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Michigan State
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dissertation entitled

"Invariant manifolds
for flows in Banach spaces"

presented by

Kening Lu

has been accepted towards fulfillment i
of the requirements for

 

Ph- D degree in _Applie.d_Math

Shui-Nee Chow

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Major professor

 

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MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

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MSU Is An Affirmative Action/Equal Opportunity Institution

 

 

*—

INVARIANT MANIFOLDS
FOR FLOWS IN BANACH SPACES

BY

Kening Lu

A DISSERTATION

Submitted to
Michi an State University
in partial ful lllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1988

ABSTRACT

INVARIANT MANIFOLDS
FOR FLOWS IN BAN ACH SPACES

By

Kening Lu

We consider the existence, smoothness and exponential attractivity of
global invariant manifolds for flow in Banach Spaces. We show that every global

invariant manifold can be expressed as a graph of a Ck map, provided that the

invariant manifolds are exponentially attractive. Applications go to the

Reaction—Diffusion equation, the Kuramoto—Sivashinsky equation, and singlular

perturbed wave equation.

ACKNOWLEDGEMENTS

It is a great pleasure to express my sincere appreciation to professor
Shui—Nee Chow, my thesis advisor, for his invaluable guidance, expert advice,
stimulating discussions and encouragement during the course of my research.

I would also like to thank professors Dennis R. Dunninger, Ronald
Fintushel, Charles L. Seebeck, and Lee M. Sonneborn for their patient reading of
my thesis and attending my defense.

A special thanks goes to my wife, Qiong, for her support and patience.

Finally, I would like to thank Darlene Robel for her superb typing job.

11

§1
§2
§3
§4
§5
§6
§7
§8

TABLE OF CONTENTS

Introduction

Notations

Linear and nonlinear integral equations

Invariant manifolds
Exponential attractivity
Semilinear evolution equations
Ck inertial manifolds

Singularly perturbed wave equation

iii

19
23
26
29
35

§1 Introduction

In the study of dynamical systems in finite dimensional spaces or manifolds,
the theory of invariant manifolds has proved to be a fundamental and useful idea.
In recent years, the theory of invariant manifolds has been generalized to flows or
semiflows in Banach spaces. See, for example, Babin and Vishik [2], Bates and
Jones [3], Carr [4], Chow and'Hale [5], Hale [16], [17], Hale and Lin [18], Henry
[21], Marsden and Scheurle [26], Wells [32] and others. On the other hand, it is
known that global compact attractors for many dissipative systems in Banach
spaces have finite capacity or Hausdoff dimensions (see, for example, Mallet-Paret
[23], Mane [25], Hale [17] Constantin, Foias and Temam [9], Babin and Vishik [2],
Hale, Magalhaes and Oliva [19]). Recently, it has been found that in many cases
these global compact attractors actually can be embedded in exponentially
attractive finite dimensional invariant manifolds which we call inertial manifolds
(see, for example, Conway, Hoff and Smoller [16], Constantin, Foias, Nicolaenko
and Temam [8], Doering, Gibbon, Holm and Nicolaenko [11], Foias, Nicolaenko,
Sell and Temam [13], Foias, Sell and Temam [14] and Mallet-Paret and Sell [24]).
This supports the believe that the asymptotic behavior of solutions of many
infinite dimensional dynamical systems resemble the behavior of solutions of finite
dimensional dynamical systems. In most cases, the inertial manifolds are shown to
be Lipschitz continuous. In Mallet—Paret and Sell [24] and Chow, Lu and Sell [7],
it is shown that for a large class of evolution equations in Banach spaces, inertial
manifolds are in fact C1 with bounded Cl norms. This smoothrzproperty is very
important in applications. The smoothness proof is not trivial even in finite
dimensional cases for the center manifold theorem (see, for example, van Gils and

Vanderbauwhede [15] and Chow and Lu [6]).

1

In this paper, we present a theory of smooth invariant manifolds based on
the classical method of Liapunov—Perron for continuous semiflows in Banach
spaces. Basic hypotheses for these semiflows will be satisfied by semilinear
parabolic equations on bounded or unbounded domains or hyperbolic equations.
Examples of these continuous semigroups from evolution equations may be found
in Bates and Jones [3]. The two basic theorems are stated for nonlinear integral
equations. One is on the existence of smooth invariant manifolds (Theorems 4.4)
and the other is on exponential attractivity of invariant manifolds (Theorem 5.1).
In fact, Theorem 5.1 is related to the squeezing pr0perty in Foias Sell and Temam
[14]. In §6 and §7, we show how our results are related the center manifold
theorem and theorems on inertial manifolds.

In §8, we consider the question of continuous dependence on parameters for
invariant manifolds or inertial manifolds. Since our existence theorem (theorem
4.4) is proved by using the uniform contraction theorem, the answer to the above
question is obviously true provided the nonlinear equations depend smoothly on
parameters. Hence, the interesting cases must involve equations which depend
singularly on some parameters. As an example, we consider the following two

scalar equations:

2 __
(1.1) c u + u —u — f(u)
(1.2) ut — u = f(u)

on the interval [0,7r] with Dirichlet boundary conditions. We will show that under

certain conditions on the nonlinear term f there are inertial manifolds If and J!

for equations (1.1) and (1.2) respectively, for all small 6. Moreover, dim .116 = dim
Jtp and If "approaches" Jtp as c 4 0. In our proof, we use an equivalent inner
product in the phase space for the damped wave equation to overcome some
technical difficulties. This inner product was first used by Mora [28] and Mora
and Sola—Morale [29]. Similar convergence results have also been independently
obtained by Mora and Sola-Morales [30]. In Hale and Raugel [20], it is shown that
the global attractor of ( 1.1) approaches that of (1.2). In fact, their results are valid

for a much larger class of equations in several space variables.

§2. Notations

Let E1, E2 be Banach spaces and U beanopen subset of E1. For any

integer k 2 0, let
Ck(U,E2) = {f |r; U -+ E2 k—times differentiable
and sup IDif(x)| < so for 1 5 i g k}

and

k .
|f|k=2 sup ID'f(X)|
i=0 xEU

where Di is the i-th differentiation Operator. Let
Ck’l(U,E2) = {r | f6 Ck(U,E2) and

3,, lef(x) — Dkf(y)|

X,yEU IX — y]
xty

<00}

 

and lflk,1= |f|k + Lip Dkf, where

k k

xty lx - Yl
x,y€U

 

Clearly Ck(U,E2) and Ck’1(U,E2) are Banach spaces with norms |-|k and

l ' lk 1-
Let Lk(E1,E2) be the Banach space of all k multilinear continuous maps

. R
from El into E2. For A E L (E1,E2,),

IIAII or nAu k
k L (£31,132)

denotes the norm of A. For notational simplicity, we will sometimes write "All for
||/\||k provided this will not cause confusion.

Let J Q IR be an interval (in most cases, we will let J = R'- = (— oo,0] or J =
lR+= [0,ao)). For any n e R and any Banach space E, we denote by C ”(J ,E) the

following Banach space
C n(J,E) = {f | f: J 4 E is continuous and
:21} e-"t|f(t)|E < co}
with norm

lfl = sup e_m|f(t)l
(300,13) m E

§3. Linear and nonlinear integral equations.
Let X, Y and Z be Banach spaces. Suppose that X g Y Q Z, X is
continuously imbedded in Y and Y is continuously imbedded in Z. Let S(t) (t
2 0) be a strongly continuous semigroup of bounded linear Operators on Z.

