Numbers of zeros on invariant manifolds in reaction- diffusion equations

Nonlinear Analysu. ?-hcory, Metho& Printed in Great Britain.

NUMBERS

& Ap$canom.

Vol.

10. Ho. 2. PP. 179493.

1986. 0

OF ZEROS ON INVARIANT MANIFOLDS DIFFUSION EQUATIONS

0362-%6X/86 13.00 + .M 19% Pcrgamon Rcu Ltd.

IN REACTION-

PAVOL BRUNOVSKQ Univerzita KomenskCho, Institute of Applied Mathematics, Mlynska dolina, 84215 Bratislava 2, &SR

and Sonderforschungsbereich

BERNOLD FIEDLER 123, Universit% Heidelberg, Im Neuenheimer

(Receioed Key words and phruses:

10 November

Feld 293, 6900 Heidelberg,

FRG

1984; received for publication 9 April 1985)

Lap number, zero number, invariant manifolds, reaction-diffusion

equation.

INTRODUCTION CONSIDER

the one-dimensional

reaction4iffusion

equation

u, = u, +f(x,u),t>O,O
(0.1)

with the Dirichlet boundary conditions u(t, 0) = u(t, 1) = 0, where fi [0, l] x R --+ R is BC’ n C”, K > 1. The equations particular case of the abstract equation

(0.2) (O.l), (0.2) can be viewed as a

du/dt + Au =f(u)

(0.3)

in a Banach space X, the basic theory of which is developed in [5]. For (O.l), (0.2), X= L2[0, 11, A is the closure of the operator defined by Au = -II" for u E C’[O, 11, u(0) = u(l) = 0, F: L’[O, l] + L2[0, l] is given by F(u)(x) = f(x, u(x)). We frequently work in the Hilbert space X’ = 9(A) = Hb([O, 11) fl H2([0, 11) with F: X1 + X1 also being C,. Let 1.1 denote the norm on X1. Applying the results of [5] one obtains that (O.l), (0.2) generates a local semiflow S on X1. The semiflow S is a continuous map of an open neighbourhood U of (0) x X1 in R’ x X1 into X1 defined by S,(u)(x) = u(t,x)

for

(t, u) E U,

where u is the solution of (O.l), (0.2), satisfying u(0, x) = u(x)

for

O
(04

It has the properties So(u) = u, Sris(u) = s,o S,(u) as long as (s, u) and (t, S,(u)) are in U [5]. In order not to obscure the formulations by technicalities we shall assume that S is a global semiflow, i.e. U = R+ X X’. This is by no means an essential restriction; sufficient conditions can be found in (5, Chapter 31. 179

I‘30

P. BRLTGOVSKYand 8. FIEDLER

The critical points of S are the stationary solutions of (O.l), (0.2), i.e. the solutions of the equation Y”+f(x, u) = 0, u(0) = u(1) = 0. The qualitative properties of S near a critical point u are determined (O.l), (0.2) at u which is the equation

(0.5) by the linearization

of

Yr = YXX+ fu (x3 G))Y

(0.6)

y(t,0)

(0.7)

= y(t, 1) = 0.

The solution D is called hyperbolic if 0 is not an eigenvaiue of the operator L = A - F’(u), i.e. (0.6), (0.7) do not admit a nontrivial stationary solution y. An important information about the global structure of the semiflow of (O.l), (0.2) is given by the orbit connections of different stationary solutions [2,3,5]. By a connecting orbit of the stationary solutions ur, u2 we understand a solution u of (O.l), (0.2) which exists for all f E (- =, =) and satisfies lim u(t, x) = ut(x), lim u(t, x) = u?(x) t--+--2 I-+% in H’[O, 11. In the terminology of [5], u(t, *) has to be in the stable manifold of uZ and the unstable manifold of ui, provided ul, uZ are hyperbolic. In this paper we obtain estimates on the number of zeros (or, more precisely, the zero number defined below) of uft, .)-ui and n(t, .)-ur. This information can be used to concrude existence and nonexistence of connections. Our approach provides an alternative to the (slightly different) zero number of u,, which was used by Hale and Nascimento [3] to soive the connection problem for f of the Chafee-Infante type (see e.g. [5, Section 5.31). For any continuous function @: [0, l] + W we define the zeru number z(q) as follows. Let n L 0 be the maximal element of l+J,U {x} such that there is a strictly increasing sequence 0 5 Q
forOsj<<:.

