- Email: [email protected]
Slow Motion Manifolds For A Class of Singular Perturbation Problems: The Linearized Equations
Slow Motion Manifolds For A Class of Singular Perturbation Problems: The Linearized Equations N. D. Alikakos*
University of Tennessee
and
University of Crete, Greece
P. W. Bates
Brigham Young University
G. Fusco I1 University of Rome, Italy
Introduction
0
Consider the Cahn-Hilliard equation Ut
as 0
=
(--E2UXX -
f(u)),,
< 2 < 1, subject to boundary conditions u, = u,,, = 0
at
x = 0,l
where F(F' = f) is a double-well function with equal depths. In 111 we rigorously established the existence of some special slow motion manifolds for (1)by carrying out for this equation the general program described in [6] and by taking into account the works of 141 and [ 5 ] . The analysis in [l]is divided naturally into three parts: (i) Spectral analysis of the linearized operator about the elements of an approximate manifold.
N . D. Alikakos, P. W. Bates and G. Fusco
2
(ii) Refinement of the first approximation via linearization. (iii) Existence of the slow motion manifold in a neighborhood of the approximate manifold by a perturbation argument. This method appears to be quite general and applicable to a large class of singular perturbation problems. In the present paper we restrict ourselves to a careful description of (ii). We choose to give a detailed presentation of this part for the bistable equation (a), for 1 layer solutions,
+
= E2UXZ f (.) u, = 0, x = 0 , l . Ut
Though part (ii) is only linear analysis, it provides a very good approximation to the true dynamics on the slow motion manifold (see Theorems A and B below in $2). We note that for (2) the existence of extremely slowly evolving solutions was established rigorously in [4], [6], [ 5 ] , 131 (see also [S] for where the phenomenon was first pointed out by formal analysis). A detailed presentation of part (iii) for (2) is given in [5].We refer the reader to Fife [7] for a large class of singular problems that includes the equations (1) and (2) and for which our methods have relevance.
1 Consider for small E
{ where
Ut
> 0 the equation
= E2UXX
u, = 0,
t f(U), 0 < z < 1, t 2 0. x = 0,l
(3)
+
f(u)= -u(u - 1)(u 1).
Let e l ( x ) , e - l ( x ) denote the monotone increasing/decreasing single node, unstable equilibria of (3). By symmetry the node is at z =
i.
Slow Motion Manifolds
1.1
3
The Approximate Manifold
M
We extend el periodically on R and denote by e ( z ) the extended function. Let d ( z ) = e(z - I), o 5 z 5 I. Fix a small number 6 > 0 and consider in L2(0,1) the set -
M =
{d/t€ I
:= (--
1 2
+ 6,-2l - 6)} .
We expect that for small E > 0 M will provide a good approximation to the connections between the equilibria -1, el and + l . Our objective is to establish that indeed this is the case. So we seek the true connection, M , as a graph over M . Note that M contains the equilibria -1, e l , +l.
1.2
The Co-ordinate System
For u in a neighborhood of M we consider the change of co-ordinates
$
in L 2 ( 0 ,1) where V I manifold M in the form:
M =W 1.3
($ := g ). We seek the 1 dimensional
) / tE I > = { 40 = + V ( W E I } .
The Reduced Flow on M
We expect that on M the solution u ( . , t ) to (3) can be represented in the form 44) = u ( t ( t > > Therefore by (3) from ut =
qt
UEi = E2U,,
+
we obtain
f(U).
(4)
N . D. Alikakos, P. W. Bates and G.Fusco
4
The reduced flow on M is given by
where b(J) is a function we seek to determine in terms of f . Hence (4)can be rewritten as
and so using the representation of M b(J)
[$+ vg] = E 2 [&
Recalling that
+ V k ] + f(d+ V ) .
(5)
+ f(d)= 0
E2&
we obtain from ( 5 ) and the condition V 1$ in L2 a system of two equations, b(J)
[$+ vg] v,
+ f(d+ V )- f(d)
= E2Vzz =
-i$
z,
z
(6)
= 0,l.
