187
Foias, G. Sell and E. Titi (1989). The inertial manifold M is said to be asymptotically
complete if the following holds:
{
For every Uo E E, there exists iio EM and r E JR,
such that IS(t)uo - Set + r)iiolE --+ 0, as t --+ 00.
(2.3)
It follows easily from the proof of Theorem 2.1 that the manifold M given by
Theorem 2.1 is asymptotically complete. Indeed (see (2.45) and (2.46)), (2.3) holds
with r = 0, iio = Yo + q,(yo); furthermore the convergence to ° of IS(t)uo - S(t)iioIE is
exponential.
2.2. Properties of T
For the sake of simplicity, we set
We start the proof of Theorem 2.1 by showing that T maps :Fb,l into itself.
Lemma 2.1. For t/J in :Fb,l,
Proof: We write
Tt/J(yo) = 1°00 eASQR(y(s) + t/J(y(s)))ds,
ITt/J(yo)I E :5 1°00 leASQR(y(s) + tP(y(s)))IEds
:5 1°00 ieASQiC(F,E) IR(y(s) + t/J(Y(S)))IFds.
Thus, by (1.3) and (1.13),
ITtP(yo)I E :5 Mok2 1°00 (Isra + A a ) eASds.
We observe that for 0', a > 0,
with
(2.4)
(2.5)