If n is finite let z(g) := n, and z(G) := ~1otherwise. As a first example consider the linearized equation has eigenvalues &, < Ai < . . . with eigenfunctions .z(+~) = k and indeed it is a classical result (see [0,

Note that we put z(O) := 0. (0.6), (0.7). The operator L = A - F’(u) $a, @i, . . . . By Sturm-Liouville theory p. 5393) that for 0 5 i
irz(+)Ij,

(OS)

whenever 4 is a (nontrivial) linear combination of $,, . . ., Gi. As a trivial illustration of our approach we prove estimate (0.8) in corollary 1.2, using the dynamic equation (0.6), (0.7). All our results depend on a basic observation, lemma 1.1, going back to Redheffer, Walter [a] and, more recently, Matano [6]. According to lemma 1.1, +(I,

.)) is nonincreasing

as a function of time t along solutions of equation condition. 1(x,0) =0

(O.l), (0.2) provided

forO
1.

that f satisfies the (0.9)

Numbers

of zeros on invariant

manifolds

in reaction-diffusion

equations

181

The proof is elementary and relies on the maximum principle for parabolic equations. For the convenience of the reader we present it in detail below. In the nonlinear case, let u be a hyperbolic stationary solution of (O.l), (0.2). Then the eigenvalues kj of the linearized equation with corresponding eigenfunctions @, satisfy for some n Z 0. Further by [5, theorems 5.2.1, 6.1.91 there A,<... C&_,
w E W”

(0.10)

w E Ws\{u}

(0.11)

for

(theorem 2.1) and z(w - u) Z dim W”

for

(theorem 3.2). Note that these estimates are suggested by the respective tangent spaces of W and W, together with the Sturm-Liouville estimate (0.8). The crucial observation of our proof is that for u = 0:

44

limlu(t)l= -for -for

@k

t-+ - x on W and some k < n t-+ + CCon W’ and some k B II, provided

(0.12)

that z(u(t, . )) is eventually finite.

Actually it is quite simple to prove (0.12) on W, as we will indicate at the end of Section 2. However, analysis on the infinite dimensional stable manifold W is quite delicate and we need detailed information on the fine structure of W before we can prove (0.12). For illustration we pursue an analogous approach to W in Section 1, as a preparation to the stable manifold case. 1. COUNTING

ZEROS

In the introduction we defined the zero number z(G) of a continuous real function @ as the maximal number of sign changes of Cp.In this section we show that z decreases along solutions u(t, *) of the parabolic equation (0.1) with Dirichlet boundary conditions, assuming that f(x,O) =0

forall

xEZ,

(1.1)

I := [0, 11. This result is essentially in 18, corollary 31 who consider f = f(t, x, u,, u,) independent of U. Similarly, Matano [6] investigates the lap number of $J, which is the zero number of C#J~ and was called “maximum order of a saw in c$” by Redheffer and Waiter [8]. Note that by definition the function z: CO@ --, N u {=} is lower semicontinuous. Further, z is constant in a Cl-neighbourhood of any Cl-function with only simple zeros. These trivial facts will become important later on.

@

182

P. BRUNOVSK? and B. FEDLER

The parabolic equation (O.l), (0.2) generates a semiflow .I?(+, = u(t) = u(r, .) on X’ Co(r), thus z(u(~)) is well defined along solutions.

C

HA C

LEMMA1.1. [6,8]. Letf(x, 0) = 0 for allx E I. Then the zero number z(~(t, .)) is nonincreasing as a function oft along solutions u(t, a) of (O.l), (0.2). Proof. With a(t, x) : = @(x, ~(t, x)))/(u(t,

x)) we write (0.1) as

U, = u, + au,

(1.2)

where a is Co. We apply the maximum principle to (1.2) to prove: if xi, xi. E I are such that u(~,x;) < 0 < u(f,x;) then there exist continuous paths yi in I x [0, r] connecting (I, xi ) to a point (0, xi), such that u < 0 (resp. u > 0) along y1 (resp. yr). To see that assume 0 < x’, < xi C 1, the case xi > xi is analogous. Let Di be the path connected component of (t, xi) in the relatively open set Kj := {(t, !$) E [O, f] X 4 (-l)‘u(r,

5) > 0).

