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Inertial Manifolds
C
h
a
p
t
e
r
Thus, if for some q < 1 the condition
3
4 KN
l-N + 1 - l-N ³ ---------(7.35)
qholds, then equation (7.32) is uniquely solvable in Cs and its solution V can be estimated as follows:
1 - ------------1 -ö ö
-1
+
.
I V I £ ( 1 - q ) æè p + M 0 æè -------l-N l-N + 1ø ø
(7.36)
Therefore, we can define a collection of manifolds { Ms } in the space 0 by the formula
Ms = { p + F ( p , s ) : p Î P 0 } ,
(7.37)
where
s
òe
F(p , s) =
-( t - t ) A Q B ( V ( t ) ,
t ) dt .
(7.38)
-¥
Here V ( t ) is a solution to integral equation (7.32). The main result of this section is
the following assertion.
Theorem 7.1.
Assume that
e 2 > m N + 1 and
4 KN
l-N + 1 - l-N ³ ---------q-
(7.39)
-
for some 0 < q < 1 , where l k = e - e 2 - m k and KN is defined by formula
(7.31).. Then the function F ( p , s ) given by equality (7.38) satisfies the Lipschitz condition
q
F ( p1 , s ) - F ( p2 , s ) £ -------------------- p1 - p2
(7.40)
2 (1 - q)
and the manifold Ms is invariant with respect to the evolutionary operator S ( t , t ) generated by the formula
S ( t , t ) U = ( u ( t ) ; u· ( t ) ) ,
t ³ s,
0
in 0 , where u ( t ) is a solution to problem (7.1) with the initial condition
U0 = ( u0 ; u1 ) . Moreover, if 0 < q < 2 - 2 , then there exist initial conditions
U0* = ( u*0 ; u 1* ) Î Ms such that
S ( t , s ) U0 - S ( t , s ) U 0* £ Cq e - g ( t - s ) Q U0 - F( PU0 , s )
1 for t ³ s , where g = --- ( lN + l N + 1 ) .
2
The proof of the theorem is based on Lemma 7.4 and estimates (7.29) and (7.30).
It almost entirely repeats the corresponding reasonings in Sections 2 and 3. We give
the reader an oppotunity to recover the details of the reasonings as an exercise.