+ K 1 (t, s, r) + K 2 (t, s, r) - K 3 (t, s, r))xds, then using (1.3), (3.7), (3.8), (3.9), (3.10), (3.11) and c t -. 6 0 , and writing. M. IY) = ° 6 3 (t - s ) ds t- p ' t. T. = M269 , P') =.
J . M a th . K y o to Univ. (JM KYAZ) 30-1 (1990) 59-91
Cauchy problem for abstract evolution equations of parabolic type By Showji KAWATSU
§ 0 . Introduction In this paper, we treat the Cauchy problem of the abstract evolution equation d — d t u + A (t) tt = f(t)
(0.1) u(0) = u
0 0, t E [0, T ]. (A3): There exists a positive constant C 3 independent both of t and s such that 0
(1.4)
11A (t)A(s) C -
3
f o r t, s e [0, To ].
(A4): F o r any t, S , T e [0, To ] (1.5)
II(A (t) — A (r) ) A(s)
- 1
1 (a(lt —r1),
Ev olution equations of parabolic type
61
where w (r) is a positive and m onotone increasing function defined in th e interval (0, cc) a n d satisfying the following c o n d itio n s (1.6) and
(1.7). T h a t is, putting 65(r) = w(r)/r,
i•or. 0ti(r)I logr Idr < cc
(1.6)
w(r)llogrl
(1.7)
0 as r
From (1.6) it is easy to see that
j• To 6 (r)d r < cc
(1.8)
Recall that in Sobolevski [1] w (r)= ra (0 < a Here we define a fundamental solution.
1).
Definition. An operator-valued function U(t, r), with values in B(E), defined strongly continuous in t, T for 0 T t To is called a fundamental solution and of the homogeneous equation d — u(t) + A (t)u(t) = 0
(1.9)
for 0 < t
To
dt
if (1): th e derivative
0
U (t, r ) exists in th e strong topology a n d belongs to
—
O t
B(E) for 0 T < tT o , a n d it is also strongly continuous in t, r for 0 T t To ; < t T0 ; (2): the range of U (t, r) is included in DA for 0 and if (3): for any x E (1.10)
0
for 0
U(t, r)x + A (t)U (t, t)x = 0
lim U (t, r)x = t it
(1.11)
T
t
T,
X.
Proposition. A ssume th at the conditions (A ,), (A 2 ), (A 3 ) and (A 4 ) hold. Then there ex ists a f undam ental solution U ( t , t ) of the hom ogeneous equation (1.9). W e now state th e assumption (F ) for the inhomogeneous term f (t ); ( F ) : f ( t ) is a continuous function with values in E a n d satisfies
(1.12)i provided that 0
f(t) — f (s) II ds -
j
0
Ev olution equations of parabolic type
65
a)(1 t
I
(2.17)
(exP(— A(t)) — exp(— 11(s)))11
s I) —
C
)1, T.
w here C denotes a positiv e constant independent all of t, s, P ro o f . Writing (A(t) — A(s))exp(— TA ()) = (A (t) — A (s))A ()
- 1
A ()ex p(— TA())
and use (1.3) and (1.5), (2.13) holds. L et us define a n operator valued function O G ) exp( — —
(t)) exp( — CA (s))A (s)
- 1
.
Then 4'(C) is continuously differentiable in th e sense o f operator n o r m . So integrating O '() from 0 to T, we have (exp( — TA (s)) — exp( — T A (t)))A (s)
(2.18)
=
exp(
—C)A(t))(A(t) — A(s))A(s)
- 1
exp( — A (s))4.