We claim that we can find elements (0,~~) E Die Otherwise, e.g. D2 is contained in the strip (0, I] x I. Replacing u by ueM does not change d, and allows us to assume a < 0, hence Au := 4.c - u,h 0 on D2. Let M := max u > 0 and choose a point (i, f) E d, with minima1 i such 9

that u(l, 2) = M. From M > 0 we conclude (f, n) E D,, hence f > 0. This implies a contradiction to the strong maximum principle: let E := D2 and apply [7, 111.2, lemma 31 to conclude u < M on dz n ((7) x I), contradicting (i, i) E D2 fl ({t} X Z). Therefore there are points (0, xi) E Di. n Invoking the Jordan curve theorem comletes the proof. As a trivial but illustrative application, we prove estimate (0.8) for finite linear combinations i @O=

kFiffk'@k

(1.3)

of Sturm-Liouville eigenfunctions $$ for the potential a(x) :=fU(x, u(x)). We use the flow (1.2), defining a solution $(t, +) with initial condition Cp(0,.) = Cp”and Dirichlet conditions. COROLLARY1.2. If the Sturm-Liouville then

potential a is continuous,

0 ~5i
i%z(@O)Zj Proof. We use the explicit representation

(1.4 k=i

of the solution +(t, o), t E R of (1.2) through Go. From (l.?), $JO+ 0 it is immediate that there exist integers k’ E (i, i + 1, . . . , j} such that [j$ in

the Cl-topology

(normalizing

@(f)/]$(f)l =

sign(ak=)$k=

]&I = l), because the & are pairwise disjoint. The @khave

Numbers of zeros on invariant manifolds in reaction-diffusion

equations

only simple zeros, hence z is constant in a C’-neighbourhood of qk. By monotonicity along solutions of (1.2) (lemma 1.1) we conclude for T > 0 sufficiently large isk’=

z(@(T, 9/\$(7-,

and the proof is complete.

91) = z($(T,

9) s z(@‘) 5 z($(-T,

183

of z

.)) = k- Sj

n

Note that the corollary holds even if j = =. 2. ZEROSONTHEUNSTABLE

MANIFOLD

In this section we prove that for any element w of an n-dimensional unstable manifold of u there are less than n zeros of w-u. On our way we investigate the fine structure of the unstable manifold. Finally we relate z(w - u) to the number of zeros of u,. Let u be a hyperbolic stationary solution of (O.l), (0.2) with eigenvalues A,<. . .
THEOREM

(i) dim W, = k + 1, and the tangent space to W, at u is spanned by &, . . ., qk; (ii) for any w E W,\ W,_,

(2.1) where the flow S, for t c 0 is defined by S_,(S,(w)) = w on W; (iii) for w E W,\W,_, and t near - 53the zero number z satisfies z(S,(w) - u) = k; and S,(x) - u has precisely k simple zeros in (0,l); (iv) for w E Wk\Wk_l, we obtain z(w - u) I k, and consequently

for all w E W z(w-u)
Note that by [5, Section 7.31, S, is weil defined for t < 0 on W. At the end of this section we outline a simple idea for the proof of theorem 2.1 which uses finite dimensionality of W. Another idea which also works for the infinite dimensional stable manifold (see Section 3) can be illustrated in the case dim W = 2. The linearization of the flow on W near u looks like Fig. 1, where #o, @i are represented by the coordinate vectors. All integral curves y(t) = ao(t)qbo + al(t)q31 which are not identically zero have the property cuo(t)cu;l (t) + 0 for t--, - ~0except of two which have al(t) = 0. Qualitatively, this picture is not destroyed by nonlinearities. The exceptional trajectories become W. in the notation of the

P. BRUNOVSKPand B. FIEDLER

Fig. 1. The strongly unstable manifold IV and a general trajectory y outside W”.

theorem.

A trajectory

y on W satisfies

‘/(0 = 6 + %(+#Jo + ~l(+#JI + C+o(t)l

+ MN

fort--,

- =.

The exceptional ones satisfy in addition cur(t) = o((Y,,(~)), all the others cam = ~((~t(t)) for t 3 - =. Consequently, (u;‘(t) (y(t) - u) mimicks $. in the first case while a;‘(r) (y(f) - o) mimicks @r in the second case for t near - m. In particular, it will have the same zero number as Go, @t respectively. We employ lemma 1.1 to conclude that (y(t) - u) does not increase with t, hence z(y(O)) 5

max(z(@0>7z(h>>

= 1.

To carry out the idea in detail we need the following. LEMMA

2.2. Consider a differential Wpx W’Jdefined by

equation

on a neighbourhood

U of the origin in R” =

i = Ax + f(x, y) i = BY +

dxt

Y)

(2.2) (2.3)

(x E R’, y E Wq). Assume that all eigenvalues of A (B) have negative real parts %a0 (ZbO, respectively), where a0 < b0 < 0, f, g are Ck, k > 0 and satisfy

c,~~o fk Y) 1(X? Y)l--l

= 0, ($iO

g(x, y) 1(x,y)I -1 = 0.

