Proc. Roy. Soc. Edinburgh, 129A, 1-45, 1999

STABILITY ANALYSIS IN ORDER-PRESERVING SYSTEMS

IN THE PRESENCE OF SYMMETRY

TOSHIKO OGIWARA AND HIROSHI MATANO

GRADUATE SCHOOL OF MATHEMATICAL SCIENCES

UNIVERSITY OF TOKYO

KOMABA, TOKYO 153, JAPAN

Given an equation with certain symmetry such as symmetry with respect to rotation or

translation, one of the most fundamental questions to ask is whether or not the symmetry of

the equation is inherited by its solutions. We �rst discuss this question in a general framework

of order-preserving dynamical systems under a group action and establish a theory concerning

symmetry or monotonicity properties of stable equilibrium points. We then apply this general

theory to nonlinear partial di�erential equations. Among other things we prove rotational

symmetry of solutions for a class of nonlinear elliptic equations and monotonicity of travelling

waves of some nonlinear di�usion equations. We also discuss the stability of stationary or

periodic solutions for equations of surface motion.

Contents

1. Introduction 1

2. Notation and main results |time-discrete systems 4

3. Notation and main results |time-continuous systems 11

4. Application (1) |Instability of periodic orbits 13

5. Application (2) - |Rotational symmetry of stable equilibria 14

6. Application (3) |Monotonicity of travelling waves 22

7. Application (4) - |Monotonicity of pseudo-travelling waves 30

8. Application (5) - |Generalized motion by mean curvature 34

Appendix |structure of totally-ordered sets 37

References 40

1. Introduction

Many mathematical models in physics, biology and other �elds possess some kind of

symmetry, such as symmetry with respect to re ection, rotation, translation, dilation,

gauge transformation, and so on. Given an equation with certain symmetry it is impor-

tant, from the point of view of applications, to study whether or not its solutions inherit

the same type of symmetry. In a typical mathematical setting, this question is formulated

as follows : Suppose that a group G acts on a space X and that a mapping F : X ! X is

G-equivariant, that is, F �g = g�F for everyg 2 G. Then can we say that solutions of the

equation F (u) = 0 are G-invariant? As is well-known, the answer is generally negative

1

unless we impose additional conditions on the equation or on the solutions. We will hence-

forth restrict our attention to solutions that are `stable' in a certain sense and discuss the

relation between stability and symmetry, or stability and some kind of monotonicity.

In the area of nonlinear di�usion equations or heat equations, early studies in this

direction can be found in Casten-Holland [2] and Matano [19]. Among many other

things, they showed that if a bounded domain is rotationally symmetric then any

stable equilibrium solution of a semilinear di�usion equation

@u

@t

= �u+ f(u); x 2 ; t > 0

inherits the same symmetry. Later it was discovered that the same result holds in a

much more general framework, namely that of `strongly order-preserving systems'. This

is a class of dynamical systems for which the comparison principle holds in a certain

strong sense, whose concept was introduced in [13], [20] (see also [32], which gives a

comprehensive survey on early developments of this theory). Mierczy�nski{Pol�a�cik [24]

(for the time-continuous case) and Tak�a�c [33] (for the time-discrete case) considered

strongly order-preserving dynamical systems with a symmetry property associated with

a compact connected group G and showed that any stable orbit has a G-invariant !-limit

set. This, in particular, implies that any stable equilibrium point or stable periodic point

is G-invariant.

The aim of this paper is to establish a theory analogous to [24] and [33] for a wider class

of systems. To be more precise, we will relax the requirement that the dynamical system

be strongly order-preserving. This will allow us to deal with degenerate di�usion equations

and equations on an unbounded domain. Secondly, we will relax the requirement that

the acting group G be compact. This will allow us to discuss symmetry or monotonicity

properties with respect to translation ; the results will then be applied to the stability

analysis of travelling waves of reaction-di�usion equations and to equations of curvature-

dependent motion of surfaces (see Sections 6 and 8).

Note that our results are concerned with properties of stable equilibrium points and

periodic points whereas [24] and [33] deal with more general stable points. Nonetheless,

our results extend the applicability of the theory to problems of much more variety.

Furthermore, our theory only requires that the equilibrium points and periodic points

be `G-stable', whose meaning will be de�ned later. This will allow us to discuss the

properties of orbitally stable (but not necessarily stable) periodic orbits or travelling

waves (see Sections 4 and 6).

Our paper is organized as follows. In Sections 2 and 3, we present notation and assump-

tions and study properties of stable equilibrium points. We treat time-discrete systems

in Section 2 and time-continuous systems in Section 3.

In Sections 4{8, we apply our results to nonlinear partial di�erential equations and

related problems. In Section 4 we prove the instability of periodic orbits in an order-

preserving semi ow (Theorem 4.2). A similar result was �rst obtained by Hirsch [13], but

our assumption is much milder. The result follows as a direct consequence of Theorem B

of Section 2.

In Section 5 we consider an initial boundary value problem for a nonlinear di�usion

equation of the form

@u

@t

= �(u

m

) + f(u); x 2 ; t > 0; (1.1)

2

where m � 1 is a constant and � R

N

is a rotationally symmetric domain that is not

necessarily bounded. We show that any stable equilibrium solution of (1.1) satisfying

suitable conditions is rotationally symmetric (Theorems 5.1, 5.5 and 5.7). The result has

been known for the case where m = 1 and the domain is bounded ([12],[19],[24],[33]),

but our general theory allows us to relax these restrictions considerably.

In Section 6 we apply our theory to so-called travelling waves for an equation of the

form

@u

@t

=

@

2

u

@x

2

+ f

�

u;

@u

@x

�

; x 2 R; t > 0;

or a system of equations of the form

8

>

>

>

>

>

<

>

>

>

>

>

:

@u

1

@t

= d

1

@

2

u

1

@x

2

+ f

1

�

u

1

; � � � ; u

m

;

@u

1

@x

�

; x 2 R; t > 0;

.

.

.

@u

m

@t

= d

m

@

2

u

m

@x

2

+ f

m

�

u

1

; � � � ; u

m

;

@u

m

@x

�

; x 2 R; t > 0;

where constants d

1

, � � � , d

m

are positive and functions f

1

, � � � , f

m

satisfy certain conditions

so that the system is of the cooperation type or of the competition type. (We assume

m = 2 in the latter case.) A solution u(x; t) is called a travelling wave if it is written in

the form

u(x; t) = �(x� ct)

for some constant c 2 R. Here we deal with travelling waves whose limiting values

lim

z!�1

�(z) = u

�

are both stable zeros of f(u; 0). Using Theorem B

0

, we will show that any stable (or

orbitally stable) travelling wave is monotone both in x and t (Theorems 6.1 and 6.7). As

a consequence, any solitary wave is unstable (Corollaries 6.2 and 6.8).

With minor modi�cations, our results extend to travelling waves for equations in higher

space dimensions such as

8

>

<

>

:

@u

@t

= �u+ f (x

1

; � � � ; x

N�1

; u) ; x 2 ; t > 0;

@u

@n

= 0 x 2 @; t > 0;

where is a cylindrical domain of the form = D�R with D being a bounded (N � 1)-

dimensional domain. A solution u(x; t) is called a travelling wave if it is written in the

form

u(x; t) = �(x

1

; � � � ; x

N�1

; x

N

� ct):

We consider travelling waves whose limiting pro�les

lim

z

N

!�1

�(z

1

; � � � ; z

N�1

; z

N

) = u

�

(z

1

; � � � ; z

N�1

)

are stable in a certain sense. We show that any stable (or orbitally stable) travelling

wave is monotone in the axial direction (Theorem 6.11). Moreover these travelling waves

inherit the symmetry properties of D provided that its symmetry group is connected.

3

In Section 7 we deal with travelling waves in temporally or spatially periodic media.

More precisely, we consider an initial value problem for the equation

@u

@t

= a(t)

@

2

u

@x

2

+ b(t)

@u

@x

+ f(t; u); x 2 R; t > 0; (1.2)

and one for the equation

@u

@t

= �(x)

@

2

u

@x

2

+ �(x)

@u

@x

+ g(x; u); x 2 R; t > 0; (1.3)

where functions a, b, f are T -periodic with respect to t while �, �, g are L-periodic with

respect to x. A nonconstant solution u(x; t) for (1.2) is called a pseudo-travelling wave if

there exists a � 2 R such that

u(x; t+ T ) = u(x� �; t); x 2 R; t 2 R;

and one for (1.3) is called a pseudo-travelling wave if

u(x; t+ �) = u(x� L; t); x 2 R; t 2 R

for some � 6= 0. Under suitable conditions on f , we show that any stable pseudo-travelling

wave for (1.2) is either monotone increasing in x or monotone decreasing in x (Theorem

7.1). Similarly any stable (or orbitally stable) pseudo-travelling wave for (1.3) is monotone

in t (Theorem 7.3). The same results hold for problems in higher dimensions. A more

extensive treatment of pseudo-travelling waves will be given in the forthcoming paper

[27].

Finally, in Section 8, we discuss an evolution equation for surfaces fS(t)g

t�0

embedded

in R

N

. The equation we consider is

V = f(n;rn; t); (1.4)

where n = n(x; t) is the outward unit normal vector at each point of S(t) and V denotes

the normal velocity of S(t) in the outward direction. We assume that f is T -periodic

in t. Under certain additional conditions on f , we prove that any T -periodic smooth

compact solution of (1.4) is unstable (Theorem 8.1). This implies, in particular, that any

smooth `compact' stationary (or T -periodic) surface for the generalized anisotropic mean

curvature ow

V = �(n; t)�+ g(n; t)

is unstable, where � denotes the mean curvature at each point of S(t) and �, g are

T -periodic functions with � > 0. Our theory also applies to a surface motion in a

cylindrical domainD�R � R

N

with appropriate boundary conditions. In the forthcoming

paper [27], we will show that any stable pseudo-travelling wave of (1.4) in D�R can be

expressed in the form of a graph x

N

= (x

1

; � � �x

N�1

; t), where (x

1

; � � � ; x

N�1

; t+ T ) =

(x

1

; � � � ; x

N�1

; t) + � for some � 2 R (See Remark 8.5).

2. Notation and main results |time-discrete systems

Let X be an ordered metric space, that is, a metric space with a closed partial order

relation denoted by �. Here, we say that a partial order relation in X is closed if u

n

� v

n

(n = 1; 2; 3; � � � ) implies lim

n!1

u

n

� lim

n!1

v

n

provided that both limits exist. We also

assume that, for any u, v 2 X, the greatest lower bound of fu; vg - |denoted by u^ v|

exists and that (u; v) 7! u ^ v is a continuous mapping from X �X into X. We write

u � v if u � v and u 6= v, and denote by d the metric of X. For a subset Y � X, the

4

expression u � Y (resp. u � Y , u � Y , u � Y ) means u � v (resp. u � v, u � v, u � v)

for all points v 2 Y . The whole argument in this and the next sections remains valid if

we assume, instead, that the least upper bound exists (see Remark 2.13).

Let F be a mapping from a subset D(F ) � X into X with the following properties

(F1), (F2), (F3) :

(F1) F is order-preserving (that is, u � v implies F (u) � F (v) for all u; v 2 D(F )),

(F2) F is continuous,

(F3) any bounded monotone decreasing orbit (a bounded orbit fF

n

(u)g

n=0;1;2;���

satisfying

u � F (u) � F

2

(u) � � � � ) converges.

Note that condition (F3) is satis�ed if F is a compact map. We will not, however,

require that F be compact.

In this paper F

n

will denote the identity mapping in the case n = 0 and the composition

mapping F � F � � � � � F

| {z }

n times

in the case n 2 N, and

D(F

n

) = fu 2 X j F

k

(u) 2 D(F ) for k = 0; 1; � � � ; n� 1g;

D(F

1

) =

1

T

n=1

D(F

n

):

Let G be a metrizable topological group acting onX. We say G acts on X if there exists

a continuous mapping : G � X ! X such that g 7! (g; �) is a group homomorphism

of G into Hom(X), the group of homeomorphisms of X onto itself. For brevity, we write

(g; u) = gu and identify the element g 2 G with its action (g; �). We assume that

(G1) is order-preserving (that is, u � v implies gu � gv for any g 2 G),

(G2) commutes with F (that is, gF (u) = F (gu) for any u 2 D(F ); g 2 G),

(G3) G is connected.

We say that an element u 2 X is symmetric if it is G-invariant, that is, gu = u for all

g 2 G. The set Gu = fgu j g 2 Gg is called a group orbit. Clearly u 2 X is symmetric if

and only if Gu = fug. In what follows, e will denote the unit element of G.

Example. The equation (1.1) for m = 1 together with the homogeneous Neumann

boundary condition @u=@n = 0 (on @) de�nes a local semi ow f�

t

g

t2[0;1)

on the space

X = C() if is a bounded domain (see Section 5). Fix t

0

> 0 and set F = �

t

0

. Then it

is not di�cult to show that (F1), (F2), (F3) hold. Now let the domain be a ball. Then

the group of rotations, G = SO(N), acts on , and this action induces a group action on

X by

(g; u) = u � g:

Conditions (G1){(G3) are easily veri�ed.

An element u 2 X is called a �xed point of F if F (u) = u. By condition (G2), if u is a

�xed point of F then so are all points in Gu.

In the rest of this section u will denote a �xed point of F satisfying the following

condition :

(E) for any �xed point u with u � u and with d(u; u) su�ciently small, there exists some

neighborhood B(e) � G of e such that gu � u for any g 2 B(e).

5

Remark 2.1. A map F is called strongly order-preserving if u

1

� u

2

implies F (v

1

) � F (v

2

)

for any v

1

, v

2

that are su�ciently close to u

1

, u

2

. It is easily seen that if F is strongly

order-preserving then any �xed point u satis�es (E). On the other hand, the converse is

not true. As we will see later, there are many important examples where F satis�es (E)

but is not strongly order-preserving.

