[universitext] the homotopy index and partial differential equations || applications to partial...

Chapter II

Applications to partial differential equations

We will now give a few applications of the theory developed in Chap

ter I. First we describe some of the types of differential operators

which generate sectorial operators.

Then we discuss local center manifolds for parabolic equations and

state an existence and approximation result for center manifolds.

Next, in Section 2.3 we prove an index product formula relating the

index of a "small" invariant set relative to t:he POE to the index of

the same invariant set with respect to the restricted flow on a local

center manifold. This formula together with the approximation result

for center manifolds permits to calculate the index in critical cases.

An example of this is given in Section 2.4.

In Section 2.5 we consider asymptotically linear parabolic equations

with nonresonance at infinity. More generally, we roughly assume that

the graph of the nonlinearity f lies, for all large values of the ar

gument, between two consecutive eigenvalue branches of the operator

A. For such equations we prove that the union of all full bounded so

lutions of the differential equation is compact and we compute its

homotopy index. Since our parabolic equation is gradient-like, this

will give us a first existence result of solutions of the correspon

ding elliptic boundary value problem.

If 0 is an isolated equilibrium of our equation then we can compute

its index and thus give conditions assuring the existence of nontri

vial solutions of the elliptic equation as well as the existence of

heteroclinic orbits for the parabolic equation. This is done in Sec

tion 2.6.

Using the homotopy index and the maximum principle we then prove (in

Section 2.7) the existence of positive solutions of second order

elliptic boundary value problems and heteroclinic orbits of the cor

responding parabolic boundary value problems.

In Section 2.8 we consider certain variational operator equations

of the form Au=f(u) where A is linear and noninvertible.

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations© Springer-Verlag Berlin Heidelberg 1987

73

By combining the Liapunov-Schmidt reduction procedure with the homo

topy invariance of the homotopy index we arrive at a continuation

principle which is similar to the coincidence degree continuation

method of Jean Mawhin.

We show that for gradient systems, the homotopy index continuation

method may work in cases in which the coincidence degree is zero.

We also give an application of this result to a periodic boundary

value problem of second order.

2.1 Sect0rial operators generated by partial differential operators

In this section we will state (without proof) a well-known result

which shows that many differential operators occurring in elliptic

boundary value problems give rise to sectorial operators.

Let ItCJRn be a bounded domain of class c 2m+ S where m>1 is an integer,

and 0<S<1. Consider the following linear differential operators

A(x,O)

Bj(X,D)

!Ct!<m. - J

b j (x)oCt Ct

(1)

Here Ct=(Ct 1 , ... ,Ctn ) denotes Ct_ Ct1 Ctn d -~/~

a multiindex, !Ct!=Ct1 + ... +Ctn '

o -0 1 ••• 0 an D.-o ox .. n J J

continuous on IT.

Moreover, a :IT+R, !Ct!~2m are Lipschitz Ct

Furthermore, if m>1 then suppose that{Bj(x,D) }j=1 represents

Dirichlet boundary conditions i.e. aj - 1

B'=---'--1 where for x E alt, v(x) is the outward normal to It at x. J avJ-

If m=1 then assume that

B(x,D)u B1 (x,D)u au a]l(x)+h(x)u(x) (2)

Here ]l:alt+Rn is a nontangential smoothly varying direction on

alt, h:alt+R is of class c 1 (alt) and u:alt+R is an arbitrary C1-function.

(2) represents the so-called "mixed-boundary conditions".

Let for k=1, ... ,r, the linear differential operators

(Ak(x,D), B~(X'D))' j=1, ... ,mk be given. Assume that these differen

tial operators satisfy the above properties.

Let f:TIxJRr +JRr , (x,u)+f(x,u), (x,u) ETIxJRr be a continuous mapping.

Moreover, assume that f is locally Lipschitzian in u E JRr , uniformly

for x E ~ .

74

Consider the following system of partial differential equations

aUk (t,x) k ~ +A (x,D)uk(t,x) fk(x,u(t,x», xES1, tEJ, k=1, ••• ,r

(3) xEaS1, tEJ, k=1, ••• ,r, j=1, ••• ,~.

If J is an interval in lR and u:Jxfl+lRr is a mapping smooth enough so

that (3) is satisfied in the classical sense (pointwise), then u is

called a classical solution of (3).

If the derivatives in Ak and B~ are interpreted in the distributional J

sense, then we speak of distributional solutions. In most applications,

due to regularity theory, distributional solutions are also classi

cal solutions. Now consider the following system (4):

k A (x,D)Uk(X)

k Bj(X,D)Uk(X,D)

fk(x,u(x» xES1, k=1, .•• ,r

o xEaS1, k=1, ••• ,r

(4)

Solutions of (4) are exactly the time independent solutions of (3).

We will now concentrate on distributional solutions of (3) and (4)

and rewrite these equations in an abstract form. For every p>1, let

D be the closure in W2m ,P(S1) of the set of all functions u E c 2m (fi") p

that satisfy the boundary conditions B.(x,D)u=O on aS1(1<j<m). J --

For u E Dp define

(ApU) (x) = A(x,D)u(x) •

Then A :D +LP (S1) is a closed operator. p p Now the following result holds:

Theorem 1.1 (see e.g. Friedman [1] or Tanabe [1]).

Suppose A(x,D) is strongly elliptic i.e.

(_1)m L aC/.(x)i;;(X>O for xEfl and i;;ERn, i;;;o10.

I (XI =m

(5)

Then the operator Ap defined ~ (5) is sectorial on Xp =LP (S1).

Moreover, A has compact resolvent and the spectrum a(A ) of A is p- - --- p-p independent of p and consists of ~ sequence {Ak }, k=0,1,2, •.•

reAO~reA1~reA2~"" of distinct eigenvalues such that IAkl+oo for

k+oo. If A2 is selfadjoint, then these eigenvalues are real.

Remark:

75

The most common example of a strongly elliptic differential operator

is given by A(x,D)=-~, where ~ is the Laplace operator. Note that

our terminology of "strong ellipticity" follows Friedman [1], while

Tanabe [1] calls A "strongly elliptic" if (_1)mA is strongly ellip

tic in Friedman's sense. If xa, a>O, is the family of fractional p

power spaces defined by Ap' then by the Sobolev embedding theorems

there is a continuous embedding.'

if 02.v<2mx-n/p.

(see Henry [1], Theorem 1.6.1). k Now given system (3)* choose for every k a Pk>O. Let A be the secto-Pk

rial operator on X defined by the differential operators Pk

(Ak(x,D), B~(X,O)) above. Let A be the product of these operators. J

r Then A is sectorial on X= IT X • (see Henry [1], p. 19).

k=1 Pk Moreover the a-th fractional power space of A

a-th fractional power spaces of the operators

is the product of the

Ak . Therefore, if we Pk

take the numbers Pk>O, k=1, ... ,r, and O<a<1 every k

in such a way that for

(6)

then XacC°(TI,~r) continuously. In the applications to (3) given in

this book, we will always assume that (6) is satisfied. .... a ~ -

Now define the operator f:X .... X by flu) (x) :=f(x!.u(x)), x E n. It is an easy exercise to prove that (by (6)) f is a well-def~ned

operator which is Lipschitzian on bounded sets in xa. f is called

the Nemitski-operator induced by the mapping f. We will often drop

the hat "~,, and use the same symbol to denote the mapping f and its

Nemitski-operator. With these definitions the system (3) (resp. (4))

can be written abstractly as

dU+A dt u flu)

(resp. Au = flu)

(7)

for u E D (A) (8) )

(7) is the abstract parabolic equation of the form (Sf) of Section 1.1.

76

2.2 Center manifolds and their approximation

In this section we recall without proof some basic existence and

approximation results for local invariant manifolds near zero with

respect to a semilinear parabolic equation. For details the reader

is referred to Henry [1], Carr [1], and Chow and Hale [1].

Theorem 2.1 (see Theorem 6.2.1 and Corollary 6.2.2 in Henry [1]).

Suppose A is sectorial in X, 02a<1, U is ~ neighborhood of zero in Xa, f:U+X is differentiable, f' is Lipschitz continuous and f(O)=O.

Let L=A-f' (0) and assume that o(L)n{A1 re \<O} is a finite set, iso---- ---- - -- - --- ---lated in 0 (L) •

Let X=X1$X2$X3 be the corresponding decomposition of X into L-inva

riant subspaces X., j=1,2,3 such that re O(L1)=0 -- - J ----- .. re O(L2)<0, re O(L3»0, (Lj=LjIXj, j=1,2,3), and let dim(x1+x2) <00. Then there exist open convex neighborhoods Vj of zero in

xj=xanx j , j=1,2,3 such that V=V1$V2$V3CU, and there exist two conti

nuously differentiable mappings ~:V1$V2+V3 and P:V1+V2 satisfying the following properties:

as x 1+x2+O in Xa

as x 1+O in Xa

Here, g(u)=f(u)-f' (0) u, and Ej is the projection of X onto

Xj , j=1,2,3, induced £y the above direct ~ decomposition.

The mapping ~ defines a so-called local center-unstable manifold

The proof of the existence of ~ follows the same lines as that of

Theorem 6.2.1 and Corollary 6.2.2 in Henry [1]. Cf. also Theorem 1

in Chapter 2 of Carr [1] and Theorem 2.11 in Chapter 9 of Chow and

Hale [1].

77

Given the mapping W, we consider, on the one hand, the equation

u+Lu = g(u) uEV ( 1 )

generating a local semiflow n on V, and, on the other hand, the re

duced equation

Since X1~X2 is finite-dimensional, (2) is an ordinary differential

equation generating a local semiflow nw on V1~V2.

Applying to this reduced equation the center manifold theorem (e.g.

Theorem 2.2 in Chapter 9 of Chow and Hale [1]) we obtain the mapping

p having the desired properties. The set {x 1+P (x 1 ) IX1 EV} is a local

center manifold for the reduced equation (2).

We thus obtain the following

Theorem 2.2

If P and Ware as in Theorem 2.1, then define

Proof:

This follows by a simple differentiation using (1) and (2) of Theorem

2.1. ~ defines the set Mc={x1+~(x1) IX1 EV1 } called a local center

manifold for (1).

The following result shows how a local center manifold can be appro

ximated up to any given order of accuracy.

Theorem 2.3

Assume all hypotheses of Theorems 2.1 and 2.2.

Let w:V 1-+X~~X~ be ~ C 1-mapping with Lipschitzian derivative w' and

such that u 1+W(U 1 ) EU ~W(u1) ED(L) for u 1 EV1 . Define

~ere, Ei is the projection onto Xi in the above direct sum decomposi-

78

tion •. If there exists ~ p>1 and an M>O such that

then, for some M>O

for all u 1 E V 1 •

Theorem 2.3 is an extension of Theorem 6.2.3 in Henry [1]. The proof

is obtained by modifying the arguments from the proof of the latter

theorem. Details are omitted.

2.3 The index product formula

Given a local center manifold Mc ={x 1 +1; (x 1 ) IX1 E V 1}' where I; is as in

Theorem 2.2 of the last section we can consider the following redu

ced equation on V1

(R)

(R) is the reduction of Eq. (1) of the last section to the center

manifold Mc.

From the center manifold theory it is known that all small invariant

sets K of Eq. (1) of the last section lie on Mc' i.e. they are go

verned by the above Eq. (R). Consequently the question arises whether

there is a relation between the homotopy index of such an invariant

set with respect to the full parabolic equation and the homotopy in

dex of the same set with respect to Eq. (R) on the center manifold.

It is the purpose of this section to show that such a relation exists.

Since (R) is a finite (usually low) dimensional ordinary differential

equation, the index relative to (R) might be easier to compute than

the index relative to (1). We will present an example of such a com

putation in the next section.

We can now state

Theorem 3.1

Let the hypotheses (and, consequently the assertions) of Theorem 2.1

of the last section be satisfied. Let I; be as in Theorem 2.2 of the

last section and TIl; be the local semiflow on V1 generated ~ ~. (R)

above.

The~ there exist open and bounded neighborhoods WjCV j of zero in

79

a x j , j=1,2,) such that the following properties hold:

(1) If KscW1 is isolated (in V1 ) and TIs-invariant, then

K={x1+s(x1) IX1 E K1 } is an isolated (in V) and TI-invariant set. More

over, both h(TI,K) and h(TIs,Ks ) are defined and

h(TI,K)=h(TIs,Ks)AIm (1)

where m=dim X2 .

(2) If KcW is isolated (in V) and TI-invariant, then Ks=E1KCW1 is iso

lated (in V1) and TIs-invariant and K={X1+s(x1) IX1 EKs }. Moreover,

h(TI,K) and h(TIs,Ks ) are defined and (1) holds.

Before proceeding to the proof of Theorem 3.1, let us notice the fol

lowing trivial, but very useful result:

Proposition 3.2

Let X and X be metric spaces and let TI(resp. TI) be ~ local semiflow

on X (resp. on X).

Suppose that there is ~ homeomorphism T:X->-X such that for every x E X

and t E R+, XTIt is defined if and only if (T (x» TIt is defined and then

(T (x» nt=T (XTIt) .

Under this hypothesis, whenever (TI,K) is admissible, then (TI,T(K» is

admissible and then

h(TI,K)=h(i,T(K» .

The proof of Proposition 3.2 is left to the reader.

A large part of this section will be devoted to the proof of Theorem

3 .1.

Let us first notice that by taking appropriate subsets of Vi' if ne

cessary, and substituting them for Vi' we may assumew.l.o.g. that g

is bounded on V. Let us further remark that since X1 and X2 are fi

nite-dimensional vector spaces it follows that X~=X1 and X~=X2 and

the norm on x~, i=1,2 induced by x a is equivalent to the norm induced

by X.

1. Step

a a a We introduce a transformation of variables T:V->-X1xX2xX3 defined as

T(u)=T(x1@x2@x3)=(x1,x2,x3 -~(x1+x2»·

a a a 0 0 0 0 Let us first prove that T (V) is open in X1 xX2xX3. Let u =X 1@X 2ElJX 3 E V.

Then there is an s>O such that if x. EX~ and ~x.-x?l<s for i=1,2,3 l l l l-

then x 1(!)x 2(!)X 3 E V. Let o<s/2 be such that if Ilxi-x~lla<o for i=1,2,

then 1~(x1+x2)-~(x~+x~la<S/2. Let O={(Y1'Y2'Y3) Ex~xx~xx~ll\yi-y~la<o

80

o 0 0 0 0 for i=1,2,3}. Here (Y1'Y2'Y3)=T(u ). Hence T(u ) EO and 0 is open in

a a a x 1 xX2 xX 3 · We show that OcT (V) . In fact, let (Y1 'Y2'Y3) EO and define

x 1=Y 1 ' x 2=Y 2 ' x3=~(x1+x2)+Y3. Since x~=Y~ for i=1,2 and 8<~/2, we

have xi E Vi' i=1, 2. Therefore x3 is well-defined and

Thus X3 EV3 , Le. u=X 1+X 2+X 3 EV. But, obviously, T(U)=(Y1'Y 2 'Y3 ) and

so T(V) is open. - -1 - -1

Let V=T(V). Define T :V~V by T (Y1'Y2'Y3)=(Y1+Y2+(~(Y1+Y2)+Y3)).