Using (1.3), (1.5) and (2.18), we get (2.14). To prove (2.15), (2.16) and (2.17) we divide exp( — T A (t)) — exp( — T A (s)) into several p a r t s . Writing = ( e x p ( — A (t)) — exp( — A (s)))A (s)
- 1
A (s)exp(— A (s)),
(t)) — exp — — A (s))), / 2 = e x p ( — A (t))(exp( — 2 2 2 we get exp(— TAW) — exp(— TA(s)) + I. Writing = exP( — A (t)) (A(s) — A(t)) A(s)
- 1
J2 = A (t)ex p( — A ( t ) ) ( e x p — A (t)) — exp( — A (s))) A (s)
2
J3 = e x p ( — A (t)) (A (t) — A(s))A(s) - 1 e x p ( — A (s)),
we get /2 = J, + J2 ± J3.
A nd writing
1
Showji Kawatsu
66
K = (A (s) — A (t)) e x p ( — A (0 )e x p ( — A (s))
K2 =
e x p ( — A (t)) (A (t) — A (s)) e x p ( — A (s))
K3 = ( e x p ( — A ( t) ) — e xp( — A (s))) A (s)exp( — - A (s )), 2
we get A(S) /1 = K1 +
K2 + K . 3
U sing (1.3), (1.5) a n d (2.14), w e g e t th e bounds II I II, IIJ1 II, IIJ2 II, IIJ3 I cw(it — sp, so we have (2.15). In view of (1.4) it suffices to consider the case s for the proof of (2.17). Using (1.3), (2.13) and (2.15), we get the bounds I
IlA ( s ) / 2 , K ill,
II K2II, II K3 I
C
w(1 t — s l)
, s o w e g e t (2 .1 7 ). T h e p ro o f of
(2.16) is sim ilar to that of (2.17).
Lemma 2.3.
Q.E.D.
The following inequalities hold for all T,
(2.19)
tA(T)) — exPl — sA(T))) A(n)
(2.20)
II
-
e [0, To ] and t, s
0.
CI t — sl
Il A ( ) texP( — tA(T)) — exl3( — sA (TM A(?1) - 1 II
CI t
—
m in(t, s)
w here C denotes a constant independent all of t, s, T, P ro o f . Since A err (exp( — tA(r)) — exp( — sA(r))) A (T)
(2.21)
=
- 1
A (t) exp(— ÇA(r))d
where n = 0, 1 and t, s > 0 when n = 1, using (1.3) and (1.4), we get (2.18) and (2.19).
Lemma 2.4. (2.22)I
Q.E.D.
The follow ing inequality hold for all t, s,
I
A ()1A (t)
(2.13)
II A(0
- 1
- 1
— A(s)
— A (s) - 1
- 1
) I 5 cw(It — sl)
II 5 Cw(lt
—
sl)
w here C denotes a positive constsant independent all of t, s, Pro o f . U sin g (1.4) and (1.5) we see that II A O ( t )
- 1
— A (s) - ) II = 1
IIAO A (t)
- 1
cw(lt — sl).
(A (t) — A (s)) A (s) -
II
Ev olution equations of parabolic type
67
S o w e get (2.22). And (2.23) is clear from (2.22) and (1.4).
Q.E.D.
Lem m a 2.5. T h e operator-v alued function A(t)exp(— T A (s)) is unif orm ly continuous in the sense of operator norm w ith respect to t, T, s w here 0 To, e T T o , 0 s T o f o r any positive number E. P ro o f . If 0
t+
To, F.
TA T
T 0 ,0 S z ls T
o
then
A (t + t) exp(— (r + AT) A(s + As)) — A(t)exp(— TA(s)) = (A (t + At) — A (t)) A (s + As) A (s + A s) exp(—(t + AT) A (s + As)) + (A (t) (exp( — (*; + A T) A (s + As)) — exp( — A (s +
s))) A (s + s) A (s + A s)ex p( — A (s + A s))) -
+ A(t)(exp(— rA(s + As)) — exp(— /1(s))) Applying (1.3), (1.5), (2.17) and (2.20), the lemma follows.