Proposition 2.2. One of the following holds :

(a) Gu = fug ;

(b) Gu is a totally ordered set and has no maximum nor minimum ;

(c) Gu 6= fug, and no pair of points w

1

, w

2

2 Gu satisfy w

1

� w

2

or w

1

� w

2

.

In the case (c), any �xed point v with v � u satis�es Gv � u.

De�nition 2.3. A �xed point u 2 X of F is called stable if, for any " > 0, there exists

some � > 0 such that

d(v; u) < � =) v 2 D(F

1

); d(F

n

(v); u) < " for any n = 0; 1; 2; � � � :

(2.1)

It is called G-stable if, for any " > 0, there exists some � > 0 such that

d(v; u) < � =) v 2 D(F

1

); d(F

n

(v); Gu) < " for any n = 0; 1; 2; � � � :

(2.2)

Needless to say, stability implies G-stability.

Theorem A. Let u be stable. Then either of the following alternatives holds :

(a) Gu = fug, that is, u is symmetric.

(b) Gu is a connected, totally-ordered set having neither the maximum nor the minimum.

Remark 2.4. As will be clear from its proof, Theorem A remains true if we relax condition

(F2) as follows :

(F2

�

) F is upper semicontinuous (that is, if a sequence fu

n

g

n=1;2;3;���

in D(F ) converges to a

point u

1

2 D(F ) and if the corresponding sequence fF (u

n

)g

n=1;2;3;���

also converges

to some point w 2 X, then w � F (u

1

)).

This weaker condition will be used when we deal with equations of surface motion (see

Section 8).

Remark 2.5. Theorem A remains true if we replace conditions (F2), (F3) by the following :

(F4) for any bounded monotone decreasing orbit fF

k

(u)g

k=1;2;3;���

there exists some �xed

point v of F such that

v � F

k

(u) for any k = 1; 2; 3; � � � ; d(v; u) � C lim sup

k!1

d(F

k

(u); u);

where C > 0 is a constant independent of u.

Condition (F4) (or (�4) which will be de�ned in the next section) is ful�lled if a bounded

decreasing orbit is known to converge in an appropriate weak sense. See Lemma 5.10 and

Remark 5.11 for details. We will use this weaker condition to discuss equations on an

unbounded domain.

6

In most application problems it is easily veri�ed that any group orbit Gu is locally

precompact. Here a subset Y � X is called locally precompact if for every w 2 Y there

exists a bounded open set U � X containing w such that the closure of U \Y is compact.

In such a case, it su�ces to assume that u is G-stable. More precisely the following holds :

Theorem B. Let u be G-stable and assume that Gu is locally precompact. Then either

of the following alternatives holds :

(a) Gu = fug, that is, u is symmetric.

(b) Gu ' R, or, more precisely, there exists an order-preserving homeomorphism from

Gu onto R.

The assertion remains true if condition (F3) is replaced by (F4).

The above theorem will play a useful role in the study of orbitally stable travelling

waves and orbitally stable periodic orbits.

As an immediate consequence of Theorem B (or Theorem A) we have the following

corollary.

Corollary C. Let u be stable (or G-stable) and assume that G is a compact group. Then

u is G-invariant, that is, u is symmetric. The assertion remains true if condition (F3) is

replaced by (F4).

Proof. By the assumption that G is compact, Gu is a compact subset of X. Since R is

not compact, alternative (b) in Theorem B does not hold, hence Gu = fug.

Remark 2.6. As will be clear from the proof of Theorem A or B, the group G need not

act on the whole space X; it only needs to act on the set of �xed points of F , and all the

results in this section remain unchanged in this weaker setting.

Remark 2.7. We do not need to assume the continuity of the group action G on X; we

simply need to assume that u 7! gu is continuous in u 2 X for every g 2 G and that

g 7! gu is continuous in g 2 G. Furthermore, as will be easily seen, we only have to

assume that conditions (F3) and (F4) hold for points u su�ciently close to u. More

precisely, (F3) and (F4) need to hold for u such that fF

k

(u)g

k=1;2;3;���

stays su�ciently

close to u. This weaker version of (F4) will be used in the study of symmetry properties

of elliptic equation in an unbounded domain (Theorem 5.7).

Remark 2.8. In the same spirit, we can also relax the assumption on the continuity of

the mapping (u; v) 7! u ^ v : X � X ! X. In fact, as will be clear from the proof of

Theorem A, it su�ces to assume that the mapping v 7! u ^ v : X ! X is continuous at

v = u, in other words, that v

m

! u implies u ^ v

m

! u (as m ! 1). In addition, we

need only require that condition (E) holds for all �xed points u such that u � u ^ hu

for some h 2 G with hu 6= u. This remark will be used in the proof of Theorem 8.1 in

Section 8, where we discuss instability of compact surfaces in a generalized motion by

mean curvature.

Now let us prove Proposition 2.2 and Theorems A, B. We begin with the following

lemma :

Lemma 2.9. De�ne

G

0

= fg 2 G j gu = ug;

G

�

= fg 2 G j gu � u or gu � ug;

G

�

= fg 2 G j gu 6� u and gu 6� ug:

7

Then the subset G

0

is closed, G

�

and G

�

are open.

Proof. From the de�nition it is easily seen that G

0

is a closed subset and G

�

an open

subset of G. Moreover condition (E) implies that G

�

is also open.

Lemma 2.10. Let G

0

, G

�

and G

�

be as in Lemma 2.9. Then one of the following holds :

(a) G = G

0

;

(b) G

�

6= ; and G = G

0

[G

�

with G

0

= @G

�

;

(c) G

�

6= ; and G = G

0

[G

�

with G

0

= @G

�

.

In the last case, any �xed point v with v � u satis�es Gv � u.

Proof. G

0

, G

�

, G

�

are mutually disjoint and

G = G

0

[G

�

[G

�

:

We �rst assume that G

�

6= ;. Then by the connectedness of G we have

@G

�

6= ;:

Since both G

�

and G

�

are open, we have @G

�

� G

0

. This means that there exists an

element h

0

2 @G

�

\G

0

. Now let g

0

be any element of G

0

and B(g

0

) be any neighborhood

of g

0

. Then, since h

0

g

�1

0

B(g

0

) = fh

0

g

�1

0

g j g 2 B(g

0

)g is a neighborhood of h

0

, we have

h

0

g

�1

0

B(g

0

) \G

�

6= ;:

It follows from this and g

0

h

�1

0

2 G

0

that

B(g

0

) \ G

�

6= ;:

This shows that @G

�

= G

0

. Now let v be any �xed point of F satisfying v � u. By

condition (E), it holds that v � g

�

u for some g

�

2 G

�

. De�ne

A = fg 2 G j gv � g

�

u; gv � ug; A

0

= fh 2 G j v � hug:

Since h 7! hu : G ! X is continuous, A

0

is a closed subset of G. In view of this and

the identity A = A

0

�1

\ g

�

A

0

�1

, we see that A is closed. (Here A

0

�1

stands for the set

fg

�1

j g 2 A

0

g.) On the other hand, since g

�

u and u are order-unrelated, neither of the

equality signs in the condition gv � g

�

u, gv � u can hold. Therefore

A = fg 2 G j gv � g

�

u; gv � ug:

It follows from this and (E) that A is also open. Thus by the connectedness of G we have

A = G, hence

Gv � u:

This proves the last statement of the lemma. We next show that G

�

= ;. Suppose that

G

�

6= ; and that there exists an element g 2 G

�

. By replacing g by g

�1

if necessary, we

may assume that gu � u. Applying the above result to v = gu, we see that Gv � u holds.

But this is impossible since

Gv = Ggu = Gu 3 u:

This contradiction shows G

�

= ;, verifying case (c).

Next we assume that G

�

6= ;. Then it follows from statement (c) that G = G

0

[ G

�

.

The assertion G

0

= @G

�

can be shown in the same manner as in (c). The lemma is

proved.

8

Lemma 2.11. The maximum of Gu exists if and only if Gu = fug. The same is true

for the minimum.

Proof. Suppose that g

0

u is the maximum of Gu. Then

gu � g

0

u for any g 2 G:

In particular, g

2

0

u � g

0

u, hence

g

0

u = g

�1

0

(g

2

0

u) � g

�1

0

(g

0

u) = u � g

0

u:

This shows that g

0

u = u, therefore u is the maximum of Gu. Consequently

g

�1

u � u for any g 2 G;

hence

u = g(g

�1

u) � gu � u:

This implies that Gu = fug. The same argument applies if Gu has the minimum. The

lemma is proved.

Proof of Proposition 2.2. Let G

0

, G

�

, G

�

be as in Lemma 2.9. Suppose that there exist

g

1

, g

2

2 G such that

g

1

u � g

2

u:

Then u � g

�1

1

g

2

u, hence g

�1

1

g

2

2 G

�

. Therefore the existence of a strictly ordered

pair of points w

1

� w

2

in Gu is equivalent to the condition G

�

6= ;. In view of this and

Lemma 2.10, we �nd that G

�

6= ; implies case (c) in Proposition 2.2. The last statement

of the proposition also follows from Lemma 2.10. On the other hand, if G

�

= ;, then Gu

is clearly a totally ordered set. The alternatives (a), (b) now follows immediately from

Lemma 2.11.

To prove Theorem A, we need the following lemma :

Lemma 2.12. Let u 2 D(F

1

) satisfy F (u) � u, and assume that the sequence fF

n

(u)g

n=0;1;2;���

is bounded in X. Then F

n

(u) converges to some point v 2 X as n ! 1. If v 2 D(F ),

then v is a �xed point of F .

Proof. Since assumption (F1) and F (u) � u imply

u � F (u) � F

2

(u) � F

3

(u) � � � � ;

it follows from (F3) that the sequence fF

n

(u)g

n=0;1;2;���

converges as n ! 1 to a point,

say v.

Next assume that v 2 D(F ). Then, by (F2) we have

F (v) = F ( lim

n!1

F

n

(u)) = lim

n!1

F

n+1

(u) = v:

Hence v is a �xed point of F .

Proof of Theorem A. Supposing that case (c) in Proposition 2.2 holds, we will derive a

contradiction. Since G is connected, the set Gu � X is connected. From this fact and

Gu 3 u, there exists a sequence fg

m

ug

m=1;2;3;���

� Gu converging to u and satisfying

g

m

u 6� u, g

m

u 6� u for all m 2 N. The inequalities

g

m

u ^ u � g

m

u; g

m

u ^ u � u

9

and assumption (F1) yield

F (g

m

u ^ u) � F (g

m

u) ^ F (u) = g

m

u ^ u � u:

Because of the stability of u, we can choose fg

m

ug

m=1;2;3;���

such that the closure of the

sequence fF

n

(g

m

u ^ u)g

n=1;2;3;���

is contained in D(F ) and is bounded for each m 2 N.

Then it follows from Lemma 2.12 that fF

n

(g

m

u ^ u)g

n=1;2;3;���

converges to some �xed

point of F , which we will denote by v

m

. By the last statement of Proposition 2.2, for any

g 2 G and m 2 N,

gv

m

� u (2.3)

holds. Since u is stable and g

m

u ^ u converges to u as m ! 1, its !-limit point v

m

converges to u as m!1. Letting m!1 in (2.3), we obtain

gu � u for all g 2 G;

which contradicts our assumption that (c) holds. Thus either (a) or (b) in Proposition

2.2 holds, and the theorem is proved.

Proof of Theorem B. Supposing case (c) in Proposition 2.2, we will derive a contradiction.

Let fg

m

ug

m=1;2;3;���

be as in the proof of Theorem A. Put u

m

= g

m

u ^ u. If

lim inf

m!1

sup

k

d(F

k

(u

m

); u) = 0;

then repeating the same argument as in the proof of Theorem A, we obtain a contradiction.

Thus we only need to consider the case where there exists an "

0

> 0 such that

sup

k

d(F

k

(u

m

); u) > "

0

for m = 1; 2; 3; � � � :

Since the mapping F is continuous, if we choose a �

0

2 (0; "

0

) su�ciently small then

d(w; u) < �

0

implies d(F (w); u) < "

0

:

By taking a subsequence if necessary we may assume without loss of generality that

d(u

m

; u) < �

0

. For each m we set

k(m) = minfk 2 N j d(F

k

(u

m

); u) > �

0

g;

w

m

= F

k(m)

(u

m

):

Then

w

m

� u; �

0

< d(w

m

; u) < "

0

: (2.4)

Since u is G-stable, d(w

m

; Gu) ! 0 as m ! 1. Hence there exists some h

m

2 G such

that

d(w

m

; h

m

u)! 0 as m!1: (2.5)

It follows from (2.4) and (2.5) that fh

m

ug

m=1;2;3;���

is bounded. By the local precompact-

ness of Gu, there exists a subsequence fh

m

j

ug

j=1;2;3;���

that converges to some point z

"

0

.

From this and (2.5), we see that fw

m

j

g

j=1;2;3;���

also converges to z

"

0

. Letting m

j

!1 in

(2.4), we get

z

"

0

� u; �

0

< d(z

"

0

; u) < "

0

: (2.6)

10

Furthermore, since each h

m

j

u is a �xed point of F and since F is continuous, the limit

z

"

0

is also a �xed point. Hence by the last statement of Proposition 2.2, it holds that

Gz

"

0

� u:

Combining this with (2.6) and letting "

0

! 0, we get Gu � u, and equivalently Gu � u.

Thus Gu = u, yielding a contradiction. Therefore either (a) or (b) in Proposition 2.2

must hold. The conclusion (a) of this theorem follows from (a) in Proposition 2.2. The

conclusion (b) follows from (b) in Proposition 2.2 and Proposition Y2, which we will give

in Appendix. The proof is completed.

Remark 2.13. It is clear that the results in this section remain true if we reverse all the

order relations that appear in the hypotheses (the greatest lower bound u ^ v by the

least upper bound u _ v, `upper semicontinuous' by `lower semicontinuous', and so on).