Consider the following assumptions:

(H1) Z= Z1622, where Z1 and 22 are invariant

linear subspaces under S(t).

(H2) PiS(t) = S(t)Pi i= 1,2,

where Pi is a projection from Z to Zi .
(H3) Fix and PiY (i = 1,2) are invariant under S(t) and S(t)Y g X ,
(H 4) S(t) can be extended to a group on Z1 .
(H5) ‘ There exist constants a,fl,7,n,M and M* such that
a>0,fl>0,0$7<1,M21,M*20 ,
(3.1) le—fltS(t)Ply|X 5 MeatlyIY for t s o, y c v,

(3.2) |e“’7"S(1)1>2x|X g Me_fit|x| X for t 2 o and x c X ,

_ (3.3) |e_mS(t)P2y|X$(Mt—7+M*)e-flt|y|Y for 1> o and y eY.

By using (3.1) and (3.2), we have that for any f(t) c Cn(lR-,Y), the
integrals

t t
[ S(t-s)Plf(s)ds and [ S(t-s)P2f(s)ds
0 -oo
exist for all t 5 0.. Hence, we can define the following linear operator
t. t
(3.4) .7f =I S(t—s)Plf(s)ds +J S(t—s)P2f(s)ds ,
0 -oo

Lem—ma}; The operator .7 defined by (3.4) is a bounded linear Operator

from C Y) to C ,X) for every t E [0,a) and the Operator norm of

.7 satisfies the following estimate

nan s K(a+c,fl—c.7)
where
(3.5) K(a,fl,7) = Me + arm) + 11*),

Proof: Obviously .7 is a linear Operator. We will show that .7 is bounded
and the estimate is valid. By using (3.1), (3.2), and (3.3), we have

f _ = "(WEN f
'5 ICU-Hm ,X) :ggle 17(t)lx
, t t
S :28 {Jule—("+‘)tS(t-s)Plf(s) [de + [file—("+6)t8(t-3)P2f(s) | de
5 {sup {M[|Jte(0’+‘)(t—5)ds + [t (t—s)'7e‘(5“)sds|]
1:50 0 -00

+ M*]‘ (was... 1} m0

-co

Wain

1 2- 7—1 * 1
S{M[—+ _ fl-c) l+M }|f| -
3+. 14% 3-7 Oman .Y) .
This completes the proof.

Let F c Ck(X,Y) and cp c ODOR—,X). Consider the following nonlinear

integral equation

t t
(3.6) 3(1)=S(1)§+ [08(t-s)P1F(cp(s))ds+[ S(t-s)P2F(<p(s))ds,

-co

Set (3(e))(t) = S(t)£ and 3((p)(t) = F((o(t)) and rewrite (3.6) in the

following abstract form:

(3-7) so= d’(5) + 315(3)» ,

where 5 c P1X.

Lemma, 3.2 If F cC1(X,Y) and K(a,fl,7)(Lip F) < 1, then there exists 0
< £0 < a such that for each 0 S c S 60 and 5 e P1X, K(a+c,fl—c,7)(Lip F) < 1
and (3.7) has a unique solution (p(§) c 071+ E(ll—,X). Moreover, (p : P1X -1
017+ c(It-,X) is Lipschitz and <p(§) is independent of e 6 [0,60].

_P_rm_f: By the continuity Of K(a,fl,7), there exists to > 0 such that
K(a+e,fl—e,7)(Lip F) < l for every t 6 [0,60]. By (3.1), we have that of is a

bounded linear Operator from P1X to 017+ (GI—,X) for every 6 6 [0,60]. Set

(33) f(soté) = 6(5) + 7(3(<P)) .

Since of and .7 are bounded linear Operators and F 6 Cl, for any (01, (02
E C 77+ 6(IF—”,X) and 6 E P1X, we have

(3.9) I }(¢1,€)-}(¢2.€) lcn+€(|R",x)

s I 3131901)) - 575002)) | Own-,X)

10
S .7 LipF cp-w -
II II( )I 1 2|CU+€(R ,X)
SK a—e,fl+e,7 LipF cp-sp -
( X )l 1 2lcn+£an ,X).

Since K(a+c,B~—c,7)(Lip F) < 1, this implies that J is a uniform
contraction with respect to the variable 5. By using uniform contraction theorem,
we have that for any 6 e P1X, )(cp,§) has a unique fixed point cp€(§) E
Cn+E(IR-,X). It is clear that J is Lipschitz continuous. Hence, 9pc“) is
Lipschitz as a mapping from P1X into Cn+c(R-’X)' Since C”+€(R-,X) Q
C "(R-,X), by uniqueness Of the fixed point Of f, we have <p€(§) = (pow) for any
6 6 [0,60]. Define 22(5) = <p0(§). This completes the proof of Lemma 3.2.

Lemma 3.3 (Fiber contraction theorem) Let E and E2 be Banach

1
spaces and U Q E1 be a closed subset. Suppose that

flzfi-iU .

szE2-iE2, xEU

and

film) ='(2(x), 4,6». x e U, y e E,

are continuous maps. Suppose that 3 is a contraction and

sup {Lip(./£x):x€U} <1 .

11

Let the unique fixed point of .2 be u and the unique fixed Of of“ be v. Then

(u,v) is an attractive fixed point Of 6'.

Prmj; See Hirsch and Pugh [22].

Lemma, 3.4 Let k z 1 be an integer and F e Ck(X,Y). 11 17 < 0,,6+(k—1)n
> 0 and K(a,fl+(k—1)17,7)(Lip F) < 1, then the unique solution <p(€) of equation
(3.7) is Ck asamapping from P1X into Chm—,X).

M: By the definition of K(a,fl,7) there exists ‘1 > 0 such that ‘1 <

a and for every 6 6 [0,51]
(3.11) K(a+c,fl+(k—1)n—c,7)(Lip F) < 1.

By using Lemma 3.2, equation (3.7) has a unique solution tp(§) E
Cn+€(lR—,X) for any :6 [0,61] and tp(§) is CO’1 from P1X to C lit-,X).
We will first show that 90(5) : P1X -+ C ”(IR-,X) is in fact Cl.

Let 112(5) : P1X 4 Cn+€1(R—,X) be continuous. Set

n+c(

t
(smite) = [0 S(t-S)P1F(¢(£)(s))ds, t s o.

The following smoothness prOperties are needed. Choose an arbitrary but fixed

infinite sequence :

cl>6l>62>-~>0.

12

Claim 1 If :1) : P1X .. Cn+61(R-,X) is Cl, then 51'” : P1X ..
_ . 1 A
Cfl+52(ll ,X) 180 .
W1: Let m) e C 17+ 5 (Ii-,X) be fixed. Assume that m) is c1
1

with respect to g. Define

t 1
(3.13) (9[(¢)-C)(t,€) = [0 S(t-8)P1D F(¢(€))(Di0(€)-C)(S)ds

where C e P1X and D111“) is the direvative of 1/1(§) with respect to 6 evaluated at
5. Let {1 and £2 E P1X be given. We have

—(n+6)t
(3.14) 1= le 2 [(«fiWLélH31¢)(t,£2)-(9}(¢)'(El—52))(t,€2))ll

511 +12 .
where
—(1r+6 )t t
1,: Ie 2 ] Sit-OiFiziisrl)
T

—F(its,t,)—Driw(s.tz))(nw(tg)(trenisndsl

e—(n+52)tjr

2 = l 03(t—s)[r(¢(s,tl)—F(ib(s,€2)

‘DF(¢(31€2))(D¢(§2) ' (€1‘§2))(S)]dsl

13

and T < 0 is a fixed constant which is to be chosen later. Since .9} is linear and
continuous in C e P1X, it suffices to show that I = O( | 51—52” as |£1—§2| —. 0. In
other words, it is sufficient to show that for any given 6 > 0 there exists 6 > 0
such that I 5 cl {1—§2| for all [{1- {2l < 6. Let c > 0 be given. Choose T < 0 so
that

(5- )T
£75291 2 2|F|1|w|1<c/2.

