

for h < s + h < t < T,

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

for s < t < s + h. Then (i)

Sh(t,s)- &(t, S)

(7.2.52)

in the norm of (E) as h -> 0, uniformly in 0 < s < t <T. (ii) DSSh(t, s) exists

and is continuous in the norm of (E), Sh(t, s)A(s) is bounded in D(A(s)) and

its closure Sh (t , s) A (s)

is continuous in (E) for h < s + h < t < T. (iii) For

every u E E and k > 0,

DSSh(t, s)u + Sh(t, s)A(s)u -* 0 (7.2.53)

as h - 0, uniformly in h < s + k < t < T. (iv) S(t, s) is (E)-continuous in

404

The Abstract Cauchy Problem for the Time Dependent Equations

0 < s < t < T, strongly continuous in 0 < s < t < T and

S(t,t)=I (0<t<T). (7.2.54)

With the use of Assumption (IV) we prove:

7.2.12 Lemma DSS(t, s) exists in the norm of (E) and is continu-

ous in 0 < s < t < T. For each s and t > s, S(t, s)A(s) is bounded in D(A(s))

and its closure satisfies

DSS(t, s) _ - S(t, s)A(s). (7.2.55)

Moreover,

II DSS(t, s )II < C/(t - s) (7.2.56)

in0<s<t<T.

We end now the proof of Theorem 7.2.5. Let u(., s) be an arbitrary

solution of (7.2.1) in s < t < T and let t be fixed in that range. Consider the

function

v(a)=S(t,a)u(a,s)

in the interval s < a < t. On the basis of definition (A) and of the properties

of S just obtained, we easily see that v(a) is continuous in s < a < t and that

v'(a) = 0 in s < a < t. Hence v(s) = v(t), that is,

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

(7.2.57)

This clearly implies uniqueness of solutions of (7.2.1); since t - S(t, s)u(s, s)

is a solution with the same initial value, we must have

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

(7.2.58)

Comparing with (7.2.57), this shows that S(t, s) is none other than S(t, s) in

disguise. Collecting all the properties hitherto proved of S(t, s) and S(t, s),

we obtain the claims of Theorem 7.2.5 in full.

We note the estimate

HS(t,s)A(s) -A(s)S(t-s;A(s))II <C/(t-s)'-K (0<s<t<T),

(7.2.59)

which is obtained for S in the same way (7.2.45) was deduced for S. It will

find use later.

*7.2.13 Example. A perturbation result (Kato-Tanabe [1962: 1]). Let

(A(t)) be a family of operators satisfying Assumptions (I) to (IV), and let

(P(t)) be another family of operators satisfying:

(V) D(P(t)) 2 D(A(t)) and

IIP(t)R(X;A(t))II<C/1X1,8 (0<t<T,XEZ),

(7.2.60)

where C >, 0, 0 < Q < 1 and both constants are independent of X and t.

7.3. Abstract Parabolic Equations: Weak Solutions 405

(VI) There exist constants C > 0, 0 < y 51 independent of s, t such

that

IIP(t)A(t)-'-P(s)A(s)-'ll<Clt-sl'y

(0Ss<t<T).

(7.2.61)

Then the Cauchy problem for

u'(t) _ (A(t)+P(t))u(t) (7.2.62)

is well posed in the sense of Theorem 7.2.5 (that is, the family (A(t)+ P(t)),

where, by definition, D(A(t)+ P(t)) = D(A(t)) satisfies all the conclusions

in Theorem 7.2.5). The propagator S(t, s) of (7.2.59) is obtained by means

of the perturbation series

E Sn(t, s),

n=0

where So = S and

(7.2.63)

S"(t,s)=ft5(t,a)P(a)S"_,(a,s)da

for0<s<t< T, n = 1, 2,... . (See Example 7.1.12.)

*7.2.14

Example. A G°° version of Theorem 7.2.5 (Tanabe [1967: 1]). Let

Assumptions (I) to (IV) be replaced by the single hypothesis: R(X; A(t))

exists for A E 1, 0 5 t

T, the function t - R(X; A(t)) is infinitely differen-

tiable in 0 5 t T for each X E

and there exist constants CO, C,,... such

that

IIDt"R(X;A(t))II <C"/IXI

(XE2,0 t,T)

for n = 0,1, ....

Then the propagator S(t, s) of (7.2.1) provided by Theorem

7.2.5 is infinitely differentiable in 0 < s < t

T (in the sense of the norm of

(E)). For every n =1, 2.... there exists a constant C, such that

IID"s(t, s)It s

s)", II Ds s(t, s)II

Cn/(t - s)" (7.2.64)

for0< s < t < T.

7.3. ABSTRACT PARABOLIC EQUATIONS:

WEAK SOLUTIONS

We have already pointed out (see Section 1.3) that strong or genuine

solutions of an abstract differential equation are by no means the only

suitable ones and that weak solutions of one sort of other may be used to

advantage many times. A prime example of this principle will be examined

in this section. In fact, we shall show that even if Assumption (IV) in the

previous section is discarded, we can conclude that the Cauchy problem for

u'(t) = A(t)u(t) (7.3.1)

is still properly posed in 0 < t

T if we substitute definition (A) of solution

406

The Abstract Cauchy Problem for the Time Dependent Equations

by that of weak solution, as formulated in the time-independent case in

Section 2.4. We shall use, however, a slightly different definition, roughly

corresponding to (2.4.10).

(B) is a weak solution of (7.3.1) in s < t < T with initial condition

u(s)=u0EE (7.3.2)

if and only if it is continuous in s < t < T, satisfies (7.3.2) and

f T(u(t), u*'(t)+A*(t)u*(t)) dt = -(uo, a*(s)) (7.3.3)

for every E*-valued continuously differentiable function u*(.) de-

fined in s < t < T such that a*(t) E D(A(t)*), A(t)*u*(t) is con-

tinuous and

u*(T) = 0.

(7.3.4)

7.3.1 Theorem.

Let A(.) satisfy Assumptions (I), (II), and (III) in

the previous section. Then the Cauchy problem for (7.3.1) is properly posed in

0 < t < T (with D = E) with respect to solutions defined by (B). The solution

operator S(t, s) is strongly continuous in 0 < s < t < T and continuous in the

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

Proof. We have already pointed out in the previous section that the

operators R, (t, s ), R (t, s ), R, (t, s ), R (t, s)

therein can be constructed

without recourse to Assumption (IV) and that some of their properties are

retained in this more general setting: in particular, Lemma 7.2.2 holds for

R (t, s) as well as its mirror image, Lemma 7.2.8, for R (t, s). We can then

construct S(t, s) and S(t, s) by means of the integral formulas (7.2.7) and

(7.2.47), respectively, and it is immediate that the continuity properties

claimed for S in Theorem 7.2.5 follow, as well as similar properties for S.

We note, however, that we cannot conclude without further analysis that

S = S, since the argument in the previous section was based on differentia-

bility of S and S which was a consequence of Assumption (IV).

We begin by proving that

u(t) = S(t, s)u (7.3.5)

is a weak solution of (7.3.1) with initial condition u(s) = u in 0 < t < T for

any u e E. To this end, let be a E*-valued function satisfying the

assumptions in Definition (B). We have

f T(u(t ), u*'(t)) dt

= f

T(S(t, s)u, u*'(t)) dt

= lim lim

(Sh(t,s)u,u*'(t))dt,

k0+ h-0+ s+k

(7.3.6)