However, the assumption that u be stable or G-stable is essential. For example, let

X = fu 2 C(S

1

) j u(x) � 0 for all x 2 S

1

(= R=Z)g;

G = fg

�

j � 2 S

1

g ' S

1

;

where g

�

u(x) = u(x � �) for u 2 X, x 2 S

1

. We de�ne the metric in X by d(u; v) =

ku � vk

L

1

(S

1

)

and introduce an order structure in X by the point-wise order relation

v(x) � u(x) (x 2 S

1

). Now de�ne F : X ! X by

F (u) =

�

Z

S

1

u(s)ds

�

u:

Then conditions (F1){(F3) and (G1){(G3) are all ful�lled. In this case any function u(x)

with

Z

S

1

u(s)ds = 1 is a �xed point of F satisfying condition (E), but the conclusion of

Theorem A or B does not hold unless u(x) � 1.

Remark 2.14. If G is not connected, then the conclusion of Theorem A or B does not

necessarily hold. For example, let X = R

2

and de�ne an order relation in X by

(u

1

; u

2

) � (v

1

; v

2

) if u

1

� v

1

; u

2

� v

2

:

Let F ((u

1

; u

2

)) = (

p

2u

1

=

p

1 + u

1

2

;

p

2 u

2

=

p

1 + u

2

2

) andG = fe; gg, where e : (u

1

; u

2

) 7!

(u

1

; u

2

), g : (u

1

; u

2

) 7! (u

2

; u

1

). Both u = (1;�1) and u = (�1; 1) are stable �xed points of

F , but Gu = f(1;�1); (�1; 1)g. A more nontrivial counterexample can be found in a non-

linear elliptic boundary value problem on a dumbbell-shaped domain with Z

2

-symmetry

([19], [23]). See also [33] for further details.

3. Notation and main results |time-continuous systems

With minor modi�cations, Theorems A, B and Corollary C carry over to time-continuous

systems. To be more precise, let f�

t

g

t2[0;1)

be a family of mappings �

t

from a subset

D(�

t

) � X to X that satis�es the semigroup property :

D(�

t

) is monotone non-increasing in t; and D(�

0

) = X;

�

0

(u) = u for all u 2 X;

�

t

1

� �

t

2

= �

t

1

+t

2

for any t

1

; t

2

2 [0;1):

For u 2 X, we call the set f�

t

(u) j t 2 [0;1)g an orbit and denote it by O

+

(u).

11

We assume that

(�1) �

t

is order-preserving for each t 2 [0;1),

(�2) �

t

(u) is continuous in u for each t 2 [0;1),

(�3) any bounded monotone decreasing orbit (a bounded orbit O

+

(u) satisfying �

t

(u) �

�

s

(u) for t < s) converges,

and that the group G satis�es (G1), (G3) and

(G2

0

) commutes with �

t

for each t 2 [0;1) (that is, g�

t

(u) = �

t

(gu) for each g 2 G,

u 2 D(�

t

), t 2 [0;1)).

A point u 2 X is called an equilibrium point if it satis�es �

t

(u) = u for all t 2 [0;1).

In the rest of this section u will denote an equilibrium point satisfying the following

condition :

(E

0

) for any equilibrium point u with u � u and with d(u; u) su�ciently small, there

exists some neighborhood B(e) � G of e such that gu � u for any g 2 B(e).

Remark 3.1. A semigroup f�

t

g

t2[0;1)

is called strongly order-preserving if the map �

t

is strongly order-preserving (see Remark 2.1) for every t > 0. It is easily seen that if

f�

t

g

t2[0;1)

is strongly order-preserving then any equilibrium point u satis�es (E

0

). The

converse is not true.

As in De�nition 2.3, an equilibrium point u 2 X of f�

t

g

t2[0;1)

is called stable if, for

any " > 0, there exists some � > 0 such that

d(v; u) < � =) v 2 D(�

1

); d(�

t

(v); u) < " for any t 2 [0;1); (3.1)

where D(�

1

) =

\

t2[0;1)

D(�

t

). It is called G-stable if, for any " > 0, there exists some

� > 0 such that

d(v; u) < � =) v 2 D(�

1

); d(�

t

(v); Gu) < " for any t 2 [0;1): (3.2)

The following are the main results for the time-continuous case :

Theorem A

0

. Let u be stable. Then either of the following alternatives holds :

(a) Gu = fug, that is, u is symmetric.

(b) Gu is a connected, totally-ordered set having neither the maximum nor the minimum.

As we remarked for Theorem A in Section 2, we can replace conditions (�2), (�3) by

(�4) for any bounded monotone decreasing orbit f�

t

(u)g

t2[0;1)

there exists some equilib-

rium point v such that

v � �

t

(u) for any t 2 [0;1); d(u; v) � C lim sup

t!1

d(u;�

t

(u));

where C > 0 is a constant independent of u.

Theorem B

0

. Let u be G-stable and assume that Gu is locally precompact. Then either

of the following alternatives holds :

(a) Gu = fug, that is, u is symmetric.

(b) Gu ' R, or, more precisely, there exists an order-preserving homeomorphism from

Gu onto R.

12

The assertion remains true if condition (�3) is replaced by (�4).

Corollary C

0

. Let u be stable (or G-stable) and assume that G is a compact group. Then

u is G-invariant, that is, u is symmetric. The assertion remains true if condition (�3) is

replaced by (�4).

The proof of these results is almost identical to that of Theorems A, B and Corollary

C except that one has to replace the notion of �xed points by that of equilibrium points.

The details are omitted. The same remarks as Remarks 2.6 and 2.7 also apply to the

time-continuous systems.

4. Application (1) |Instability of periodic orbits

In this and the following four sections, we present applications of the results in Sections

2 and 3. The �rst application, which we give in this section, is concerned with periodic

orbits of semi ows in rather an abstract setting.

Let f�

t

g

t2[0;1)

be a semigroup of mappings as in Section 3. We assume that �

t

(u) is

continuous in t as well as in u. Such a semigroup of mappings is called a local semi ow

on X. It is called a semi ow if we further have D(�

t

) = X for every t � 0. By de�nition,

any local semi ow satis�es (�2).

An orbit O

+

(u) = f�

t

(u) j t 2 [0;1)g is called a periodic orbit if there exists a � > 0

such that �

�

(u) = u. In this case the point u is called a periodic point or, more precisely,

a � -periodic point. Note that the quantity � need not be the minimal period in this

de�nition. A periodic orbit O

+

(u) is called a closed orbit if u is not an equilibrium point.

De�nition 4.1. A closed orbit O

+

(u) = f�

t

(u) j t 2 [0;1)g is called orbitally stable if

for any " > 0 there exists a � > 0 such that

d(v;O

+

(u)) < � =) d(�

t

(v); O

+

(u)) < " for any t 2 [0;1):

It is called stable if for any " > 0 there exists a � > 0 such that

d(v; u) < � =) d(�

t

(v);�

t

(u)) < " for any t 2 [0;1):

Clearly stability implies orbital stability.

We consider local semi ows satisfying the following condition :

(P) for any � > 0 and any � -periodic points u, v 2 X satisfying u � v, there exists a

� > 0 such that

�

t

(u) � �

s

(v) for any t; s 2 [0; �]:

We are now ready to state our main result of this section :

Theorem 4.2. Let f�

t

g

t2[0;1)

be a local semi ow satisfying conditions (�1), (�3) in

Section 3 and condition (P) above. Then any closed orbit is orbitally unstable (hence

unstable).

Proof. Let O

+

(u) be an orbitally stable closed orbit with period � . Denote by P

�

the set

of all � -periodic points of the semigroup f�

t

g

t2[0;1)

and let F = �

�

. Then fF

n

g

n=0;1;2;���

de�nes a discrete semigroup on X, and P

�

coincides with the set of �xed points of F . It

is easily seen that conditions (F1), (F2), (F3) in Section 2 are all ful�lled. Furthermore,

since each u 2 P

�

is a periodic point of f�

t

g

t2[0;1)

, �

t

(u) can be de�ned for all t 2 R

13

and we clearly have �

t

(P

�

) = P

�

for any t 2 R. Thus f�

t

g

t2[0;1)

is extended to a one-

parameter group acting on P

�

. Denote this group by G. Then conditions (G1), (G2),

(G3) can easily be checked. Conditions (P) and (�1) imply condition (E

0

). Furthermore u

is a G-stable �xed point of F such that Gu = O

+

(u) is a compact subset of X. Applying

Theorem B and Remarks 2.6, 2.7 we see that either of the following holds :

(a) Gu = fug ; (b) Gu ' R.

Since Gu is compact, case (b) is excluded. This means that u is an equilibrium point

of the semigroup f�

t

g

t2[0;1)

, contradicting the assumption that O

+

(u) is a closed orbit.

The theorem is proved.

Example. The above Theorem applies, for example, to semilinear parabolic equations

of the form

8

>

<

>

:

@u

@t

=

N

X

i;j=1

a

ij

(x)

@

2

u

@x

i

@x

j

+ f (x; u;ru); x 2 ; t > 0;

u = 0; x 2 @; t > 0;

where is a domain in R

N

. This result has been known if is a bounded domain (see

Hirsch [13]), but our theorem also covers the case where is unbounded, provided that

@f

@u

(x; 0; 0) � ��; x 2

for some � > 0.

5. Application (2) - |Rotational symmetry of stable equilibria

Some of the example in this section have already been discussed in Mierczy�nski{

Pol�a�cik [24] and Tak�a�c [33], but in view of their importance, we restate them (Theorems

5.1 and 5.4), and will indicate how our results improve on theirs.

Let G be a connected subgroup of the rotation group SO(N) and � R

N

be a G-

invariant domain with smooth boundary @. Here we say that a domain is G-invariant

if gx 2 for all x 2 , g 2 G. A typical example of such a domain is a disk or an annulus

in the case N = 2 ; a ball, a spherical shell, a solid torus or any other body of rotation in

the case N = 3.

First let us consider an initial boundary value problem for a single equation :

8

>

<

>

:

@u

@t

= �u+ f (x; u;ru); x 2 ; t > 0;

u = 0; x 2 @; t > 0;

u(�; 0) = u

0

; x 2 :

(5.1)

Here f (x; u; p) : � R � R

N

! R is a C

1

function satisfying a certain growth condition

to be speci�ed below. The stationary problem corresponding to (5.1) is

�

�u+ f (x; u;ru) = 0; x 2 ; (5.2a)

u = 0; x 2 @: (5.2b)

In order that problems (5.1) and (5.2) be G-invariant, we assume that

(f 1) f(gx; u; gp) = f(x; u; p) for every x 2 , u 2 R, p 2 R

N

, g 2 G.

14

For example, in the case where G = SO(N), condition (f 1) holds if f is written as

f = h(jxj; u; jruj; x � ru).

We discuss the problems (5.1), (5.2) in the space

X = C

0

() = fw 2 C() j w = 0 on @g

or in the space

X = C

0

() \ �

�

(); 0 < � < 1;

where �

�

() is the so-called `little H�older space of exponent �', which consists of functions

w 2 C

�

() satisfying

lim

�!0

sup

jx�yj<�

x;y2

jw(x)� w(y)j

�

�

= 0

and is endowed with the usual C

�

() norm. It is well-known that �

�

() coincides with

the closure of C

1

() |or, equivalently, that of C

1

()| in the space C

�

(). For problem

(5.1) to be well-posed in X, we assume the following growth condition on f :

(f 2)

�

there exist a constant 0 < � <

1

1� �

and a continuous function H(u) such that

�

�

�

�

@f

@u

(x; u; p)

�

�

�

�

< H(u)(1 + jpj

2�

); jr

p

f (x; u; p)j < H(u)(1 + jpj

�

)

for all x 2 , u 2 R, p 2 R

N

.

Here we set � = 0 if X = C

0

(). Condition (f 2)

�

(resp. (f 2)

0

) guarantees that (5.1) is

well-posed in X = C

0

() \ �

�

() (resp. X = C

0

()). This can be shown, for example,

by combining the general result of [35] with the standard estimate of the semigroup e

t�

in the space C

0

() \ �

�

() or C

0

() (see, for example, [25] for such estimates). The

stability of solutions of (5.2) will be discussed in the topology of X.

Let f�

t

g

t2[0;1)

be the local semi ow on X generated by (5.1). In other words, the map

�

t

on X is de�ned by

�

t

(u

0

) = u(�; t) for each t 2 [0;1); (5.3)

where u(x; t) is a solution of (5.1) with initial data u(�; 0) = u

0

. A function u(x) is a

solution of (5.2) if and only if it is an equilibrium point of f�

t

g

t2[0;1)

. We say that a

solution u(x) of (5.2) is stable if it is a stable equilibrium point of f�

t

g

t2[0;1)

. The action

of G on induces a group action on X by

g : u(x) 7! u(g

�1

x):

De�ne an order relation in X by

u

1

� u

2

if u

1

(x) � u

2

(x) for x 2 : (5.4)

Then the greatest lower bound of u ^ v exists for any u; v 2 X and is given by

u ^ v(x) = minfu(x); v(x)g; x 2 :

It is also easily seen that the operation (u; v) 7! u^ v is continuous if X = C

0

()\�

�

()

or X = C

0

(). (Note that this is not the case if the little H�older space �

�

() is replaced

by the standard H�older space C

�

(). The same remark applies to the continuity of the

map g : u(x) 7! u(g

�1

x)).

15

It follows from the maximum principle ([29]) that condition (�1) in Section 3 holds.

The standard parabolic estimate shows that (�2), (�3) are ful�lled. Clearly (G1), (G3)

are also ful�lled, and condition (f 1) implies (G2

0

). Further the strong maximum principle

shows that equation (5.1) forms a `strongly order-preserving dynamical system' (see [13],

[19], [32]). Hence, as we noted in Remark 3.1, every solution u of (5.2) satis�es condition

(E

0

). (See also Lemma 5.2 below for a more direct veri�cation of condition (E

0

).)

Applying Corollary C

0

, we obtain the following :

Theorem 5.1. Let be bounded. Then any stable solution u of (5.2) is G-invariant,

that is, u(gx) = u(x) for all x 2 , g 2 G.