(V,$) = 0
(7 )
for V(J,z) and b ( [ ) , where (,) denotes the L2 inner product. Note that V(0,z) = 0 and b ( 0 ) = 0. (8)
1.4
The Iteration Scheme
We change independent variables in (6) so that the boundary conditions become homogeneous. For this purpose we set 20
= V - [Ax2
where
A = A ( [ )=
1 2
-
[?&O)
+ Bz] - $(1)]
B = B ( [ ) = -?&(O).
Slow Motion Manifolds
5
(6), (7) and (8) now take the form (A' := b([)
3)
[$+ U J ~+ A'z2 + B'z] = E'w,,
+ f(d+ w + Ax2 + B z ) - f(d)+ E
w,(F,O) =
+
~ ( A Bz),,, ~ ~
%(t,1) = 0,
(w+ Ax2 + B z , $ ) = 0
w(0,z) = Az2
+ Bz,b(O) = 0.
The system above can be written abstractly in the form
where F i ( w , b ) = E'w,,
-+f (J + w + (Ax2+ B z ) ) - f(d)
+ + A'z2 + B'z] + & ( A z 2+ Bz),, F 2 ( ~ , b= ) + Ax2 + B z , 4 ) -b[$
W<
(W
applying the standard Newton scheme to this equation we have (w",Cn)
= (wn-1,cn-I)
We set Wn
wo(t,z)
- [F'(wn-l,c"-l]-l
= Wn-l - h,
= -(Az2 + Bz),
F(wn-1,cn-l).
cn = en-' - I;, CO([)
= 0,
n = 1 , 2 , . ...
More explicitly h = h ( [ , z ) ,k = k([) were determined at each step by
+ f'(d+ tun-' + Ax2 + B z ) h - cn-'---dh = k[$ + w:-' + A'z2 + B'z] + E ~ w , " , ~ +f(d+ wn-l + Ax2 + Bz)- f(d) -cn-'[< + wy-' + A'z2 + B'z] + E'[Az' + Bz],,.
E2hz,
(9)
N . D.Alikakos, P. W . Bates and G. Fusco
6
with boundary and initial conditions
and
Note that for n = 1 the equation (9) is elliptic, while for n 2 2 it is parabolic with ( playing the role of the time variable. We expect this scheme to be well defined and convergent (cf. Fusco [ 5 ] ) . In the present paper we restrict our study to the first approximation, n = 1.
2
The lStApproximation
We note that for n = 1 the equation in terms of the original variable V takes the form E2VZX
vx (V,+
+
f’(U<)V = cut -4
(10)
=
-uz, 4 5 = 0,l.
(11)
= 0.
(12)
This equation can be obtained directly by (formally) linearizing (6). We seek to determine V = V([,z), and c = c([) so that (10)) (11) and (12) are satisfied. We will also obtain precise estimates on c ( ( ) and on the difference Ib([) - c(()l. In fact we will show that c([) is the first term in the asymptotic expansion of b ( [ ) . More precisely we will establish the following two theorems:
Theorem A. There is a unique pair (c([))V‘) that solves ( l o ) , (11)) (12), and moreover the following estimates hold for small E :
Slow Motion Manifolds
7
where p1 := p1 is the first eigenvalue of the linearized operator C,,t := f'(d)with homogeneous Neumann conditions, 11 11 the L2-norm.
~~g +
Theorem B (cf [6]) The following estimate holds for small
E:
We note that E $ ( O ) , & ( O ) are transcendentally small in E and so Theorem B above establishes (see part(i) of Proposition A) that the speed b(<) is transcendentally small.
2.1
Preliminaries
Consider the operator
with homogeneous Neumann boundary conditions at x = 0 , l . We will be concerned with properties of the spectrum of C,X. The associated eigenvalue problem is
E~H;,, H;, = 0,
+
f'(d)H; = &"H; z = 0,1, i = 1,2,...