It is obvious that T- 1 is the inverse of T, and so T is a homeomor

phism. We denote by (x1 ,x2 ,Y3 ) generic points of V=T(V).

Moreover, define the local semiflow 11 on V as the "image" of 11 under

T, i.e. (Tu)~t=T(unt).It is easy to see that 11 is the local semiflow

on V which is generated by the solutions of system

X1+L 1X1 N1 (x 1 ,x2 ,Y 3 )

x 2+L 2x 2 N2 (x 1 ,x2 ,y3 ) L (2) f y 3+L3Y3 N3 (x 1 ,x2 ,Y3 )

-1 . Here, Ni(x1,x2'Y3)=EigT_ (x 1 ,x2 ,Y 3 ) for 1=1~2 '-1 E3 :=E1+E2 and

N3(x1,x2'Y3)=~'(x1+x2)[E3g(x1+x2+~(x1+x2))-E3gT (x 1 ,x 2 ,Y3 )] +

-1 + E3gT (x1,x2'Y3)-(E3g(x1+x2+~(x1+x2)) .

2. Step

Define the transformation

S:V1~V2~X1xX2 by S(x1~x2)=(x1,x2-P(x1))·

Proceeding exactly as in the 1st step we prove that Y=S(V1~V2) is

open in X1 xX2 and S is a homeomorphism of V1~V2 onto Y.

We denote by (x 1 'Y2) g:neric points of Y. Let ~~ be the imaqe of 11~ under the map S. Then 11~ is easily seen to be the local semiflow on

Y generated by the solutions of the system

where

L f

(3)

81

3. step

Since O(L)n{AlreO(A)<O} is a finite set, isolated in o(L), it follows

that for some y>O, reo(L 2 )<-y and reo(L3»~.

Therefore, by Theorem 1.5.3 in Henry [1], there are constants C,y>O

such that

for x 2 E X2 ' t<O (4)

(5)

Here,1 lis the norm on X.

Define the following homotopies (for T E [0,1]):

By the definition of N.,M. and the convexity of V., i=1,2,3, it fol-J J 1.

lows that N. (T) (resp. M. (T)) is a well-defined locally Lipschitzian J ~ J

mapping from V into Xj , j=1,2,3, (resp. from Y into Xj , j=1,2).

Consider the equations

x 1+L 1X 1 N1 (T) (x 1 ,x2 'Y3)

X2+L 2X2 N2 (T) (x 1 ,x2 'Y3) } X3+L3Y3 N3 (T) (x 1 ,x2 'Y3)

x 1+L 1x 1 M1 (T) (x 1 'Y2)

Y2+ L2Y2 M2 (T) (x 1 ,Y2 ) }

We then have the following

82

Lemma. 3.3

There are open balls Wi at zero in x~, i=1,2,3 and constants Cl ' ~>O suc~ that Wi cV i' i=l, 2,3, W=W1 xW2 xw3cV and such that for all ,E [0,1 ],

t, to E. JR, t.:.t o ' the following properties hold:

(1) If s-+(x l (s), x 2 (s), Y3(s» is ~ solution of (6,) on [to,t] lyingm

W for s E [to,t], then -)l(t-t )

IIY3(t>lla~cle °IIY3(to )ll a • (8)

(2) If s-+(x 1 (s), Y2(s» is ~ solution of (7,) on [to,t] lying in

W1 XW 2 for s E [to ,t], then +)l(to-t)

1iY2(to)ll~cle 1iY2(t)~. (9)

- --Moreover, if xi EWi , i=1,2 then q,(x 1 +x 2 ) EW 3 and p(x 1 ) C:W2 •

Proof of Lemma 3.3:

Since g, q, EC 1 and g(O)=O, q,(0)=0, g' (0)=0, q,' (0)=0, it follows that

for every E>O there is a O(E), 0<O(E)<E/2 such that the function

E-+O(E) is increasing, lim O(E)=O and such that E+O

(writing B. (a)={xEx':lllxll <a}) : 1- 1- a

for xi EBi(O(E», i=1,2 (10)

IIp'(xl)II~E for xl EB 1 (0(E» (11 ) _ _ _ 3 _

~g(xl+x2+x3)-g(xl+x2+x3)~~E iIllxi-xi~a

for xi,xi EBi(o(E», i=1,2,3. (12 )

Let E>O be fixed and such that Bi(E)cVi for i=1,2,3. Later on, we

will impose additional conditions on E.

Let O:=O(E). There is a 01 <0/2 such that q,(x 1 +x 2 ) EB 3 (0/2) and

p(x l ) EB 2 (0/2) for x i EB i (ol)' i=1,2. It follows that if x i EB i (Ol)'

i=1,2, Y3EBi(01)' then x1+x2+(1-')Y3EV and from (10) and (12):

~

~II q,' (xl +x 2 ) II a ·11 E311 ·11 g (xl +x2 +q, (xl +x 2 ) ) -g (xl +x2 + (1-,) Y 3 +

+ q, (xl +x 2 ) ) II + II E311 ·11 g (xl +x2 + (1-,) y 3 + q, (xl +x 2 ) ) -

2 --g (xl +x2 + q, (x 1 +x 2) ) II ~ E II E 311 . IiY 311 a + Ell E31111 Y 3 11 a . ( 13)

83

Moreover, if x 1 EB 1 (o1) and Y2EB2(o1) then x 1+(Y2+ P (x 1» EV1(l!V2' so

(x 1 'Y2) EY, and from (11), (13) we obtain (using the equivalence of

the norms II II and I I on x<;=x., i=1,2 and assuming that e:<1 a. 1. 1.

~M2(T) (x1 'Y2) 1=IM 2(x 1 , (1-T)Y2) 11.::.IIp' (x 1 ) ~ iE1~ !g(x 1+P(x 1 ) +

+.(x1+P(x1»)-g(x1+P(x1)+(1-T)Y2+.(x1+ (1-T)Y2+P(x1»)~+

+~E2~~g(x1+P(x1)+(1-T)Y2+·(x1+(1-T)Y2+P(x,»)-g(x1+P(x,)+

+. (x, +P (x,) ) ) 11~2e:211 E11111Y 2 ~ a. +2e:1 E211 ~ Y 211 o.~ (de: 2 11 E111 +de: II E211) IIY 211

Here d is such that

a. a. for all y E x 1 +X2

Define Wi =Bi (01) , i=1,2,3.

Now let_s+(x,(s), x 2 (s), Y3 (s», sE [to,t], be a solution of (6 T ) ly

ing in W for all s E [to ,t]; also let s+(x1 (s), Y2 (s», s E [to ,t],

be a solution of (7 T) lying in w1xW2 for sE [to,t].

Then, from the variation-of-constants formula we obtain, using (4),

(5) and Theorem '.4.3 in Henry [1],

_ -y (t-to) 1IY3 (tH o.~ce 1IY3 (to) II a. +

('4 )

and

_ +y(to-t) t +y(to-s) 1IY2 (tb) II~Ce IY2 (tl ~ +c[ e ~~(T) (x1 (s) ,

to +y(t -t) t +y(t -s)

~ Ce 0 ~Y2(t)~+e:"l e 0 ~Y2(s)~ds

Y2 (s» lids <

(15 )

to 2 -

Here, to,t are arbitrary elements of [to,t], e:'=e: IIE3~+e:lE31, e:"=de:2~E1~+de:~E21, and Co. is a constant depending only on L3 and a..

Now, apply Gronwall's Lemma to (15); moreover, apply Lemma 7.1.1.

in Henry T1] to u(t>=eYtIIY3 (t) ~o.. Then we obtain constants C1 ' q>O

84

such.that

-y(t-to) q(t-tO) 1Y3 (t)ii a < c11iY3(tO)~ae e ( 16)

-y (t-to) e:" (t-to) < c1~Y2(t)ie e ( 17)

Here C1=C 1 (a,C) and q=(ca ·e: ' .r(1-a» 1/'-a, where r is

the gamma function. Choose e: so small that y-e:">~>O and y-q>~>O for

some ~.

Then (16) and (17) imply (8) and (9), respectively. The lemma is

proved.

Lemma 3.4

-Let W be as in Lemma 3.3. Suppose that N (resp. N<ji) is ~ closed set a -- - - - -in X1xX2xX3 (resp. in X1xX2), NcW (resp. N<jicw1xW2), and let nIT)

(resp. TI<ji(T» be the local semiflow ~ V (resp. ~ V1xV2) generated EY the solutions of (6 T) (resp. (7 T».

Then N is {~(Tn) }-adrnissible (resp. N<ji is {~<ji(Tn) }-admissible) for every sequence {T }c[O,1]. _ n

Furthermore, nIT) does not explode in N (resp. TI<ji(T) does not explo

de in N<ji) for TE[O,1].

Finally, let K(T) (resp. K<ji(T» be the largest ~(T)-invariant set in

N (resp. the largest TI<ji(T)-invariant set in N<ji)' Then K(T)cW1xW2x{O} (resp. K<ji(T)cW1x{O}).

Proof of Lemma 3.4:

a - - -Let N be closed in X1xX2xX3' New, and let N<ji be closed in

X1x,X2 , N<jicW1xW2. Take an arbitrary sequence h n }c[O,ll and set

nh =1f(xn )· We will first show that N is

and {z }CN be n n

zn nn t= (x 1 (t) ,

such that t +00 n n n

x 2 (t), Y3(t»,

{n }-admissible. In fact, n _

and znnn[O,tn]cN. Writing we obtain from (8):

-~t

~y~(tn)la ~ C1e n~y~(O)~a <

let {tn }cJR +

It follows that ~y~(tn) ~a+o. Moreover, since X1xX2 is finite-dimen

sional and both {x~(tn)} and {x~(tn)} are bounded, it follows that

there is a subsequence of {(x~(tn)' x~(tn»}' denoted by the same

symbol {(x~(tn)' x~(tn»}' which converges to some (x1,x2) EX,xX2 •

Consequently, zn~tn+(x1,x2'O) and as N is closed in X1xX2xx~, we have (x1 ,x2 ,O) EN. This proves that N is {n(Tn)}-admissible. That N<ji

85

is {TI~('n)}-admissible :ollows trivially from the fact that X1xX2 is

finite-dimensional and N~ is bounded.

The second assertion of the lemma follows immediately from Theorem

I. 2. 4.

Now let K(,) be~the largest TI(')-invar~ant set in N, and let K~(') be the largest TI~(,)-invariant set in N~. We will first prove that

K(,)cW1 xW2 x{o}. In fact, if ZEK(,) then there is a full solution

t+a(t) of (6,) such that a(O)=z and a(t)c:i(,)c:N for all t ElR. Writing

a (t) = (x1 It), x 2 (t), y 3 (t» we obtain from Lemma 3.3 for every

to' t f lR, to <t

Letting to +_00 we see that IIY3(t)IIa,=O for all tElR, thus proving

K ( , ) cW 1 xW 2 x { 0 }.

~imilarly, let t+a(t) ~be a full solution of (7,) such that

a(t)=(x1 (t), Y2(t» EK~(') for tElR. Taking to,t arbitrary, to<t,

we obtain from (9)

Taking t++oo we obtain 1IY2 (to )ll=o. Hence K~(,)cW1x{O}, as claimed.

The lemma is proved.

4. Step

We can now complete the proof of Theorem 3.1:

Choose W. to be small open balls at zero in X~ for which CIW.cV., ~ ~ ~ ~ -~ ~

i=1,2,3, such that T(CIW1~CIW2~CIW3)cW and S(CIW1~CIW2)cW1xW2' and

such that ~(W1~W2)cW3 and P(W1 )cW2 •

Let W=W1~W2~W3.

We first prove (1) of Theorem 3.1:

Let K~cW1 be an isolated (in V1 ), TI~-invariant set. Then, by Theorem

2.2. it is easily seen that

is TI-invariant.

Being isolated in V1 , K~ is, by definition, closed in V1 • But since

the closure CIW1 of W1 in X1 is in V1 , it follows that K~ is closed

in X1 • Hence there exists a set B, closed in X1 , BcW1 , such that B

is an isolating (in V1 ) neighborhood of K~.

86

Let N=B~CIW2~CIW3. Then N is closed in xa, and, by our assumptions,

K is in the interior of N relative to the topology of Xa. Hence K is

in the interior of N relative to V. To prove that N is an isolating

neighborhood of K, it therefore suffices to show that K=K', where K'

is the largest TI-invariant set in N. Obviously KcK'. By the defini-- a -tion of T and TI it follows that N=T(N) is closed in x1x~2xx3:K'=T(K')

is a TI-invariant set and NeW. Now use Lemma 3.4: Since TI(O)=TI, it

follows that K'=K' (0)eW1xW2x{OL Thus whenever x 1+x 2 +x 3 EK', then

x 3 =<jl(x1 +x 2 ). Therefore K~={x1+x2Ix1~x2+<jl(x1+x2) EK'} is TI<jl-in~ar~an= and K~eB~CIW2. Obviously S(B~CIW2)=N<jl is closed in x 1 xx 2 and N<jlew 1 xW2 .

Again from Lemma 3.4 we conclude that K~=S(K~)eW1x{OL For every

x 1 +x2 E K~ it follows that x 2=p (x 1 ) and consequently

Kk={x1Ix1+P(x1)+<jl(x1+P(x1)) EK'} is TI~-invariant. But since KkeB, it

follows that Kk=K~. It follows that K'=K and this proves that N is

an isolating neighborhood of K. - - -Let K(T) be the largest TI(T)-invariant set in N=T(N). By Proposition

3.2 K(O)=K=T(K). By Lemma 3.4 K(T)eW1xW2x{0} and so K(T) is easily

seen to be independent of T, i.e. K(T)=T(K)=K for every T E [0,1].

Therefore N is an isolating neighborhood of K(T), for every T E [0,1].

Using this fact, Lemma 3.4 and Theorem 1.2.4 it is trivially seen

that (1) and (2) of Definition 1.12.1 are satisfied for the map

~:T+(TI(T), K(T)). Hence ~ is S-continuous. Theorem 1.12.2 implies

that

h(TI(O), K(O)) (18 )

~e~ K<jl={x 1+P(x 1 ) IX1 EK~} and K1={(x1,P(x1)~lx1 EAK~L Then, o~viously, K=K1x{0}. Moreover, it is immediate that TI(1)=TI<jlxTI3, where TI<jl is the

image of TI<jl under the homeomorphism x 1@x 2+(x 1 ,x2 ), and TI3 is the lo

cal semiflow on V3 generated by the solutions of

Using (5) we conclude from Theorem 1.11.1 that h(TI 3 ,{0})=I O. Hence

by Theorem 1.10.5

A

h(TI (1), K1 x{O}) (19 )

o Here we used the obvious fact that (Y'YO)A(S ,sO) is homeomorphic to

(Y,yO)' for every pointed space (Y,yO).