The conclusion of Theorem 5.1 remains true if we replace the Dirichlet boundary con-

dition by the Neumann boundary condition

@u

@n

= 0 (5.5)

or the Robin boundary condition

@u

@n

+ �u = 0; (5.6)

where � > 0 is a positive constant. In these cases, we set X = C().

In most of the applications treated in this paper, much of the argument is devoted to the

veri�cation of condition (E

0

). The following lemma gives a direct veri�cation of (E

0

) for

problem (5.2) without using the strongly order-preserving property of (5.1). Variations

of this lemma will be used later in other applications.

Lemma 5.2. Let be a (possibly unbounded) domain in R

N

and let u, u be solutions of

(5.2) (resp. (5.2a), (5.5) or (5.2a), (5.6) ) satisfying

u(x) � u(x); u(x) 6� u(x); x 2 (5.7)

and let u

j

(x) (j = 1; 2; 3; � � � ) be solutions of (5.2) (resp. (5.2a), (5.5) or (5.2a), (5.6) )

converging to u(x) locally uniformly on as j ! 1. Then for any R > 0 there exists

j

0

2 N such that

u

j

(x) � u(x); x 2

R

; j � j

0

;

where

R

= fx 2 j jxj < Rg. If, in particular, is bounded, then u

j

(x) � u(x) on

for su�ciently large j.

Proof. First we consider (5.2). Assumption (5.7), the strong maximum principle and the

Hopf boundary lemma ([29], [14]) yield

u(x) < u(x); x 2 ; (5.8)

@u

@n

(x) >

@u

@n

(x); x 2 @: (5.9)

Here @v=@n(x) denotes the outward normal derivative of v at each point of x 2 @. Since

@ \

R

is compact, there exists � > 0 such that

@u

@n

(x) >

@u

@n

(x) + �; x 2 @ \

R

: (5.10)

16

Now, by the elliptic a priori estimate, u

j

! u locally uniformly on implies u

j

! u in

C

1

(

R

), hence

@u

j

@n

(x)!

@u

@n

(x) (as j !1) uniformly on @ \

R

:

The conclusion of the lemma now follows from this and (5.8), (5.10).

Next we consider (5.2a), (5.5) and (5.2a), (5.6). In these cases we have

u(x) < u(x); x 2

instead of (5.8), hence

u(x) + � < u(x); x 2

R

for some � > 0. Therefore the conclusion of the lemma obviously holds. The proof is

completed.

Remark 5.3. All the results in this section carry over to the Sobolev space setting : the

space C

0

() can be replaced by W

1;q

0

() and C() by W

1;q

()with q > 1, provided that

the growth condition (f 2)

0

is suitably modi�ed. No modi�cation is necessary if q > N .

Next we consider systems of equations of the form

8

>

<

>

:

@u

1

@t

= d

1

�u

1

+ f

1

(u

1

; u

2

); x 2 ; t > 0;

@u

2

@t

= d

2

�u

2

+ f

2

(u

1

; u

2

); x 2 ; t > 0;

(5.11)

where we assume either

@f

1

@u

2

> 0;

@f

2

@u

1

> 0 for a.e. (u

1

; u

2

) 2 R

2

(cooperation system)

or

@f

1

@u

2

< 0;

@f

2

@u

1

< 0 for a.e. (u

1

; u

2

) 2 R

2

(competition system):

We impose the boundary condition (5.2b), (5.5) or (5.6) on each of u

1

; u

2

. Under these

hypotheses, it is known that (5.11) de�nes a strongly order-preserving dynamical system

on the space X = C

0

()�C

0

() (for (5.2b)) or on the space X = C()�C() (for (5.5)

or (5.6)) ([13], [22], [32]). Therefore the following analogue of Theorem 5.1 holds :

Theorem 5.4. Let be bounded. Then any stable equilibrium solution of a cooperation

system or a competition system of the form (5.11) is G-invariant.

As mentioned earlier, Theorem 5.1 - |or more precisely its simpli�ed version| �rst

appeared in Casten{Holland [2] and in Matano [19]. Later Mierczy�nski{Pol�a�cik [24] and

Tak�a�c [33] treated these results in a more general framework of strongly order-preserving

dynamical systems.

In the rest of this section we deal with two problems to which the existing theory

of strongly order-preserving systems ([24], [33]) do not apply : the case of degenerate

di�usion and the case where the spatial domain is unbounded. We begin with the

following initial boundary value problem for a degenerate di�usion equation :

8

>

<

>

:

@u

@t

= �(u

m

) + f(x; u); x 2 ; t > 0;

u = 0; x 2 @; t > 0;

u(�; 0) = u

0

; x 2 ;

(5.12)

17

wherem > 1 is a constant, and the domain is as above. We assume that f : �[0;1)!

R is a C

1

function satisfying (f 1) and

(f 3.1) f(x; 0) = 0 for every x 2 ,

(f 4.1)

@f

@u

(x; 0) < 0 for every x 2 , or

@f

@u

(x; u) � 0 for every x 2 and every u 2 [0; "

0

],

where "

0

> 0 is some positive number.

Typical examples include f (u) = �u+ u

p

and f(u) = u

p

, where p > 1.

The equation in (5.12) is called degenerate, since the di�usion coe�cient mu

m�1

van-

ishes where u = 0. As is well-known, problem (5.12) does not in general possess a classical

solution. Thus we must consider (5.12) in the framework of weak solutions. A function

u(x; t) is called a (weak) solution of (5.12) if it has the following properties :

(i) u(x; t) is a nonnegative continuous function on D

T

0

= � [0; T

0

) for some T

0

> 0,

(ii) u(x; t) = 0 on @� [0; T

0

),

(iii)

ZZ

D

T

0

f(u(x; s))

m

��(x; s) + f(x; u(x; s))�(x; s) + u(x; s)�

t

(x; s)gdxds

= �

Z

u(x; 0)�(x; 0)dx

for every test function � 2 C

2;1

(D

T

0

) vanishing on @� [0; T

0

) and on � fT

0

g.

Given an equilibrium solution u of (5.12), we set

X = fu 2 C

0

() j u(x) � 0 in g

with the metric de�ned by d(u; v) = ku � vk

L

1

()

and the order relation by (5.4). It

is known that (5.12) generates a local semi ow on X, which we denote by f�

t

g

t�0

(see

[1], [30]). The following theorem holds :

Theorem 5.5. Let be bounded. Then any stable equilibrium solution of (5.12) is G-

invariant.

Proof. Let u be a stable equilibrium solution of (5.12). Obviously conditions (G1),

(G2

0

) and (G3) are ful�lled. Furthermore, it is known that (�1), (�2) hold (see [1]).

Condition (�3) follows from the Lebesgue convergence theorem. We show that u satis�es

condition (E

0

). Once this is proved, the conclusion of the theorem follows immediately

from Corollary C

0

.

Let v 2 X be an equilibrium solution of (5.12) satisfying v � u. Since u and v are

solutions of the elliptic equation

�

�(u

m

) = �f(x; u); x 2 ;

u = 0; x 2 @

and since f(x; u); f(x; v) 2 L

1

(), we get u

m

; v

m

2 C

1

() by the standard L

p

estimate

and the Sobolev embedding theorem. Therefore f(x; u) is H�older continuous on , hence

u

m

; v

m

2 C

2

() \ C

0

() by the Schauder estimate ([11]).

First we consider the case where f

u

(x; u) � 0 (0 � u � "

0

for some "

0

> 0) for every

x 2 . In this case, f(x; u)+ku

m

is strictly increasing in u 2 [0; kuk

L

1

] for every x 2 if

we choose the constant k > 0 su�ciently large. Then the function w = u

m

� v

m

satis�es

�w � kw = �(f(x; u) + ku

m

) + (f(x; v) + kv

m

) � 0:

18

Applying the strong maximum principle and the Hopf boundary lemma, we have

(v(x))

m

< (u(x))

m

; x 2 ;

@(v

m

)

@n

(x) >

@(u

m

)

@n

(x); x 2 @:

Arguing as in the proof of Lemma 5.2, we see that there exists some neighborhood B(e) �

G of e satisfying

(gv)

m

= g(v

m

) � u

m

for all g 2 B(e). This implies gv � u for all g 2 B(e).

Next we consider the case where f

u

(x; 0) < 0 for every x 2 . In this case there exists

an " > 0 such that f (x; u) is strictly decreasing in u 2 [0; "]. We prove only the case where

the set fx 2 j u(x) > 0g is connected ; other cases can be proved as a consequence of

this. De�ne a subset

"

of by

"

=

�

x 2 j (u(x))

m

>

"

m

2

�

:

It follows from the strong maximum principle that (v(x))

m

< (u(x))

m

for all x 2

"

.

Hence there exists some neighborhood B(e) � G of e such that if g 2 B(e) then

(gv(x))

m

< (u(x))

m

; x 2

"

;

(gu(x))

m

< "

m

; x 2 n

"

:

We show that

(gv(x))

m

� (u(x))

m

; x 2 (5.13)

holds for all g 2 B(e). Suppose that (5.13) does not hold for some g

0

2 B(e). Put

w(x) = (u(x))

m

� (g

0

v(x))

m

. Then the function w attains its negative minimum at some

point, which we will denote by x

0

2 . Since x

0

2 n

"

and since f(x; u) is monotone

decreasing in u 2 [0; "], we have

�w = �f (x; u) + f (x; g

0

v) < 0

in a neighborhood of x

0

. Applying the strong maximum principle, we are lead to a

contradiction. This means that condition (E

0

) is ful�lled. The theorem is proved.

Remark 5.6. Theorem 5.5 remains true if we replace u

m

in (5.12) by a more general non-

linearity �(u) 2 C

1

([0;1)) satisfying �(0) = 0, �

0

(u) > 0 for u > 0, �

�1

(w) 2 C

�

loc

([0;1)).

We can also relax the conditions on f in Theorem 5.5 ; the theorem's conclusion holds true

if f is a locally Lipschitz continuous function satisfying either of the following conditions :

(f 4.1a) there exist an " > 0 and a k > 0 such that f (x; u) + ku

m

is strictly increasing in

u 2 [0; "],

(f 4.1b) there exists an " > 0 such that f(x; u) is strictly decreasing in u 2 [0; "].

Finally we discuss the case where the domain is unbounded. Let G be a connected

subgroup of the rotation group SO(N) and � R

N

be a G-invariant unbounded domain

with smooth boundary @. Consider the initial boundary value problem

8

<

:

@u

@t

= �u+ f (x; u;ru); x 2 ; t > 0;

u(�; 0) = u

0

; x 2 ;

(5.14)

19

under the Dirichlet, the Neumann or the Robin boundary condition. Here f(x; u; p) : R�

R

N

! R is a C

1

function satisfying (f 1), (f 2)

0

and

(f 3.2) f(x; 0; 0) = 0

(f 4.2)

@f

@u

(x; 0; 0) � �� � for some � > 0 and for every jxj su�ciently large.

By an equilibrium solution u of (5.14), we mean a function u 2 C

1

()\C

2

()\L

1

()

satisfying

�u+ f(x; u;ru) = 0; x 2

along with the above-mentioned boundary conditions.

We assume that

(D) there exists a constant l such that any points x; y 2 can be joined by a polygonal

arc contained in and of length d � ljx� yj.

Our result is the following :

Theorem 5.7. Under condition (D), any stable equilibrium solution u of (5.14) satisfying

u(x)! 0; jru(x)j ! 0 as jxj ! 1

is G-invariant.

We will prove this theorem by using Corollary C

0

. To do so, we must �rst specify

the space X. Let C

unif

() denote the space of bounded uniformly continuous functions

on . Set X = C

unif

() \ C

0

() in the case of the Dirichlet boundary condition and

X = C

unif

() in the case of the Neumann or the Robin boundary condition. In both

cases, X is endowed with the L

1

()-topology. Condition (D) and the growth condition

(f 2)

0

guarantee that (5.14) generates a local semi ow on X (see [25], [35]). It is easily

seen that g : u(x) 7! u(g

�1

x) is a homeomorphism on X for every g 2 G. On the

other hand, the map g 7! u(g

�1

x) : G ! X is not necessarily continuous. However, it is

continuous if u(x) converges to a constant as jxj ! 1.

To apply Corollary C

0

, all we have to check are conditions (�4) and (E

0

). In view

of Remark 2.7, one needs to check condition (�4) for u su�ciently close to u(x) in the

L

1

()-topology. This follows immediately from Lemma 5.10 below. And (E

0

) follows

from the following lemma :

Lemma 5.8. Let u be as in Theorem 5.7. Then, for any equilibrium solution v of (5.14)

su�ciently close to u and satisfying

v(x) � u(x); v(x) 6� u(x); x 2 ;

there exists some neighborhood B(e) � G of e such that

gv(x) � u(x); x 2 ; g 2 B(e):

Proof. We consider only the case of the Dirichlet boundary condition. The case of the

Neumann or the Robin boundary condition can be treated analogously.

By condition (f 4.2), there exist � > 0, " > 0 and R

0

> 0 such that

@f

@u

(x; u; p) < �� for juj < "; jpj < "; jxj > R

0

: (5.15)

20

We then choose a su�ciently large R > R

0

such that

ju(x)j <

"

2

; jru(x)j <

"

2

for jxj > R: (5.16)

Now let v 6= u be an equilibrium solution of (5.14) satisfying v(x) � u(x), jv(x)�u(x)j <

"=2 for all x 2 . Then, by Lemma 5.2, we see that there exists some neighborhood

B(e) � G of e such that if g 2 B(e) then

gv(x) < u(x); jxj � R; x 2 : (5.17)

Note also that

u(x)� gv(x) = u(x)� gu(x) + g(u(x)� v(x)) � u(x)� gu(x);

hence

lim inf

jxj!1

(u(x)� gv(x)) � lim inf

jxj!1

(u(x) � gu(x)) = 0:

Combining this, (5.17) and Lemma 5.9 below, we see that

gv(x) � u(x); x 2 ; g 2 B(e):

The lemma is proved.