Q. E. D.
Lemma 2.6. The operator-valued functions (A (T) — A (t))exp(— (t — T)A (t)), (A(T) — A(t))exp(—(t — 'OA (t)), exp( — (t — A ( 0 ) , exp( — (t — T)A(T)) are uniformly continuous in th e sense o f operator norm w ith respect to t, T provided t — T E, t, T E [0, To ], f o r any positive number e. P ro o f . T h e continuity o f (A(T) — A(t))exp(—(t — t) A (r)) a n d ( A ( r ) — A(t))exp(—(t — T)A(t)) is obvious from lemma 2.5. W riting exp(— (t — r)A(r)) = A (T) exp(—(t — r) A (r)) A ( r ) ' and exp( — (t — r) A (t)) = A (t) exp( —(t — r) A (t)) A ( t ) ', the continuity o f these tw o functions is obvious from (1.3), (2.23) and lemma 2.5. L em m a 2 . 7 . F o r a n y XE E , t h e fu n c tio n A (t)exp(— T A (s)) A () continuous with respect to t, T, S, E [ 0 , T o ].
-1
x is
P ro o f . Writing 1, =(A(t + At)— A(t))A(s + As)
1 -
exp(— (T - A T) A (s + A s)) A (s+ A s) A ( +
I , = A (t)exp(— (r+A r)A (s+4s))A ()
-1
(A (0— A (
A (+ A 0
= A (t)(exp(— (t+ A T) A (s+ A s))— exp(— (r+ A T) A (s)))A () 1 = A (t)A (s)
1
(exp(—(r + AT) A(s)) —exp( — T A(s))) A (s) A()
-1
-1
-1
,
we get A (t + A t) exp(
+ A T) A (s + A s)) A ( + A 0
-1
— A (t)exp(— r A (s)) A ()
-1
= + / 2 + 13 + Using (1.3), (1.4), (1.5) and (2.16) we see that 11/ C o)(14t1 ), II /2 II Cco( A O and 111 C o ) ( 1 4 .9 1 ) . A s f o r 1 , from the strong continuity of the semigroup exp 1
3
4
Show ji K aw atsu
68 (
A.(s)) and (1.4), for any x e E as A T
/4.x11 —>
0
Thus w e have proved this lemma.
Q.E.D.
Corollary. T h e operator-v alued functions exp(—(t —T ) A ( r ) ) , exp(—(t — A(t)), (A(r) — A(t))exp(—(t — r)A(r))A(r) a r e strongly continuous i n t , T, w hereO r .5_t To . - 1
-
L em m a 2.8. If 0 t — T (50. U(t, T ) , which is given by the formula (2.6), defines a B(E)-valued fu n c tio n . And U(t, r) is strongly continuous with respect to t , T. Morever if 0 < t — T So , U(t, T ) is continuous in th e sense o f operator norm. P ro o f . From the convergence of the series (t, r) in th e sense of operator n o rm a n d lem m a 2.6, w e conclude that U(t, r ) is continuous in th e sense of operator norm provided 0 < t — T 60 . From th e corollary to lem m a 2.7 and
exP1 (t — s)A (s))0 (s,
< co ,
—
we conclude that U (t, r) is strongly continuous in t, T for 0
t —T
So .Q.E.D.
§ I I I . Local strong differentiability o f a fundamental solution In this section we prove that, for any x e E, the strong derivative
—
U (t,
r)x
.0 t
6 0 , th a t U(t, r)x e D , T for 0 < t — T 0 R e m a r k t h a t t h i s i s local a n d t h a t — U(t, r)x + A(t)U(t, r)x = 0 holds.
exists and is continuous with respect to t,
A
Ot
differentiability near the diagonal set. For the proof of global differentiability, we need a lemma on the uniqueness of solution of homogeneous equation, which will be proved in th e next section. L et us consider a n operator-valued function U p (t, r) for e t —T So ,
(3.1)
Up(t,
r) = exp(—(t — r) A(r)) +
exp( (t — s)A (s))0 (s, r)ds, —
where E is any fixed positive number and 0 < p < v1 3 . By virtue of the continuity o f th e continuity o f 0(s, r ) a n d exp(—(t — s)A (s)) a n d th e differentiability of analytic semigroup, w e se e th a t, fo r a n y x e E, U p (t, r )x is differentiable with respect to t. A nd U p (t, r) satisfies
(3.2)
0
Up(t,
r)x = — A (r) exp( —(t — r)A(r))x + exp(
—
p A(t — p))0(t — p, r)x
69
Ev olution equations of parabolic type t-p t
A ( s ) exp ( —(t — s) A (s)) (s, -r)x ds.