IftZT,it is clear thatI=O(|§1-{2|)as |§1-§2| 40. Lett< T<0. Wehave

11 S IIte(O~—62)(t—s)-i-(251— 2)s
T

(3.15) 2|F|1.

”DWQWLI [61-52 I dSI

(Plx’Cfl-l' 51(R—3X»

6—
e(, 2n

53—1-52 2|F|1|¢|1|€1-€2|

S§l§1-€2| .

Since T < 0 is finite, it is not hard to see that I2 < g |{1—§2| if |§1—{2| is
sufficiently small. This proves the claim.

Claim 2: If (b : P1X -i Ci77+5iak ,X) is C], then flit/2 : P1X -»

14

C. (Ii—,X) is Ci for i= 1,2, ---k.

"W’Hl

PM We will prove the claim by induction. Assume that for i
= 1,2,o . -,k—1 the claim is true. By induction, we can compute the (k—1)th
derivative of Bid with respect to 5, Dk-l(.7ii/)). It is not hard to see that
Dk_1(.7lz,b) has the same integral form as .7i. Using the same argument, we have
Dk-1(.7i¢) is C1 from P1X to Ckn(ll,X). This proves the claim.

Let i/( f) : P1X -» C (II—,X) be continuous. Similarly set

”+51
t
321/45) = ] S(t-8)P2F(¢(s.€))ds .

claimsii m) is Ci from P1X to ci R‘,X) then ~72)pr is Ci

n+6i(

from P1X to C (ll—,X) for 1$i$k.

i”Hi1“
Claim 3 is similar to claim 2 and the proof is omitted.
Now, we will now prove that the unique solution tp : P1X -+ C "(IR-,X) is
Cl. Since differentiability is a local property, it is sufficient to show that (p : B -i
cam-,X) is c1 for any fixed but arbitrary bounded ball in P1X. Let E1 =
CO(B,Cn(R—,X)) and E2 = CO(B,L1(P1X,Cn(R-,X))). Let 3 e E1 and 11: e E
Define

2.

t t
3(¢)(t,€) = S(t): + [0 S(t—8)P1F(¢(£))ds +] S(t-8)P2F(¢(€))ds.

-co

and

15

J¢(\II) = S(t)£ + It S(t-s)P1DF(¢v(s,£))\IIds + It S(t—s)P2DF(¢(s,())\IIds .
0 -no

In the definition Of «($010, we assume that for every C e P1X, J¢(\II)-C E
Co(B,C”(R-,X)) and is defined by

t
We). C)(t.£) = S(t): + [o S(t-8)P1DF(¢(£))(\I'- <)(s.t)ds

t
+ ] S(t-6)P2DF(¢(§))(‘I'-C)(s.£)ds.

Since (Lip F)K(a,fl,7) < 1, J¢(-) is a uniform contraction. Hence, J¢(-) has
a unique fixed point wtli for every w E E1.

By Lemma 3.2, 3 is a contraction in E1. Let (p(§) be the unique fixed
point of Band <I> E E2 the unique fixed point Of .1500). We claim that (D =

D {in To prove our claim, let

By fiber contraction theorem, (WP) is an attractive fixed point of 6’. This says
that for every (0 e E1 and \II 6 E2, we have

SHRIMP) -1 (<p,<I>) as n -v oo.

where 3’“ denotes the nth iterate of 6’.

16

Fixed d e 0103,an (ll-,X)). By claims 2 and 3, :7 d e
1

C1(B,Cq+62(R—,X)) and
t
(D at!» that) = [0 S(t-6)P1DF(1IJ(§))(D¢(€)°C)(s)ds

t
+] S(t-6)P2DF(¢(£))(D¢(£)-()(S)ds .
This implies that D.7¢v e E2 because C 'H’ 6(R-,X) g C ”(IR-,X) for all 6 > 0. Thus,
“(l/1,1310): (3(2),x,)(0¢))= (gimme);

62(34):») = (showmwovmw) = (32(2).D.92(¢));

and
- n O- . -O o =
i5’n(1/1,13¢)-(3 (“Mgr—1w) ”3(2) 03(2)) whammy».
We note that,

3n_1(¢)0- . -OJ3(¢)OD3(¢) E E2 .

17

By the attractivity Of the pair ((0,9), we have Snub) -+ cp and D3n(¢) -i (D as n —.
00. This implies c = Dcp and (p is 01..
Next we assume the theorem is true up to k—l and we will use induction.

By claims 1 and 2, we have
nine) e c°(B.c,,,(n'.x», n, = in—ilelk'l,
l

for i = 1, k—l. Let E1; = CO(B,Lk-1(P1X,Cn(R—,X))) and 1312‘ =

CO(B,Lk(P1X,Cn(R-,X))). By differentiating .A' and .2 formally, we define for

w e E1; and fl 6 El; the following functions:

I:
cam) = [0 S(t—slrlnridolmom

t
+ [0 Sit-slplnirisolid—1)Dntlwicwitlvdtllds

k—l ‘
+ R1 +[ S(t—s)P2DF(<,o(§))Q(§)ds

t
+ [ S(t—s)P2D2F(tp(§))[(k—1)D<p(£)w(€)+w(€)D80(€)ldS + R3271
and

_ ‘ k—l
aim) - [0 S(t-S)P1DF(<p(€))p(€)ds + R,

18

where Rlihl, i = 1,2,3, is an apprOpriate term involving derivatives of (p with

respect to C of order at most k—1. By using the same arguments as in the case k
= 1, we have that Dk_1<p : P1X -» Lk'1(P1x, CO(B,C”(R,X)) is t continuously
differentiable.

This completes the proof Of Lemma 3.3.

§4. Invariant manifolds.

Let F c Ck(X,Y). Consider the formatting-integral equation -

t
(4.1) x(t) = S(t-t0)x(to) + [t S(t—t0)F(x(s))ds
0

where x(t) is a map from an interval J g R to X.

Definition 4.1: If x: J .1 X is continuous and satisfies (4.1) for all to, t E
J, t0 5 t, then we call x(t) a solution Of (4.1) on J. For x0 5 X, we denote by
x(t,x0) a solution of (4.1) which equals to x0 at t = 0.

Lemma 4.2 Let 1) < 0. Assume that (H1)-(H5) are satisfied. Let x(t) be

a solution Of (4.1) on 11-. Then the following prOperties are equivalent

(i) P2x(t) e 0001':sz .
(ii) x(t) e Cfl(R—,X) .
(iii) x(t) can be expressed as

x(t) = S(t)x(0) + J;S(t-s)PlF(x(s))ds + [t S(t—s)P2F(x(s))ds ,

10

Proof. First we prove that (i) implies (ii). Since

19

20

C°(R".X) g 0,,(IR‘ZX)

we have that (i) implies P2x(t) c C ”(R—,X). Since x(t) is a solution Of (4.1) on
51-, By using (H 4) and (4.1) we have

t
(4.2) P1x(t) = S(t)Plx(O) + [03(t—s)Plr(x(s))ds .