Lemma 5.9. Let � > 0, " > 0, R > 0 be such that

@f

@u

(x; u; p) < ��; juj < "; jpj < "; jxj > R:

Let u be an equilibrium solution of (5.14) satisfying ju(x)j, jru(x)j < " for x 2 n

R

and let v be an equilibrium solution of (5.14) satisfying jvj < " for x 2 n

R

, where

R

= fx 2 j jxj < Rg. Suppose that

u(x) � v(x); x 2 (@

R

) \ ;

lim inf

jxj!1

x2

(u(x)� v(x)) � 0:

Then u(x) � v(x) in n

R

.

Proof. Let �

0

= @ \ fx 2 R

N

j jxj � Rg, �

1

= \ fx 2 R

N

j jxj = Rg. Then,

@( n

R

) = �

0

[ �

1

. The function w(x) = u(x)� v(x) is a time-independent solution of

the linear parabolic equation

@w

@t

= �w +

N

X

i=1

�

i

(x)

@w

@x

i

+ �(x)w; x 2 n

R

; t > 0 (5.18)

under the boundary condition

�

(5.2b) or (5.5) or (5.6), x 2 �

0

; t > 0;

w = u� v; x 2 �

1

; t > 0;

(5.19)

where

�

i

(x) =

Z

1

0

@f

@p

i

(x; v(x); �ru(x) + (1� �)rv(x)) d�;

�(x) =

Z

1

0

@f

@u

(x; � u(x) + (1� �) v(x); ru(x)) d�

21

and x = (x

i

), p = (p

i

) 2 R

N

. On the other hand, the function r(t) = �"e

��t

is a

subsolution of (5.18), (5.19), that is,

8

>

<

>

:

@r

@t

� �r +

N

X

i=1

�

i

(x)

@r

@x

i

+ �(x)r; x 2 n

R

; t > 0;

(5.19) with w replaced by r and = by �, x 2 �

0

[ �

1

; t > 0;

since �(x) < 0 for x 2 n

R

and @r(t)=@n = 0, r(t) < 0 for x 2 @, t > 0. Note also

that

r(t) < 0 � lim inf

jxj!1

x2

w(x); t > 0:

Clearly r(0) � w(x) for x 2 n

R

. Hence it follows from the comparison theorem that

r(t) � w(x); x 2 n

R

; t > 0:

Letting t!1, we get

0 � w(x); x 2 n

R

and the proof is completed.

Lemma 5.10. Let u(x; t) be a uniformly bounded solution of (5.14) which is monotone

decreasing in t. Then u(x; t) converges locally uniformly to an equilibrium solution v(x)

as t!1 and

ku(�; 0)� vk

L

1

()

= lim

t!1

ku(�; 0)� u(�; t)k

L

1

()

:

Proof. By the uniform boundedness and the monotonicity of u(x; t) in t, the limit

v(x) = lim

t!1

u(x; t)

exists for any x 2 . The standard parabolic estimate shows that this convergence takes

place in C

2

loc

(). Consequently v is an equilibrium solution. The limitting equality is

obvious.

Remark 5.11. Condition (�4) is also satis�ed (with C = 1) if X is a Hilbert space |

or, more generally, a re exive Banach space | and if �

t

: X ! X is weakly continuous.

This observation allows us to deal with equations in the Sobolev space setting when only

weaker parabolic estimates are available.

6. Application (3) |Monotonicity of travelling waves

As another application of our theory in Sections 2 and 3, we will discuss in this section

the monotonicity of stable (or orbitally stable) travelling waves arising in some reaction-

di�usion equations. The �rst result in this section is already known ; in fact, it follows from

the standard linearization arguments as well as from our theory. However, the linearization

method cannot be easily generalized to degenerate di�usion equations, while our method

works equally well for some degenerate di�usion equations. Degenerate equations will be

discussed in a forthcoming paper ([28]). To clarify our ideas, we will start with the case

of a single equation.

First let us consider an equation of the form

@u

@t

=

@

2

u

@x

2

+ f

�

u;

@u

@x

�

; x 2 R; t > 0; (6.1)

22

where f(u; p) : R� R ! R is a C

1

function satisfying the same condition as (f 2)

0

. This

growth condition guarantees that equation (6.1) generates a local semi ow on the space

X = C

unif

(R) (see [25], [35]), where X = C

unif

(R) denotes the space of bounded and

uniformly continuous functions on R endowed with the L

1

(R)-topology.

A nonconstant solution u(x; t) of (6.1) is called a travelling wave if it is written in the

form

u(x; t) = �(x� ct);

where c is a constant and �(z) is some function. We call the constant c the speed of the

travelling wave.

Here we restrict our attention to travelling waves that satisfy the condition

lim

z!�1

�(z) = u

�

(6.2)

for some constants u

+

and u

�

. By a phase plane analysis, it is not di�cult to show that

(6.2) implies �

0

(z)! 0 as z ! �1. Therefore these constants must satisfy f(u

�

; 0) = 0.

A travelling wave is called a solitary wave (or a travelling pulse) if u

+

= u

�

, a travelling

front if u

+

6= u

�

.

In what follows we assume that

(f 4.3)

@f

@u

(u

�

; 0) < 0:

Condition (f 4.3) implies that both u

+

, u

�

are linearly stable equilibrium points of the

ordinary di�erential equation

dr

dt

= f(r;0); t > 0:

A travelling wave �(x� ct) = u(x; t) is called stable if for any " > 0 there exists a � > 0

such that

ku(�; 0)� �k

L

1

< � =) ku(�; t)� �(� � ct)k

L

1

< "; t 2 [0;1) (6.3)

for solution u(x; t) of (6.1). It is called orbitally stable if

ku(�; 0)� �k

L

1

< � =) d(u(�; t); O(�)) < "; t 2 [0;1); (6.4)

where O(�) = f�(� � cs) j s 2 Rg � X and d denotes the distance function on X. The

right-hand side of (6.4) is equivalent to saying that

inf

s2R

ku(�; t)� �(� � cs)k

L

1

< "; t 2 [0;1):

Clearly stability implies orbital stability.

We say that a travelling wave �(x� ct) is monotone if �(z) is a non-decreasing function

or a non-increasing function of z.

Theorem 6.1. Any stable (or orbitally stable) travelling wave of (6.1) is monotone.

The following is an immediate consequence of Theorem 6.1.

Corollary 6.2. Solitary waves of (6.1) are orbitally unstable (and hence unstable).

Remark 6.3. As a converse to Theorem 6.1, monotone travelling waves are known to be

stable. More precisely, Fife and McLeod [7] (for the case f = f(u)) and Sattinger [31]

23

show that if there exists a monotone travelling wave �(x � ct) for (6.1), (6.2), then any

solution with initial data satisfying

u

�

< u

0

(x) < u

+

; lim

x!�1

u

0

(x) = u

�

approaches to this travelling wave in the following sense :

lim

t!1

ku(x; t)� �(x� ct� h)k

L

1

(R)

= 0

for some h 2 R. In other words, this monotone travelling wave is `stable with asymptotic

phase'. Chen [3] generalizes their result by allowing the equation to contain nonlocal

terms (but requiring that the comparison theorem hold for the equation). >From this

stability result follows the uniqueness of the travelling wave : More precisely, if

~

�(x� ~ct)

is a travelling wave for (6.1), (6.2) with

u

�

�

~

�(z) � u

+

; z 2 R;

then

~

�(z) � �(z � h) for some constant h 2 R. Volpert et al. [34] studies stability of

monotone travelling waves for systems of equations (see Remark 6.9). In a forthcoming

paper [27] we also study this problem in a more general setting including pseudo-travelling

waves.

Remark 6.4. Note that the above-mentioned uniqueness results of [7], [31], [3] do not

exclude the possibility of existence of another travelling wave for (6.1), (6.2) whose range

is not con�ned in the interval [u

�

; u

+

]. For example, if f = f(u) has exactly four zeros

u

�

< � < u

+

< � with f

0

(u

�

) < 0, f

0

(�) > 0, f

0

(u

+

) < 0, f

0

(�) > 0 and if

Z

u

+

u

�

f(u)du < 0;

Z

u

�

u

�

f (u)du > 0 for some u

�

> �;

then one can show by a phase plane analysis that

(i) there exists a monotone travelling wave for (6.1), (6.2) with speed c

1

> 0;

(ii) there exists a nonmonotone travelling wave for (6.1), (6.2) with speed c

2

, where

0 < c

2

< c

1

; its pro�le �(z) is monotone increasing in �1 < z < z

�

for some z

�

2 R

and monotone decreasing in z

�

< z <1.

Before proving these results, let us �rst rewrite equation (6.1) with the moving coordi-

nate z = x� ct :

@u

@t

=

@

2

u

@z

2

+ c

@u

@z

+ f

�

u;

@u

@z

�

; z 2 R; t > 0: (6.5)

Denote by f�

t

g

t2[0;1)

the local semi ow generated by (6.5) on the space X = C

unif

(R).

It is easily seen that

~

�(x � ct) is a travelling wave for (6.1) if and only if

~

�(z) is an

equilibrium point of f�

t

g

t2[0;1)

. It is also clear that a travelling wave

~

�(x� ct) is stable

if and only if

~

�(z) is a stable equilibrium point of f�

t

g

t2[0;1)

. In the case where c 6= 0,

one can show that

~

�(x� ct) is orbitally stable if and only if

~

�(z) is a G-stable equilibrium

point, where G = fg

�

j � 2 Rg ' R is the group of translations

g

�

: u(z) 7! u(z � �) (6.6)

acting on the space X.

24

Proof of Theorem 6.1. Let �(x � ct) be a stable (or orbitally stable) travelling wave.

De�ne an order relation in X by

u � v if u(z) � v(z) for z 2 R; (6.7)

and a group G ' R as above. Clearly assumptions (�1), (�2), (G1), (G2

0

) and (G3) are

ful�lled. As in the previous example, Lemma 5.10 holds for (6.5), hence condition (�4) is

satis�ed. Clearly � = �(z) is a stable (or G-stable ) equilibrium point of f�

t

g

t2[0;1)

, and

the group orbit G� is locally precompact since G ' R. Further � satis�es condition (E

0

),

which can be shown by using an analogue of Lemma 5.8, with u replaced by �. (The proof

of this modi�ed version of Lemma 5.8 is almost identical to that of Lemma 5.8 itself, and

is therefore omitted.)

Applying Theorem B

0

, we obtain the conclusion of the theorem.

Remark 6.5. The same results as in Theorem 6.1 and Corollary 6.2 hold if we set X =

fw j w � � 2 H

1

(R)g instead of C

unif

(R) ; see Remark 5.11.

Remark 6.6. Using Corollary C

0

, we can also show that spatially periodic travelling waves

are unstable. To see this, set X = C(R=�Z), where � is the period of the travelling wave,

and let G ' R=�Z be the group of translations on R=�Z.

The above results for single equations easily extend to systems of certain types. Let us

consider a system of equations of the form

8

>

>

>

>

>

<

>

>

>

>

>

:

@u

1

@t

= d

1

@

2

u

1

@x

2

+ f

1

�

u

1

; � � � ; u

m

;

@u

1

@x

�

; x 2 R; t > 0;

.

.

.

@u

m

@t

= d

m

@

2

u

m

@x

2

+ f

m

�

u

1

; � � � ; u

m

;

@u

m

@x

�

; x 2 R; t > 0;

(6.8)

where d

1

; � � � ; d

m

> 0 are constants and f

1

; � � � ; f

m

: R

m

� R ! R are C

1

functions

satisfying a growth condition similar to (f 2)

0

. Here we assume that (6.8) is a system of

the `cooperation type', namely that

@f

i

@u

j

� 0; i 6= j: (6.9)

A solution u(x; t) = (u

1

(x; t); � � � ; u

m

(x; t)) of (6.8) is called a travelling wave with

speed c 2 R if it is written in the form

u

i

(x; t) = �

i

(x � ct); i = 1; � � � ;m

where �

i

(z) are some functions such that at least one of which is nonconstant. For

simplicity, we consider only the case where system (6.8) is irreducible, more precisely, the

case where the m�m matrix H = (h

ij

) with each component de�ned by

h

ii

= 0; h

ij

= sup

z2R

@f

i

@u

j

�

�

1

(z); � � � ; �

m

(z);

@�

i

@z

(z)

�

for i 6= j

is irreducible.

Here we restrict our attention to travelling waves that satisfy the condition

lim

z!�1

�(z) = u

�

= (u

�

1

; � � � ; u

�

m

); (6.10)

25

where u

+

i

, u

�

i

are constants. A travelling wave is called a solitary wave (a travelling pulse)

if u

+

= u

�

, and a travelling front if u

+

6= u

�

.

Condition (6.10) and a vector-valued version of (f 2)

0

imply �

0

(z)! 0 as z !�1. In

fact, a vector-valued version of (f 2)

0

implies for a su�ciently large C > 0

j�

i

00

j � C(1 + (�

i

0

)

2

); z 2 R;

and hence

�

�

�

�

�

i

0

�

i

00

1 + (�

i

0

)

2

�

�

�

�

� C j�

i

0

j ; z 2 R: (6.11)

Now let fz

k

g

k=1;2;3;���

� R be any sequence such that z

k

! 1 (resp. �1) as k ! 1.

Then, by (6.10) there exists some sequence fy

k

g

k=1;2;3;���

such that y

k

! 1 (resp. �1),

�

i

0

(y

k

)! 0 as k !1 and that �

i

0

is nonnegative or is nonpositive in the interval [z

k

; y

k

]

or [y

k

; z

k

]. Integrating both sides of (6.11) over this interval, we have

1

2

�

�

�

�

log

1 + (�

i

0

(z

k

))

2

1 + (�

i

0

(y

k

))

2

�

�

�

�

� Cj�(z

k

)� �(y

k

)j:

Letting k !1 and using (6.10), we get �

i

0

(z

k

)! 0. This shows �

0

(z)! 0 as z ! �1.