Using (2.2) a n d (2.7), we see that Up(t, r)x = — A (t) exp( — (t — A (T))x
(3.3)
+ (exp(— pA (t — p)) — 1)0(t, -r)x + (1— exp(— p A (t — p))) f eX P (
—
eX P (
—
-p
0 1 (t, s)0(s, -r)xds
p A (r — 0(01(t — P, t) 0 1 ( t, T ) ) x t p -
p A (t — p)) f
(0 (t — p, — 01(t, s)) 0(s, t)x ds 1
t-p
J
— A(t) e x p ( — ( t — s)A (s))0(s, r)xds = / 1 ± / 2 ± / 3 ± / 4 ± / 5 — 16.
N ote th a t / — 1
= — A(t) U p (t, T) so w e have
a p
-ox + A (t)U (t, -r)x = 1 + 1 + 1 + 1 .
(t,
p
2
3
4
5
a
To prove uniform convergence of —U (t, r)x as p I 0 with respect to t, T for e et t — T S o , w e need the following lemma 3.1 a n d lemma 3.2. L e m m a 3. 1 . II/2 I 111311/11,x11, 111411/11x11, 11/511/11-xll uniformly with respect to t, T for E t — T 6 0 . ,
tends t o 0 a s p 10
Lemma 3 .2 . / 6 tends to a lim it as p 0 uniformly with respect to t. T for E t — To
.
Admitting lemma 3.1 and lemma 3.2 for the moment, let us finish the proof of local differentiability. S i n c e A (t) is a closed operator, it follows from lemma 3.2 that
f t
(3.4)
exp(—(t — s)A (s))0(s, -c)xdse D ft—P
exp(—(t — s)A (s))0(s, -r)xds
lim A(t) pj0
= A (t) f exp( — (t — s)A (s))0(s, -r)xds. t
70
Showji Kawatsu
a at
In view of lemma 2.6, it is easy to see that U ,(t, r)x and — U (t, r)x are uniformly continuous with respect to t, T for E t - T . And it is clear that U (t, r)x tends to U(t, r)x as p . 0. Moreover, from (2.6), (3.3), (3.4), lemma 3.1 and lemma p
a at
3.2, w e see that U(t, x)xe D A and — U (t, r)x tends to — A (t)U(t, r)x uniformly with respect to t, T for e t
—T
6 0
.
Thus we conclude that U(t, r)x e D A a n d
a —u(t -ox exists and is continuous with respect to t, T for 0 < t —T at
So a n d that
the equation a
— at u(t,T)x+
A (t)U(t, x)x = 0
holds. Before beginning the proof of lemma 3.1 an d lemma 3.2, we divide 0 (s, 2) (t, T ) and 0(s, x) — 0(t, r) into several parts and get the estimate of each part, which will be used in the proof of lemma 3.1 and lemma 3.2, and we state these estimates a s lemma 3.3. In what follows the letter C denotes various positve constants independent of t, T, p , b u t possibly depending o n So a n d e. Using (2.2), we get 1
-r) — 0 1 (t, r) = (A(t) — A(s))exp( —(t — x)A(r )) -
-
+
(3.5)
(A(r)
—
A (s))(exp( — (s — x) A(x)) — exp( — (t — x)A(r)))
= J i (t, s, -r) + J 2 (t, s, r) and using (2.7) and (3.5), we get (3.6)
0(s,
(t,
T) -
= (0
1
(s, 2 )
-
(01(s,
-
01(t, r) ( r , r)dr
,
-
J2(t, S ,
= 11t,
1(t, s, r) 0 (r, x)dr + 01 (t , r) ( r , r)dr
=
-
— 01 (t r))0 (r, r)dr
+
•
(t, r))
(t, • K i(t, S, - K 3 (t, S,
J2(t, S , K 2(t, S ,
J 2 (t, s, r) 0(r, -r)dr
71
Ev olution equations of parabolic type Lemma 3 . 3 . If O
r< s< t T
o
w e have the following inequalities.