Using Proposition 3.2 we obtain from (18) and (19)

87

"'" "" -v .....

:onsider now the map ~:~+(TI~(T) ,K~(T)) where K~(T) is the largest

TI~(T)-invariant set in N~=S(BffiC1W2). As before, it follows that

K~(T)CW1x{0} and therefore K~(T) is independent of T, i.e.

K~(T)=K~(O)=K~=S(K~)=K~x{O}. It follows again that ~ is an S-conti

nuous mapping. Theorem 1.12.2 implies

( 21)

But TI~(1)=TI~XTI2' where TI2 is the local semiflow on V2 generated by

the solutions of

Using (4) we conclude from Theorem 1.11.1 that h(TI 2 ,{0})=Im where

m=dim x2 .

Now we obtain from (20), (21), Proposition 3.2, and Theorem 1.10.5

(20) and (22) imply

i.e. formula (1).

Part (1) of Theorem 3.1 is proved.

To prove part (2), let KCW be an isolated (in V) and TI-invariant set.

Thus, by definition, K is closed in V. But since C1WcV, it follows

that K is closed in xa. Hence there is a set N, closed in xa, NcC1W, ~

such that N is an isolating neighborhood of K. Let N=T(N) and K=T(K).

Using the same arguments as those in the proof of part (1), we prove

that for every x 1+x 2+x 3 EK, it follows that x 2=P(x 1 ) and

x3=<!> (x 1 +P (x 1)), i.e. x 1 +x2+x3=x1 +~ (x 1). It follows that K~=E1K is a

TI~-invariant set, isolated in V1 ' and K={x1+~(x1) IX1 EK~}.

Therefore the rest of part (2) follows from part (1) and the theorem

is proved.

If x1={0} in Theorem 3.1, then ° is a hyperbolic equilibrium, i.e. a

so-called saddle-point property is satisfied (see e.g. Henry [1]).

Then h(TI~,{O}) is obviously equal l.0. Using the formula

88

h(1T,{O}) = rn . If A has compact resolvent, then this latter result is valid under

more general assumptions on f:

Theorem 3.5

Suppose that A is sectorial in X and has compact resolvent. Let

O<u<l and assume that U is a neighborhood of zero in xu, f:U+X, is

locally Lipschitzian, f(O)=O and f' (0) exists. Let L=A-f' (0) and

assume that there is ~ decomposition X=X 29X3 into L-invariant sub

spaces xi' i=2,3 with re a(L2 )<-0<0 and re a(L3 »0>0 for some 0>0

and L. :=Llx., i=2,3. --]. ].

Then dim X2=:m<oo. Moreover, K={O} is an isolated invariant set,

h(1T,{O}) is defined and h(1T,{O}=Im.

Proof:

We may assume that U is a ball at zero of radius p>O, and that f is

Lipschitzian and (hence) bounded on U. Define gT:U+X to be

gT(U) = (l-T) (f(u)-f' (0) u), TE[O,l],

and 1TT to be the local semiflow on U generated by

u+Lu = gT(U)

We will prove that the map ¢(T)=(1T ,{O}) is a well-defined S-conti-T

nuous map from [0,1] into S=S(U). Assuming this for the moment, we

infer from Theorem 1.12.2 that h(1T O,{O})=h(1T l ,{O}). However, (SO) is

our original equation, so 1T 0 =1T. Furthermore, (Sl' is a linear equa

tion to which Theorem 1.11.1 applies. By that theorem h(1T l {O})=Im.

This proves Theorem 3.5, except for our claim that ¢ is S-continuous.

To prove our claim, it is only necessary to show that there is a

closed set NeU, such that for every T E [0,1], N is an isolating

nei~hborhood of K={O}, relative to 1T T . Then Theorems 1.2.4 and 1.4.4

immediately imply all hypotheses of Definition 1.12.1, i.e. ¢ is S

continuous. If such a set N does not exist then there is a sequence

Tn E [0,1] and for every n, a full solution un:IR+U of 1Tn:=1TT with n

c n sup II un (t) II of 0 , tEIR U

c <II u (0) II + 1 n n U

89

and c +0 as n+=. Let gn=g . n Tn --1

Then for all n large enough, the maps gn:V+X, gn(v)=cn gn(cnv) are

well-defined and Lipschitzian, hence, bounded on V, where

v={uExC(lllull <2}. Let v (t)=c- 1u (t). Then vn is a full solution of _ _ C( n n n TIn where TIn is the local semiflow generated on V by solutions of

v+Lv

Let N={vEXC(1 Ilvll <1}. Then Theorem 1.4.4 implies that N is strongly C(-{n }-admissible. In particular, {v (O)} >1 contains a convergent sub-n n n sequence. We may assume that v (O)+v O as-n+oo • Now v (0) EA- (N) for n n TI _ _ _ n

all n. Since gn+O as n+oo uniformly on V, TIn+TIO where TIO is the re

striction to V of the linear semiflow TI1 above.

Now Theorems 1.2.4 and 1.4.5 imply that v 0 E A (N) naN. However, by TI1

Theorem 1.11.1 A (N)={O}, a contradiction. TI1

This completes the proof of Theorem 3.5.

Remark:

The decomposition x=x2ex3 with the properties listed in Theorem 3.5

exists if and only if 0(L)=02u03 where re 02<-0<0 and re 03>0>0.

Moreover, m is necessarily the total algebraic mUltiplicity of eigen

values A of L with re A<O. (cf. Corollary 1.11.2 and Henry [1]).

2.4 A one-dimensional example

As an application of our result in the last section, we will now com

pute the homotopy index of K={O} for a one-dimensional Dirichlet

boundary value problem.

More precisely, we cons.ider the following equation

dU at

u(t,O) = u(t,1f) O.

x E (0,1f) L J

Here f: JR +JR is a locally Lipschitzian function.

Let us write equation (1) in the form (Sf). To this end, let 2 2 1 X=L (0,1f), D(A)=H (O,1f)nH O(O,1f),

A:D(A)+X , Au

( 1 )

90

where a: is the derivative in the distributional sense. A is a secto

rial operator with compact resolvent and a(A)={(r+1)2[r=0,1,2, ... }

with corresponding (normalized) eigenfunctions e (x)=V2/n sin(r+1)x, r xE (O,n). Write A =(r+1)2, r>O. r -As in section 2.1, it follows that f defines a corresponding locally

A 1/2 Lipschitzian Nemitski operator f:X +X. Also, it is well-known that

x 1 / 2=H6(0,n) .

It follows that we can apply our theory to equation (1). First assume

the nonresonance case, i. e. suppose that ]1=f' (0) exists and ]1 ft' a (A) ,

say Ar <]1<Ar +1 , where r~-1 and 1._ 1 :=-00. Then a(A-f' (O))=a(A)-]1. There

fore there are exactly r+1 negative eigenvalues of a(A-f' (0)) and all

the other eigenvalues are positive. All eigenvalues being simple,

Theorem 3.5 implies that h(n,{O}) is defined and

We now consider the resonance case. Assume f to have a Lipschitzian

derivative.near zero and let ]1=f' (O)=Ar for some r~O. We shall compu

te the index of {a} under some (generic) hypotheses on f. More pre

cisely, let

f (s) (2)

where a#O, K~2 is an integer, and S(s)=0(sK+1) for s+O.

Let g(s) :=f(s)-Ars and L=A-Ar . Let x 1=span{er }, x 2=span{e O, ... ,er _ 1 },

x3-span{ei[i~r+1}. Then x=x1ex 2ex3 , the spaces Xi are mutually ortho

gonal and L-invariant, and writing Li=L[X i , we see that L1=0, i.e.

o

Hence there is a one-dimensional local center manifold close to zero

which can be described by a mapping ~:v1+v2ev3 where Vi is a neigh

borhood of zero in x:/2, for i=1,2,3. K l

Set (u 1 )= ·v, u 1 EX 1 with vl.er , vED(A) to be determined la-

ter.

Then for x E (O,n)

K K g (u1 +<1> (u1 ) ) (x) = a [er (x) + v (x)] +

K +S«u1,er>er(x)+<u1,er> v(x))

Since [[ is an equivalent norm on X1 ' it follows that

(3 )

91

(4)

Computing /:,.(u1) by means of the formula in Theorem 2.3 we obtain

K K II IIK+1 -a<u1,e > <e ,e >e +0(uu1 ) r r r r

KKK I II K+1 = -<u1,e > {Lv-ae +a<e ,e >e }+O( u 1 ) r r r r r (5)

Let us now choose vle in such a way that the expression in braces r K K

is zero. Since L=A-Ar and < aer -a<er,er>er,er>=O, such a v can be determined in a unique way.

Using (5) and Theorem 2.3 we therefore conclude that I - n nK+ 1 I c;, (u1) - (u1) II x1 /2.::Mn u 1 11 for all u 1 E V l' Thus the reduced equation on the center manifold reads

U1 (6)

K K+1 TIf . K+1 Here o=<e ,e >=(VZ7i) s~n (r+1)xdx. r r 0

Writing y=<u1 ,er >, we obtain from (6) an equivalent scalar equation

K K+1 Y = a·oy +O(y ) =: h(y). (7)

If K is odd or if K is even and r is even, then 0>0. Hence is this

case,

sign h(y)

for small yf.O.

K sign ay

This implies that the index hc;,(TI,{O}) is given by the index of {a}

with respect to the scalar equation

K Y = ay (8)

However, this latter index is trivial to compute and thus we obtain

the following

92

Proposition 4.1

Let f(S)=A r s+asK+0(sK+1) as s+O, with a~O and K>2. Then if K is odd

or if K is ~ and r is even, then {a} is an isolated invariant set

of 'IT f •

Furthermore, the index h('IT~,{O}) on the center manifold is given ~

0 if K is even and r is even, -----h('IT~,{O}) { I1 if K is odd and a>O -----

l.0 if K is odd and a<O -----The index of {a} with respect to the full semiflow 'ITf is given ~

o if K is ~ and r is even,

{ l.r+1 if K is odd and a>O

if K is odd and a<O

Proof:

The formula for h('IT~,{O}) is the result of the preceding remarks; the

formula for h('ITf,{O}) follows from the index product formula (Theorem 3.1)

by noticing that m=r.

We now consider the critical case R even, r odd. Here it seems pos

sible that the index depends on the higher order terms in the expan

sion of f. To show this, we will restrict ourselves to a special case

f (s) (9)

Here K is even, r is odd, a~O and b is arbitrary. Define g(s)=f(s)-Ars.

K 2K-1 Let ~(u1):=<u1,er> v+<u1 ,er > w where v is as before and w~er is to be determined later.

It follows for x E (O,'IT),

Computing ~(u1) we obtain therefore

2K-1 K-1 K 2K-1 -<u1 ,e> {Lw-aKe v+aK<e ,v>e -be + r r r r r

+ b<e2K- 1 ,e >e }+0(lu1~2K). r r r

As before, wIer can be found such that the expression in braces is

zero.

Theorem 2.3 implies that

Consequently, the reduced equation on the center manifold reads

93

= <u1 ,e >2K-1(aK<eK,v>+b<e 2K- 1 ,e »e +0(llu1112K) • u 1 r r r r r (10)

Writing y=<u1 ,er >, we obtain an equivalent scalar equation

( 11)

If the expression in braces is ~O, then the index of {oJ with respect

to (11) exists and is equal to the index of {a} with respect to

where

Since

fact,

y = ny2K-1 (12)

K 2K-1 n=aK<er,v>+b<er ,er >·

v is independent of band

depend on b.

2K-1 <er ,er»O, this index does, in

Now, it is easily seen that (with ~:=r+1)

1 xf . K+1 1 . K+2 v(x)=CO-cos ~x s~n ~sds-CO-Z ·s~n ~x,

~ 0 ~ (K+1)

Co := a(v'277T)K

No~ a simple integration yields

where C1>O is a constant.

This shows that the first term in n is negative, and we obtain the

following

94

Proposition 4.2

Let K be even and r be odd. Moreover, let f be as in (9) above.

Then the index of {O} with respect to TI f exists if either b<O or else

if b is positive and large enough.

Furthermore,

{ if b'::'O in (9) ,

if b is positive and large enough

2.5 Asymptotically linear systems

If X is a normed space and TI is a local semiflow on X, then by Koo we

will denote the union of all full bounded orbits of TI. In other words,

a point x E X is in Koo if and only if there is a full solution a: JR .... X

of TI, a(O)=x, and a[JR]is bounded. Koo is obviously an invariant set,

and, in general, Koo is unbounded. There are situations, however,

where Koo is bounded. This is so, for example, if TI is dissipative in

some sense (a concept which will be defined later on). In such a

case, Koo is compact and connected and attracts all solutions, i.e.

every solution tends to Koo as t+oo.

In this section we will encounter another class of semiflows for

which Koo is bounded (and in fact compact) but Koo is not an attractor,

in general. This is so for asymptotically linear systems with non

resonance at infinity.

More precisely, we have the following

Theorem 5.1

Suppose that A is sectorial in X and has compact resolvent. Let

0'::'a<1 and assume that f:Xa .... X is locally Lipschitziah, mapping boun

ded sets in Xa into bounded sets in X and asymptotically linear, i.e.

such that there exists a bounded linear map B:Xa .... X such that

(f(u)-Bu)/~u~ .... 0 as lu~ .... 00. a - a Let L=A-B and assume that there exists ~ decomposition X=X2EDX 3 into

L-invariant subspaces Xi ,i=2,3, with re a(L2 )<-O<0 and rea(L3 »o>0

for some 0>0 and L. :=Llx., i=2,3. Then dim X2 =:m<00. Moreover, if TI ----- -- ~ ~ --is the local semiflow on x a generated by the solutions of

u+Au flu) ( 1 )

and Koo is the union of all full bounded orbits of TI, then (TI,Koo) ES(Xa )

Proof:

This theorem is "dual" to Theorem 3.5 and has a "dual" proof. We

therefore only sketch the details.

Define gT:xa+x, TE[O,ll to be the map

gT(U) = (1-T) (f(u)-Bu)

95

and let TI be the local semiflow on Xa generated by the solutions of T

u+Lu = gT(U)

Let KT be the union of all full bounded orbits of TI r'. We will show

that the map ~:[0,1]+S, ~(T)=(TIT,KT) is well-defined and S-continuous.