Then, there exists a positively invariant neighbourhood R of 0 and a p-dimensional Ck submanifold W of R through (0,O) tangent to the subspace y = 0 at (0,O) such that each solution (x(t), y(t)) of (2.2), (2.3) with (x(O), y(0)) E C&W satisfies iim ly(r)l-’ x(r) = 0.

(2.4)

Numbers of zeros on invariant manifolds in reaction-diffusion

185

equations

Proof. For the finite dimensional case considered here, it is easy to prove (2.4) directly from (2.2), (2.3), choosing suitable scalar products on RP, Iwq and deriving a differential inequality for q(r) := ]x(t)l*/]y(t)l*. H owever, we give a different proof which carries over without change to an infinite dimensional situation occurring in the stable manifold (see lemma 3.1 and its proof in the appendix). The existence of R and an invariant manifold W tangent to the subspace y = 0 at (0,O) follows from [4, lemma 4.1 and corollary 5.1, chapter IX]. If R is chosen sufficiently small, W can be represented as the graph of a Ck function h from some neighbourhood of 0 in the xspace into Rq with h’(0) = 0. It follows from [4] that if one introduces in R new coordinates u = x, v = y - h(x) then the (II, v)-representation a,: (u, v) += (u,, vi) of the time one map of (2.2), (2.3) satisfies ui = Au + U(u, v)

(2.5)

vi = Bv + V(u, v)

(2.6)

with U, V having similar properties as f, g in (2.2), (2.3) and, in addition, V(u, 0) = 0. The time one map of a differential equation maps initial values of its solutions into their values at time one. By choosing suitable norms 1.1 in the u-, v-spaces we can assume /Au] < (a + 0) 1~1 IBv] ’ (b - 0) Iv1 where a := exp ao, b := exp b. and 0 < 8 < (b - a)/2, 8 < b. Also, there is a positive function K(P) on some right neighbourhood of zero such that K(P) + 0 for p + 0 and

Iwb v>l< K(P)(I4 + Ivl>,IVh VI < f4P)lVI if lul + Iv/ < p. Let now (u, v) E S2 and let S2 be so small that ]ui] < Iu], lvil < Iv\. Then, we have

14

ix

(0 + 0) I4 + K(P)(IuI+ 14)= a + 0 + 4~) I4 b-o-K(p)jV/+b-&-K(p)’ (b- 0) Id - K(P) b'i

K(P)

(2.7)

Let

We have lininp(p) = 0 and there exists a p. > 0 such that a + 8 +

K(P)

<

a(b -

8 - K(P))

for

any p < po. From (2.7) we have for E E (0,l - CY)

(u11 I4 C (CY + E)i;;i 1% Choose any (uo, vo) E such that luk] < ylvk] for Since (uk, vJ --, 0, there P(pl)/& < y. From (2.8)

I4 p(p> E .

(2.8)

as soon asm >

R with v. # 0 and any y > 0. We prove that there exists an N > 0 all k > N where (uk, vJ = @(u,, vo). Indeed, assume the contrary. exists an No such that lukl + lvkl < pi d p. for all k 2 LI,, where it follows that

Iuk+A < YIv~+J asSoOnas I4 + Ivd <

~1

and

/uA < ~1~4

(2.9)

186

P. BRUNOVSK~ and B. FIEDLER

If lukl> ylvklfor k 1 No then by (2.8) also

bkl

~
I%0I

fork2NN,

which is impossible. Thus, there exists an N B N,, for which luNl < yjv,vj. By (2.9), we have ]ukj < yjvkJ for all k > N. Since y was arbitrary, ,lili Ivkl-+rk = 0. For the differential equation (2.2), (2.3) this means that if (x(t), y(r)) is its solution with (x(O)7 Y(0)) E s2\w ( or, equivalently, y(0) # h(x(O))), then

lx(k>l

(2.10)

kizcr b(k) - hMk))l = ” We have

IWI _

bWl

b(k)- h(xW)l

IWI

IYW- h(k))1

IWI

bb(W)1 IXWI ’ MN - h(xW)l ( 1 + IuWl >

(2.11)

’

lx(k>l

luW>l( 1_

x(k)I ) ’ ly(k) h(W)l’ b(k)- h(xW)l IxWl

Ix(k)1

lh(xW)l

I

Since h(x) = o(lxl), from (2.10), (2.11) we obtain (2.12) Let now k d t < k + 1. By standard formula we obtain:

Gronwall

estimates

/(x(t), y(r))1 5 Cl(x(k), y(k))] = : p

and the variation

of constants

for all k and t E [k, k + 1)

with some C Z 1. Again by Gronwall and variation of constants we obtain

MOI5 C,(lx(k>lf f(p). (IWI + lu(Wl>) MOI2 C,(lWl - R(P).(lx(k)1+ luW>l)) for all k E N, t E [k, k + l), suitable constants C1, Cz > 0 and a function I;-(p) satisfying z

*@) = O

Numbers of zeros on invariant manifolds in reaction-diffusion

equations

187

Thus we have (for all k E N, t E [k, k + 1)) fi

s

t

IWl - IuWl-’ - (I+ WI) + R(P)

>.