By using (3.1), we have that Plx(t) c C 7lat-,X). Hence (i) implies (ii).
Next we show that (ii) implies (iii). By (4.1), we have (4.2) and

t
(4.3) P2x(t) = S(t-t0)P2x(t0) + [t S(t—s)F2(x(s))ds .
0

-(fl+ NH)
(44) |S(t—t0)P2x(to)lx 5 Me " ° |x(t0)lx

-{fl+fl)(t-t0)+flt0
5 Me IXIC"(R—,X)

—flt -(fl+n)t
5 Me 0 'XICn(R-,X) .

Letting t0 —. -co in (4.3) and using (4.4), we have

t
P2x(t) =[ S(t—s)P2F(x(s))ds .

21

Hence,

1’.
x(t) = S(t)P1x(0) + [05(t—s)1>1r(x(s))ds

t
+[ S(t—s)P2F(x(s))ds .
‘1!)
Finally, we show that (iii) implies (i). Since F is bounded, we have

.132...“ s |F|0{M[t (tern-(HXfl-"lds + N] e—(HW-fllds}

s |F|0{M Bid—arm + N33,}

Hence,

P2x(t) c com-£2 ).
This completes the proof.

Theorem 4.4. Let 17 < 0. Assume that (HQ—(H5) are satisfied. If F c
Ck(X,Y), 5+(k—1)17 > 0 and

K(a.fi+(k-1)n.7)(Lip F) < 1,

then there exists a Ck invariant manifold J! for the flow defined by (4.1) and

J! satisfies

22
(i) I: {xol x(t,x0) is defined for allt c [R— and P2x(t,x0) e C0(lR—,X)}
(ii) .3: {t + h(€)|§ e P1X}

where h: PIX-oP x is Ck.

met: Let

2

hit) =]° S(t-6)P2F(so(£)(8))ds,

where «2(5) is the unique solution of (3.7). By using Lemma 3.4, we have h(§) =
<p(§)(0) - 8(0): is Ck from P1X to P2X. To prove that J! is invariant, let x0 6
A Since F is globally Lipschitz and x0 6 oil, the unique solution x(t,x0) is defined
. for all t c R and x(-,x0) e C ”(R—,X). Furthermore, x(t,x0) E J! for all t _<_ 0. Let

t1 > 0.. Since x(t,x0) is a solution of (4.1), y(t) = x(t+tl,x0) satisfies

t
y(t) = S(t-t0)y(t0) + ]t S(t-5)F(y(s))ds
0

for all -00 < t.0 _<_ t S 0. Since x(-,x0) E C”(R_,X), y() = x(-+t1,x0) E Cn(lR_,X).
Hence, y(0) = x(tl,x0) e Jt. This completes the proof of the theorem.

§5. Exponential attractivity

In this section, we will prove that the invariant manifold J! Obtained in

Theorem 4.4 is exponentially attractive. More precisely, we have the following.

fIfhmrgm 5.1 Let n < 0. Assume (Hl)-(H5) are satisfied. If F c
Cl(x,Y). K(a.zi.7)(Lip F) < land

MK( a.fl.7)Lip(F)

(5-1) <

 

*
then for any solution x(t,x0) of (4.1) on [0,oo), there exists a unique x0 6 all

such that

_ a):
:33 e m|(x(t,x0)—x(t,x0)|x < +00 .

*
Proof: By Theorem 4.4, J! is a C1 invariant manifold. Let x and x be
*
any two solutions Of (4.1) on (0,00) and w = x —-x. Hence, w(t) satisfies the

following equation

(5.2) w = S(t—t0)w(t0) + I: S(t—s)(F(w+x)—F(x))ds .
0

As in §4, it can be shown that if w is a solution of (5.2) then w E C "(IR-IX) if

and only if

23

24
w = S(t)tt22 + [;S(t-8)P2(F(w+x)-F(x))ds

+ JtS(t-s)Pl(F(w+x)—F(x))ds

where (.112 = P2w(0) = P2x*(0)-P2x(0) = 5; — (2.
Let 07002) = S(t)trl2 and

t
y(wx) = [05(t—s)P2(F(w+x)—r(x))ds

+ ItS(t—s)P1(F(w+x)—F(x))ds.

Clearly a? is a bounded linear Operator from P2X to C ”(R+,X) and }
takes C”(R+,X) into itself. For any w1 and W2 6 Cfl(lR+,X), we have

le"t(l(w1.X)-}'(w2,x))lx

t
s Iem[08(t—S)P2(F(wl+x)—F(wl+x)—F(w2+x))ds I x
t
+ IemJOS(t—s)P1(F(w1+x)—F(w2+x))ds | X

s {Mlfi + 36“] + M*/r1}(Lip F)lw,—— w?) C (R, X)
n i

25

Hence, 1(w2,w,x) = «((112) + }' (w,x) is a uniform contraction with respect to x
and ($122. By using uniform contraction theorem, we have that for any (.22 c P2X,
and any solution x(t) Of (4.1) J has a unique fixed point w(x,w2).

Furthermore, if L01 = P1w(x,w2)(0), then

(5.3) wl = I;S(-s)P1[F(w(x,w2)+x(s))-F(x))]ds = g(x,w2).

k—l - a: :1: a:
C and w = x —x. Let Plx (0) = (1 and P1x(0) = (1. Thus,

Note that g is
* * *
x c J! if and only if {2 = h(§1), where h is given in Theorem 4.4. By using (5.3),

x* f .3 if and only if

(54) t] = 61 + g(x.h(€:) — :2) .

Since Lip(g) < 1 and Lip(h) < 1, by using condition (5.1) we have that for every
solution x(t) of (4.1) on [0,oo) equation (5.4) has a unique solution 5:. This proves

the theorem.

§6 Semilinear evolution equations

As a simple application of the results in §4, we will show how one can
Obtain Ck global center unstable manifolds for abstract semilinear evolution
equations in Banach spaces. We will not prove the existence of local Ck center
unstable manifolds since they can be Obtained by using a cut off function. We
refer the readers to Carr [4] for more detail.

Consider the following semilinear evolution equation in the Banach Space Z
= Z1922, where Z1 and Z2 are subspaces Of Z.

x + Ax = f(x,y)
(6.1)
{5' + By = 30%)

where x e Z1 and y e Z2, A and B are linear Operator from their domains 9
(A) and 9 (B) into Z1 and 22 respectively, and f and g are nonlinear
maps. We assume that B is a sectorial Operator [21]. For 0 5 7 S 1, let B7 be
the 7—fractional power Of B. The domain Of B7 is .9 (B7) = 23. It is well
known that z; is a Banach Space with norm |x| 7 = |B7x|. Note that Z3 =
22.

Let 05 7<1 befixed. Assumethat f: Z1 x 23-» Z1 and g: Z1 x Z34Z2

satisfy the following conditions

itx.y)=0(lx12+)y)§) and gix.y)=0<lx12+lyli)

26

27

as (x,y)-v (0,0). Assume that the spectra 0(A) and 0(B) of A and B satisfy

the following conditions

(6.2) {IRMAD s *1 .<_ o

Re(a(B)) > A2 > 0
We also assume
(6.3) A : X1 -+ X1 is bounded.