Proposition A For E sufficiently small the spectrum of C,,,,PI'' > pf"
(13)
> . . . satisfies
(ii) There is a positive constant C = C'(S), independent of E and [ such that $," 5 -C < 0. Proposition B Let IT1 be the principal eigenfunction of JC,,~,L2 normalized so that IlHlIl = 1. Then we have
N . D. Alikakos, P. W. Bates and G. Fusco
8
where F ( E )satisfies
and where u = u(n)is the heteroclinic orbit of u
+ f(u)= 0,
connecting -1 to +l. We will assume part (ii) of Proposition A and give proofs of the remaining statements. We refer to [4]for the original proofs of general versions of Propositions A and B and to the appendix in [l] for still another approach. Proof of Proposition B Let 5 = - [ax2 bz] where a = ;[E$,(l) and note that U , satisfies
+
6
+ f/(d)v =
E2EXZ
-
w, = 0,
- eX(0)],b = 6,,(0),
+
+
- [ 2 a ~ ~f ’ ( d ) ( a z 2 bz)]=: Q 2 = 0,l.
(15)
By considering (13) above for i = 1 and subtracting (15) we obtain E2(H1-
E)xz
+ f’(d)(Hl - =) = p1H1 - Q (HI
where
p1 E p;”.
We can express
- 5),= H1
0,
2
= 0,l
(16) (17)
- U , in the form
Hi-E=CiHi+R,
(18)
with R I Hl(in L2 sense) and c1 a constant (depending on E and [) to be determined. Substituting this expression in (16) and (17) we obtain c2Rzz f ‘ ( d ) R= p1(1 - c l ) H ~- Q R, = 0 (R,Hl)= 0
+
From this we easily obtain
9
Slow Motion Manifolds
and from part (ii) of Proposition A it follows that
Going back to (18) and reverting to
~
from which we obtain Hi(1 - ~
1= )
+ R - [ a x 2 4- 6x1.
Therefore by taking L2 norms and using (19) we arrive at I1 - c11 =
IIGII + 0 (Gm)
(Note that by the symmetry o f f a = -b = -&).
By a well known
-&
= -&~Z(E) we obtain the first estimate and therefore setting in (14). With an extra argument that utilizes (20) we can obtain the second estimate. Proof of part (i) of Proposition A We begin with the identity
1. Set z = &ii$.
By Proposition B established above
N . D. Alikakos, P. W. Bates and G. Fusco
10
On the other hand
-
J’
{&2
1’{
[(H’
- Z)J2
0
+
0
+2Jo =: I
- f’(d)(Hl - 2 ) 2 }
dz
+
- E ~ ( z , ) ~ f’(d).”} da:
1 { E 2 ( H 1 - Z)&
+ 11+ I I I .
-f
‘ ( w f 1
- +} dx
It is not difficult to see that
11 = 0 (by symmetry),
2. We will assume that Therefore
p1
> 0, a fact that
is easy t o establish.
and so P1 [ 1 + 2
J,’ H 1 z d z ] = 0 ( ( & ( 0 ) ) 2 /& )
*
(23)
From (23), using once more (22) we obtain the desired result.
Remarks on the Cahn
- Hilliard Operator
Our approach above for studying the spectrum can be summarized as follows: First part (ii) of Proposition A is established and then information on the size of the small eigenvalue is established, indirectly,
Slow Motion Manifolds
11
via its corresponding eigenfunction whose shape we approximately know (Proposition B). Our strategy in [l]for studying the spectrum of the Cahn-Hilliard operator is similar, thanks to the variational characterization of eigenvalues due to Bates and Fife [a]. We now describe a different approach for estimating the (unstable) eigenvalues for the Cahn-Hilliard problem, which gets directly to the eigenvalue via a reduction t o the bistable case, and which does not presuppose the analog of part (ii) of Proposition A for the Cahn-Hilliard operator. For perturbations with zero average the appropriate eoigenvalue problem for the Cahn-Hilliard equation takes the form - ~ 2 h ; x x x-x ( f ' ( 8 ) h i )
xx
= A:"h;
hi, = hixrx = 0 , x = 0,1, i = 1 , 2 , ... h;dx = 0 ,
Jt
where 8 stands for a multilayer analog of d.The important points about $ are: First that p;'( > 0 and second that information on the smallness of &", the principal eigenvalue of the corresponding eigenvalue problem for the bistable operator, is available. Proposition
o < A;" 5 (Const.) &2 Proof
1. Set
p1
E E := pl' ,h := h l ,
H(x)=
I
X
h(x)dx.