Assuming this and using the fact that K1={0} by Theorem 1.11.1 we

immediately get the assertion of the theorem.

To prove that ~ is well-defined and S-continuous, it is only necessa

ry to show that there is a bounded set NCXa such that KTcN for all

T E [0,1]. Suppose this is not true.

Then there is a sequence Tn E [0,1] and a sequence of full bounded

solutions t+u (t) of n :=n with n n Tn

c := sup~u (t)~ +00 n n a as n+oo

and ~u (O)i>c -1. n -1 n a - 1

Let vn(t)=cn un(t) and gn:x +X be defined as gn(v)=c~ gTn(cnv),n E~,

Then gn is locally Lipschitzian for n E ~. We will show that for

every P2:.0,

suii:' I g (v) I + 0 IVi O. Then there is an r such that if lui >r then ~f(u)-Bull<Elul • a - a Moreover, there is an M=M(r)2:.IIBII·r such that if lui <r then a-

II f (u) II~M. It follows that

and so

Ign(v) II < 2Mc- 1 n if c . ~vl < r n a-

96

and

if c ·11 vii > r . n a

This implies the desired claim.

Now proceeding exactly as in the proof of Theorem 3.5 we get that a

subsequence of vn converges to v=O. However, this contradicts our as

sumption that

as n->-oo.

The theorem is proved.

The assumptions of Theorem 3.5 are in particular satisfied for sys

tems cr partial differential equations discussed in Section 2.1, if

we assume that the function f: TI. JRr ->-JRr has the property that

(f(x,s)-A*s)/llsll->-o (2)

as lis 11->-00, s E JRr uniformly for x E TI and A * It a (A). In other words, the

nonlinearity is asymptotically linear with nonresonant slope.

If the operator A is self-adjoint (as it is often the case in the

applications) and if fIx,s) is the gradient of a function F(x,s), i.e.

f(x,s)=gradsF(x,S), then Theorem 3.5 may be improved. Before discus

sing such an improvement we need the following important concepts.

Definition 5.2

Let X be a metric space and TI be a local semiflow on X. A point

Xo E X is called an equilibrium of TI, if the constant function

a (t) =xO' t E JR, is a solution of TI.

A continuous function V:X->-JR is called a Liapunov-function for TI, if

for every x EX, the function t->-V (XTIt) is nonincreasing for t E [0, wx ) .

TI is called gradient-like with respect to V, if V is a Liapunov-func

tion for TI and whenever a. is a nonconstant full solution of 1T, then

t->-V(o(t)) is not a constant function.

The following proposition holds:

Proposition 5.3

+ Let V:X->-JR be a Liapunov function for TI. If J=JR (resp. J=JR and

o:J->-X is a solution of 1T with o[J] relatively compact, then V is con

stant on w(o) (resp. on w*(o)). If in addition, TI is gradient-like

with respect to V, then w(o) (resp. w*(o)) contains only equilibria

of TI.

97

The proof is a trivial exercise using the fact that w(cr) (resp. w*(cr))

are invariant sets.

As an obvious corollary we obtain

Proposition 5.4

If 'IT is gradient-like with respect to V and cr:JR+X is ~ nonconstant

full solution of 'IT with cr[JR] relatively compact, then w(cr) and w*(cr)

~ nonempty disjoint sets containing only equilibria of 'IT.

Consequently, a nonconstant full solution cr of a gradient-like local

semiflow with cr[JR] compact joins two disjoint sets of equilibria.

Such a solution cr is also known as a heteroclinic orbit.

We now have the following

Theorem 5.5

Assume the following hypotheses:

(1) (H,< , » is ~ Hilbert space;

(X, ~ II) is ~ real Banach space; X is ~ JR-subspace of H, and the inclusion XcH is continuous.

(2) DcR; A:D+H is ~ self-adjoint linear operator ~ H bounded below,

OcDnx, A[O]cX, and A:=Alo, A:O+X, is ~ sectorial operator in X, de-a. II I 0 1 fining the fractional power spaces (X'I a.) .0~o.~1, X =X, X =0.

Moreover, A has compact resolvent.

(3) 1 /2~o.<1; fm:Xo.+X, mE IN, is ~ sequence of locally Lipschitzian mappings, and G :Xo.+lR , mE IN, is ~ sequence of Frechet-differentiable

-- m mappings, such that:

(3.1) for every u,h E xo., mE IN

OGm (u) (h) = <fm (u), h>

(3.2) for ~ M>O and all mE IN , u E Xo.

Ilf (u) II < Mdull +1) m - a.

(3.3) for some v,o>O, and all mElN, uEO

where ~'IH is the norm of H. Under all these hypotheses, there exists an L>O such that

98

for every mE:JN, and every full bounded solution t+u (t) of

u+Au = fm(u)

sup ~u(t)~ < L • tEJR 0.-

Proof:

Let TIm be the local semiflow on Xo. generated by (4m).

1. Step:

Since A is self-adjoint and bounded below, it follows that for some - -k>O, A+kI~O and re cr(A+kI»oO>O. Write A1=A+kI, A1=A+kI. -1/2 - 1/2 1/2 It follows that A1 is well-defined, A~ /2u=A1 u for u E X and

since o.~ 1/2, i. e. xo.c:x 1 / 2 it follows that the map xo. 3 U+A~ / 2 u E H is

a bounded linear map.

Define V :xo.+JR as m

V () l«A-1/2 A- 1 / 2 > k< » G ( ) m u = 2 1 u, 1 u - u,u - m u

If t+u(t) EXo. is differentiable for tE (t1 ,t2), then

-<fm(u(t», u(t». (5)

Hence, if t+u(t) is a solution of TIm' u(t) ED, for t E (t1 ,t2 ), then

(5) implies

= -<AU+fm (u) ,-Au+fm (u) > = -II-:Au+fm (u) II~

It follows that TIm is gradient-like with respect to Vm•

2. Step

(6)

We claim that there is an LO>O such that if mE:JN and u is an equili

brium of TI , then II ull <LO• m 0.-

In fact, if this is not true, we can assume w.l.o.g. that there is a

sequence {un} of equilibria of TImn such that cn=llunllo.+co.

Let

Let

v =c- 1u • n n n

f :Xa.->-X be n

- -1 defined as fn(v)=c n fm (cnV). Let TIn be the local

n semiflow generated by v+Av=f (v).

- n -1 By hypothesis (3.2), Ilf (v) II <M(llvll +c ). n - a. n

Since Au =f (u), i.e. (A+kI)u =f (u )+ku it follows that n_ mn n n mn _n n

(A+kI)Vn=fn(vn)+kvn i.e. Vn=(A+kI)-1 (fn(vn)+kvn ). Since A has com

pact resolvent, we conclude that, w.l.o.g., vn converges in X to

some v E xa.. Hence II v n -v II H ->- 0 as n-HlO. By hypothesi s (3.3),

99

-1 n O.,::vllvn II H- o·cn ' so o.,::vlvII H, i.e. v=O, a contradiction since II vnlla.=1. This proves our claim.

3. Step

Suppose now that the theorem is not true.

Then there is, w.l.o.g., a sequence t->-un(t), of full bounded solu

tions of (TImn) such that

and

c = sup ~u (t) i ->-00 n tElR n a.

Ilu (0) I > c -1 > 0 n a. - n

as n .... oo

Let v (t)=c-1u (t). Let fn:Xa. .... X be defined as above, i.e. n n n - -1 fn(v)=cn fn(cnv).

Notice that in hypothesis (3) we can assume w.l.o.g. Gm(O)=O, since,

otherwise, we can replace Gm by Gm-Gm(O).

- a. - -2 Define Gn:X .... lR, Gn (v) =cn ·Gm (cn v) • _ n_

It is easily seen that DG (v)h=<f (v),h> for v,hEXa.. Therefore, n _n guing as in Step 1, we V :xa. .... lR ,

n

see that TIn is gradient-like with respect

Vn(v) ~«A~/2v, A~/2v>-k<V'V»-Gn(V) •

1 1

ar-

to

Moreover, since G (v)=fDG (t·v) (v)dt=f<f (t·v),v>dt, we obtain, with mOm 0 m

some constant C>O

100

Now let un be an equilibrium of TI Then, by Step 2 IIi:! II <LO and, mn n a-- -1-

for vn:=cn un' we have

- - 1 2 1/2 2 - 2 2 - 1-Iv (v ) I <-2c - (11A1 U IIH+kllu IIH)+c- IG (u) I<c- C n n - n n n n m n - n (7) n

with some constant C>O, independent of n and of the equilibrium un.

Since t+vn(t) is a full bounded solution of TIn' its a- and w-limit -1 -

:ets contain only equilibria of TIn' i.e. elements vn=cn ·un ' where

un is an equilibrium of TIm n

Therefore, for every t 1<t2 we obtain from (7)

However, by hypothesis (3.3),

- -Vn(vn(t1»-Vn(vn(t2) f1: t (v (v (t»)dt

t n n 2

f211-Av (t) +f (v (t» II~dt > t n n n 1

t > J 2 11 ,; (t) II H ( v II v (t) II H - 0 • c -1 ) d t >

t n n n 1

> Bn·~vn(t2)-vn(t1) ~H '

where B :=inf (vllv (t) II H-o.c- 1). n tE[t1,t2 ] n n

Hence by (8) and (9)

Fix E:>O.

If Ilvn (t) IIH2.v-1 (C~1 o+E:) for t E [t1 ,t2 ] then Bn2.E:, therefore

(8)

(9)

(10)

Since the a-limit set of vn consists of equilibria v of TIn' satisfy

ing, by (3.3), the property

II -II -1 II-II -1-1 o 2.v v H-ocn i.e. v H2.v c n 0

101

It follows that whenever Ilv (t 2 ) IIH>v -1 (c~10+£) for some t2 E JR, there

is a t 1 <t2 such that Ilv(t 1 ) II H=v- 1 (c~10+E:) and Ilv(t) IIH2:.v-1 (C~10+E:)

for t E [t1 ,t2l.

Hence, by (10)

It follows that

II II --1 -1-1 v (t) H<2£Cc +v (c 0+£) n - n n for t E JR, n E IN . ( 11 )

Now let N={v E xal iv~ <1}. a-By hypothesis (3.2) there is a B>O such that II f (v) ~ <B for all n E IN n -and all v EN. Therefore Theorem 1.4.4 implies that N is strongly

{; }-admissible. n

It follows that a subsequence of {vn(O)}, again denoted by {vn(O)}

converges in Xa to some point wOE Xa with II w 0 II a = 1. However, (11) im

plies that IlwoIIH=O, i.e. wO=O, a contradiction which proves the theo

rem.

As a corollary to the proofs of Theorems 3.5 and 5.5 we obtain the

following

Theorem 5.6

Assume hypotheses (1) and (2) of Theorem 5.5. Moreover, let 1/2~a<1

and U be ~ neighborhood of 0 in Xa. Let f:U+X be locally Lipschitzian

and assume that f(O)=O and that f' (0) 'exists and f' (0)=0.

Suppose there is ~ Frechet-differentiable map G:U+JR such that for

every u E U and hE Xa,

DG (u) (h) = <f (u) ,h>

Consider the equation

for an arbitrary A E JR .

Then every point (A*, 0) E JR xU with 1,* E a (A) is ~ bifurcation point

of (T A), i.e. there exists ~ sequence (An,un ) E JR xU such that

(An,un)+(A*,O) as n+oo , unfO and un is a solution of (T A ) for nE IN n

Moreover, every point (1,*,0) with 1,* ~ alA) is not ~ bifurcation point.

102

Proof:

Let).. E lR be arbitrary. Then U o E U is a solution of (T)..) if and only

if Uo is an equilibrium of the local semiflow TI).. on U generated by

equation

u(t)+Au = )..u+f(u)

Let now ).. * I/. cr (A). Then, proceeding as in the proof of Theorem 3.5

one can easily show that there is an £>0 and the neighborhood N of 0

in Xa, NCU, such that K)..={O} is the largest TI)..-invariant set in N for

all)" E ()..*-£,)..*+£). This proves that ()..*,O) is not a bifurcation

point of (T)..), establishing the second part of our assertion.

Now let )..* Ecr(A). Suppose ()..*,O) is not a bifurcation point. Then,

there exist an £>0 and a closed bounded neighborhood N of 0 in xa, NcU, such that whenever U o EN is a solution of (T)..),).. E ()..*-£,)..*+£),

then uO=O.

Now define V)..: U+lR ,).. E lR as

1 1/2 1/2 V)..(u) = Z«A1 u, A1 u>-k<u,u»-)..<u,u>-G(u)

As in the proof of Theorem 5.5 (see equations (5), (6) in that proof)

it follows that TI).. is gradient-like with respect to V)... Now let

).. E ()..*-£,)..*+£) and t+u(t) be a full solution of TI).. contained in N.

Then u[lR] is relatively compact. If u(t)=uO' i.e. if Uo is an equi

librium of TI).., then by our assumption, u(t)=uO=O. If t+u(t) is not a

constant solution, then, by Proposition 5.4 w*(u) and w(u) are disjoint, non

empty and contain only equilibria of TI)... Since w*(u)Uw(u)cN, it fol

lows that w*(u)=w(u)={O}, a contradiction.

This argument proves that N is an isolating neighborhood of K)..={O}

relative to TI).., for).. E ()..*-£,)..*+£). It is therefore clear that the

map ~:)..+(TI)..,{O}),).. E ()..*-£,)..*+£) is well-defined and S-continuous.

Therefore Theorem I.12.1 implies that h(TI)..,{O})=const. for

)..E ()..*-£,)..*+£). We may assume that ()..*-£,)..*+£)ncr(A)={)"*}. Let

)..1=),,*-£/2')..2=),,*+£/2.

Then Theorem 3.5 and the remark following the proof of that theorem

imply that

m. ". ~ l. , i=1,2

where mi is the total algebraic multiplicity of all eigenvalues

AEa(A) with A<A i , i=1,2.

103

But m2 is m1 plus the algebraic mUltiplicity of 1.*. Hence m1 <m2 and m1 m2

so I ~I , a contradiction.


Let us now recall that a global semiflow TI on a normed space X is

called Eoint-dissipative, if there exists a bounded set BcX with the

property that for every x E X there is a to=to (x) such that XTIt E B

for t~tO' TI is called conditionally completely continuous for t~to'

if for any bounded set BcX there is a compact set B*cX such that for

any t~to and any xEX for which XTI[O,t]cB, it follows that

XTI[tO,t]cB*. Note that if TI is conditionally completely continuous

for t~to with some to>O, then every bounded set NcX is strongly TI-ad

missible. By well-known results on dissipative systems, if TI is

point-dissipative and conditionally completely continuous for t~to

with some to>O, then the union J of all full bounded orbits of TI is

a compact and connected global attractor (see e.g. the proofs of

Theorem 3.1 and Lemma 3.3 in Chapter 4 of Hale [1]).