It follows that the limitting values u

�

satisfy f

i

(u

�

; 0) = 0. Now we assume that

(f 4.4) u

�

are both linearly stable equilibrium points of the ordinary di�erential equation

8

>

>

>

<

>

>

>

:

dr

1

dt

= f

1

(r

1

; � � � ; r

m

; 0); t > 0;

.

.

.

dr

m

dt

= f

m

(r

1

; � � � ; r

m

; 0); t > 0:

(6.12)

A travelling wave �(x � ct) is called stable if for any " > 0 there exists a � >

0 such that (6.3) holds, and it is called orbitally stable if (6.4) holds. Here we set

X = C

unif

(R)� � � � � C

unif

(R)

| {z }

m times

endowed with the metric kvk

L

1

= max

1�i�m

kv

i

k

L

1

for v =

(v

1

; � � � ; v

m

) 2 X.

We say that a travelling wave �(x � ct) is monotone if �

i

(z) (i = 1; 2; � � � ; m) are all

non-increasing functions or all non-decreasing functions.

Theorem 6.7. Any stable (or orbitally stable) travelling wave of (6.8) is monotone.

Corollary 6.8. Solitary waves of (6.8) are orbitally unstable (and hence unstable).

Remark 6.9. In the case where f

i

= f

i

(u

1

; � � � ; u

m

), Volpert et al. [34] show that any

monotone travelling waves are `asymptotically stable with shift'. (In our terminology this

basically means `stable with asymptotic phase' ; see Remark 6.3). They also prove that

nonmonotone travelling waves are not asymptotically stable with shift provided that the

pro�le (�

1

; � � � ; �

m

) possesses at most a �nite number of extrema (component-wise). Our

results show that such travelling waves are not even orbitally stable ; furthermore we do

not need to make any assumptions on the number of extrema of the pro�le.

Proof of Theorem 6.7. Let �(x� ct) be a stable (or orbitally stable) travelling wave.

26

De�ne an order relation in X = C

unif

(R)� � � � � C

unif

(R)

| {z }

m times

by

(u

1

; � � � ; u

m

) � (v

1

; � � � ; v

m

) if u

i

(z) � v

i

(z) for z 2 R; i = 1; � � � ; m;

and de�ne a group G ' R by

G = fg

�

j � 2 Rg;

where g

�

u(z) = u(z��) = (u

1

(z��); � � � ; u

m

(z��)). We denote by f�

t

g

t2[0;1)

the local

semi ow on X generated by

8

>

>

>

>

>

<

>

>

>

>

>

:

@u

1

@t

= d

1

@

2

u

1

@z

2

+ c

@u

1

@z

+ f

1

�

u

1

; � � � ; u

m

;

@u

1

@x

�

; z 2 R; t > 0;

.

.

.

@u

m

@t

= d

m

@

2

u

m

@z

2

+ c

@u

m

@z

+ f

m

�

u

1

; � � � ; u

m

;

@u

m

@x

�

; z 2 R; t > 0:

(6.13)

Then, as in the proof of Theorem 6.1, we see that conditions (�1), (�2), (�4), (G1),

(G2

0

), (G3) are ful�lled. Further � = �(z) is a stable (or G-stable) equilibrium point of

f�

t

g

t2[0;1)

and G� is locally precompact. We will show that � satis�es condition (E

0

).

Once this is proved, the conclusion of the theorem follows immediately from Theorem B

0

and a time-continuous version of Remark 2.7.

Since u

�

and u

+

are linearly stable equilibrium solutions of (6.12), we can choose

constants �

�

ij

such that

@f

i

@u

j

(u

�

1

; � � � ; u

�

m

; 0) < �

�

ij

; i; j = 1; � � � ;m

and that (0; � � � ; 0) is stable in both of the linear systems

8

>

>

>

<

>

>

>

:

dr

1

dt

= �

�

11

r

1

+ � � �+ �

�

1m

r

m

; t > 0;

.

.

.

dr

m

dt

= �

�

m1

r

1

+ � � �+ �

�

mm

r

m

; t > 0;

(6.14)

8

>

>

>

<

>

>

>

:

dr

1

dt

= �

+

11

r

1

+ � � �+ �

+

1m

r

m

; t > 0;

.

.

.

dr

m

dt

= �

+

m1

r

1

+ � � �+ �

+

mm

r

m

; t > 0:

(6.15)

Take " > 0 su�ciently small so that

@f

i

@u

j

(u

1

; � � � ; u

m

; p) < �

�

ij

; ju� u

�

j < "; jpj < ";

@f

i

@u

j

(u

1

; � � � ; u

m

; p) < �

+

ij

; ju� u

+

j < "; jpj < "

with u = (u

1

; � � � ; u

m

) 2 R

m

. Now let v = (v

1

; � � � ; v

m

) 2 X be an equilibrium point of

f�

t

g

t2[0;1)

satisfying v � � and kv � �k

L

1

< "=2. Choose a large R > 0 such that

j�(z)� u

�

j <

"

2

;

�

�

�

�

@�

@z

(z)

�

�

�

�

<

"

2

; z 2 (�1;�R);

27

j�(z) � u

+

j <

"

2

;

�

�

�

�

@�

@z

(z)

�

�

�

�

<

"

2

; z 2 (R;1):

Then, it follows from condition (6.9) and the condition of irreducibility of system (6.8)

that a vector-valued version of Lemma 5.2 holds. Furthermore, Lemma 5.9 remains true

for system (6.13) with little modi�cation in the proof, except that we have to discuss

(�1;�R] and [R;1) separately since the linearized equation at x = �1 and the one

at x =1 may be di�erent, and that the comparison function r(t) = (r

1

(t); � � � ; r

m

(t)) is

now taken as a solution of (6.14) (or (6.15)) with initial data (�"; � � � ;�� "). Therefore,

combining these lemmas, we see that Lemma 5.8 holds, and hence condition (E

0

) holds.

The proof is completed.

Remark 6.10. Remark 6.6 remains valid for system (6.8). To be more precise, spatially

periodic travelling waves of (6.8) are unstable.

We remark that Theorem 6.7, Corollary 6.8 and Remark 6.10 remain true for systems

of competition type with two species (that is, m = 2 and @f

1

=@u

2

, @f

2

=@u

1

� 0). To see

this, simply observe that such systems are of cooperation type with respect to (u

1

;�u

2

).

Note that in this case we say that a travelling wave �(x� ct) = (�

1

(x� ct); �

2

(x� ct)) is

monotone if �

1

(z) and ��

2

(z) are both non-decreasing functions or both non-increasing

functions.

In the special case where f

1

(u

1

; u

2

) = u

1

(1� u

1

� u

2

), f

2

(u

1

; u

2

) = u

2

(�� �u

1

� u

2

),

(6.8) is known as the Lotka-Volterra competition system. Under certain assumptions on

the coe�cients �; � and , Kan-on [16], [18] proves the existence of a stationary solution

(�

1

; �

2

) that satis�es

0 < �

1

< 1; 0 < �

2

< �; (u

�

1

; u

�

2

) = (0; �): (6.16)

He further proves its instability by using spectral analysis. Since our general theory

applies to this system with

X = f(w

1

; w

2

) 2 C

unif

(R) � C

unif

(R) j w

1

� 0; w

2

� 0g ;

his instability result is a special case of our Corollary 6.8.

Now let us consider travelling waves for equations in higher space dimensions. To be

more precise, let us consider the equation

@u

@t

= �u+ f(x

1

; � � � ; x

N�1

; u); x 2 ; t > 0 (6.17)

with x = (x

1

; � � � ; x

N

). Here is a cylindrical domain of the form = D�R whereD is a

bounded (N�1)-dimensional domain with smooth boundary, and f(x

1

; � � � ; x

N�1

; u) : D�

R! R is a C

1

function. We impose the Dirichlet, the Neumann or the Robin boundary

condition on @.

A travelling wave u(x; t) is a solution of (6.17) that satis�es

u(~x;x

N

; t) = �(~x; x

N

� ct); x = (~x;x

N

) 2 = D � R; t 2 R

for some constant c and some function �(~z; z

N

) of z = (~z; z

N

) 2 . Here we deal with

travelling waves such that

�(~z; z

N

)! u

�

(~z) as z

N

!�1 uniformly in ~z 2 D:

We assume that

28

(f 4.5) u

�

(~x) are linearly stable solutions of

~

�u

�

+ f(~x;u

�

) = 0; ~x 2 D (6.18)

under the above mentioned boundary conditions. Here

~

� denotes the (N � 1)-

dimensional Laplace operator.

A travelling wave �(~x; x

N

� ct) is called stable if for any " > 0 there exists a � > 0 such

that (6.3) holds, and it is called orbitally stable if (6.4) holds.

Theorem 6.11. Any stable (or orbitally stable) travelling wave of (6.17) is either mono-

tone increasing in x

N

or monotone decreasing in x

N

.

Proof. Let �(~x; x

N

�ct) be a stable (or orbitally stable) travelling wave. SetX = C

unif

().

De�ne an order relation in X by (5.4) and a group G ' R by

G = fg

�

j � 2 Rg;

where g

�

u(z) = g

�

u(~z; z

N

) = u(~z; z

N

� �). Denote by f�

t

g

t2[0;1)

the local semi ow on X

generated by the equation

@u

@t

= �u+ c

@u

@z

N

+ f(~z; u); z 2 ; t > 0 (6.19)

with z = (~z; z

N

) 2 = D � R. Then, as in the proof of Theorem 6.1, we see that

conditions (�1), (�2), (�4), (G1), (G2

0

), (G3) are ful�lled. Further � = �(~z; z

N

) is a

stable (or G-stable) equilibrium point of f�

t

g

t2[0;1)

and G� is locally precompact. We

will show that � satis�es condition (E

0

).

Since u

�

(~z) and u

+

(~z) are stable solutions of (6.18), we can choose functions �

�

(~z)

such that

f

u

(~z; u

�

(~z)) < �

�

(~z); ~z 2 D

and that 0 is linearly stable in both of the linear equations

@r

@t

=

~

�r + �

�

(~z)r; ~z 2 D; t > 0 (6.20)

under the above mentioned boundary conditions. Take a small " > 0 satisfying

f

u

(~z; u(~z)) < �

�

(~z) for ku(~z)� u

�

(~z)k

~

X

< ";

f

u

(~z; u(~z)) < �

+

(~z) for ku(~z)� u

+

(~z)k

~

X

< ";

where

~

X = C

0

(D) in the case of the Dirichlet boundary condition and

~

X = C(D) in other

cases.

Now let v be an equilibrium point of f�

t

g

t2[0;1)

satisfying v � � and kv� �k

L

1

< "=2.

Choose a large R > 0 satisfying

j�(z)� u

�

(z)j <

"

2

; z = (~z; z

N

); z

N

2 (�1;�R);

j�(z)� u

+

(z)j <

"

2

; z = (~z; z

N

); z

N

2 (R;1):

Then, clearly a higher dimensional version of Lemma 5.2 holds for (6.17). Furthermore

Lemma 5.9 remains true with little modi�cation except that we have to deal with D �

29

(�1;�R] and D� [R;1) separately and that we have to replace r(t) by a solution r(~z; t)

of (6.20) with initial data

min

�

0; inf

�1<z

N

<�R

w(~z; z

N

)

�

or min

�

0; inf

R<z

N

<1

w(~z; z

N

)

�

for ~z 2 D. Thus a higher dimensional version of Lemma 5.8 holds for (6.17), hence

condition (E

0

) is satis�ed. The proof is completed.

Remark 6.12. Applying an analogue of Theorem 5.7, one sees that if D is

~

G-invariant for

some connected subgroup

~

G of SO(N � 1) and if

f(g~x; u) = f (~x;u); ~x 2 D; u 2 R

for all g 2

~

G then any stable (or orbitally stable) travelling wave is

~

G-invariant. For

example, if D is an (N �1)-dimensional ball with N � 3 and if f = f(u), then any stable

travelling wave u(x; t) is written in the form u(x; t) = �(j~xj; x

N

�ct), where � is monotone

in the second argument.

7. Application (4) - |Monotonicity of pseudo-travelling waves

In the previous section we showed that stable (or orbitally stable) travelling waves are

monotone in the axial direction, hence in t. Our method is also applicable to the so-called

pseudo-travelling waves in temporally or spatially periodic media, thereby establishing

monotonicity result either in x or in t. In this paper we will give only an outline of the

proof. See the forthcoming paper [27] for a more comprehensive treatment of this subject,

including references on the existence of pseudo-travelling waves.

First we study pseudo-travelling waves in temporally periodic media. Let us consider

the non-autonomous equation

@u

@t

= a(t)

@

2

u

@x

2

+ b(t; u)

@u

@x

+ f(t; u); x 2 R; t > 0; (7.1)

where a : R! R and b; f : R� R! R are T -periodic with respect to t. We assume that

a(t) > 0 for all t 2 R, so that (7.1) is strictly parabolic. We also assume some regularity

conditions on a, b, f | for instance, a(t) is H�older continuous, b(t; u), f (t; u) are C

1

| so

that the initial value problem for (7.1) is well-posed in C

unif

(R).

A nonconstant solution u(x; t) of (7.1) is called a pseudo-travelling wave if it satis�es

u(x; t+ T ) = u(x� �; t)

for some constant �. Obviously any travelling wave is a pseudo-travelling wave, but the

converse is not true. The ratio �=T is called the e�ective speed or the average speed of

the pseudo-travelling wave. In what follows we focus our attention on pseudo-travelling

waves u satisfying

u(x; t)! u

�

(t) locally uniformly in t as x! �1;

where u

+

and u

�

are some functions of t. It is not di�cult to see that u

+

and u

�

are

T -periodic solutions of the ordinary di�erential equation

dr

dt

= f(t; r); t > 0: (7.2)

Here we assume that

30

(f 4.6) both u

+

(t) and u

�

(t) are linearly stable periodic solutions of (7.2).

