- ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY 291
We choose 'f/ so that 5. + 'f/ < 0, and we let C 71 be the weighted space of functions
f such that t---+ e^71 t f(t) is continuous and bounded. We define an operator
A: B(O, R) c C~([O, oo); Ee) n C 71 ([0, oo); Ei+e)---+ C 71 ([0, oo); Ee EB Ei+e)
byA(u) = (u' - F(u), uo),
where R < Ro. It follows from maximal regularity theory (see for example the
results of Da Prato and Grisvald [91]) that the linear initial value problem
v' = F'(O)v,v(O) = vo
has a unique solution and that the linea r operatorA' (O)v = ( v' - F' (O)v, vo)
is a n isomorphism.
For nonlinear equations, we can prove the following proposition. We assume
that we can rewrite (35.13) in the form
x' =Ax + f(x(t)),x(O) = x o,
where A is linear, and that the following conditions hold:
(1) f E C^1 (Ei+e,Ee).
(2) A is sectorial.
(3) f'(x) is lo cally Lipschitz continuous; i.e., there exists Ro such that
llf'(x) - f'(y)llL(£ 1 ,£ 0 )
1.f
sup llx -yllei < oo llxlle 1 , llYll£ 1 <Ro.PROPOSITION 35 .19. There exists r > 0 so that if lluoll£i+ 8 ::::; r, then th e
initial value problem (35.13) (subject to conditions (1)- (3) enumerated above) has
a unique solution
u E C~([O, oo); Ee) n C 71 ([0, oo); Ei+e).
Moreover,
t-+oo lim lle^71 tu(t,uo)llc <-1+8 =^0
uniformly with respect to u 0.
PROOF. The claim is proved essentially by an implicit function theorem argu-
ment. We clarify the argument using fixed point theory.
Consider a solution in the weighted spaceY = C 71 ([0, oo); Ee+1) n C~([O, oo); Ee),
where 'f/ is chosen so that 'f/ + w < 0, with w = sup{Re .A : .A E a-(A)} < 0. We look
for a fixed point of the map I'(x) = ~' which maps x to the solution of the linear
initial value problem((t) =A~+ f(x(t)),
~(O) = x o.
If u E B(O, Ro) CY, then the assumptions on f imply that f(x(t)) E C 71 ([0, oo); Ee).
It then follows from the linear bound (35.12) that
llI'(x)llY::::; C llf(x)llc,,([0,00);£ 8 )'