Here, by a global attractor we mean that for every x E X, dist(xTIt,J)->-O

as t->-oo.

Let A be sectorial on X with compact support and f:Xu->-X, 0~u<1, be

locally Lipschitzian and map bounded sets in XU into bounded sets

in X. Then, by Theorem I.4.3, the local semiflow TI generated by

u+Au=f(u) is conditionally completely continuous for t~to with to>O

arbitrary.

We now have the following

Theorem 5.7

Assume hypotheses (1) and (2) of Theorem 5.5. Furthermore, suppose

that A has compact resolvent. Let Ai' i~O be the eigenvalues of A

and 1._1 be an arbitrary (finite) number <1. 0 ' Suppose 1/2~u<1 and that

f:Xu->-X is a locally Lipschitzian mapping satisfying the following

condition:

(E) there are €,0>0 and k>-1 such that for all u E XU

where A*=~(Ak+Ak+1) and W=~(Ak+1-Ak)-€' Let m* be the total algebraic multiplicity of eigenvalues A of A

:,:,ith A~Ak'

104

Suppose also that there is ~ Frechet-differentiable mapping G:Xa7lli

such that for all u,h E Xa

DG(u)· (h) <f (u) ,h> .

Finally, let TI be the local semiflow generated ey solution of

u+Au = flu)

and Koo be the union of all full bounded orbits of TI. Then (TI,Koo ) ES

and

Furthermore TI is ~ global semiflow, and if k>1 (i.e. if m~O) then

is not point dissipative and Koo has ~ nonempty unstable manifold.

Proof:

It is an immediate consequence of condition (E) and Corollary 3.3.5

in Henry [1] that TI is a global semiflow.

Define the following homotopy

forTE[O,1J.

Let TIT be the semiflow generated by

u+Au = gT(U)

and KT be the union of all full bounded orbits of TI.

We will prove that the map <!>(T)=(TIT,KT) is well-defined and S-conti

nuous. In fact, for this to be true, it is only necessary that there

exist a common bound (in Xa ) for all sets KT,T E [0,1]. Suppose that

such a common bound does not exist. Then there is a sequence Tm E [0,1]

and a sequence urn of full bounded solutions of TI with Tm

c =sup II u (t) II 700 as m7oo . m tElli m a

Let fm:=gT and Gm:Xa7lli be defined as Gm(U)=(1-Tm)G(u)+(1/2).Tm:\*<U,u>. m

Then hypotheses (1), (2)aand (3.1), (3.2) of Theorem 5.5 are clearly

satisfied.

By our assumption, :\* E P (A) and hence (11.-:\*) -1 :H7H is a Hermitian

operator. Hence II (A-:\*) -111.:::.max{ 1]11 1]1 E (J ( (A-:\*) -1)}. Since (11._:\*)-1

105

is compact, by hypothesis (1), it follows that whenever ].1 E cr (A-A*) -1,

].1fO, then ].1 is an eigenvalue of (A-A*)-1, hence 1/].1 is an eigenvalue

of A-A*. This implies for every u E D

Hence

where cO=min{ I A I I A E cr (.A-A*) }.

Now ].1Ecr(A-A*) if and only if A*+].1 E cr(A) if and only if ].1=Aj-~(Ak+Ak+1) for some j2:.0.

Therefore 1].112:.~(Ak+1-Ak) and so co>w.

Now we obtain for all u E D and m E IN

II-Au+f (u) II =i-Au+A*u-A*u+f (u)" >1I-Au+A*ull -lif (u)-A*ul m H m H- H m H

This proves hypothesis (3.3) and so Theorem 5.5 immediately leads to

a contradiction. Hence, indeed, the map ,+(TI"K,) is well-defined

and S-continuous. Since (TIO,KO) = (TI,Koo) , it follows that

by Corollary 1.11.2. We shall now prove the last assertion of the theorem. Let m>O. Then

Im is the homotopy type of a connected set. Therefore an application

of Corollary 1.11.9 yields the existence of a full solution t+u*(t)

with supllu*(t)II <00 and supllu*(t) 11=00. t<O a t>O

This shows on the one hand that Koo is not an attractor for the solu-

tion u* as t+oo. In particular, TI cannot be pOint-dissipative. On the

other hand, w*(u*) is bounded and invariant, so w*(u*)CKoo ' Therefore

106

the solution u* lies on the unstable manifold of Koo.

The proof is complete.

As a simple consequence we obtain

Corollary 5.8

Under the assumptions of Theorem 5.7, there exists a solution U o ED (A)

of

Au f (u) • (12 )

Proof:

,m* -In fact, h(1T,K oo)=L ;to. It follows that Koo;t0. Let Vo EKoo. Then either

Vo is an equilibrium of 1T, i.e. a solution of (12), or else there

exists a nonconstant full bounded solution t+u(t) with u(O)=vO. By

Proposition 5.4 w(u) and w*(u) are nonempty and contain only equili

bria, i.e. solutions of (12).


We shall now apply Theorem 5.7 and Corollary 5.8 to parabolic equa

tions considered in section 1 of this chapter. We will only treat

the case r=l, i.e. a single equation, leaving it to the reader to

formulate and prove the corresponding results for systems (r>l).

Theorem 5.9

Consider the equations (1.3) and (1.4) with r=l. Assume that the li

near differential operators in (1.3) and (1.4) satisfy the conditions

listed in section 2.1.

Let Ap be as in (1.5) and assume that A2 is self-adjoint.

Let AO<A 1 < ... be the (common) eigenvalues of the operators Ap and

A_l be an arbitrary (finite) number <A O.

Let the nonlinearity f 1=f:TIX1R+1R in (1.3) and (1.4) be continuous

and locally Lipschi tzian in s E 1R, uniformly for x E TI.

Suppose that f satisfies the following condition (E*):

(E*) There are p>O, s>O and k>-l such that whenever x E TI and s E 1R

with Isl~p, then

Consider the abstract versions (1.7) (resp. 1.8) of (1.3) (resp.

(1.4)) where p>n, 1/2<a<1, X=LP(~), H=L2(~), A=A ~A=A2 and f:Xa+X - p is the Nemitski operator defined Qy the function f.

107

Let TI be the local semiflow on x a generated by the solution of (1.7).

Then the following properties hold:

(1) TI is ~ global semiflow.

(2) if Koo is the union of all full bounded orbits of TI, then Koo is

compact, h(TI,K oo ) is defined and

where m* is the total algebraic multiplicity of all eigenvalues \ of

A with \.:5.\k'

Moreover:

(2.1) if k=-l (i.e. if m*=O), then TI is point-dissipative.

(2.2) if k>-l (i.e. if m*>O), then TI is not point-dissipative and

Koo has ~ nonempty unstable manifold.

(3) There exists at least one (classical) solution of equation (1.8).

Proof:

Let G: TIxlR -+lR be defined as

s G(x,s) = jf(x,t)dt (13)

° and ~:xa-+lR be the corresponding Nemitski operator, i.e.

G(u) (x) G(x,u(x) )

Using the imbedding XacCO(TI) it is an easy exercise to show that all

hypoth~ses of Theorem 5.7 are satisfied with G, and f replaced by G

resp. f in that theorem.

Therefore Theorem 5.7 implies properties (1), (2) and (2.2). Moreover,

Corollary 5.8 and regularity theory implies property (3). It there

fore remains to prove property (2.1).

Let k=-1.

We use arguments from the proof of Theorem 5.7. Since A is self-ad

joint and A*<\O' it follows that 0(A-\*)~CO>O where Co

CO=min{ I \ I 1\ E 0 (A-\*) }>.l(\ -\ ). -2 ° -1

Hence for every u E H and t~O,

(14 )

Let t-+u(t) be a solution of TI on Xa. Then for t>O, u(t) ED(A) and

108

t+u(t), t>O, is differentiable in X. Hence t+u(t) is differentiable

as a mapping into H and for t>O ,

. -u(t)=-Au(t)+f(u(t))

Fix t>O, h>O arbitrarily.

Then by (14)

- (A-:\*) u (t) +f (u (t) ) -:\*u (t)

111 -(A-:\*)h +E (u(t+h)-u(t))-(e u(t)-u(t)) ~H

Taking h+O+ we get

+11 (-AU(t)+f(u(t)) )+(A-:\*)u(t) IIH

However, our assumptions imply

where W=~(:\k+1-:\k)-E and 0>0 is a constant.

Therefore

This inequality implies that the function t+llu(t) IIH is decreasing as

long as Ilu(t) IIH is large enough. Therefore t+llu(t) IIH must be bounded.

Now a standard bootstrapping argument using the Sobolev inequalities

(see e.g. Theorem 1.6.1 in Henry [1]) implies that t+u(t) is bounded

in X=LP(~). An application of Theorem 1.4.3 now proves that t+u(t)

is bounded in Xa. Now it is easy to show that TI is point-dissipative.

Let B be a bounded neighborhood of Koo und Uo E xa be arbitrary. Since

the solution u(t) :=uOTIt is bounded in Xa for t>O, it follows from

109

Theorem I.4.3 that u(t), t~E>O lies in a compact set CcXu , for arbi

trary E>O. It follows that w(uO) is a nonempty invariant set for TI.

Thus, by the definition of Koo' w(UO)cK oo ' But this implies that

u (t) E B for some to>O and all t~tO'

Now our remarks preceding the statement of Theorem 5.7 complete the

proof of property (2.1).


Remark

Note that in the extension of Theorem 5.9 to systems (r>l) we must,

in addition, to an analogue of condition (E*) assume the existence - r r ClG - r of a function G:rlxJR ->-JR such that f(x,s)=a:s(x,s) for all xErI,sEJR.

2.6 Estimates at zero and nontrivial solution of elliptic equations

If f(x,O)=O in Theorem 5.9, then u=O is a solution of equation (1.8),

so (3) of Theorem 5.9 is not very informative in this case. There

fore it is useful to have conditions for the existence of nontrivial

solutions of equation (1.8). In this section we will present some

such conditions. We begin with a theorem which is an improvement for

gradient-like systems of Theorem 3.5.

Theorem 6.1

Assume hypotheses (1) and (2) of Theorem 5.5. Furthermore, suppose

that A has compact resolvent. Let AO<A 1 < ... be the eigenvalues of A

and let A_l<AO be arbitrary.

Suppose that 1/22 u<1 and let f:V->-X be locally Lipschitzian on ~ neigh

borhood V of zero in XU with f(O)=O and such that the following con

dition is satisfied:

(F) There are "E>O and 1>-1 such that for all u E V

where ~*=1/2(Al+Al+l) and w=1/2(Al+ 1-Al)-"E.

Let q* be the total algebraic multiplicity of all eigenvalues A of A

with A2A l'

Suppose also that there is ~ Frechet differentiable mapping G:V->-JR

such that for all u E V, hE XU

DG (u) (h) <f(u),h> .

110

Finally,le: TI be the local semiflow generated ~ solutions of

u+Au = flu)

Then

Proof:

(TI, {a}) E StU) and

h(TI, {a}) = Iq * •

Theorem 6.1 is dual to a part of Theorem 5.7 and has a dual and even

simpler proof. Therefore we shall only sketch the arguments.

Define the homotopy

gT (u) = (1-T) f (u) +Tll*U for T E [0,1], u E U •

Let TIT be the semiflow generated on U by

u+Au = gT (u) •

Using the arguments from the proof of Theorem 5.7 we see that Theo

rem 6.1 will be proved if we can show that there is a closed bounded

set N in Xu, NCO, such that for every T, N is an isolating neighbor

hood of {a}, relative to TI • If this latter statement is not true, T

then the gradient-like nature of TIT implies that there is a sequence

T m E [0, 1] and a sequence urn E U, umf 0, such that urn +0 in XU as m+oo ,

and urn is an equilibrium of TITm for mE:IN •

Proceeding as in the proof of Theorem 5.7 we obtain that for u ED

where cO-w>o.

It follows for every m that

o = II-Au +(1-T )f(u )+T ll*U ~H m m m m m

=~-AU +(ll*U )+(1-T )(f(u )-ll*(U »11 > m m m m m H

>~-AU +ll*U i -(1-T ) ~f(u )-ll*U II > - m mH m m mH

It follows that Ilum~H=O i.e. um=O a contradiction.


111

Corollary 6.2

Suppose that all hypotheses of both Theorem 5.7 and Theorem 6.1 are

satisfied. Let k~l.

Thenthe following properties hold:

(1) There exists ~ nontrivial solution u o of

Au f (u)

(2) There exists ~ nonconstant full bounded solution t~u(t), t E lli

of

u+Au f (u)

such that either u(t)~O as t~oo, or u(t)~O as t~-oo.

Proof:

k~l implies that h('IT,{O})~h(1T,K,:O>. Since OEKoo' it follows that there

exists a v O~O, v 0 E Koo. If v 0 is an equilibrium of 'IT then property

(1) holds for uO=vO.

Otherwise there exists a nonconstant full bounded solution t~v(t)

of 1T with v(O)=v O. Now by Proposition 5.4 w*(v) and w(v) contain

equilibria of 'IT, thus completing the proof of (1). * Since h('IT,Koo)=Im , ('IT,K oo ) is irreducible by Theorem 1.11.6. Now Theo-

rem 1.11.5 implies that there exists a solution t~u(t) EKoo of 'IT

joining K1={0} with some set K2cKoo ' 0 rt K2 . This implies property (2)

and completes the proof.

Theorem 6.1 and Corollary 6.2 have immediate applications to parabo

lic and elliptic equations.

Theorem 6.3

Suppose that all hypotheses of Theorem 5.9 hold except (possibly)

for condition (E*). Instead, assume that f satisfies the following

condition (F*):

(F*) f(x,O)::O for all xED, and there are p, s>O and 1>-1 such that

whenever xED and s E lli with 0 < I s I.:::.p then

Then ('IT,{O}) is defined and

112

where q* is the total algebraic multiplicity of all eigenvalues A of

A with A2.AI.

If, in addition, f satisfies condition (E*) of Theorem 5.9 with k11, then the following properties hold:

(1) There exists ~ nontrivial solution uo of equation (1.4).

(2) There exists ~ nonconstant solution t+u(t) of (1.7) such that either u(t)+O ~ t+oo, or u(t)+O as t+-oo.

The proof is a simple application of the preceding results. We will

now treat the resonance case at zero. We need the following result:

Proposition 6.4

Let N1 ,N2 be two compact subsets of ]RI , I~O, N2cN1 • Then

whenever v,m~O are such that v<m or v>m+l.

Proof:

Write X=N 1!N2 and XO=[N2]. Then (X,XO) is a compact pointed space.

For m~O, let (Sm,sO) be the m-dimensional unit sphere with a base

point.