Let «It be an invariant manifold Of (7.1). J! is called a global center
unstable manifold if it is the graph of a C1 map h: Z1 -+ Z2 which satisfies h(0) = 0
and Dh(0) =0.

Since A is bounded and B is sectorial, the linear Operator

—A 0
O-B

generates an analytic semigroup S(t) on Z. Set X = Z1 9 z; and Y = Z. It

 

can be shown [21] that (H1)—(H4) are satisfied. We will see that (H5) is also
satisfied. Since (6.2) and (6.3) are true and B is a sectorial Operator, there
exists a constant w2 > 0 such that for every small “’1 > 0 there exists M 2 1, such

that

—wt
(6.4) le-tAI sMe 1, tgo

28

-wt
(6.5) Ie-tBISMe 2, t_>_0
—wt
(6.6) |B7e-tB|$Ml—7e 2, t>0
t

Let “’1 < 17< w , a: n—wl > 0, andfl= w2—n> 0. Wehavethat (H5) is
satisfied. By using Theorems 4.4 and 5.1, we have the following center unstable
manifold theorem.

Theorem 6.1: Assume that conditions (6.2) and (6.3) are satisfied. Assume
that 0 < 7 < 1. For any integer k _>_ 1, if

chk(Z Oz; ,,zl) gch (z 392% H22)

kn < «)2 and K(n-w1.w2-kn.7)(Lip(f) + Lip(g)) < 1 .

then (6.1) has a Ck global center unstable manifold .14 Furthermore, if
|Lip(f)+Lip(g)| is sufficiently small, then J! is exponentially attractive.

Remark 6.2 In Theorem 6.1, we do not require Ck norms of f and g to be
small.

§7 ck inertial manifolds.
Consider the following equation in the Banach Space Z

dz
(7.1) {37, '1' AZ '1' R(Z) = 0
z(0) = Z0

p
where A is a sectorial Operator on Z, R(z) is a nonlinear map from X 1 to

Xp2 where the exponents )01 and p2 satisfy either 1 2 ,01 2 p2 > 0, or 1 > p1 2
P2 2 0-

An invariant manifold «ll of (7.1) is called an inertial manifold of (7.1) if it
is a finite dimensional Lipschitz manifold and is globally attractive. In this section
we will applied the results obtained in §4 and §5 to the abstract nonlinear
evolution equations (7.1) tO Obtain Ck inertial manifolds. Applications will also
considered.

Assume that the spectrum of A, 0(A), satisfies the following conditions

(7.2) 0(A) = 01(A) U 02(A) ,
(7.3) A1 = sup{ReA : A c 01(A)} < inf{ReA : A c 02(A)} = A2 .
(7.4) 01(A) consists Of only eigenvalues with finite,

multiplicities and is a finite set

29

30

(7.5) _ Rea(A) > o .

Let P1 be the projection associated with 01(A) and P2 = I — P1. Then

there exist constants M 2 1, all > 0 and (.122 > 0 such that

w ltl

(7.6) IPIe-At'xl gm 1 |x| , for all tclR

”1 ”2
—At “”2"
. < >
(7 7) IP2e x|p1_Me lepl, t_0
_ p -p -w t

(7.8) [P2e Atx| 5 Mt 2 1e 2 |x| , t> 0

”1 l’2

Let "’1 < 1) < 322, a = 17—611, 3 = Luz-17, and 7 = pl—pz. Then
hypotheses (Hl)-(H5) are satisfied. By using Theorems 4.4 and 5.1, we have

72 )0
Thgzrem 7.1: If R c Ck(X 1,X 2) and there exists 1) > 0 such that
“’1 < n < k1) < (.02, Lip(F)K(a,fl—(k—l)1),7) < 1 and

MK(a.fl.7)Lip(F)
1 - K(a.)6.7)Lip(F)

 

then (7.1) has a (3k inertial manifold.

Exa_l_nple 7.2 Let Z be a Hilbert space. Consider the following problem [14]:

{g%+Au +R(u)=0

u(0) = no

31

where u e Z, A is positive self—adjoint linear Operator with domain 3 (A) dense

in Z,
R(u) = Cu + B(u,u) + f

where C is linear, B is bilinear and f e Z is fixed.. Assume A has a compact
inverse A-l. Hence, the spectrum Of consists Of only eigenvalues Ai, i = 1,2,...,

satisfying:
AISAzs'...'SAi-’m, 3814:!)

Let ei e Z, i = 1,2,..., be the eigenvector Of A corresponding to eigenvalues ’\i'
Let N > 0 be an integer and P1 be the projection from H into span{el, ,eN}
and P2 = I—Pl. Furthermore, we assume f c D(A1/2), C and B satisfy the

following conditions
(7.10) |A1/2B(u,v)| 5 61 I All] |Av| for all u,ch(A1/2)
(7.11) [Al/2Cu|5c2|Au| forall ucD(A1/2)

where c1, c2 2 0 are constants.
Since A is a positive self—adjoint Operator with compact inverse in the

Hilbert space Z, we have the following prOperties.

A Itl
eN

(7.12) IPIe—Atl 5 , for tth

32

A ltl
(7.13) [Alfie—At?” 5 Ali/2e N , for t clR
—A t
(7.14) |P2e-Atl 5e N+l , for t2 0
-A t
(7.15) |A1/2P2e-Atl 5 (Fl/2+ Alufk N“ , for t> o .

For many equations in applications, e.g., 2D Navier—Stokes equations [9],
the flows are dissipative, i.e., there exists a bounded ball in an appropriate
function space such that every solution will eventually enter the bounded ball and
stay there for all future time. Hence, the study Of asymptotic behavior Of solutions

can be reduced to the study of a modified equation:
(7.16) 3% + Au + F(u) = o

where

M) = 06(IAUI)R(u). 048) = 4%). 0(8) . 030100 s 0(8) s 1
0(3) =1 for [s] 51, 0(3) = 0 for Is] 22.

and c > 0 is some constant. Since 0 c C(8(R) and the norm of Hilbert space is

smooth,

F(u) c Ck(Zl,Z1/2)

33

for any integer k _>_ 0. For more detail, see [8], [13] and [14].
By Theorem 7.1, equation (7.16) has a Cl inertial manifold provided

Lip(F)K(a,fl.7) < 1-

Since Lip(F) is only a finite number, we need to have K(a,fl, 7) small. Recall that

we may choose

.\ -,\
0:3: N+12 N

 

It is not difficult to see that K(a,fl,1 2) _. 0 as A1 /2-A1/2 -+ on. This says that if
N+l N

the gap (All; «(I-21V 2) is sufficiently large, then equation (7.16) has a C1 inertial
manifold.
Example 7.3 Consider the Kuramoto—Sivashinsky equation [12], [13] and

[27] :

(7.17) gut- + $1- + §3- + ugui = 0 in [0,7r]><fll+

with boundary conditions
(7.18) u(0,t) = u(7r,t) = 0

and

34

(7.19) —giu(o,t) = $2 u(7r,t) = O
Let A = 3:1 , B(u,v) = 11ng and Cu = $7.
The Operator A with boundary condition (7.18), (7.19) has eigenvalues

_ 4 _
Ak—k , k— 1,2,3,... .

It is not hard to see that Example 7.2 is applicable in this example provided
the flow is dissipative.