2

W)(l + IxWl *IuWl-‘1

I-

and (2.12) readily implies (2.4), completing the proof of the lemma.

n

Proof of theorem 2.1. A neighbourhood V of u in Wu can be considered as an open subset 52 of R”, the coordinates z = (z,,, . . . , z,_~) chosen in such a way that zk(w) = Ek(w - u) for w E Wu near u. Then, locally at U, the restriction of (O.l), (0.2) to WYhas the form (2.13)

i=Cz+q(z),

where C = diag {-A,, . . ., -A,_l}, Consider the associated system

q is CX and q’(0) = 0.

dz/dt = -Cz - q(z) which is obtained from (2.13) by time reversal t = --t. This system satisfies the assumptions of lemma 2.2 with x = (zo, . . . , z,-~), y = z,-~. We denote by wa_, the submanifold W the existence of which is asserted in lemma 2.2. It is given by an equation z,-1 = k-&o,.

. . , zn-2),

ZER

where hn_2 is C” and satisfies (2.14)

h,,_*(O) = 0. By lemma 2.2, if z(t) is a solution of (2.13) with z(0) E Q, zn-i(0) f h,&zo,

. . .,

z,-21,

(2.15)

* 4L.2(O)l

=a

(2.16)

= 0

(2.17)

then ,~~~lzn-‘(~)l-l/(zo(~),. From (2.14) it follows that Jim= lGl-l(a

I(zo(4

* * f 9 L-.2W1

if (2.15) does not hold. Since W” is tangent to the unstable space of L, from (2.16), (2.17) it follows respectively c$nm ILI(&W for w E V\w,,_,

- u>I-’IV - J%-1)(%(w) - u)l- 0

(2.18)

and

,zm=

I&-l

+ ES)

cw)

-

u)l I(1 -

E,pl - ES)@,(w) - u)I-’ = 0

(2.19)

for w E w,,-2. We define W,_, = {SI(W~_2)]t B 0). By [5, theorem 6.1.91, W,,_, is an invariant submanifold of W”. The properties (2.18), (2.19) obviously extend to w E Wy\Wn_2, W,,_2, respectively.

P. BRUNOVSKY and B. FIEDLER

188

On W,,_*, the differential equation is again of form (2.13) with C = diag {-I.+ . . . , -An-J. Applying lemma 2.2 to the equation on W,_? we obtain an (n - 2)-dimensional submanifold wn-3 of wa-? represented by z,_z = hn_3(Zg,. . . , z,-3) with (2.20)

h,_,(O) = 0 such that ,“m= Iz,-Z(Ol-l

I(zo(r), . . .f Zn-3W)l

= 0

(2.21)

for all solutions z(t) with z,_z(O) # hn_z(z,,, . . . , z,,._~). Again, we extend l@,_s to an invariant submanifold of W,_, by W,,_; = {S,(#‘n_3), t Z 0). F rom (2.19) and (2.21) it follows that ,lim%)E,_@,(w) for w E W,_,\W,_,

- u)j-’

. j(Z - En-,)(&(w)

- u)l = 0

while for w E W,,_, it follows from (2.19) and (2.20) that

lim in$E,(S,(w) P-+--r k=O

- u)l-‘l

(I-l$lEk(S,(w)

- u)i =O.

In this way we may proceed further and after n - 1 steps obtains all the (k + 1)dimensional manifolds W, such that for w E W,\W,_, we have ,f&

](I - E&W)

- ~>l/]&(Uw)

- u>I = 0.

This in turn implies for w E W,\W,_, that ,!mZ (&(w) - n)/l&(w)

- 01 = *

@k.

(2.1)

Recall that the above limit is considered in X1 C C’(Z), and $k has only simple zeros with z(@k) = k. By our remark preceding lemma 1.1 this implies z(S,(w) - u) = k for t near -=. Now we invoke lemma 1.1 for z(u(t)),

u(t): = S,(w) - u. Note that u satisfies an equation

u, = u, +i(x, u), u(t, 0) = u(t, 1) = 0, wheref(x,u):=f(r,u-+u(x))

-f(x,u(x)).H

encef(x, A 0) = 0 and lemma 1.1 implies for tnear --jc

z(w - U) = z@(O)) d z@(t)) This completes the proof of theorem 2.1.