Then, by definition, [N 1!N2 ]Al.m is the homotopy type of the pointed spaee XASm. Moreover, SmASk is homeomorphic to Sm+k (see Proposition

6.2.15 in Maunder [1]).

Moreover, by a restricted associativity property of the smash product, (XASm)ASk is homeomorphic to XA(SmASk) (see Theorem 6.2.23 in Maunder

[1]) •

It follows that for m,k~O, XASm+k is homeomorphic to (XASm) ASk.

Let (Hq)qE Zbe an unreduced cohomology theory. Then by Proposition 7.16 in Switzer [1] Hq(X,{XO}) is isomorphic to Hq +1 (XAS 1 ,{*}) for

all q E Z. Here, * is the base point of XAS 1. (Actually, Proposition

7.16 is a result about homology theory. However, the dual result

about cohomology trivially follows by "dualizing" the proof of Pro

position 7.16, i.e. by reversing the arrows in all diagrams involved).

Consequently, obvious induction implies that

(1 )

Here q E Z, k~O and * is base point of XASk

Now take (Hq)qE Z to be the Alexander-Spanier cohomology theory. Let

113

v,m~O be such that v<m or v>m+l. Then (1) implies

(2)

v-m { If v<m, then H (X, xJ)=O, of course.

Moreover, since the dimension of N1 is ~l, it follows from results

in Spanier [1]

for q>l. (See, in particular, exercise D. 4, Chapter 6 in Spanier

[ 1 ] ) •

Hence, in both cases,

o • (3)

In particular, since HV(SV,{so}):]R, it follows that XASm cannot be

homotopy equivalent to (SV,sO).

This proves the proposition.

Theorem 6.5

Assume all hypotheses of Theorem 5.9 except (possibly) for condition

(E*) which is replaced ey the following condition

(R) f (x, 0) =0 for all x E TI; the partial derivative ~; (x, s) exists for

all I s I small and all x E TI, is continuous in (x, s) and continuous in

s uniformly for x E TI. Furthermore, there is s>O and k'::0,r~-1 such that either ----

(R1 ) r>k and af - for x E ri, else ag(x,O)::"k-E: or

(R2) r<k and af -ag(x,O)~"k+E: for x E ri.

Under these hypotheses, if {a} is an isolated equilibrium of TI, then

K={O} is an isolated invariant set for TI, h(TI,{O}) is defined and

m h(TI,{O}) ~ L r

Here, m_ 1 :=0 and for r>O mr is the total algebraic multiplicity of

all eigenvalues" of A with "~"r'

Proof:

If 0 is an isolated equilibrium of TI, the gradient-nature of TI imme

diately implies that K={O} is an isolated invariant set. Using hypo-

114

thesis (R) and the fact xaceO(TI) with continuous inclusion, it is

easily proved that the Nemitski operator f:Xa~X is differentiable

in a neighborhood U of 0 and f' is Lipschitz continuous on U.

Let L=A-f' (0). Since A has compact resolvent, L has compact resol

vent as well. Therefore the set {A E oIL) Ire A~O} is a finite set of

eigenvalues of L, isolated in O(L). Also, there exists a direct sum

decomposition X=X 1@X 2@X 3 , with dim(X 1ffiX 2 ) <00, such that Xi is L-inva

riant for i=1,2,3. Moreover, if L. :=Llx. then L l. is a sectorial ope-l 1

rator on Xi' i=1,2,3 such that re O(L 1 )=0, re O(L2 )<0, re O(L 3»0.

(cf. Theorem 1.5.2 in Henry [1]).

Consequently, all hypotheses of Theorem 3.1 are satisfied.

Thus

h(n,{O})

where m=dim X2 .

(4 )

We recall that n~ is the "restriction" of n to a local center mani

fold.

Let l=dim X1 . Since n~ is a local semiflow on an open subset of - 1 X1 =lli , we obtain, using an appropriate compact isolating block B

for K~={O} relative to n~,

(5)

where N1=B, N2=B-.

Now (5) and Proposition 6.4 imply that

h(n,{O}) f IV (6)

whenever v<m or v>m+l.

Now suppose that A* E P (A). Then A* E P (A) and there are (A-A*) -inva

riant subspaces X+,X_, X=x+@x_, dim X <00 and such that O(A+-A*I+»O

and O(A_-A*I_)<O where the subscript + or - denotes the restriction

of the corresponding operator to X+ and X_' respectively.

We claim that for u E D (A), UfO

and

<0 if u E X «A-A*)U,U> {

>0 if uEX+ (7)

115

0 if u E X1 <Lu,u> { < 0 if u E X2 (8)

> 0 if u E X3

Since dim X_<oo and dim(X 1$X2 )<00 and since A-A* and L are symmetric,

the inequalities (7) and (8) follow immediately for u E X_UX1 UX 2 •

Now let k>O be such that a(A-A*+k»O.

For u E XCI. define

1 - - 1/2 - - 1/2 -V(u) = Z«(A-A*+k) U,(A-A*+k) u>-k<u,u».

It follows that V is continuous on XCI., Let uo;tO, Uo ED (A) nx+. If

t+u(t), t>O is the solution through Uo of

u = - (A-A*I) u (9 )

then, by well-known results (cf. Theorem 1.5.3, Henry [1]), u(t)+O

exponentially in xCI., as t+oo. But since dV~~(t)) = -II (A-A*I)u(t) II~

for t>O, it follows that

V(uO) > V(O) = 0

Le. (7) follows for uED(A)nX+.

Now define ~:H+H by

of (~u) (x) = ag(x,O) 'u(x)

Then ~ is a well-defined bounded linear operator, ~(u)=f' (0) (u) for

u E XCI., L=A-~ is self-adjoint and L (u) =L (u) for u E XCI..

Let k>O be such that a (L+k) >0 and define for u E XCI.

- 1 - - 1/2 - - 1/2 -V(u) = Z«(L+k) u,(L+k) u>-k<u,u»

Using V and analog ous arguments as above we obtain

for UO ED(A)nX 3'{0}, Le. (8) holds for uED(A)nXr The claim is proved.

Now suppose that (R1) holds. Let A* E P (A) be arbitrary with

116

Ak-S<A*<Ak , A*>Ak _ 1 (A_ 1 :=-00).

Then for u ED (A)

<Lu,u> > «A-A*)U,U> .

Hence, (7) and (8) obviously imply that

Consequently

and therefore dim X1+dim X22dim X .

In other words (since dim x_=mk _ 1 , m_ 1 :=0)

Now (6) implies the result in case hypothesis (R1) holds. Now suppose

(R2) holds.

Choose A* E P (A), Ak <A*<Ak +S, A*<Ak +1 .

Then for u E D (A)

<Lu,u> < «A-A*)U,U>

(7) and (8) imply in this case that

Hence,

Therefore dim X_2 dim X2 ' i.e. mk2m.

Therefore, mr<mk2m and (6) again implies the result.


Corollary 6.6

Assume all hypotheses of Theorem 5.9. Let f (x, 0) =0 for all x E I2 and

and suppose that ~ ~ (x, s) exists for I s I small and x E I2 is continuous

in (x, s) and continuous in s, uniformly for x E TI •

Finally, suppose that the following conditions hold:

(1) if k>O then

either af

(a) a-s(x,O)'::=)\k-€ for all xETI

or at (b) a-s(X,O)':'Ak +1 +€ for all xETI

(2) if k=-1, then af -a-s(X,O).:.A O+€ for all xETI.

117

Under these hypotheses there exists ~ nontrivial solution of (1.4).

Proof:

If ° is not an isolated equilibrium of TI, then there are infinitely

many (nontrivial) equilibria, i.e. solutions of (1.4) and we are done.

S.o assume that 0 is an isolated equilibrium of TI.

By Theorem 6.5

m h(TI,{O}) f. L k

However, Theorem 5.9 implies that

It follows that ° E Koo and Koof. {O}. As in the proof of Corollary 6.2

this implies that Koo contains a nontrivial equilibrium of TI, i.e. a

nontrivial solution of (1.4).


2.7 Positive heteroclinic orbits of second-order parabolic equations.

In this section we will consider the following equation of type (Sf)

u+Au f(u)

We make the following assumptions:

(Q1) ro is a bounded domain of JRq with orientable boundary of class

c2 +S, 0<6<1. Moreover, for x E n q . . q .

A(x,D) :=- L a .. (x)D1.DJ_ L b. (x)D1.-c (x) i,j=1 1.J i=1 1.

(2)

is a uniformly strongly elliptic differential operator, the functions

118

a .. ,b. ,c:TI +JR being Lipschitz continuous·. The matrix (a .. (x» is 1J1 1J

symmetric and c (x).::O for x E TI.

(Q2) For x E a~ a B (x,D) =O.a):i + h (x) (3)

is a boundary operator with the following properties:

(i) ~:a~+JRqis a smoothly varying nontangential direction on ~ poin

ting outward, and h is of class c 1 and nonnegative.

(ii) either 0=0 and h=1 or else 0=1.

(iii) if 0=1 and h=O then ctO.

(Q3) For p>1, let A :D +LP(~) be the sectorial operator induced by p P

(A(x,D), B(x,D» (see Section 2.1). Then A2 is self-adjoint. More-

over, A=A for some fixed p>n, and f:Xu+X, U>-21 is the Nemitski opera-p -tor generated by a function f:TIxJR+JR. The function f is Lipschitzian

on compact subsets of TI x JR .

We have the following

Proposition 7.1

Suppose that (Q1)-(Q3) hold.

Moreover, assume that

(Q4) f(x,s»O for xETI, s<O.

Let JcJR be an open interval and u:J+Xu be ~ solution of (1f) on J.

Then the following properties hold:

(1) u is a classical solution of (1 f ).

(2) if u(t) (x).?O for all (x,t) E ~xJ, then either u=O or else

u (t) (x) >0 for all (x,t) E ~xJ.

(3) if J=JR and u is bounded, then u(t) (x)~O for all (x,t) E ~xJR .

Proof:

1. Step

u is a classical solution of (1 f ): In fact, by Theorem 3.5.2 in d U Henry [1], the map t'dtu(t) EX, t E J is well-defined and continuous.

Since XU imbeds continuously into CV(TI), where 0<v<2u-~, we conclude p

that the function u:TIxJ+JR, u(x,t) :=u(t) (x) is well-defined, Holder

cO!'ltinuous with exponent v in x E TI and C 1 in t with the derivative d v -. dtu(t) E C W) for t E J.

Moreover, for t E J, the function x+f (x, u (x, t» is easily seen to be v - v -an element of C (~). Thus, by (1f) Au(t) E C W) for t E J.

By well-known results from the theory of linear elliptic PDEs, there

is a unique solution v E C 2 +V (TI) of the equation

Av = g

with g(X)=f(x,u(x,tll-:tu(t) (x).

Therefore v=u(t) and so u(.,t) EC2 +V (n"). All this proves that

u(x,t)=u(t) (x) is a classical solution of (1 f ).

2. Step

119

(4)

Let u(x,t)=u(t) (x)~O for all (x,t) E nxJ. To prove (2) we may assume

w.l.o.g. that u is bounded. Let M be a bound or lu(x,t) I (x,t) EITxJ. Then, by (Q3) there is an L>O such that whenever XEQ, s,sE [-M,M],

If(x,s)-f(x,s) I < Lis-si (5)

Let v(t) (x)=eLt u(t) (x). A simple calculation shows that

v+Av eLt (f (u) +Lu) • (6)

From (5) we get, observing f(x,O)~O (by (Q4» and u(x,t)~O that

f(x,u(x,t»+Lu(x,t) > 0 (7)

on QxJ.

Consequently, v+Av>O.

Suppose that u(xo,tO)=O for some (xo,tO) E nxJ. Then, applying the strong maximum principle (pp. 173-174 in Protter and Weinberger [1])

to w=-v, we get v(x,t)=v(t) (x)=O for xEIT, t.s.to ' tEJ. Thus u(t)=O

for all such t. The uniqueness of solutions of (1f) now implies that u(t)=O for all tEJ, proving (2).

3. Step

Suppose J=lR and u is bounded. Suppose E={ (x,t) E nxlR lu (x,t) <ohl~

and let -M=inf u. Then -M<O. By assumption (Q4) and step 1 we get

u (t) (x) + (Au (t) (x) > 0 for (x,t) E E.

If there is a point (xo,tO) E nxlR with u(xo,tO) =-M then u has its

minimum at (xo,tO)' Therefore we easily obtain that Xo E an. Now using the strong maximum principle again we obtain

120

Now, by (Q2)'

o

Hence 0=0 and h(xO)=O, a contradiction to (Q2).

It follows that there is no (xo,tO) E r/xlR with u(xo,tO) =-M. Thus there is a sequence (x ,t ) EE with u(x ,t )~-M and It I~oo. We may assume n n n n n xn ~xO E Q. Since u is bounded, {u (t) It E lR} is relatively compact in XCI.,

so we may also assume that u(tn)~. Since TIf is gradient-like (A2 being assumed self-adjoint) it follows that w is an equilibrium of TI,

i.e. a solution of

A(w) = f (w) •

Thus -M=min {w(x)lxEQ}.

Applying the strong maximum principle to w we obtain a contradiction.

Hence E=~, which proves (3). The proposition is proved.

Theorem 7.2

Assume hypotheses (Q1)-(Q4). Moreover, suppose that there are constants Y1 'Y2 E lR, p,£>O such that one of the following properties holds:

ei ther (Loo):

Y1+£<"0-£ and

y1+E.::f(x,s)/s.::"0-£ and all s~p or (L.!,)

"0+£<Y2-£ and

"0+£.::f(x,s)/s'::Y2-£ and all s~p.

for all x E Q

for all x E'IT

Here, "0 is the smallest eigenvalue of A.

Under these hypotheses the union K; of all nonnegative full bounded

orbits of TIf is bounded, h(TIf,K;) is defined and

{ t,O l, if (Loo) holds

o if (L~) holds.

Proof:

Notice that by results in Amann [1], "0>0 and "0 is a simple eigenvalue of A which has an eigenvector U o with U o (x) >0 for x E'IT.

Now suppose that (Loo) holds.

Let f:QxlR~lR be Lipschitzian on compact sets in QxlR and such that

f(x,s)=f(x,s) on nx[-p/2,oo), f(x,s)<O for s>O and f(x,s)=-ys for

121

TIx(-oo,-p]; where y>O is such that -Y~Y1'

Let Ki be the union of all nonnegative full bounded orbits of TIr. Since (1f) and (1f) define the same equations as long as u(t) (x»-p/2

f n bt· h + + h ( +). d f· d· ff h ( +). or x E '" we 0 a~n t at Kf=Kf , TIf,Kf ~s e ~ne ~ TIf,Kf ~s

defined and then the two indices are equal. +

An application of Proposition 7.1 shows that Kf is, in fact, the union

of all full bounded orbits of TI f . Now Theorem 5.9 completes the proof of the theorem in the first case.