A pseudo-travelling wave u(x; t) is called stable if for any " > 0 there exists a � > 0

such that

ku(�; 0)� u(�; 0)k

L

1

< � =) ku(�; t)� u(�; t)k

L

1

< "; t 2 [0;1) (7.3)

for solution u(x; t) of (7.1). It is called stable-with-shift if

ku(�; 0)� u(�; 0)k

L

1

< � =) inf

�2R

ku(�; t)� u(�+ �; t)k

L

1

< "; t 2 [0;1):

(7.4)

Obviously `stability-with-shift' is a weaker property than `stability'.

Given a constant � 2 R, let us de�ne a family of mappings fF

n

g

n=0;1;2;���

on the space

X = C

unif

(R) by

F (u

0

)(x) = u(x+ �; t

0

+ T ):

Here t

0

2 R is an arbitrarily �xed constant and u(x; t) denotes a solution of (7.1) for

t > t

0

satisfying u(x; t

0

) = u

0

(x) 2 X. It is easily seen that u(x; t) is a pseudo-travelling

wave with e�ective speed �=T if and only if u(x; t

0

) is a �xed point of F and that u(x; t)

is a stable (resp. stable-with-shift) pseudo-travelling wave if and only if u(x; t

0

) is a stable

(resp. G-stable) �xed point of F , where G = fg

�

j � 2 Rg ' R is the group of translations

de�ned by (6.6).

Applying Theorem B, we obtain the following :

Theorem 7.1. Any stable (or stable-with-shift) pseudo-travelling wave of (7.1) is either

monotone increasing in x or monotone decreasing in x.

Outline of proof. To see that Theorem B applies, we only have to check condition (E).

All other conditions are easily veri�ed.

Note that the parabolic version of Lemma 5.9 holds for pseudo-travelling waves u(x; t),

v(x; t) of (7.1) with e�ective speed �=T if we replace by f(x; t) j x 2 ; t > 0g and

R

by f(x; t) j x 2

R(t)

; t > 0g. Here we take � = �

�

(t), R = R(t), where � = �

�

(t),

R = R(t) are smooth functions such that

�

�

(t+ T ) = �

�

(t); R(t+ T ) = R(t) + �; t 2 R;

@f

@u

(t; u; p) < �

�

(t); ju� u

�

(t)j < "; jpj < "; t 2 R

and such that 0 is a stable solution of

dr

dt

= �

�

(t)r; t > 0:

Thus the time-discrete version of Lemma 5.8 holds. Condition (E) follows from the time-

discrete version of Lemma 5.8.

Remark 7.2. The same result as in Theorem 7.1 holds for problems in higher dimensions.

Next we consider pseudo-travelling waves in spatially periodic media. Let us consider

an initial value problem for the equation

@u

@t

= �(x)

@

2

u

@x

2

+ �(x; u)

@u

@x

+ g(x; u); x 2 R; t > 0; (7.5)

where � : R! R and �, g : R�R! R are L-periodic with respect to x. We assume that

�(x) > 0 for all x 2 R, so that (7.5) is strictly parabolic. We also assume some regularity

31

conditions on �, �, g | for instance, �(x) is H�older continuous, �(x; u) g(x; u) are C

1

|

so that the initial value problem for (7.5) is well-posed in X = C

unif

(R).

A nonconstant solution u(x; t) of (7.5) is called a pseudo-travelling wave if there exists

a � 6= 0 such that

u(x; t+ � ) = u(x� L; t); x 2 R; t 2 R:

The ratio L=� is called the e�ective speed or the average speed. Here we restrict our

attention to the pseudo-travelling waves that are asymptotically constant as x ! �1,

or, more generally, `asymptotically periodic' as x! �1. To be more precise, we assume

that there exist functions u

+

(x), u

�

(x) such that

u

+

(x+ L) = u

+

(x); u

�

(x + L) = u

�

(x); x 2 R;

lim

x!�1

ju(x; t)� u

�

(x)j = 0 locally uniformly in t � 0:

It is easily seen that both u

+

and u

�

satisfy

�(x)

@

2

u

�

@x

2

+ �(x; u

�

)

@u

�

@x

+ g(x; u

�

) = 0; x 2 R: (7.6)

In what follows we assume that

(f 4.7) both u

+

and u

�

are linearly stable equilibrium solutions of (7.5).

In the special case where u

�

are constants, equation (7.6) amounts to

g(x; u

�

) = 0; x 2 [0; L];

and condition (f 4.7) is satis�ed if

g

u

(x; u

�

) < 0; x 2 [0; L]:

We say that a pseudo-travelling wave u(x; t) is stable if for any " > 0 there exists a

� > 0 such that (7.3) holds. Similarly we say that u(x; t) is orbitally stable if, instead of

the right-hand side of (7.4),

inf

s2R

ku(�; t)� u(�; t + s))k

L

1

< "; t 2 [0;1)

holds.

Theorem 7.3. Any stable (or orbitally stable) pseudo-travelling wave of (7.5) is either

monotone increasing in t or monotone decreasing in t.

To prove Theorem 7.3, given a pseudo-travelling wave u(x; t) with e�ective speed L=� ,

we de�ne a mapping F by

F (u

0

)(x) = u(x+ L; �);

where u(x; t) denotes a solution of (7.5) with initial data u(x; 0) = u

0

(x). Denote by

E the set of all �xed points of F , and by f

t

g

t2[0;1)

the local semi ow on the space X

generated by (7.5). As is easily seen, E coincides with the set of pseudo-travelling waves

| or, more precisely, their values at t = 0 | of (7.5) with e�ective speed L=� . Since

F (

t

(u

0

))(x) = u(x+ L; t+ �) =

t

(F (u

0

))(x);

we have F �

t

=

t

� F , hence

t

(E) = E for any t � 0:

32

Furthermore, since any pseudo-travelling wave can be extended globally on the time in-

terval �1 < t < 1, the restriction of

t

on E forms a one-parameter group of home-

omorphisms on E. Denote this group by G. Applying Theorem B and Remark 2.6, we

obtain the conclusion of the theorem.

The same result as in Theorem 7.3 holds for a higher dimensional problem de�ned on

a periodically undulating cylindrical domain . More precisely, let us consider the initial

boundary value problem of the form

8

>

>

<

>

>

:

@u

@t

= �(x)�u+

N

X

i=1

�

i

(x; u)

@u

@x

i

+ g(x; u); x 2 ; t > 0;

@u

@n

= 0; x 2 @; t > 0;

(7.7)

where is a domain with smooth boundary @ such that there exists a family of bounded

domains D(x

N

) � R

N�1

satisfying

= f(~x;x

N

) j ~x 2 D(x

N

); x

N

2 Rg;

D(x

N

) = D(x

N

+ L); x

N

2 R

for some constant L > 0. Here the functions � : ! R, �

i

, g : � R ! R are also

L-periodic with respect to x

N

. We assume that �(x) > 0 for all x 2 , so that (7.7)

is strictly parabolic. We also assume that �; �

i

; g satisfy regularity conditions similar to

those for (7.5).

A nonconstant solution u(x; t) = u(x

1

; � � � ; x

N

; t) of (7.7) is called a pseudo-travelling

wave if there exists a � 6= 0 such that

u(x

1

; � � � ; x

N�1

; x

N

; t+ �) = u(x

1

; � � � ; x

N�1

; x

N

� L; t); x 2 ; t 2 R:

Here, as in the one-dimensional problem (7.5), we restrict our attention to the pseudo-

travelling waves that are asymptotically periodic as x

N

! �1. To be more precise, we

assume that there exist functions u

+

(x), u

�

(x) such that

u

+

(x

1

; � � � ; x

N�1

; x

N

+ L) = u

+

(x

1

; � � � ; x

N�1

; x

N

); x 2 ;

u

�

(x

1

; � � � ; x

N�1

; x

N

+ L) = u

�

(x

1

; � � � ; x

N�1

; x

N

); x 2 ;

lim

M!1

sup

�x

N

�M

x2

ju(x; t) � u

�

(x)j = 0 locally uniformly in t � 0:

It is easily seen that both u

+

and u

�

satisfy

8

>

>

<

>

>

:

�(x)�u+

N

X

i=1

�

i

(x; u)

@u

@x

i

+ g(x; u) = 0; x 2 ;

@u

@n

= 0; x 2 @:

In what follows we assume that

(f 4.8) both u

+

and u

�

are linearly stable equilibrium solutions of (7.7).

Theorem 7.4. Any stable (or orbitally stable) pseudo-travelling wave of (7.7) is either

monotone increasing in t or monotone decreasing in t.

33

The proof of this theorem is almost identical to that of Theorem 7.3. The map F is

now de�ned by

F (u

0

)(x

1

; x

2

; � � � ; x

N

) = u(x

1

; x

2

; � � � ; x

N

+ L; �);

where u(x; t) is a solution of (7.7) with initial data u

0

.

8. Application (5) - |Generalized motion by mean curvature

Let fS(t)g

t�0

be a family of time-dependent hypersurfaces embedded in R

N

, whose

motion is governed by the equation

V = f(n;rn; t); (8.1)

where n = n(x; t) is the outward unit normal vector at each point of S(t) and V denotes

the normal velocity of S(t) in the outward direction. A typical example of (8.1) is

V = ��+ g(t);

where � = (1=(N � 1)) tracern is the mean curvature at each point of S(t). In the case

g(t) � 0, this equation is known as the mean curvature ow equation. An anisotropic

version of the above equation, namely,

V = �(n; t)�+ g(n; t);

is also a well-known example of (8.1).

We consider (8.1) in the framework of generalized solutions. The notion of such solutions

was introduced by Evans and Spruck [6] and independently by Chen, Giga and Goto [4].

Let f (p; Z; t) : S

N�1

�M

N

� [0;1) ! R be a smooth function that is T -periodic in t

and satis�es

(f 5) for each M > 0, there is a constant � 2 (0; 1) such that

lim inf

"#0

1

"

ff(�p;�Q

p

(Z

1

+ "Z

2

); t)� f(�p;�Q

p

(Z

1

); t)g � � traceQ

p

(Z

2

)

for any (p; Z

1

; t) 2 S

N�1

� S

N

� [0; T ], Z

2

2 S

N

satisfying jZ

1

j �M , Z

2

� 0.

Here M

N

is the space of N � N real matrices, S

N

the space of N � N real symmetric

matrices, and S

N�1

denotes the N � 1 dimensional unit sphere in R

N

. Let Q

p

(Z) be

de�ned by

Q

p

(Z) = (I � p p)Z (I � p p)

for p 2 S

N�1

, Z 2 M

N

with I 2 M

N

being the identity matrix. For Z 2 S

N

, we write

Z � 0 if Z is positive de�nite. The equation (8.1) is called strictly parabolic if condition

(f 5) is ful�lled.

Let us de�ne a metric space X by

X =

8

<

:

(�;D)

�

�

�

�

�

�

D is a bounded open set in R

N

and

� is a compact set such that

@D � � � R

N

nD

9

=

;

equipped with the metric d de�ned by

d((�; D); (�

0

;D

0

)) = h(�;�

0

) + h(D [ �;D

0

[ �

0

):

34

Here, given compact sets K

1

and K

2

, the symbol h(K

1

;K

2

) denotes the Hausdor� metric

between K

1

and K

2

if K

1

;K

2

6= ;, while h(K

1

;K

2

) =1 if K

1

6= ; and K

2

= ; or K

1

= ;

and K

2

6= ;, and h(K

1

; K

2

) = 0 if K

1

= K

2

= ;.

Following [4], we de�ne a generalized solution of (8.1) as a curve f(�

t

; D

t

)g

t�0

in X,

where �

t

and D

t

are, respectively, the level surface fw = 0g and the super level set

fw > 0g of a viscosity solution w(x; t) of the parabolic equation

w

t

= H(rw;r

2

w; t);

where

H(p; Z; t) = jpj f

�

�p;��

Q

p

(Z)

jpj

; t

�

with p = p=jpj.

In this paper we will call a solution fS(t)g

t�0

of (8.1) compact if S(t) is compact for

each t � 0, and smooth if S(t) is a smooth hypersurface for each t � 0.

Theorem 8.1. Any smooth compact solution of (8.1) that is T -periodic is unstable.

Corollary 8.2. Any smooth compact stationary surface is unstable.

Proof of Theorem 8.1. Suppose that there exists a stable smooth compact solution

fS(t)g

t�0

that is T -periodic. Then u = (S(0); intS(0)) is a stable �xed point of the

mapping F de�ned by

F ((�;D)) = (�

T

;D

T

);

where (�

t

;D

t

)

t�0

is a solution of (8.1) with initial data (�

0

;D

0

) = (�;D) (ref. [6], [4]).

Here intS denotes the region enclosed by a surface S.

We de�ne an order relation in X by

(�

1

;D

1

) � (�

2

; D

2

) if D

1

� D

2

and D

1

[ �

1

� D

2

[ �

2

;

and a group G ' R

N

by

G = fg

�

j � 2 R

N

g;

where g

�

(�;D) = (� � �;D � �) = f(x � �; y � �) j x 2 �; y 2 Dg. Clearly conditions

(F1), (F3) and (G1){(G3) are ful�lled. It is known that (F2) does not hold; one can,

however, easily show that condition (F2

�

) holds.

In what follows we will verify condition (E). Once this is done, then in view of Remark

2.4 one can apply Theorem A, to obtain

gu � u or gu � u for any g 2 G; (8.2)

which contradicts the fact that S(0) is a compact surface, and this contradiction proves

the theorem.

Now let u = (�;D) be a �xed point of F satisfying u � u ^ hu for some h 2 G

with hu 6= u (see Remark 2.8), and let f(�

t

;D

t

)g

t�0

be the corresponding T-periodic

generalized solution of (8.1). Then

D [ � � intS(0) \ ( intS(0) � �) for some � 6= 0;

where intS(0) = intS(0) [ S(0). Therefore there exists a smooth surface S

1

satisfying

D [ � � intS

1

( intS(0);

D � intS

1

( intS(0):

(8.3)

35

This means that u � u

1

� u, where u

1

= (S

1

; intS

1

). The local existence theorem

for classical solutions of (8.1) shows that there exists a family of smooth hypersurfaces

fS

1

(t)g

t2[0;T

0

)

solving (8.1) with initial data S

1

(0) = S

1

(see, for instance, [8]).

