290 35. STABILITY OF RICCI FLOW
is a unique solution of the initial value problem. (See Theorem 6.10 in [84] for a
detailed proof.) So this says that if the spaces chosen are the interpolation spaces
Ei+e and Ee and if the sectorial operator A: Ei+e ---+ Ee extends to a sectorial
operator A : E 1 ---+ E 0 , then indeed the formal expression (35.11) is the unique
solution of the initial value problem.
We can use this expression, as defined in terms of these interpolation spaces,
to derive estimates controlling the so lutions of linear initial value problems and
then proceed to prove stability for nonlinear evolution. We carry out this two-step
procedure relying on the following two propositions:
PROPOSITION 35 .18 (Estimates for linear initial value problems). Let u(t) be asolution of (35.10) with f E C([O, T]; Ee). If w = sup{Re >. : >. E a(A)} < 0 and if
'T) satisfies 'T) + w < 0, then
(35. 12 ) supiiet^17 tu'(t)lit: 9 +supit ie^17 tu(t)lit: 1 + 9 :::;Csupit ie^17 tf(t)lit: 9.PROOF. Define
E2 = { x E Ei : Ax E Ei} ,
Ei+e = Ie(E2, Ei).If supa(A) < 0, then llxllc '-1+9 = llAxll" c,9. If 'T] = 0, then
llu(t)llt: 1 +9 = llAullt: 9 =supi1e-eA~>0^2 e~Au(t)li'-0 ":::; sup 11e-e lot A es+;-· A Aes+;-· A f (s) dsl 1£0
r= e-e
:::; 21
-es~p Jo (~+a) 2 -edallfllt: 9:::; C llf(t)ilt: 9 ·The result follows since u is a so lution of (35.10). For general 'T), the proof is similar,
using Ji = e^17 t f and u 1 = e^17 tu. D
For nonlinear initial value problems, we apply the estimates of Proposition 35 .18
to the linearization of the equation and then invoke an implicit function theorem
to draw conclusions about the so lutions of the nonlinear equation. We sketch a
proof of asymptotic stability for nonlinear evolution of the following form. (See
Theorem 2.2 in [92] and the proof of Theorem 9.1.2 in [214] for more details.) We
work with
(35.13) u'(t) = F(u(t)), t 2 0,
u(O) = uo,
where F E C^1 (E1 and Eo), F E C^1 (Ei+e,Ee) for some e < 1 , where F(O) = 0
(so 0 is a fixed point), and where for each u in an open ball B(O, Ro) c Ei+e
of radius Ro around 0, one has appropriate hypotheses on F'(u) : Ei+e ---+ Ee.
(Assume, for example, that it generates an analytic semigroup and has bounded
norm.) If the supremum >. of the real parts of the eigenvalues of F' is negative,
then one can show that there exists an r > 0 so that if lluoll" '-1+ 9 < r , t hen u E
C 17 ([0, oo)Ji+e) n C~([O, oo);Ee) (where the C 17 spaces are defined below), and it
decays exponentially.