= z&(w)

- u) = k.

H

From our theorem we deduce a relation between the number of changes of monotonicity of a hyperbolic stationary solution u (some “lap-number”, cf. [6]) and the zero number z(w - u) on the unstable manifold of u.

Numbers of zeros on invariant manifolds in reaction-diffusion 2.3. Let u be a stationary hyperbolic w E W” be in its unstable manifold. Then

COROLLARY

equations

189

solution of (O.l), (0.2), U, C 0, and let

z(w - 0) < z(u,). Proof. Due to theorem 2.1 it suffices to prove that II:= dim W” S z(u,). The function y : = u, solves the linearized equation

Yxx + f&1

G))Y

= 0.

On the other hand, the eigenfunction Qn_t has II + 1 zeros on the closed interval [0,11. By the comparison theorem, between any two consecutive zeros of $J~-, there has to be a zero of n u,~.By u, 9 0, all zeros of u, are simple. This implies z(u,) Z n and the proof is complete. We outline an alternate proof of theorem 2.1, (iv) which works only for W”, as far as we know. Consider any trajectory u(r) on W”\(u) and let y(f) : = u(t)/lu(t)l be its projection onto the unit sphere. Then obviously lim E’y(f) = 0. I--r--m Since W” is finite dimensional,

we may thus pick a sequence fk* - r: such that (2.22)

exists in X1 C C’(I). But @ is in the unstable eigenspace of u, hence Section 1 implies for fk near --CD z(W -

U) =

without any intermediate

Z(U(0))

6 Z(U(tk))

COnStrUCtiOn

= Z(y(tk))

= Z(Q)

<

n =

dim w”,

of wk.

3. ZEROS ON THE STABLE MANIFOLD

We turn to investigate the zero number z(w - u) on the stable manifold W’ of the hyperbolic stationary solution u of (O.l), (0.2), keeping the assumptions and notations of Section 2 in effect. Similarly to the unstable case we need the following lemma on the fine structure of W. LEMMA 3.1. Assuming hyperbolicity of u above and f E C”, K 2 2, there exists a decreasing sequence Ws = W, 3 Wn+, 3.. . of invariant C”-submanifolds of the stable manifold W through u such that (i) the tangent space to wk at u is spanned by $k, &+r, . . . (ii) for any w E wk\wk+r

!& (‘dw) - u)/ls,(w) - u( = We defer the proof of this lemma to the appendix.

2 +k.

(3.1)

P. BRUNOVSKQ and B. FIEDLER

190

As an immediate consequence u(t): = S,(w) - u satisfies

of lemma 3.1 we can conclude for w E Wk\Wkcl, k 2 n, that

= Z(Cq5k) = k,

by lower semicontinuity of z and monotonicity z Z n on all of IV, if for example n

of z (lemma 1.1). However, this does not imply

Wk

f

bh

k?zn

To remedy this point we use the following alternative which is proved in [l]: (i) either z(u(t)) stays infinite for all t 20; (ii) or z(u(&,)) < ~0for some t,, 2 0, and u(t) has only simple zeros for an open dense set of t E [to, co). Using this fact, we will conclude below that /-, w, c {wl.?(w - 0) = =} u {u}. ktn

THEOREM 3.2. Let u be a hyperbolic wEWkCw, w#uweobtain

stationary

solution of (O.l), (0.2) as above. Then for

z(w - u) Z k and in particular for all w E W’\(u) z(w - u) Z dim W”. Proof. With the preceding remarks it is sufficient to prove for w # u z(w - u) 2 k

forallwE

Wk+l,

k L n.

Obviously we may assume that z(w - u) < =. Then, by [l, theorem], there exists a l L 0 such that u(t, .) = s,(w) - u has only simple zeros. Because wk+l has codimension 1 in wk we may then choose li E wk\wk+, such that z(zi) = z(u(t)) (just choosing I/u - ~(r)ll~~~~small enough). But by the remarks above z(a) e k, thus monotonicity

of z (lemma 1.1) yields z(w - u) = z(u(0)) L z(u(t)) = z(a) 2 k

and we are done.

q

4. APPENDIX We give a proof of the fine structure of the stable manifold claimed in lemma 3.1. To this end we first construcf an invariant manifold corresponding to a tine, splitting the spectrum of the linearization. We use a general analytic semigroup setting