C
11J 2 (t, s, '011
(3.8)
s)
a* C t— —
1(t, s, -011 5
(3.7)
(t — s)6(s — r) t— fs
(r — r )
di
li K i (t, s, t) 11 -- CW(t — S) j , t — r
(3.9)
Cs (7 )(s —
IlK 2 (t, s, "OM
(3.10) (3.11)
gt
r )6 (r —
r)
— s)
i
dr
t—r
T
63(t — r)c (r — t)dr
C
11K3(t, s, '011
d r
-
S
Writing
Proof o f lem m a 3.3. ( 3 .7 ) follows from (2.13). J2(t , s,
1
")
1 = (A (s) — A (r))A ery A
(-0 exp( —( — T)A (T))C", 2
and using (1.3) a n d (1.5), we have r) II 5 cw (s — —
=C
f s
1 —
)2
(
lc
t s (75 (s — r), t—
thus we get (3.8). Using (3.7), (3.8) and (2.12), we get (3.9) and (3 .1 0 ). And (3.11) follows from (2.10) a n d (2.12). W e proceed now to th e proof of lemma 3.1 a n d lemma 3.2. Proof of lemma 3.1. At first, let us estimate 12 S i n c e .0(t, t)x is uniformly continuous with respect to t, T for E t — t 15_ , for any positive number y, there exists a finite set T o = {(t , T,),•••,(t , TN )} of points of the set F = { (t, T y e t — 6 , t T O such that, for any (t, r)e T , w e can take (t e , r e)e T o which satisfies •
N
1
0
0(ti, T )x — 0(t, r)xll
(3.12)
i
Y 2(C + 1) 2
where C i s th e co n sta n t in (1.3). B y lem m a 2.7, exp(— A (ii))x is uniformly e [0, To ], so there exists a positive n u m b e r su c h th a t, if p < continuous for then 2
(3.13)
II(exP(— PA (t —P)) —
(t
x
2
for any t e [E, To ] and i = 1,2,•••, N . Therefore, if p < .5 then for any (t, T) E F we can take (t e , T ) E F o which satisfies (3.12) and we see from (3.13) that for any t, r i
72
Showji Kawatsu 111 211 15 11(exP( — p A(t — p)) — 1)(0(t, 1.)x — 0(ti, + (exP( — P A(t P ) ) —
0
(t1, ti)x
YY — 2 2 provided s t — r S o . So we conclude that II 1 2 11tends to 0 as p 0 uniformly with respect to t, T for e t — 60. To estimate 1 3 , we use (1.3), (2.10), (2.12) and e t — T 60 , p < e l3 . If we p u t t — s = r, we see from (1.8) that .
II1 311
di(t — s)(76(s —-c)ds x
C
C
t —p
co(60) P w (r)dr x . 2e/3J 00
So we conclude that 111 3 II / II XI tends to 0 as p 0 uniformly with respect to t, e t— 60. To estimate 14 , we use (3.5). Then we have 11411 = 1 exp( — p A (t — p))(J 1 (t, t — p, T) Using (3.7), (3.8), e
t—T 1
60
J 2 (t,
T
for
T
for
t — p, T))x
and p < e/3, we see that
4II 5 c(
w (p )
(6 0 )
a) ) x + p2e2/3
E
•
So we conclude that II 1 4 II/IIX II tends to 0 as p 0 uniformly with respect to t, e t — t (50 . To estimate 15, we use (3.5) and (3.6). Then we have 15 = exp( — p A(t — p))(K i (t, t — p,
K2(t, t — p, -c))x.