Example 7.4: Consider the following reaction—diffusion equation

(7.20) ut-uxx = f((u) 0 S x 5 77
with boundary condition
(7.21) u(0,t) = u(7r,t) = 0 .

For simplicity, we assume f E C1(L2(0,7r),L2(0,7r)). Since the eigenvalues Of

Operator A = —6‘2/(9x2 with boundary condition (7.20) are An: 112

a II = 1127' ' ° 7
Example 7.2 is again applicable in this case provided the flow (7.20) (7.21) is

dissipative. I

§8. Singularly perturbed wave equation.

In this section, we will consider a scalar semilinear parabolic equation in

the interval [0,1]:

Ut-Uxx=f(U) 05x57
(8.1) U(t,0) = U(t, 7r) = 0
U(0,x) = U0(x) .

and a singularly perturbed scalar semilinear wave equation in [0,7]:

2
ri-u“+ “t -uxx =f(u), 03x _<_ 7r
(8'2) [mm .= u(t,7r) = o
“(0.10 = 1100‘). ut(0,x) = 1110‘) -

where U0 6 L2(0,7r), 110 E H3(0,7r), 111 e L2(O,7r) and f is C1 from ll into itself. In
this section, we will show that under some conditions, for sufficiently small 5 (8.2)
has an inertial manifold .16 which "approaches" to an inertial manifold J! Of (8.1)
as c approaches 0. Precise convergence statements are given in Theorems 8.6 and
8.8.

Recently, it is shown by Hale and Raugel [20] and Babin and Vishik [2] that
under some mild conditions on f, for all e > 0 there exists a compact (global)
attractor for equation (8.2). Moreover, for sufficiently small 6 > 0, these

attractors are uniformly bounded. Thus, we may assume without loss Of generality

3S

36

that equations (8.1) and (8.2) are modified equations (see §7). Hence, we assume
f e Cl(L2(0.7r),L2(0.x)), i.e, the mapping v(x) -. f(v(x)), o 5 x 5 7r, is O1 as a
mapping from L2(0,7r) into itself and has bounded C1 norm.

We will rewrite equation (8.2) as a system of first order equations. For

technical reasons, we consider the following change Of variables:

11 = — 26—211 + 26—1v, and w = (u,v).

t

We can rewrite equation (8.2) as a system:
(8.3) wt = Cew + 26-1 f(w),
where

C =—2e‘21+2e‘1 o 1 , A=—62/6x2 and f(w)= o
[sol

Let X = H6(0,W)XL2(0,7I’) and N > 0 be an integer. Set

__ sin px 0 , _
(8.4) xN _ span{( 0 ), (sin px) . P _ N+1,N+2,...}

.1. _ sinx sin Nx 0 0
(8.5) XN—span{( 0 ),...,( 0 ),(3inx),...,(sin Nx)}

Clearly x = xN e xltl, xN is orthogonal to x11} and dim x;I = 2N.. Moreover,

both XN and XN are invariant subspaces Of the Operator Cf. We also note that

37

the spectrum of C c consists of only eigenvalues.

Define an equivalent inner product in H(l)(0,ir) by

1 1

<u, v>- — ((A+(12 -2(N+1)2 ))2 u, (A+(:21 (—2N+1)2 )) 2v) L2
6

where ( , ) 2 is the usual inner product in L2. By using the above inner product
L

in H(1)(0, 7r), we define the following equivalent inner product in the product space

X: 111(0, 7r)xL2 (0, 7r) by

<< wl,w2 >> = < u1,u2 > + (vl,v2)L2

where wi = (ui,vi), i = 1,2. The norm induced by <<.,.>> will be denoted by
ll ° ”-

Lemma 3.1 There exist an 6 dependent decomposition X = XN e XE e
X§ with projections PN, PE, P; respectively, where XN is as in (8.4) and XN Q
X113 (see (8.5)) such that

(i) XN, XE and X3} are invariant subspaces Of C c

-2+2(1- 3 2(N+1) 2,)1/2

06‘; C
(n) lle Pane , tzo

 

 

c —2+2(1-r2(N+1)2)1/2,
t 2 °
ue ‘ PNII s IIPNIIe ‘ , tzo

38
—2+2(1-62N2)1/2,

Ct 2 "

ue ‘Pltusnrltue ‘ . tso

where [I - M denotes the Operator norm in the Hilbert space (X,<< - , . >>).

(iii) ||PN|| = 1 and there exists 60 > 0 such that for every 0 < c < 60,
"Pg" 5 2 and ”Pt" 5 2.

m We have that X = XN 6 Xfi and XN’XN are invariant subspaces

Of X. By restricting C e to X111, we find that the eigenvalues Of C e | X‘ are:
N

 

At = —2:l:2(1-62n2)1/2

n 2 ’
e

(8.6) n = 1,2,...,N

and corresponding eigenvectors are

 

sin nx ], n =1,2,...,N .

:t .
Ansmnx

Let

x§=span{[ sin nx ]:n=1,...,N}

An 8111 nx

xi} = span {

 

sin nx J :n =1,...,N} .

+.
Ansmnx

39

Obviously, KN = XE e X'N1' and XE, XN are invariant subspaces Of C E and

(8.7) << sin mx >> = 0 for m 1: n.

 

sinnx ],

 

 

Ad:

:1:
n A

inx inmx
sn ms

Note that XE is not orthogonal to XII}. Hence, X = XN e XE e x; and (i)
holds.

Let PE, and PN be the corresponding spectral projections [31] and PN be the
unique orthogonal projection onto XN. Obviously, we have ||PN|| = 1. By using
(8.7) we have that

 

-2+2(1— —£ 2(N+1) )21/2
Ct 62
“9 PN"- < "PNlle t for t Z 0

and

C --2+2(1—e2N2)1/2 t
t 2
He ‘ 13;" g||1>§ue ‘ , for tgo.

Now we consider C 61XN° For any w e XN

<<(—22(1-e2(N+1)2)1/ 21 +2
6

 

 

=----%§(1«52(N+I))1/2[((r1\+(€-2-'2(1‘1+1)111111L2+ We]

40

"LL.“

+ 1.,r—(N+1)2)(u,v)L2

5 — 22 (1—£2(N+1)2)1/2[<u,u> + (v,v)L2]
C

+ 22 (1—e2(N+1)2)1/2[<u.u> + (v.v)L2)

This says that the Operator:

— 22(1—t2(N+1)2)1/2I +2- 0 I
f
c_2-A 0

is dissipative (see Pazy [31]). By the Lumer—Phillips theorem [31], the above linear

Operator generates a contraction semigroup. Thus, we have

 

§- 0 I t
6—2-A o 26—2(l—£2(N+1)2)1/2t
IR "Se , 120
Hence
—2+2(1—c2(N+1)2)1/2,
C t 2 "

IR ‘PNHSe ‘ , 120

41

We will now get the estimates for PE and Pfi. For any w 6 X113, w = w2 + w3
where w2 e XE and w3 e x§. We claim that

0 << W2,W3 >> 0 0
°°3 "—llwgllllw3ll " ‘7

 

where 0 is the angle between w2 and W3. Suppose

W2 = sinnx , W3 =
An sin nx

 

sinmx ,
A: sin mx

Then cos 0:0 if natm. Ifn=m, then

<< W2,W3 >>
lwgllwgl

 

cos 0:

<sin nx , sin nx> +A;A';

 

(<sin nx , sin nx>+(A‘I1l')2)1/2 (<sin nx , sin nx>+(Apz)1/2

n2+ E-2—2(N+1 )2-1-4 5an

<

(n2+ 17 —2(N+1)2+(—2+2(;_E
C C

 

2112 1/2

 

)2)1/2(n2+ 17 —2(N+1)2+ 17;)1/2
E E

40 mic-+0-

This proves our claim. Since XE and X3} are finite dimensional vector spaces,

42
there exists (0 > 0 such that if 0 < c 5 £0 Icos 0| 5 %. Hence
<< w,w >> = << W2,W2 >> + << W3,W3 >> + 2 << w2,w3 >>
2 << W2,W2 >> + << w3,w3 >> — ||w2|| ||w3||
2 %(<< w2,w2 >> + << W3,W3 >>)
This implies (iii) and completes the proof.