Following Langer we note the identity A1
Jt H 2 d x --_ Jt [E2(hx)2- f ' ( d ) h 2 ]d x . Jth2dx
Jo]
h2dx
(24)
N . D.Alikakos, P. W. Bates and G. Fusco
12
By the hypothesis X I > 0; it follows that E2
I’
1
1
(hx)2dz 5
0
f‘(8)h2dz 1
5 m i h2dx
Where m = m a x t (-f’($(z))),
and therefore
X
L1(h,)2dz 5
&
J’o h2d x .
2. The inequality ( 2 5 ) shows that h is special (as expected) with oscillation that can be measured in terms of The following lemma makes precise this intuition.
5.
Lemma
Consider the Fourier expansion for h, h = C y c k d k . Set B = and let ko be the largest integer k satisfying
2
a k 2 5 2B. Then
1 -lIh1l2
4B
5 IP1I2
7
where llh112 := s,’ h2 dz.
Proof of Lemma Consider the sine/cosine Fourier expansions. By ( 2 5 ) M
and so
M
13
Slow Motion Manifolds
Therefore
Hence ko+l
1
1
On the other hand
and so by the estimate above
2 llh1I2 3. Conclusion. From (24) via the lemma above we obtain ’
X1(Const.)E2
5
-
s,’ [&2(h,)2+ f ’ ( 8 ) h 2 ]d x s,’ h2dx
Ji [&2(h,)2+ f ’ ( 8 ) h 2 ]dx
Jt h2 d x
E9
= Pl . t
1
We note that the estimate given in the proposition above is not included in the general spectral comparison principles in [ 2 ] . We also note that the estimate is not sharp. In fact the optimal estimate is
and though this improvement is not really needed in the study of the 1-dimensional problems, it would be very important for higher dimensional considerations. For the reader who may wonder where we missed we note that our derivation is based on very general considerations and is not taking into account the special shape of the eigenfunc tion.
N . D.Alikakos, P. W. Bates and G. Fusco
14
2.2
Solution of the 1”‘ Approximate Equation [Proof of Theorem A]
Consider (9) for n = 1 and recall the connection with system (lo), (11)) (12). Dropping the superscripts we obtain
{ for
+
+
+
E ~ w , , f’(T8)w = - E ~ [ ABz],, ~ ~ - f’(T8)[Az2 B x ] w, = 0) x = 0 ) l ( w ,&pa$) = -([Ax2 B x ] ,&pa$)
+
C<
+
= F ( E )as defined above;
T$ = -3 was used.
(26)
1. Reduct ion of orthogonality condition
Set
w=w
+ { (Ax2+ B z , &iI 6) + (w,&p 6- HI)} H I
where llHlll = 1. Then (26) takes the form E
~
+ f’(T8)E E ~ =~p ( E - W ) - E ~ [ A+xBzIXz ~ - f‘(d)[Az2 + B z ] + cT$
w, = 0) 2 = 0) 1 (rn,Hl) = 0
(27)
2. The map w +. W
Ti? = w
+ (w,&p$
+
-
+(Ax2 B x , & ~ ~ ) H I =: Q ( w ) ( A z 2 B x ,& p < ) H l .