Now assume (L~).

Consider the following homotopy:

(1-T)f(x,s)+Tg(s) (7)

for T E [0,1].

Here g(S)=(AO+E) S+E for s~O and g(S)=-S+E for s<O.

Let K~ be the union of all full bounded orbits of TI :=TIf • , T T

Then, by Lemma 7.3 below there is a closed bounded set NCXu such that

K cInt N for all T E [0,1]. T

Thus h(TIT,KT) is defined and

However, (TIO,KO)=(TIf,K;) by Proposition 7.1. Therefore the theorem

will be proved if we can show that K1=~'

Indeed, if K1f~' then the fact that TI1 is gradient-like implies that

there is an equilibrium u of TI 1 , i.e. a solution of

Au = g(u)

By Proposition 7.1, u~O so

Let < ,> be the scalar product in L2 (rl).

Then

(AO+E)<U,UO>+<E,UO>

AO<UO'U> .

(9)

(10)

122

Since<A2 is self-adjoint and XacL2 (n),

(Aa+E)<U,Ua>+<E,Ua > = Aa<Ua,U>

Hence

Since <E,Ua>=~EUa(X)dx>a, we get a contradiction.

Thus, indeed K1=~ and the theorem follows.

Lemma 7.3

( 11)

( 12)

There exists ~ constant L>a such that whenever , E [a, 1] and t+u (t)

is ~ full bounded solution of 1T,' then II u (t) II a2L for all t E lR •

Proof:

By (L~), there is a constant M>a such that

( 13)

for all, E [a,1]' x En, s~a.

If the lemma is not true, then there is a sequence 'n E [a,1] and a

sequence {u } of full bounded solutions of 1T :=1Tf where fn:=f, ' n n n n

such that c =sup Ilu (t) II +00 as n+oo and c -Iu (a) ~ <1 for all n E If'J • n tElR nan n a

-1 Let Vn (t) =cn ·un (t) •

vn is a full bounded

By Proposition 7.1 v (t) >a for all t E lR, n E IN • n:;; -1

solution of (If ) where fn(x,s)=cn fn(x,cns). n

Now we obtain from (13)

-<Av (t) ,ua>+<f (v (t» ,ua > n n n

-1 > -A <v (t) .uO>+ (AO+E) <v (t) ,uO>-c <M,uO>. o n < n n

Write SnIt) :=<vn(t) ,uO>· Then by (14)

This implies that

( 14)

( 15)

123

(16 )

for t>O.

Since SnIt) is bounded, we obtain

E

o < S (O)-~ < 0 - n E

(17 )

Now, by admissibility, we may assume (taking subsequences if neces

sary) that v (0) -+v O for some v O E XCI.. Moreover, by our assumptions n

1 • (18 )

However, <vO,uo>=lim B (0)=0 by (17). n-+co n

Since vO~O and vO~O by (18) we get a contradiction to the fact that

u o (x) >0 for all x E~.


Theorem 7.4

Assume hypotheses (Q1)-(Q4)'

Moreover, suppose f(x,O):=O for xETI, and there is an E>O such that

either

(LO) : f (x, s) <AOs for all O<S<E , x E TI ,

or

~

(La) :f(x,s) >AOs for all O<S<E , x E ~

Under these hypotheses, {a} is an isolated TIf-invariant set,

h(TIf,{O}) is defined and

Proof:

if (LO) holds ,

if (L') holds. o

Let y,E>O be arbitrary. Define

g (s) -ys if (LO) holds, and

s>O g(s)

-s+s s<O if (La) holds.

124

Consider the homotopy

(1-T)f(x,s)+Tg(s) ( 19)

for T E [0,1] •

Let N be a closed bounded neighborhood of zero in XU such that

I u (x) I <~/2 for all u E N and x E n (remember that XU imbeds continuous_. 0-

ly into C (n).

Let TI =TIf and K be the largest TIT-invariant set in N. We claim that T T T

K ={ O} for T E [0,1]. T

Assuming this for the moment we get that h(TIT,KT) is defined and

(20)

Since u=O is an equilibrium of TIf=TIO' we get that

(21)

If (LO) is satisfied, we get K1={0} and h(TI 1 ,K1)=L o by Corollary

1.11.2; and if (Lb) holds, then as in the proof of Theorem 7.2, K1=~ so h(TI 1 ,K1 )=O.

This proves the theorem, except for the claim.

If the claim is not true, then for some T E [0,1], there is a solution

v"lO, Iv(x) 1.2.~' xEn, of

Av = fT (v)

By Proposition 7.1, v(x) >0 for x E n. Let u o be as in the proof of Theorem 7.2. Evaluating <Av,uO> and

<v,AUO> and keeping in mind that A2 is self-adjoint, we obtain

(22)

(23)

Suppose that (LO) holds. Then (f(v)-AOv) (x)~O and (g(V)-AOV) (x)~O

for all x En.

If (La) holds then (f(v)-AOv) (x).:.O and (g(V)-AOV) (x).:.O for all x dL Hence in both cases, either

f(x,v(x» == AOV(X) for all x En

125

or else

g(v(x» = AOV(X) for all x En

However, this contradicts the fact that v (x) >0 for x E n and the defi

nition of g.

The claim is proved.

Theorems 7.2 and 7.4 now imply the following existence result:

Theorem 7.5

Assume hypotheses (Q1)-(Q3). Assume also that f(x,O):::O for x En.

Moreover, suppose that either (LO) and (L~) holds or else that (La)

and (Loo) holds.

Under these hypotheses, there exists ~ positive equilibrium w of (1f)

i.e. ~ solution of

Aw f(w) (24)

with w(x) >0 for x En.

Furthermore, there exists a nonconstant full bounded solution u:m+xa

of (1f) with u(t) (x) >0 for x E n. Both w and u are classical solutions of the corresponding equations.

The proof of Theorem 7.5 is obtained by first redefining fIx,s) for

s<O, so that f(x,s)=-ys for s<O, and then applying our previous re

sults. Details are left to the reader.

2.8 A homotopy index continuation method and periodic solutions of

second-order gradient systems

In this section we will study the periodic boundary value problem for

second order systems.

To be more precise, let n be the bounded interval [O,T], T>O, and

f:m xmm+mm be a continuous mapping where m>O is an arbitrary inte

ger. We assume that f(x+T,u):::f(x,u) for xEm, uEmmand that f is

locally Lipschitzian in u, uniformly for x Em. We are looking for

T-periodic solutions x+u(x) of the following second order system of or

dinary differential equations.

u"(x)+f(x,u(x» = ° (1 )

D = {UEH2 W,mm)lu(0)=U(T), u'(O)=u'(T)}.

126

Define A:D .... X , Au:=-u" • (P)

Here, the derivative is understood in the distributional sense.

A is self-adjoint and bounded from below. Therefore, A is sectorial

(cf. Henry [1]). A has compact resolvent. Moreover, o(A) consists of

eigenvalues Ak : =e;·k y, k=O, 1,2, •••

If x a , a>O, are the fractional power spaces generated by A, then the - A 1/2 A

Nemitski operator f:X .... X, flu) (x)=f(x,u(x)) is well-defined and

Lipschitzian on bounded sets in x1/2.

Consequently, T-periodic solutions u of (1) can be regarded as equi

libria of the local semiflow TI on x1/2 generated by the solution of

u+Au flu) (2) •

To obtain existence results for solutions of (1) we will first state

a basic continuation principle for the homotopy index, which is ana

logous to and motivated by the continuation principle of the

Leray-Schauder degree due to Mawhin.

Consider the following hypotheses:

(Hyp 1) X is a (real) Banach space, A:D(A) .... X is a sectorial operator

on X generating the family xB, B~O of fractional power spaces, 0~a<1, and f:Xa .... X is a locally Lipschitzian mapping.

(Hyp 2) The local semiflow TI on Xa generated by the solutions of

. u+Au f (u) (3)

is gradient-like with respect to some function v:xa .... m •

(Hyp 3) A has compact resolvent. Moreover, there is a 0>0 such that

O(A)={O}Uo', re 0'>0.

Let X1 :=ker A1 and P:X .... X be the projector onto X1 associated with

this spectral decomposition.

(Hyp 4) There is a set fcX a , open in Xa , bounded in X and f[f] is

bounded in X. We write N for the closure of f in xa.

(Hyp 5) For every A E (0,1), if u: m .... Xa is a solution of

u -Au+A(I-P)f(u)+Pf(u)

such that u[m]c::N, then u[m]cf.

127

(Hyp 6) Let r 1 ={ u E: r I Pu=u} =rnx 1 . Let TI be the local flow on Xl ge

nerated by the ODE

(D)

and K be the largest invariant set in Nnx1 . Then Ker 1 and the homo

topy (Conley) index h(TI,K) is non-zero.

Theorem 8.1

If (Hyp 1) - (Hyp 6) are satisfied, ~ there exists a solution Uo E: N

of

-Au+f(u) o (4)

Proof:

Ci. For A E: [0,1] define the following mappings fA:X +X as

{ (l-ZA) (I-P)f(u)+Pf(u) if 0~A~1/2

fA (u) P f (Pu + (2 - 2 A) (I - P) u) if 1 /2~A~1

Since for u E: XCi. , Pu E XCi., (I-P) u E XCi. and since the two definitions

agree for A=1/2, fA -is well-defined. Moreover, it is an easy exercise

to show that fA is locally Lipschitzian for all A (hence in particu

lar, it makes sense to speak of solutions of (SA) and (Hyp 5) above).

Finally,if An+A in [0,1] then fA (u)+fA(u) locally uniformly in XCi.. n

Consider the equations

u = -AU+fA(u) for A E [0,1 ]

Let TIA be the corresponding local semiflow on XCi. generated by the so

lutions of (T A). Theorem I.2.4 implies that whenever An+A, then TIA ' 'n

and TI does not explode in N. Moreover, (Hyp 3), (Hyp 4) and Theorem

I.4.4 imply that for every sequence {An} in [0,1], N is {TI A }-admisn

sible.

In particular, (taking the constant sequence TIA =TI A) we see that N is n

strongly TIA-admissible for every A E (0,1]. Let KA be the largest in-

variant set in N relative to TI A. Since N is strongly TIA-admissible

it follows that KA is compact in XCi., and therefore KA is closed. If

128

we can show that KAcInt N for every A E (0,1] (i.e. N is an isolating

neighborhood of KA, relative to TI A) then the mapping ~(A)=(TIA,KA) is

a well-defined S-continuous mapping of h=(0,1] into S (in the sense

of Definition 1.12.1).

So let us show that KAcInt N for A E h. For 0<A<1/2 this follows from

(Hyp 5) since rcInt N. Let 1/2:::_A':::'1 be arbitrary and UOEK A. Then

there exists a solution u:JR-'XCi. of (T A) with u[JR]cN and u(O)=uO. Let

u 1 (t) : =Pu (t) u 2 (t) : = (I-P) u (t). Then U1 (t) : =pf (u 1 (t) + (2-n) u 2 (t))

u 2 (t)=-A2u 2 (t). Here A2 is the restriction of A to X2=(I-P)X. Since

A2 is a sectorial operator on X2 (see Henry [1], Theorem 1.5.3) with

0(A2 )=O', from the fact that re 0'>0, and using simple estimates we

get that u 2 (t) =0. It follows that u1 (t) =Pf (u 1 (t)) for all t E JR, so

u 1 :JR+X1 is a solution of the ODE (D) in (Hyp 6) with u 1 [JR]cB, where

B:=Nnx1 . By (Hyp 6) u 1 [JR]cf 1 . Thus u(t)=u1 (t) EfcInt N for tEJR.

In particular, u (0) =uO E Int N. This proves our claim. Hence, indeed,

the map ~:h+ S, ~(A)=(TIA,KA) is well-defined and S-continuous.

Thus h(TIA,KA)=const from the continuation invariance of the homotopy

index (Theorem 1.12.2).

Our argument above proves that for A~1/2,KA=K where K is as in (Hyp 6) .

For A=1 the system (T A) is unccupled:

{

Thus we can view TI1 as the product of TI with the semiflow TI' on X~ generated by U2 =-AU 2 . As a consequence, we have

,....... A. A

h(TI1,K1)=h(TIXTI' ,Kx{O})=h(TI,K)Ah(TI' ,{O}). However, since 0(A2 »0>0 it

follows from Corollary 1.11.2 that h(TI' ,{O})=l.°. Consequently, AA 0 AA AA_

h(TI,K)Al. =h(TI,K) implies that h(TI1,K1)=h(TI,K)fO (by Hyp 6)). Hence

for every A E (0,1], h(TIA,KA)fO and therefore KAf~. By admissibility,

taking a sequence An+O, we see that KOf~ (although KO may have points

on aN). Choose v 0 E KO' Since (TO) is just Eq. (1), there is a solu

tion u:JR+XCi. of equation (1) with u(O)=v and U[JR]CKOCN. Since TI is

gradient-like and KO is compact, Clu(JR) contains an equilibrium Uo of TI, i.e. there exists a Uo EN with -AUO+f(uO)=O as claimed.


Remark:

In (Hyp 3) we may, more generally, assume tliat a(A)={O}Ua'Ua" where

re a'>o>O and re a"<-o<O. In this case a" necessarily consists of a

129

finite number of eigenvalues with total algebraic multiplicity equal

to some p~O. In (Hyp 6) we then have to assume that l.PAh(;,~) is non

zero, where l.P is the homotopy type of a pointed p-sphere. Under these

modified assumptions, Theorem 8.1 remains valid with essentially the

same proof.

If the set B in (Hyp 6) of Theorem 8.1 is an isolating block for TI,

then h(TI,K)=[B/B-, [B-]].

The next proposition gives a criterion for [B/B-,[B-]] to be nonzero.

Proposition 8.2

!-et TI be ~ local semiflow on ~ metric space X and let B be an isola

ting block for TI. Assume that B is strongly TI-admissible. If the num

ber r 2 of connected components of B is finite and greater than the

number r 1 of connected components of B, then the homotopy ~ of

(B/B-, [B-]) is nonzero.

Proof:

Let Hq , q~O be any unreduced cohomology theory (with coefficients in

Z). Theorem I.3.7 and Proposition I.10.8 imply that

Hq(B/B-,[B-])~Hq(B,B-). There is a long exact sequence

Now take Hq to be the Alexander-Spanier cohomology theory. Let q=O

and assume that H1 (B,B-)={O}. Then exactness implies that

Img=ker f=Hq(B-), so g is surjective. Since there are only finitely

many components in B (resp. B-), all these components are open in B

(resp. B-). By Corollaries 9 and 6 in Section 5 of Chapter 6 of

Spanier [1], we have

and

Zr 1 to zr 2 , Since r 2 >r 1 there is no surjective homomorphism from a

contradiction. Hence H1 (B,B-)~{O} which proves that the homotopy type

of (B/B-,[B-]) is nonzero.

