- ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY 289
One defines
etA ='= -
1
-j e^0 '1"R (>. A) d>.. 27ri -y ' '
where 'Y is a contour around the spectrum. If A is sectorial, then t -* etA is a
strongly continuous, analytic semigroup. That is, limt-+O 1 letAx - xi I = 0 for all
x E Ea, the map t-* etA is analytic, and e(t+s)A = etAesA.
If f E C([O, T]; E1), it is not hard to show that formula (35.11) admits a rigorous
interpretation. In this case, u(t) E C([O, T]; E 1 ) n C^1 ([0, T]; E 0 ) solves the initialvalue problem (35.10). However, without such a strong restriction on f , things
may not work so smoothly. Assume that we only have f E C([O, T] ; E 0 ). In this
case, we are only able to conclude that Sf defined by (35.11) satisfies
Sf E C^8 ([0, T];E 0 ) n C([O, T]; Eo),
where Ee is (for the chosen Banach spaces E 1 c Eo and for a ny e E (0, 1)) the
corres ponding "interpolation space", as defined below. One is not able to claim that
u( t) = Sf is a solution of the initial value problem. Let us see why. We may assumethat w = sup{Re >. : >. E cr(A)} < 0 without loss of generality since otherwise we
may consider A - wj. Then one has the operator bound jjAketAllt:o :::; C/tk, and
one calculates that
jjerAu(t) - u(t)jj£o = lllat (e(r+t-s)A -e(t-s)A) f(s) dsll£o
= lllat for Ae(a+s)A f(t - s) dcrds L o
:::; Cmaxllf(s)llt:a f t r dcrds
s lo lo er+ s
:::; Cmaxlif(s)llt: s o T log (~)T.
(See Lemma 6.9 of [84].) Note that in this calculation, the constant C dep ends upon
the time limit T. This dependence is what allows us to eliminate the dependence
on t of this quantity. So I lerAu (t) - u (t) I lt:o is only o( T^8 ) for e < l. One can show
that u E C^8 ([O, T], Eo) similarly. Unfortunately, this result is optimal.
One way to handle this is to assume more regularity on f. An alternative
approach, which does not require enhanced reg ularity, is the maximal regularity
method, developed by D a Prato and Grisvard in [91]. For this method, we nee d
to carefully d efine interpolation spaces Eo. For e E (0, 1), let J/^1 denote the little
HOlder space defined as follows : If A generates a strongly continuous se migroup,
then for u E E 0 and u(t) = etAu,
u(-) E h^8 ¢:? lim t..J,O r^0 llu(t) - ullt: 0 = 0.
Define
Ee= Ie(EoJ1) ~ { u E Ea: etAU E h^0 }.
(See Subsection 2.1 b elow for more comprehensive definitions of the little Holderspaces.) Da Prato and Grisvard h ave proved t he rem arkable result that if f takes
values in Ee, then
u(t) =lat e(t-s)A f(s)ds E C^1 ([0, T]; Ee) n C([O, T]; E1)