Using (1.3), (3.9) and (3.10), we have r— P
(3.14)
/511
C x (1)(P)f
C o'(r — r-r) d t
t
r
—
C x P f t - P (13(t
tir)rd)(r
As for the first term of the right hand side of (3.14), put t — T and use e T • J o , we see from (1.7) that 0
4)
f
t— P
t
=
d r
T and Y —T
65(r — -r) Idr =Tco(p) - P Tc ' 6—( r 'r') dr' t—r o = —1 co(p) I T T 0 1
—(
(r ' p
(
T—r' ± W
)
(7)(0)dr' no
UW ( 0 ) 00)(1 1 0 001
65(r)dr')
110gP1) 0
0 as p 0,
=
Ev olution equations of parabolic type
73
where the convergence is uniform with respect to t, T for E t — T = T So . As for the second term of the right hand side of (3.14), p u t t — t = T , then we have,
15 1 = p .
p f = T j,
t - P
(7)(t — p — r)(7)(r t—r
-c)
Cojt — p — r)co(r — t—r
T)p
dr
dr +
f t
-
P
—
(7)(t — p — 06) (r — t)dr,
T
p u t t — p — r = r' a n d apply (2.9) to the second term and use e T < E13, then w e have 6
w ( 0)
15 . 1
E
C
6
T - P
p +
o 6
r ', w e have -
M 5 (p, p') -= -I I T
P'
1 —
p s'
2 w (6 °)
CISI 0 '
0 -
2
1
r
1 '
& I
= 2 a)
-(
's '
c ls '
0
s'
s
0
6 (r)dr' f
e
) 0o) . { P ' o
I' 1
-ds' / s'
65(01logr' 1 dr'
'
=
> 0 as
—
T (5 1 3 .
r+ L
t—p
(P, P )
r')dr'
Co(r')dr'9
uniformly with respect to T for e __ t To estimate M4(P, P'), writing
M4.1
'
) w ( r ) w (T
1
T - r'
T and use 0 < p < p'
,
00) f
e
,+
T =
3 6 (s - r)6 (r - r)
ds
J
t-p t—P
M 4 .2(p, p') =
ds
s
d r
r
t
r )6 (r -
—
, ,
t
—
3
we see that M4(P1 p ) = '
(P P').
(P P')
+ M 4 .2
r
w (s - r ) c o ( 6 0 )
M 4 .1
,
,
2
As for M 4 1(p, p'), use t - r then w e have .
M4.1
(P,
W
(c O)E
E, S —
/
and put r -
=
r',
3
P) f 6 .5 ( r ) d r ' - - - * 0
as p , p ' 0
2e2/9
As for M 4 .2 (p, p'), use co(r then w e have
T)
w(60 ), r -
T
- 3
and put t - r = r ', s - r = s',
Showji Kawatsu
78 „
n
s
< w(6o)u
s
i
(7)(s — r)
t-p t '
ff
' ( r) (7)
dr
r
dr' ds' --> 0 a s p' 10,
D
where D = { (r', s')Ir' t — t _ _ So , s' 0, 0 r' — s' p'}. Thus we conclude that M 4 (p, p') tends to 0 as p, p' 1, 0 uniformly with respect to t, T for £ t - T < T h i s completes the proof of lemma 3.2. § IV. Uniqueness o f fundamental solution I n th is section w e prove global existence and uniqueness o f fundamental so lu tio n . W e first prove local uniqueness near the diagonal set. Definition. A function u(t) with values in E is called a strong solution of the initial value problem (4.1) o n [T, d u(t) + A (t)u(t)= 0 —
dt
(4.1)