Numbers of zeros on invariant manifolds in reaction-diffusion

equations

191

du ;i; + Au =Au) in a Banach space X with norm 1.1,where A is sectorial linear X-* X fi C!- X is C’. where LI is a neighborhood of OinX”,~~l,OBcu<1;f(O)=O. Let L: =A -f(O) have spectrum o(L). By u(t; ua) we denote the solution of (4.1) with initial data ~(0; ~0) = The following lemma is well known in the finite dimensional case. It replaces [4, lemma 5.1 and corollary 5.1, chapter IX] in the proof of the infinite dimensional version of lemma 2.2. Its proof is modelled in close analogy to [5, theorem 5.2.11. Nevertheless, for the convenience of the reader we give a detailed proof. LEMMA4.1. Assume y > 0 is such that a(L) = u, U u2, u, = a(L) n {Re A < y}, uz = u(L) n {Re E. > y} is a decomposition of u(L) into spectral sets. Let X = X, @ X2 be the decomposition of X corresponding to the decomposition of a(.C,) and let Et and El be the spectral projections onto X, and Xr respectively, Et $ E2 = I. Then there exist p > 0, M > 0 and a local invariant C” submanifold S of the ball {]ul. < p/2M} such that: (if 5 is Cz diffeomorphic under E2/s to an open neighborhood of 0 in X; : = X2 17 X”; (ii) S is tangent to XT at 0; (iii) if 1E2u(0)lo < p/2M and lu(f)/ ,erCpforallr~Othenu(0)ES; (iv) if u(O) E S then sup ]u(r)]. e” < 0. r10 Proof Without loss of generality assume u(A) C {Re h > O}. By Lt. Lz denote the restrictions of L to Xi, Xr respectively, let T,(t) : = exp( -+L.,t) be the semigroup on X, generated by L, and u,..= E,u the X,-component of u. Note that dim X, < =J, L, is bounded and there exist 0 < B < y < 6 such that IT,(r)] S Me-p, IAT~(~)&A-*~

IA”T,(t)j

5 Mee8’

5 Mee6’, IAT,(

fortSO,

5 Mr-“e-6’

for f 2 0.

(4.2)

Write g(u) :=f(u) -f’(O)u with components g,:= Eg. Then there exists a positive function k on (0, po), p. > 0 such that k(p) + 0 for p+ 0 and Mu’) - g(uZ)I 5 k(P) Iul - f121d as soon as ]u’]. < p, j = 1.2. By [5, lemma 3.3.21, u(r) solves (4.1) iff u(t) solves the variation of constants version of (4.1)

UlO)= ~,Ob,@)+

IfT,O- Sk, (44) b

(4.1)’

0

u20)

=

~2Oh*@)

+

t T2(t I 0

-

dg2(44)

d.T.

Assuming that the solution u(f) satisfies

I49l. e”isboundedast+-r,

(4.3)

we conclude that for f+ = I~~(-r)~,(r)/,~~e~l~~(r)lD~O which implies u,(O) = -

* T,(-r)gr(u(r)) I0

dr,

and, again by (4.1)‘, we obtain u(r) = T,(t)a + I’ T*(r 0

dg,(uW)

d.7 -

I-

T,O

,

-

4g*(uW)

d.f

(4.4)

where n : = E,u(O) E X2. We show that for p > 0 sufficiently small integral equation (4.4) has a unique solution u,(f) satisfying lu,(r)l&’ < P provided 101,< p/2M.

192

P. BRL’NOVSKY and B. FIEDLER

Let R, be the set of continuous functions u: [0, =) + X” such that Ilu(.)il:=supiu(t)l.er~p I10

is finite. The set R, endowed with the metric generated by Ij.11ISa complete metric space. We claim that for p small enough and ]/u/(, C p/ZM, a E X5 the map F, defined by F.(u(.))

(r) := T*(+J + j’ 120 - r)gz(n(s)) ~ - I’ T,(r - r)g,(u(r)) 0 I is a contraction R,,-) R,. Indeed ]]FO(u(.))]I d sup e”l Tr(t)a], + sup LZO

+

sup rs”

110

= e”lA”T,(f

’ e7’lAeT2(t

j0

- s)l.jg,(u(s))l

I,

5 Mjal, + lEzl sup

,zo I’0

- I)/.ig2(u(s))i

b

ds

ds

e”M(f - ,)-“e-6(‘-r)k(P)~u(s)/=

d.s

r +

IE, I sup i er’iMe-B(i-‘lk(p)lu(S)I, ds rzo ,

5

Mlal, + IE,I.%f.k(p)