We shall apply Theorem 8.1 to system (1). Let A be the sectorial ope

rator defined in (P) at the beginning of this section.

Let P=P1:X~X be the projector defined as

T pu=l Ju(x)dx, and write P2 =I-P.

-T 0

Then X=X 1$X2 , X1 =P 1X, X2 =P 2X and the sum is orthogonal. If Ai as the

130

restriction of A to Xi' i=1,2, then A1=0

-A t where 8>0 is any number with sectorial on X2 with ije 2 ijx <e- 8t , t>O

2 VO<2; .Moreover, for every a~O there is a constant Ca such that

-A t ijA~e 2 Ilx ,2.C a t-a e- 8t (cf. Theorem 1.4.3 in Henry [1]).

2 We have the following

Lemma 8.3

Let g:ill+X2 be any bounded locally Holder continuous mapping. Let

M be abound on g, and u:ill+X2 be ~ bounded solution (defined on ill!)

of the equation

Then u is bounded in x 1 / 2 and 2

OOI -1/2 -8s Ilu(t)11/2~C1/2MS e ds<oo

o

Proof:

1 Let a="2' to E ill be arbitrary, and L be a bound on u. Then for all n>O

we have

Thus

-a -8n n -a -8s LC n e +C MIs e ds <

a a 0

Taking n+oo we get

OOI -a -8s C M s e ds < 00 •

a 0


set S=C'C 1 s-1/2e -8s where C' is a constant such that 1/20 '

Ilull ::.. c'lluL cO ([O,T],:ffim) ~

for u E x~

Here we use the fact that x~ imbeds in CO([O,T],:ffim)

We now obtain the following theorem:

Theorem 8.4

131

Let f::ffi x:ffim ->-:ffim be ~ mapping satisfying the properties listed at the

beginning of this section. Moreover, assume that f the gradient of

some function F::ffix:ffim->-:ffi, i.e. f(X,u):~~(x,u) for all (x,u)E :ffix:ffim.

Assume, in addition, that f is bounded on :ffi X:ffimQy some constant M>O.

Suppose that there is an open bounded set Gc:ffim and ~ class T of C 1_

functions V::ffim ->-JR such that for all VET Gc{ v E :ffim Iv (v) <O} and for

every u E aG there is ~ V=V u E T such that V (u) =0 and whenever h E :ffim

and Ilhll::"T1/2MS then <grad V(u), f(x,u+h»;tO for every xE:ffi. -Then B=C1G is an isolating neighborhood of an invariant set KcB rela-

tive to the local flow n generated ~ the equation

u = g(u)

1T where g(u) :=Tff(x,u)dx.

°

(5)

If the homotopy index h(n,K);tO, then there exists ~ T-periodic solu-

tion x->-u (x) of (1) with 11 E Int B and II;; (x) 112T 1 /2MS for all x E :ffi

where

1 T T f u(x) dx

° u u-u

Proof:

Let f:x1/2->-x be the Nemitski operator generated by the function f.

We will verify (Hyp 1)-(Hyp 6) of Theorem 8.1. (Hyp 1) and (Hyp 3)

follows from our remarks preceding the statement of Lemma 8.3 and

those at the beginning of this section. Since f is gradient of F, n

is easily seen to be gradient-like with respect to the map v:x1/2->-:ffi

defined as

V(u) 1/2<A1/ 2 1

1/2 T u, A1 u>-k/2<u,u>- fF(x,u(x»dx

° Here, <, > is the scalar product in X=L 2 (0,T,:ffim) and A1=A+kI for

some k>O. This is verified similarly as in the proof of Theorem 5.5.

This proves (Hyp 2).

132

Now let G'=int CI G=Int B. Furthermore let

r-{ '/21 'II II '/2 ~ -'/2 -8s } - u E X P,u E G , P2u '/2<2MT C'/2Js e dx. o

Then r is open and bounded in Xu. Moreover, f[r] is bounded. This

proves (Hyp 4).

Now let N=CI r and let u:m+x'/2 be a bounded solution of (SA) with

u[m]eN. Write ui=Piu, i=',2 for uEX. Since IIA(I-P)f(U)+Pflu)IIx-'::MT'/2

it follows from Lemma 8.3 that

(6)

Ilu 2 (t)II 0 <MT'/2 B C ([ 0, T] , mm) -

(7)

If u(tO) It r for some to' the~ by (6) u 1 (to) It G'. It follows that

u, (to) E dG. Then there is a VET with V(U, (to)) =0 and - m <grad V (u, (to))' f (x,u, (to) +h) >;l0 for all x E m and hEm with

II h ~'::M T' / 2 B. In particular, by (7) <grad ~(u, (to))' f(x,u, (t o )+u2 (tO) (x))>;lO for

all x Em.

Integrating we obtain <grad V(U, (to))' P,f(u(tO))>;lO.

Since u,=P,f(u), it follows that :tV(u, (t)) It=t ;l0, implying that _ 0

for some small s>O, V(U, (t)) >0 for all t E (O,d or all t E (-£,0).

For all such t, u(t) il CI r=N, a contradiction.

This proves (Hyp 5). Note that this argument is also valid for A=O,

hence any full solution of (3) lying in N for t Em, lies in r.

Now r,=rnx,=G' and Nnx, =CI G'=CI G=B. Thus TI=TI and K=K. The same

arg~ent shows th~t Ker, proving that B is an isolating neighborhood

of K relative to TI, and that (Hyp 6) is satisfied. Now Theorem 8.'

and the remark just made yield a solution Uo E r, of (4). Classical

regularity implies that Uo EC2 ([O,Tl,mm) and UO(T)=UO(O), uO(T)=UO(O).

Hence extending Uo periodically for all xE m and using the estimate

(7) we get the desired result.


Remark:

An analogous result can also be proved for the system

uxx-Au+f(x,u) O,U T-periodic (8)

where A E a (A) .

In fact, let Au=-u +AU with D(A) as before. Then o(A)={O}Uo'Uo" xx

where re 0'>6>0 and re 0"<-6<0 for some 6.

Now use the remark following the proof of Theorem 8.1.

As a corollary to Theorem 8.4 we obtain

Theorem 8.5

Let f and F satisfy the assumptions of Theorem 8.4, except that f

may be unbounded this time.

133

Suppose that (;ClRm is an open bounded set in lRm and there exists ~ y>O and ~ class T of c1-functions V:lRm .... lR with Gc{ulv(u)<O} for all

VET and such that whenever u E oG, then there is ~ V=Vu E T such that

V(u)=O and <grad V(u), f(x,u+h»fO for all xElR and Ihl~Y. Let B=C1G

and K be the largest invariant set in B with respect to the local

flow TI generated £y the solutions of the ODE

where

~(t) g(u(t))

T g(u) T f f(x,u) dx

o

(9)

Then KcInt B. If the homotopy index h(TI,K)fO, then there exists an

EO>O such that whenever O<E<E O' then the equation

uxx+Ef(x,u(x)) o

has ~ T-periodic solution u with

1 T u = T fu(x)dxEInt B, iu(x)-ull < EM, xElR,

o

where M is some constant independent of u and E.

Proof:

(10)

First assume that f is bounded. Let Ilf(x,u)II<M. Choose EO such that

EOMT 1/ 2 B=Y.

Let O<E~EO' and f(x,u)=Ef(x,u),

F(X,u) = EF(x,u) , M EM.

Then for every u E oG there is a VET with V(U) and

<grad V(u), f(x,u+h» f 0 for ~hll<MT1/2B ~ Y (11)

134

Let TI pe the local flow generated by

where

u g(u)

1 T g(u) := T JI(x,u)dx .

o

( 12)

Then (11) implies that B is an isolating neighborhood of the largest -invariant set K in B relative to TI. Moreover, t+u(t) is a solution

of (9) if and only if t+U(Et) is a solution of (12). In particular,

a set B is an isolating block for TI if and only if it is so for TI

with the same sets of ingress, resp. egress resp. bounce-off points.

It follows that K=K and h(;,K)=h(TI,K)FO.

Now Theorem 8.4 gives the desired result if f is bounded.

If f is not bounded, take some big ball B1=Br (0) of radius r 1 with

B=C1Gdnt B 1 • 1

Now let <p:lRn+lR be a COO-function such that

<p(u) if Ilull.9r1

<p(0) 0 if ~ui~3r1

- - 13F Let F(x,u)=<p(u). F(x,u) and f(x,u)=a-(x,u). Apply the theorem to f. _ u

Then there are M>O, EO>O such that whenever O<E<EO then

u +Ef(x,u(x» 0 xx

has a T-periodic solution u with uEInt Band IU(X)-U!<EM for all - 1 x E lR. Let E1<EO be such that E1M~2r1. Thus, if 0<E<E 1 , then for all

x E lR

Thus f(x,u(x»=f(x,u(x», xElR, so u satisfies the original equa

tion. The proof is complete.

Remark:

There is an analogous result for T-periodic solutions of the system

where

135

(cf. the remark following the proof of Theorem 8.4).

If the Brouwer degree d(g,G,O)~O in Theorem 8.5, then the result is

well-known and can be obtained by the coincidence degree method (see

Mawhin [1], Gaines-Mawhin [1]). In this case, f needs not be a gra

dient.

However, as we shall see now, there are situations with f gradient

in which d(g,G,O)=O, so the coincidence degree method does not work,

but the homotopy index h(~,K)~O and so Theorem 8.5 can be applied.

Also, it will follow from results of section 3.3 in Chapter III,

that whenever d(g,G,O)~O in Theorem 8.5 then also h(~,K)~O. In other

words, for gradient systems, whenever the degree method works, then

so does the homotopy index, but not conversely.

Proposition 8.6

Define H: JR3 +JR !?y

H(u)

and g(u) = grad H(u)

3 Define the functions Vi:JR +JR, i=1, ... ,4 as

Let G={UEJR3IV2(u)<O for i=1, ... ,4L Then B=C1G is an isolating block for the equation

u = g(u)

with

(13 )

If K is the largest invariant set in B relative to the local semi

flow ~ generated !?y (13) then h(~,K)= homotopy type of (B/B-,[B-])~O.

On the other hand d(g,G,O)=O.

136

Remark: It is easily seen that B has the form

The arrows describe the direction of the flow.

Proof:

1 It follows that g(u)=O if and only if u 1=±3'u2 =u3 =0. Furthermore, 3 .

det Dgfu)=2u1 if g(u)=O.

Moreover, ( ±1,0,0)T E G. Hence d(g,G,O) is defined and equal zero, as

claimed.

We will now show that if u E ClG and Vi (u) =0, then

<grad Vi(u), g(u»>O if i=1 or 2

and

<grad Vi(u), g(u»<O if i=3 or 4 .

So let uE ClG:

1. Case: V1 (u)=0.

{5 Then u 1=3.

<grad V1 (u), g(u»

2. Case: V2 (u)=0.

Since V4(u)~0,

<grad V2 (u), g(u»

where h (y) =_ ~ y2 -2y + _2_ V5 91/5

Now h' (y)<O for y>-

h' (y) >0 for y<-

Therefore

V5 T V5 T

i.e.

h(u1) .:. min (h(- vJ), h(O)) >0 •

u 1 2 Since 1/5+1~3>0, we get <grad V2 (u), g(u»>O.

137

The remaining claims are proved in a similar fashion. Thus, indeed, B is an isolating block for (13) and since B-= {u E ClG I V 1 (u) =O} u{ u E ClG Iv 2 (u) =O} has two connected components whereas B is connected, -it follows from Proposition 8.2 that the homotopy type of (B/B-,[B-]) is nonzero, as claimed.

This concludes the proof of the proposition.

Remarks:

If e:lR +lR3 is any continuous T-periodic function with

T fe(x)dx = 0 , o

then define F(x,u)=H(u)+<e(x) ,u> where H is as in the above example.

Then F and f(x,u) :=~~(x,u) satisfy all hypotheses of Theorem 8.5 The degree continuation method cannot be used for our choice of G.

However, a different choice of G can make the degree method work again. For instance, if G contains just one of the equilibria then

d(g,G,O)fO. However, by a modification of the previous example we can arrange that no choice of G with d(g,G,O)fO is possible.

To wit, let 6>0 be small. Let B: lR +lR be a COO-function such that

B(s)=B(-s), 5 E lR,

138

(3(s) if Is I.::;

if I s I~;+o

1 1 0<(3 (s) <1 and (3' (s) >0 if '3<S<'3+o.

DefiIle

H(u)

where

Then for g=grad H we have

gi (u) g i (u), i=2, 3

where g is as in the previous example.

If the Vis are defined as above, and 0>0 is sufficiently small, then

it is easy to prove that for u E dG

<grad Vi(u), g(u»>O if Vi(u)=O and i=1,2

<grad Vi(u), g(u»<O if Vi(u)=O and i=3,4

Hence u=g(u) has the same behavior on the boundary of G as u=g(u) •

In particular, B is again an isolating block with the same set B-,

so the homotopy index is nonzero again.

However, let us show that for any bounded set UcE3 with g(u)#O for

.u E dU we have d(g,U,O) =0. In fact, g(u) =0 if and only if

IU 1 I.::;, u 2=0=u3 · Hence the zero set of g is a line segment J. Hence

either unJ=0 in which case d(g,U,O)=O or else JcU. In the latter case,

consider a perturbation g of g, g=g (U)-(E,O,O)T where E>O is a small E E _

number. Since g(u)#O on dU, we have for all E small, gE(u)#O on au

and d(gE,U,O)=d(g,u,O). However, gE(u)=O if and only if

2 1 (3 (u1 ) (u1 - 9) = E

Our assumptions imply that for each small E there are exactly two

such solutions

139

1 < I Y (E) I < 1+ 6 •

Computing the Jacobian at the equilibria we get

det D gE(u)

If U 1=+Y(E) then S(u 1»O, S'(u1»O, so det D gE(~»O. For U 1=-Y(E),

this determinant is negative. Now for E~O, Y(E)~, so the two equi

libria are both in U for E small.

This implies d(gE'U,O)=O for E small, and our claim follows.

The important feature of the above example is the fact that B is sim

ply connected. Due to the Poincare Bendixson theory, no such example

with B simply connected can be given in two dimensions.

On the other hand, there exist simple two-dimensional examples in

which B is, say, a ring.

In fact, define G:IR2 ~IR as G(u)=lluI1 4-81Iuf and let B={uE ]iI1~llui~3}. If g=grad G, then B is easily seen to be an isolating block for u=g(u)

with B-=oB.

Thus h(K)'O. On the other hand, it is not hard to prove (by suitably

perturbing g), that d(g,U,O)=O, where U=Int B.

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