Lemma 3.2 Let

1 1

* l
K (c,N) -— 2(0-“11 + «22-0)

 

 

(1—2e(N+1)2)U2

where

 

 

 

_2—2(1-r2N2)1/2 _2-2(1—e2(N+1)2)1/2 _(N+1)2+N2
‘2 ‘ 2 . “’2" 2 . ”— 2 '

l 6 6

Then there exist 60 > 0, 0 < c < 1 and an integer N > 0 such that
*
(8.8) K (6,N)Lip(f) < c < 1 .

Prmf We have that a -) N2 and fl -» (N+1)2 as c -+ 0. This implies

43

1 l

+ , as 6 -¢ 0.
(N+1)2—N2 (N+1)2-N2
2 2

 

 

*
K (£,N) -l 2

 

 

We can chose N so large that the above limit is strictly less than one. Thus, the
*
lemma follows directly from the continuity Of K .

W If f e Cl(L2,L2) and N > 0 satisfies the following gap

condition:

1 1

1
(8.9) + < ,
(N+1)2-N2 (N+1)2--N2 2711319111

2 2

 

 

 

 

then there exists £0 = 60(N) > 0 such that for every 60 > f > 0 equation (8.3) has
a (:1 inertial manifold .476 with dim .36 = N.

hoof By (8.9) and Lemma 8.2, there exists (0 = c0(N) > 0 such that
condition (8.8) is satisfied for all 0 < c < to. Let a = 17—321 and fl = w2—1]. It is
not hard to see that hypotheses (H1)—(H5) are satisfied because “PR” and 11PN“
are uniformly bounded in 0 < e < 60. Next we note that if w = (u,v) E X, then by
the definition of the norm || - I] if

1
(1—260(N+1)2)1/

 

c1= 2

then In] 2 S tclllwll. This implies that if wi = (ui,vi), i = 1,2, then
L

Ila-1iiiwl)-i<w2))n=ue‘1 o n
f(u1)-—f(u2)

44
< e‘lLip(f)lu1-u2|L2 s clLipmnwl—wzu.

This says that the coefficient {'1 Of the nonlinear function f in equation (8.3) will
be canceled by the 6 term in the norm || - ||. Hence, by using the above estimate,
Lemma 8.1, Theorems 4.4 and 5.1 (with k = 1, M = 3, M* = o and 7 = o in
condition (H5)) and the gap condition (8.9), we Obtain the desired result. We note
that K*(€,N) takes the role Of K(a,fl,7) in Theorems 4.4 and 5.1

We note that the inertial manifold «ItE in Theorem 8.3 can be written as:
+ +
.16 = {PNa-i- h€(PNa): 06 X}

where

.069,
(8.10) MP 0): [0 e (PN+P—)f(w (P a))ds

and w ”P a)( ) is the unique solution Of equation (3. 6) with §= PNa, S(t) =
exp(C€t),F = f, P1 = P; and P2 = PN+PN.

The following follows from Example 7.4.

Thggrgm 3.4 If f c 01(L2,L2) and N > 0 satisfies the gap condition (8.9),
then equation (8.1) has a C1 inertial manifold

sap = {U0 : U(t,U0) e C”(R-,L2) and satisfies (8.1)}

45

= {f+h(f) = fe QNL2}

where QN is the orthogonal projection from L2 to span{sin x,- - -,sin Nx} and

(8.11) hit) = [0 eAs(I-QN)f(W(€))dS

where W(§)(o) is the unique solution Of equation (3.6) with S(t) = (At,

2 _ _ _
QNL , F — f, P1 — QN and P2 — I-QN..
W Suppose that the conditions in Theorem 8.4 are satisfied. For

{6

each R > 0, there exists MI > 0 such that if [5| L2 _<_ R and g E QNL2, then

(i) [(31711Ut(t,£+h(£))|L2 5 M1 for t e IR-
(ii) [emUtt(t,£+h(§))|L2 5 M1 fort 6 IR‘
(iii) lemAl/2U(t.§+h(€))l 2 5 M1

L .

where U is the unique solution of equation (8.1) with U0 = §+h({) and h is given
by (8.11)
firm; For each U0 = §+h(§), we have

U(taUO) = UN(119€)+h(UN(ti€))

where UN(t,€) is the solution Of the following initial value problem:

46

(8.12) (UN)t = AUN+QNf(UN+h(UN)) UN(0) = g

Since equation (8.12) is finite dimensional and f is globally Lipschitz, UN exists for
all t. From our choice of N and the Spectral prOperty Of A] QN, we Obtain from

Gronwall's inequality and equation (8.12) that there exists Mi such that

2
(8.13) |e’11UN(t)|L2 3 Mi for all |€|L2 s R and g e QNL .

Since h is Cl, we have

11,6310) = (UN),(t.t)+Dh(UN)-(UN),(t.t).

By (8.12) and (8.13), we have

le”‘v,lL2 s (1+Lip(h))(MiIIAQNII+IIQNII Iflo) .

Since QN is an orthogonal projection, IIQNII = 1. Thus, (i) follows from the above
inequality.

Next, Ut satisfies the variational equation:

Wt = AW + Df(U)W
{w(o) = Ut(0)

The above equation is linear and nonautonomous. If we consider Df(U)W as a

47

perturbation to the autonomous equation:

then by condition (8.9) one can prove exactly as in §4 and §5 the existence of a
time varying C1 finite dimensional invariant manifold (see Henry [21]). Thus, we
can prove (ii) by using exactly the same method as in (i).

To show (iii), we note that AU = Ut—f(U) E Cn(R-,L2). Since U(t) e L2
and AU(t) E L2 for every t g 0, by a well—known interpolation theorem (Adams
[1], p.75) we have Al/2U E C”(R_,L2). This completes the proof.

Let
:7“) = {(U0,Ut(0,U0)) : U0 e 7p} and
(8°14) ERR = {(anUtmlUO» 2 U0 = (+11“) 5 JP, '61 L2 < R}

where U( - ’U0) is the unique solution of the initial value problem (8.1) and R is an
arbitrary constant. We have the following theorem.
Thflrem 3.3 Suppose that f c C1(L2,L2) and N > 0 satisfies the gap

condition (8.9). Then for each R > 0, we have

lim{ sup (inf [Wo-wl 1 2)}=1im{ sup dist(W0,./It€)}=0.
6-70 “1024,11 we 6 0x 75-9 “1064,11

48

where “l: is the inertial manifold given by (8.10) and WpR is as in (8.14).