(4.5)

1’ reme(r-@ dt.l]u(.)]] 0

with some constant C independent of p. Thus, if ]alo < p/2M and p > 0 is small enough that k(p). C < p/2M, then F, maps R, into R,. Also, repeating the same steps as in (4.5) we find

llF.(u’(.)) - FG,(U?(~))llL5 ~llU’(~)- u?(.)ll as soon as ]]uj(.)]I S p, j = 1,2, so F. is a contraction in R,. Consequently, F. has a unique fixed point u( .) E R, which solves (4.4). The map (u( .), u) + F.(u( .)) is C” on R, x ({Iaim < p/2M} fl XT). Indeed, the map is linear in a and estimating as in (4.5) one obtains

;~fe~l~-‘(~du(.) + EU(.))(~ - F,(u(.))W) - j’ T2(f - ~)&(4S)MJ) ds+ j= T,(r - Sk; (u(s)Ms)4, + 0 0

for E-+ 0.

i

Therefore (u(.), b)+

Tz(r)b + j’ T,(r - s)g;(u(s))u(r) 0

ds - I’ T,(r - s)g; (u(s))+)

dr

(4.6)

,

is the Glteaux differential of the map u( .), a) -t F,(u( .)). Since the map (4.6) is continuous in (u( .), b), the differential is Frechet and (u( .), a) + F, I u( .)) is Cr. To obtain C” we iterate the arguments above. By [5, 1.2.61 the fixed point u,(.) of F. is a C”-function of a in {la]. < p/2M} n Xp. Consequently the map h: {Ial* < p/2M} fl Xp +X, defined by h(a) := u,(O) = 0 - j‘ r,(-S)gt(u.(s))

d.r

0

is C” and, since E&a)

= E2a = a, has a C” inverse on its image 5. Thus, h: {]a]= c p/2M} n

xf 4x,

is a C”-diffeomorphism. This proves (i) and, using g;(O) = 0, as a direct consequence (ii). By definition of R,, (iv) holds. By construction and (4.4). ?I IS mvanant wttn respect to the semmow (4.1). It Ibzu(U)la < p/21M and iu(f)lner < P

Numbers of zeros on invariant manifolds in reaction-diffusion

equations

193

for all t Z 0, then we have shown that u( .) satisfies (4.4). Since u( .) E R, and u(t) = u,(r) with a := E>(O) we conclude # u(O) E S. Thus (iii) holds and the proof is complete. Proof of lemma 3.1. Existence of the manifolds W, as claimed in lemma 3.1 follows from lemma 4.1, choosing y with &_, < y < E.,. Using existence of the manifolds W,, we apply the proof of lemma 2.2 successively for each k on a neighborhood U of u := 0 (w.1.o.g.) in W,, with coordinates y = E,u and x = 2 E,u as in the notation of Section 2. Note that the /‘k proof of lemma 2.2 carries over to analytic semigroups without the assumption that x is finite dimensional. Now lemma 2.2, together with u(r) = S,(w) + 0 and lemma 4.1, (ii) imply

and the proof is complete. ‘Acknowledgemenrs-The

n

authors are indebted to W. Alt for helpful criticism.

REFERENCES 0.

1. 2. 3. 4. 5.

ATKINSONF. V., Dticrere and Continuous Boundary Problems, Academic Press (1964). BRL‘NOVSKT P. & FIEDLERB., Simplicity of zeros in scalar parabolic equations, preprint (1984). H~\LE J. K., Topics in Dynamic Bifurcation Theory, CBMS Regional Series in Mathematics No. 47, Am. Math. Sot., Providence, RI (1981). HALE J. K. & DO NASCIMENTO A. S., Orbit connections in a parabolic equation, preprint. HAR~~AN P., Ordinary Differenriu[ Equations, 2nd edition, Birkhluser, Boston (1982). HENRYD., Geometric theory of semilinear parabolic equations, Lecture Notes in Marhemarics 840, Springer, Berlin (1981). MATANOH., Nonincrease of the lap number of a solution for a one dimensional semilinear parabolic equation, Pub. Fuc. Sci. Uniu. Tokyo Sec. lA, 29, 401-441 (1982). PROTTERM. gi WEINBERGERH., Murimum Principles in Diflerential Equntions, Prentice Hall, Englewood Cliffs, NJ (1967). REDHEFFER R. M. & WALTER W., The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains, Muth. Ann/n 209, 57-67 (1974). SMOLLERJ., Shock Waves and Reaction Diffiion Equations, Springer, New York (1983).