A REMARK ON THE EXISTENCE OF SUITABLE VECTOR
FIELDS RELATED TO THE DYNAMICS OF SCALAR
SEMILINEAR PARABOLIC EQUATIONS
FENGBO HANG AND HUIQIANG JIANG
Let be a bounded smooth domain in Rn , be the outer normal direction of @
R Rn ; R . The in…nite dimensional dynamical system de…ned
and f 2 C 1
by
ut = u + f (x; u; ru) ;
@u
@ = 0 on @ ;
(1)
x 2 ; t > 0;
on suitable Sobolev spaces has attracted much interest (see [A, H] and more recent
references at the end of this note). Many e¤orts have been made to show the
complexity of its dynamical behavior (see some survey papers and recent articles
[DP, P1, P2, P3, Pr, PR, R] and the references therein). In particular, the following
nice result was proven in [P2]: if there exists a smooth vector …eld on , =
( 1;
; n ) such that
rank ( (x) ; @1 (x) ;
@
@ = 0 on @ ,
; @n (x)) = n for all x 2 ;
then for any smooth vector …eld X on Rn , there exists a smooth function f , such
that span fP1 ;
; n g is invariant under (1) and for any integral curve of X, c =
n
c (t), u = i=1 ci (t) i (x) is a solution to (1). Moreover, it was shown that such
kind of vector …eld always exists on a starshaped domain. The main result of this
short note is a classi…cation of all the domains on which one may …nd this type of
vector …elds. More precisely, we have
Theorem 1. Let
Rn (n 2) be an open bounded smooth domain, then the
necessary and su¢ cient condition for the existence of a smooth map F : ! Rn
with
rank (F (x) ; @1 F (x) ;
; @n F (x)) = n for any x 2 ;
(2)
@F
=
0
on
@
,
@
is that
is di¤ eomorphic to B 1 or B 2 nB1 .
Remark 1. In fact, if is di¤ eomorphic to B 1 , then any solution to (2), F , must
have exactly one zero in . If
is di¤ eomorphic to B 2 nB1 , then any solution to
(2), F , does not vanish at all. These conclusions will follow from the arguments
below.
First we reduce the existence of such a vector …eld to the existence of vector …eld
with less restrictions.
Lemma 1. Let
a smooth map G :
Rn (n 2) be an open bounded smooth domain, if there exists
! Rn such that
rank (G (x) ; @1 G (x) ;
; @n G (x)) = n for any x 2
1
2
FENGBO HANG AND HUIQIANG JIANG
and
n
dim span G (x) ; im ( Gj@ )
;x
o
= n for any x 2 @ ;
here ( Gj@ ) ;x denotes the tangent map of Gj@
map F : ! Rn satisfying (2).
at x, then we may …nd a smooth
Proof. Let " > 0 be small enough such that the map
: dist (y; @ )
: @
[0; 3"] ! y 2
: (x; t) 7! x t (x)
3"
is a di¤eomorphism. Let P = G
. For any t 2 [0; 3"], let Pt (x) = P (x; t) for
x 2 @ , then we may assume " is small enough such that for any t 2 [0; 3"],
n
o
dim span Pt (x) ; im (Pt ) ;x = n for all x 2 @ :
Let
: R ! R be a smooth function such that
"; when t
t, when t
(t) =
and
0
(t)
"=2;
3"=2;
0 for all t. De…ne
Qt (x) = Q (x; t) = P (x; (t)) for x 2 @ , 0
Then it is clear that for any t 2 [0; 3"],
n
dim span Qt (x) ; im (Qt )
Let
;x
o
t
= n for any x 2 @ :
G (y) ; if y 2 ;dist(y; @ ) 2";
1
Q
(y) ; if y 2 ;dist(y; @ )
then it is clear F satis…es all the requirements.
F (y) =
Corollary 1. Assume 1 is di¤ eomorphic to
to (2), then we may …nd a solution to (2) for
3".
2,
and for
2 too.
1
3";
we may …nd a solution
To derive the necessary condition for the existence of a vector …led satisfying
(2), we will need
Lemma 2. Let
a smooth map H :
Rn (n
! Sn
1
2) be an open bounded smooth domain, if there exists
such that
rank (@1 H (x) ;
; @n H (x)) = n
and
dim im ( Hj@ )
then
;x
=n
1 for any x 2
1 for any x 2 @ ;
is di¤ eomorphic to B 2 nB1 .
Proof. First we claim that each path connected component of @ is di¤eomorphic to
S n 1 . This is clear when n = 2. If n 3, since Hj@ has full rank everywhere and
@ is compact, Hj@ : @ ! S n 1 is a covering map (see [M]). Since S n 1 is simply
connected, we see each path connected components of @ must be di¤eomorphic to
S n 1 . Indeed, the restriction of H to such a component serves as a di¤eomorphism.
