203
hence
N+I
c(N + 1)P ~ cnb + 2rl L cajpa,
j=no
2r1C a (N + l)1+ p a
-
0
l+pa
If N + 1 ~ 2 1 / P no, (3.16) is proved. If not we infer from (3.17) that
and (3.16) follows as well.
(3.17)
Remark 3.2. Lemma 3.1 can be extended to the case where enough informations are
available on the second term of the asymptotic expansion of An,
An = cn P (l + En), as n --+ 00.
The dimension of the inertial manifold n is then larger than or equal to no, such that
lenl ~ 1/2 for n 2: no·
Remark 3.3. The following remark (A. Debussche) related to Remark 2.2 and Lemma
3.1 is useful.
Assume, as is often the case for boundary value problems, that (1.1) is of the form
du
dt + vAu = R(u),
(3.18)
with v > 0, and A self-adjoint as before. Let J-ln, n 2: 1, denote the eigenvalues of A
with (3.11), and let An = VJ-ln.
Then as in Lemma 3.1, if (3.13) holds, for any rl, rz > 0 given, we can find and
estimate n's such that (3.14) and (3.15) hold. Indeed we first find n. depending on A,
but not on v, such that
Then we determine no = no(v) 2: n. such that
\ I-a _
I-a I-a> I-a cJ-lon
>
( P) I-a
An
- V
J-ln
_ V
2
_ rz,