Let v(x; t) and v

1

(x; t) denote the signed distance function of S(t) and S

1

(t), respec-

tively, namely,

v(x; t) =

�

minfjx� yj j y 2 S(t)g if x 2 intS(t);

� �minfjx� yj j y 2 S(t)g if x 62 intS(t);

and v

1

is de�ned in the same way as v. By the comparison theorem for generalized

solutions of (8.1), v

1

(x; t) � v(x; t) holds for t 2 [0; T

0

), x 2 R

N

. As is shown in [8], there

exists some " > 0 such that v and v

1

solve the equation

v

t

= H(rv;r

2

v(I � vr

2

v)

�1

; t) + tracer

2

v(I � vr

2

v)

�1

(rv rv)

in U

0

= f(x; t) 2 R

N

� (0; T

0

) j jv(x; t)j < "g and U

1

= f(x; t) 2 R

N

� (0; T

0

) j jv

1

(x; t)j <

"g, respectively. Applying the strong maximum principle, we �nd that either

v

1

(x; t) < v(x; t) in U

0

\ U

1

or

v

1

(x; t) � v(x; t) and U

0

= U

1

;

the latter being impossible by virtue of (8.3). Thus, for any t 2 (0; T

0

), we have S(t) \

S

1

(t) = ;. Hence for an arbitrarily �xed t

�

2 (0; T

0

) there exists a neighborhood B(e) � G

of e such that

g(S

1

(t

�

); intS

1

(t

�

)) � (S(t

�

); intS(t

�

)); g 2 B(e):

Therefore, by the comparison theorem for generalized solutions, we have

g(�

t

�

; D

t

�

) � g(S

1

(t

�

); intS

1

(t

�

)) � (S(t

�

); intS(t

�

)); g 2 B(e):

Again by the comparison theorem, we obtain

gu = gF (u) = g(�

T

; D

T

) � (S(T ); intS(T )) = F (u) = u; g 2 B(e):

This veri�es condition (E) in its weaker form mentioned in Remark 2.8. The proof is

completed.

Remark 8.3. The above result can be generalized to a spatially-inhomogeneous equation

of the form

V = f(x;n;rn; t); (8.4)

provided that f is translation-invariant in some direction, namely

f(x+ s�

0

; p; Z; t) � f (x; p; Z; t)

for some �xed �

0

2 R

N

nf0g and any s 2 R, and provided that the local existence theorem

for smooth classical solutions holds for (8.4). The proof is identical to that of Theorem 8.1

except that the group G in (8.2) now represents the group of translations in the direction

�

0

. Note that, while stationary plane curves of (8.1) (for N = 2) are all convex by virtue

of condition (f 5), equation (8.4) can possibly possess nonconvex stationary plane curves.

36

Remark 8.4. There are some related works in the literature. Ei and Yanagida [5] consider

(8.1) (and (8.4)) in the special case where f is independent of t and derive the same result

as Theorem 8.1 and Remark 8.3 for this special case. Their argument, however, cannot

easily be generalized to time-dependent problems. Giga and Yama-uchi [10] consider

(8.1) and derive the same result as Theorem 8.1 for possibly noncompact surfaces under

certain additional hypotheses on the second fundamental form of the surface. Because of

this additional hypotheses, their result does not apply, for instance, to nonconvex plane

curves (when N = 2) and therefore cannot easily be generalized to equation (8.4). The

methods of [10] and [5] are quite di�erent from ours. The former relies on an explicit

construction of super- and subsolutions and the latter is based on linearlization arguments

and spectral analysis. Our method, on the other hand, relies less on technical calculations

and treats the problem in a more general frame work. Incidentally, existence of convex

periodic solutions is known for V = �(n)�+g(t) in the case N = 2 by Giga and Mizoguchi

[9].

Remark 8.5. Our theory also applies to a surface motion in a cylindrical domain D�R �

R

N

with appropriate boundary conditions. We can show that any stable pseudo-travelling

wave of (8.1) in D �R can be expressed in the form of a graph x

N

= (x

1

; � � � ; x

N�1

; t),

where the function satis�es (x

1

; � � � ; x

N�1

; t + T ) = (x

1

; � � � ; x

N�1

; t) + � for some

constant � 2 R. Here a solution fS(t)g

t�0

of (8.1) is called a pseudo-travelling wave if

S(t+ T ) = S(t) + �e

N

; t 2 R

for some � 2 R, where e

N

= (0; � � � ; 0; 1) 2 R

N

. The details will be discussed in the

forthcoming paper [26].

Appendix |structure of totally-ordered sets

In this appendix we discuss the structure of totally-ordered connected sets in a metric

space X. One of the propositions we present here is used in the proof of Theorems B and

B

0

in Sections 2 and 3. These propositions are rather standard if the metric space X has

a linear structure, but they do not seem to be well-known if X has no such structure ;

at least the authors could not �nd relevant references in the literature. Since the results

seem to be interesting in their own right, we will state them in full generality and give an

outline of the proof.

Let X be an ordered metric space endowed with metric d and order relation � as in

Section 2. Let Y be a subset of X such that

(A1) Y is a totally-ordered set ;

(A2) Y is connected ;

(A3) Y is locally precompact (that is, Y is locally compact).

Then the following propositions hold :

Proposition Y1. Let Y � X satisfy (A1), (A2), (A3). Then for any pair of points

a; b 2 Y with a � b, the set

[a; b]

Y

= fx 2 Y j a � x � bg

is homeomorphic and order-isomorphic to the unit interval [0; 1] � R. More precisely,

there exists a homeomorphism ' : [a; b]

Y

! [0; 1] that is order-preserving.

37

Proposition Y2. Let Y � X satisfy (A1), (A2), (A3) and suppose that Y has neither

the maximum nor the minimum ; more precisely suppose that for any x 2 Y there exist

points y; z 2 Y satisfying y � x � z. Then Y is homeomorphic and order-isomorphic to

R.

Outline of the proof of Proposition Y1. The proof consists of several steps.

Step 1 (Connectedness).

As is easily seen, (A1) and (A2) imply that [a; b]

Y

is a totally ordered connected set.

The details are omitted.

Step 2 (Closedness).

Let x

�

2 X be an arbitrary accumulation point of [a; b]

Y

and set

K = [a; b]

Y

[ fx

�

g:

It is easily seen that a � x

�

� b and that K is totally-ordered. Now suppose x

�

62 [a; b]

Y

.

Then x

�

6= a and x

�

6= b, hence x

�

is neither the minimum nor the maximum of the totally

ordered set K. It follows that K n fx

�

g =[a; b]

Y

is not connected, contradicting the result

in Step 1. Thus we have x

�

2 [a; b]

Y

. Therefore [a; b]

Y

is a closed set.

Step 3 (Compactness).

Suppose that [a; b]

Y

is not a compact set. Then there exists a sequence fx

k

g

k=1;2;3;���

�

[a; b]

Y

which does not possess a convergent subsequence. Replacing fx

k

g

k=1;2;3;���

by its

subsequence if necessary, we may assume without loss of generality that fx

k

g

k=1;2;3;���

is a

strictly monotone sequence. Assume, for instance, x

1

� x

2

� x

3

� � � � . (The monotone

decreasing case can be treated analogously.)

De�ne

Y

1

= fx 2 [a; b]

Y

j x � x

k

for some k 2 Ng;

Y

2

= fx 2 [a; b]

Y

j x � x

k

for every k 2 Ng:

Clearly [a; b]

Y

= Y

1

[ Y

2

, and Y

1

, Y

2

are not empty since a 2 Y

1

, b 2 Y

2

. It is also clear

that Y

2

is a closed set, hence

Y

1

\ Y

2

= ;: (a.1)

Next we show that

Y

1

\ Y

2

= ;: (a.2)

Suppose that (a.2) does not hold. Then we can choose a sequence fy

j

g

j=1;2;3;���

� Y

1

that

converges to a point z 2 Y

2

. Now �x k 2 N arbitrarily. Since

lim

j!1

y

j

= z � x

k+1

� x

k

;

and since [a; b]

Y

is totally-ordered, we have

y

j

� x

k

for su�ciently large j. On the other hand, for each y

j

2 Y

1

we have

x

k

0

� y

j

for su�ciently large k

0

. In view of these and by taking a subsequence if necessary, we may

assume without loss of generality that

x

1

� y

1

� x

2

� y

2

� x

3

� y

3

� � � � :

38

Since fx

k

g

k=1;2;3;���

has no convergent subsequence, there exists some � > 0 such that

d(x

k

; z) > � for su�ciently large k: (a.3)

On the other hand, by the de�nition of fy

k

g

k=1;2;3;���

,

d(y

k

; z)! 0 as k !1: (a.4)

Since each [x

k

; y

k

]

Y

is connected as stated in Step 1, there exists, for any " 2 (0; �), a

sequence fz

k

g

k=1;2;3;���

that satis�es

z

k

2 [x

k

; y

k

]

Y

; d(z

k

; z)! " as k !1: (a.5)

Choose " > 0 su�ciently small. Then, by (A3), we can chose a subsequence of fz

k

g

k=1;2;3;���

converging to some point ~z 2 X. Here we have ~z 2 [a; b]

Y

since [a; b]

Y

is closed. From

the latter statement of (a.5) we obtain

d(~z; z) = ": (a.6)

On the other hand, the inequalities

y

k

� z

k+1

� y

k+1

imply

z � ~z � z;

hence z = ~z, which contradicts (a.6). This contradiction establishes (a.2). The combina-

tion of (a.1) and (a.2), however, contradicts the connectedness of [a; b]

Y

. Thus [a; b]

Y

is

compact.

Step 4 (Construction of a homeomorphism).

For each integer k 2 N, we de�ne a function �

k

(x) on [a; b]

Y

by

�

k

(x) = sup

a�x

1

�����x

k

�x

(d(a; x

1

) + d(x

1

; x

2

) + � � � + d(x

k�1

; x

k

)):

It is easily seen that �

k

: [a; b]

Y

! R is a continuous function satisfying

0 � �

k

(x) � kM; (a.7)

x � y =) �

k

(x) � �

k

(y); (a.8)

where M = max

y;z2[a;b]

Y

d(y; z). Considering that [a; b]

Y

is compact and totally ordered, one

can also show that

x � y =) �

k

(x) < �

k

(y) for su�ciently large k (a.9)

(the proof is omitted). Now de�ne a function f : [a; b]

Y

! R by

f(x) =

1

X

k=1

1

k

3

�

k

(x):

By (a.7), this series is uniformly convergent on [a; b]

Y

, therefore, f (x) is continuous. By

(a.8) and (a.9), f(x) is strictly increasing. This means that f is a one-to-one continuous

mapping from [a; b]

Y

onto the interval [0; f (b)]. Since [a; b]

Y

is compact, f is a homeo-

morphism. Thus, f(x)=f(b) gives an order-preserving homeomorphism from [a; b]

Y

onto

[0; 1] � R.

39

Proof of Proposition Y2. Fix an element x

0

2 Y and set

Y

�

= fx 2 Y j x � x

0

g; Y

+

= fx 2 Y j x � x

0

g:

We will show that Y

+

is homeomorphic and order-isomorphic to the interval [0;1), and

Y

�

to the interval (�1; 0]. The conclusion of the proposition will then follow immediately.

In what follows we will only consider Y

+

, since Y

�

is treated similarly.

First we take a sequence

x

0

� x

1

� x

2

� x

3

� � � �

satisfying the following conditions :

d(x

k

; x

k+1

) � 1; k = 0; 1; 2; � � � (a.10)

if Y

+

is an unbounded set, and

d(x

k

; x

k+1

) �

1

2

supfd(x

k

; y) j y 2 Y

+

; x

k

� yg (a.11)

if Y

+

is bounded. We show that this sequence has no upper bound in Y

+

. In fact, if

y

�

2 Y

+

is an upper bound of fx

k

g

k=0;1;2;���

, then by the compactness of [x

0

; y

�

]

Y

and the

monotonicity of fx

k

g

k=0;1;2;���

, this sequence converges. This contradicts (a.10) in the case

where Y

+

is unbounded. In the case where Y

+

is bounded, this and (a.11) imply that

lim

k!1

x

k

is the maximal element of Y

+

, contradicting the assumption of the proposition.

Since fx

k

g

k=0;1;2;���

has no upper bound in Y

+

, we have

Y

+

= [x

0

; x

1

]

Y

[ [x

1

; x

2

]

Y

[ [x

2

; x

3

]

Y

[ � � � : (a.12)

Since each [x

k

; x

k+1

]

Y

is homeomorphic and order-isomorphic to the unit interval [0; 1], it

is easily seen that Y

+

is homeomorphic and order-isomorphic to the interval [0;1). The

proof is completed.

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41

Note added in proof: After completing this work, one of the authors had a chance to

talk with Prof. Wei-Ming Ni and found that the basic idea in the paper

Y. Li and W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic

equations in R

n

, Comm. P.D.E. 18 (1993), 1043-1054.

is very similar to what we have used in the proof of our Lemma 5.8. The above paper

considers the elliptic equation (5.2) on R

n

and proves radial symmetry of positive solutions

by using the so-called moving plane method. Our Lemma 5.8, on the other hand, veri�es

condition (E') for solutions of (5.2). On examining the apparent similarity between the

idea of proofs of the two results, we realized that our condition (E') is deeply related,

at least in its spirit, to a condition that guarantees the moving plane method to work;

thus the similarity was not surprising at all. As a consequence, some of the symmetry

or monotonicity results proved in the present paper for stable solutions can be easily

modi�ed to apply to positive solutions or solutions lying between certain values; they

include Theorems 5.5, 5.7, 6.1, 6.7, 6.11, 7.1, 7.3, 7.4.

42