To proceed, we observe that from the assumption on H, it follows from implicit
function theorem that for any 2 S n 1 , H 1 ( ) is a smooth one dimensional
submanifold of , moreover H : ! S n 1 is a smooth …ber bundle (see [M]). Fix
THE EXISTENCE OF SUITABLE VECTOR FIELDS
a point x0 2 @ , let 0 = H (x0 ) and = H
(see Theorem 6.7 of chapter VII in [B])
n 1
( ; x0 ) !
n 1
; x0 !
1
3
( 0 ), then we have an exact sequence
n 1
Sn
1
;
0
!
n 2
( ; x0 ) :
If n 3, then both n 1 ( ; x0 ) and n 2 ( ; x0 ) vanishes, this shows n 1 ; x0 =Z
e
and hence is di¤eomorphic to B 2 nB1 . If n = 2, then since 1 ( ; x0 ) vanishes and
; x0 is again isomorphic to Z, this shows must
0 ( ; x0 ) is …nite, we see 1
be di¤eomorphic to B 2 nB1 .
Now we are ready to prove the theorem.
Proof of theorem 1. First if = B1 or B2 nB 1 , then G (x) = x satis…es the assumption in the Lemma 1, half of the theorem follows from the lemma and Corollary 1.
On the other hand, assume for some , we may …nd a smooth map F satisfying (2).
For x 2 @ , choose a base for the tangent space of @ at x, namely e1 ;
; en 1 ,
then
rank (F (x) ; @1 F (x) ;
; @n F (x))
= rank (F (x) ; F e1 ;
; F en 1 ; F )
= rank (F (x) ; F e1 ;
; F en 1 ) = n;
we see F (x) 6= 0 on @ . Moreover, it follows from the fact
rank (F (x) ; @1 F (x) ;
; @n F (x)) = n for any x 2
that the zeroes of F in must be isolated, hence only …nitely many, say x1 ;
Then for " > 0 small enough, let
U= n
we know
rank (F (x) ; @1 F (x) ;
and
Let
m
[
; @n F (x)) = n for any x 2 U
H (x) =
rank (@1 H (x) ;
B" (xi );
i=1
n
dim span F (x) ; im ( F j@U )
then clearly
; xm .
;x
o
= n for any x 2 @U:
F (x)
for x 2 U ,
jF (x)j
; @n H (x)) = n
1 for any x 2 U
and
dim im ( Hj@U )
;x
=n
1 for any x 2 @U:
It follows from the Lemma 2 that U must be di¤eomorphic to B 2 nB1 , hence
be di¤eomorphic to either B 1 or B 2 nB1 .
must
Acknowledgment: The research of the …rst author is supported by National
Science Foundation Grant DMS-0209504. We would like to thank Professor WeiMing Ni and Professor Peter Poláµcik for very helpful discussions.
4
FENGBO HANG AND HUIQIANG JIANG
References
[A]
H. Amann. Existence and regularity for semilinear parabolic evolution equations. Scuola
Morm. Sup. Pisa Cl. Sci. 11(4):593–696, 1984.
[B] G. E. Bredon. Topology and geometry. Graduate Texts in Mathematics, 139. SpringerVerlag, New York, 1997.
[DP] E. N. Dancer and P. Poláµcik. Realization of vector …elds and dynamics of spatially homogeneous parabolic equations. Mem. AMS, 140(668), 1999.
[H] D. Henry. Geometric theory of semilinear parabolic equations. Lecture notes in mathematics,
Vol. 840, Springer–Verlag, New York, 1983.
[M] J. W. Milnor. Topology from the di¤erentiable viewpoint. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997.
[P1] P. Poláµcik. Complicated dynamics in scalar semilinear parabolic equations in higher space
dimension. Journal of Di¤ erential Equations. 89(2):244–271, 1991.
[P2] P. Poláµcik. Imbedding of any vector …eld in a scalar semilinear parabolic equation. Proc.
AMS. 115(4):1001–1008, 1992.
[P3] P. Poláµcik. Reaction-di¤usion equations and realization of gradient vector …elds. In International Conference on Di¤ erential Equations (Lisboa, 1995), 197–206. World Scienti…c
Publishing, River Edge, NJ, 1998.
[Pr] M. Prizzi. Perturbation of elliptic operators and complex dynamics of parabolic partial
di¤erential equations. Proc. Roy. Soc. Edinburgh Sect. A, 130(2):397–418, 2000.
[PR] M. Prizzi and K. P. Rybakowski. Some recent results on chaotic dynamics of parabolic
equations. In Proceedings of the Conference "Topological Methods in Di¤ erential Equations
and Dynamical Systems" (Kraków-Przegorzaly, 1996), no. 36, 231–235, 1998.
[R] K. P. Rybakowski. The center manifold technique and complex dynamics of parabolic equations. In Topological methods in di¤ erential equations and inclusions (Montreal, PQ, 1994),
Volume 472 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 411–446. Kluwer Acad. Publ.,
Dordrecht, 1995.
E-mail address : [email protected]
Department of Mathematics, Michigan State University, East Lansing, MI 48824
E-mail address : [email protected]
School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St.
S.E., Minneapolis, MN 55455