ELQQI For each W0 E 7123’ we have W0 = (U0,Ut(0,U0)) and
Ut = Uxx + f(U), U(O) = U0
Define
W(t) = (U(t) sum—111(7))

where U(t) = U(t,U0). Thus W(t) = (U(t),V(t)) satisfies the following
perturbation of equation (8.3): '

-1“
Wt=C£W+2c f(W)+§-

 

0 .
Utt
Let w e .16 be a solution of (8.3) and 0 < f < 60 (see Theorem 8.3). Let z(t) =

W(t) — w(t). Hence, z(t) satisfies the following equation:

_1 * *
zt = sz + 2c {f(z+w€)—f(w£)} +5-

 

0 .
Utt
By (i), (ii) and (iii) of Lemma 8.5, we have that z E Cn(R-,X). By Lemma 4.2, we

have

49

C f(t—s

) — ‘ “ e
P§{2€ 1[f(z+w€)—f(w€)]-+-2- O

C
z(t) = e 6tP'ISzm) + I; e }

U

0}
vl

tt

 

 

tt

9 (PN+P§){26_1[1(Z+Wc)-1(We)]+§

'1!)

+ [t C ((t—s)

 

Since die is a graph over the finite dimensional subspace P§X (X =
H5(0,7r)xL2(0,7r)), we may choose w£(0) so that Pnz(0) = 0. Note that

II[ o ]u= IU,,IL2
U
tt

 

Hence,
t C (t—s) _ . .
121Cn(R_,X) =28 emlljoe f P§{2c l[f(z+wE)—f(wc)]+§ U0 J}
tt
C . .
+ [t e f(t-S)(PN+P§){26_1[f(z+w c)—f(w (n+5

 

0 H!
s,]

t

s Lip(f)K*(c,N) 1210,7011) + %M1K*(6,N)c. (Lemma 8.5)

*
By Lemma 8.2, K (c,N)Lip(f) < c < 1 for all 0 < c < 60, where c is some fixed

constant. It follows from the above inequality

50

cM

1
IZI _ S 6
Cfl(R ,X) 2112113111111

Hence,

cM
1
< <
IzionflgflL2 - "2(0)" - We ,p .

This implies the theorem.

Lemma 3.7 Assume f(0) = 0. Assume that f is C1 from L2 into itself and N
> 0 satisfies the gap condition (8.9). Then for any R > 0, there exists M2 > 0 such
that if "(ll 5 R and C e P§X, then the following inequalities are satisfied by any
solution w(t) of (8.3) on the inertial manifold J“, 0 < c < ‘0 (see Theorem 8.3):

(8.15) llemw(t)ll 5 M2 t s o
(8.16) Ilemwt(t)ll 5 M2 t s o
where w(O) = C+h 610 and he is given by (8.10).

Prmf: Since w is a solution of (8.3) on the inertial manifold If, by Lemma

4.2, we have

c t .
(8.17) w(t) = e ‘ c+ [ e P'N1'%f(w)ds

51

t (t—s)C€ 2 .
e (PN+P§) E f(w)ds .

+
‘——fi

-00

Since f(0) = 0, we obtain as in the proof of Theorem 8.3 that
(8.18) lle—1f(w)ll s e‘lLipm IuIL2 s c,Lip(f)nwn

where w = (u,v) and

_ 1
c1 ’ 2 1/2'
(l-2£0(N+l) )

 

. ' Ill
By the gap condition (8.9), we have Lip(f)K (e,N) < c < 1 for some fixed c
*
(Lemma 8.2). By Lemma 8.1, (8.18), the definition of K and equation (8.17), we

have
. 1 1
IWI _ ssllClI +c Lip(f)2(—+—_-)|W| _
0,701 ,x) 1 W1 “’2 ’7 Cn(R ,x)
*
s sucu + Lip(nK (c,N)IWI _ .
C (R ,X)
7)
Hence,
Iva 513311

C 71( R_,x)

52

This implies (8.15). Since Jte is invariant and is the graph of h c’ we have w(t) =

((t) + h €(C(t)), ((0) = C, C(t) e P§X for all t. Furthermore, ((t) satisfies

2.
(8.19) c, = 0.4 + P§[-.-f(<+h.(o)1 .

We note that P§X is invariant under C e and (8.19) is finite dimensional. By
(8.18), (8.19) and (iii) of Lemma 8.1, we have

ICtl s {no.1 + u + 4c1Lip(f)(l+Lip(h,))}lC|
PNX

ova-,X) cum-,X) -

Since IICCI + II 5 Sup {IA-{l} and A? 4i as c -+ 0 (see (8.6)), there exists a
PNX ISiSN
constant M3 independent of c 6(0,c0) such that

IC I _ 5M .
t C (R ,X) 3
17

This implies (8.16) and completes the proof.

Let

7‘ = {u : w = (u,v) c If for some v 6 L2}

and

53

(8.20) 7.3 = {u e 7t. = w = (u,v) = <+h.(<), ucn .<. R}.

Theorem 8.8 Assume f(0) = 0. Suppose that f c Cl(L2,L2) and N > 0
satisfies the gap condition (8.9). Then for each R > 0, we have

lim{sup (inf I)lu-—U| )}=lim{sup dist(u,.lt)}=0.
6-10 uEJZ'R L2 640 ueJigR p

where .ltp is the inertial manifold given by Theorem 8.4 and 7‘ R is as in (8.20).

Prmf Let w0= (u0,v0) and 110 6 76,1? Let w(t) = (u(t),v(t)) bet the
unique solution of (8.3) with w(O) = wo. Since .116 is invariant, u(t) satisfies the

following equation:

ut + Au = f(u) -i—utt
is):

Let Z(t) = U(t) - u(t), where U(t) is the unique solution of (8.1) on the inertial
manifold JD with initial data U(0) = U0 6 1.2. Thus, Z(t) satisfies

zt + AZ -.- f(Z+u(t))-f(u(t))- in“

Since U() E 1p and w(-) e If, we have Z( -) 6 Cn(R—,L2). By using Lemma 4.2
and (8.16), we have

54

-At t-A(t—s) 62
Z(t) = e QNZ(0)+ (0e {QN[f(Z(S)+U(S))-f(u(8))l-:1- uttlds

t —A(t—s) £2
+J e {II-0N][f(Z(s)+u(s))-f(u(s))]-:1—u,,}ds -

-ao

As in the proof of Theorem 8.6, we may assume without loss of generality that

QNZ(0) = 0. As in the proof of Theorem 8.6, we have

M22

Z S c

This completes the proof.

Bgmgk 8.2 Consider the damped sine—Gordon equation

12

(8.21) If utt + ut — uxx = sin u
with boundary conditions
(8.22) u(t,0) = u(t,7r) = 0.

Theorem 8.3 is not applicable in this case because f(u) = sin n is not a C1 map

2 into itself (see Henry [21]). However, (8.21) (8.22) defines a C0 nonlinear

from L
semigroup on (Han$)xH(l) (Hale [17], Theorem 7.5 in Chapter 4) and f(u) = sin u

is C1 from H3 into itself. If we define the following inner product in (Hanngé:

55
l 2
<<< wl,w2 >>> = (A(A + ?—2(N+l) )u1,u2)L2 +(AV1,V2)L2

where wi= (ui,vi), i = 1,2, 6 (H2nH3)le, then we can get the same results as
Theorem 8.3, Theorem 8.6 and Theorem 8.8 for the damped sine—Gordon equation
(8.21) (8.22) by using the same arguments.

 

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