+
+
The linear map Q is a small perturbation of the identity. In fact
I I Q b ) - wII L IIwIl II&p6
-
HI/I = 41) l l w l l ~
(28)
where at this point we made a very mild use of Proposition B. Therefore I P - wll = ~(1>11~11 +O(~(O>>. (29)
Slow Motion Manifolds
3 . The map c
15
+ w(c)
Now we return to (26) and ignoring for the time being the angle condition (w,&iI$) = - ( [ A x 2 B z ] , & p $ ) , we study the dependence of the solution w on (arbitary) c = c ( 0 . By Proposition A for small E zero is not in the spectrum of L,,J and therefore there is a unique w for any given c. Clearly
+
where
k l , 162
satisfy
E2k2zz
k2,
Let
CT
+
= 0,
fl($)k2
+
= - E ~ [ ABz],, ~ ~ - f'(d)[Az2
2
=0,1.
+ Bz]
(31)
.
+ BE],, - f'($))[Az2 + Bz].
:= -c2[Az2
Estimation of kl 00
kl = ~ ( k l , H i ) H i ,IIHiII = 1, i = 1 , 2 . . . . i=l
Therefore dropping superscripts in the pf", Czl p i H i ( k 1 , H i ) = X z i ( $ , H i ) H i , and SO ( k 1 , H ; ) = ($, H i ) / P i . Hence
Therefore, by Proposition A, part (ii)
N . D. Alikakos, P. W. Bates and G . Fusco
16
Estimation of
k2
Equation (31) can be written in the form
{
c2k2Xx
k2z
+f’(d)k2 =
= 0, x = 0 , l
(0, Hi)Hi
+
0 ’
where gL is the projection of u in the orthogonal complement of the eigenspace generated by H I . Thus P1
where by Proposition A, part (ii)
4. Satisfying the angle condition
Next we show that c can be chosen so that the orthogonality condition (W,H1) = 0 is satisfied. From (27) we see that the requirement on c takes the form 4
C(U[,
H1) = -P1@
-
w,H1)
+ ( c 2 ( A z 2t Bx),,
+ .f’(d)(Ax2 + Bx),Hi).
(34)
Note that the right hand side of (34) depends (linearly) on c. We need to estimate the coefficient of c on that side and show it is negligible with respect to ( $ , H I ) . From (29), (30) we have IIW -
41 = o ( ~ > I I ~ 1 I I I+c lo ( W 2 l l + Wm))
where in fact the o(1) term is O(~x(0)/& (see (28) and recall estimate (14)). Therefore the first term on the right of (34) can be estimated as follows:
Slow Motion Manifolds
17
which by (32) (and also (33)) equals
From this computation it is clear that (34) is solvable for c. Notice also that in obtaining this conclusion we used only the very weak estimate (consequence of (14))
5 . Calculating c
Consider equation (34). We begin with the coefficient of c:
-($,fh) = ($+l) = (&
H1 - & P ( 4
6) + &P(E)(&&
Next we examine the second term on the right of (34), (.,HI) (see definition of CT below (31)): Integrating by parts we obtain
+
(.,HI) = - P I ( I I I , A z ~ B z ) - (2A t B)H1(1) t BHl(0) and therefore
(.,Hi) = $(1)H1(1)
-
$(O)
- p1(H1,Az2t
Bz).
(36)
Note that (a,H1) It=o= 0. We will show that the term involving is negligible. For this purpose we need to calculate further. First by the symmetry o f f , ?$(l)= -?$(O), & ( O ) = -&(1). Next from
E2Hlzz t f’(+)Hl = PlH1 H I , = 0,
z =
0,l
N. D . Alikakos, P. W. Bates and G. Fusco
18
and
E2(g)xx + f’(d)$= 0
we obtain by simple manipulations the identity
Utilizing this in (36) we arrive at (031)
= - p16(o)
[J,’ HIT& dx - gx(0) J, H1(x2 - x ) dx] . 1
&X(O)
Finally, employing estimate (14) we obtain
Now combining (37) with the estimate on ( $ , H I ) at the beginning of step 5 and with estimate ( 3 5 ) we obtain that
6. The V-estimate
Consider (27) for the special value of c for which the orthogonality condition holds. Recall also that
where
Q(w)= w
+ ( w ,f i F $
- H1)Hi.
By estimate (as),Q is a small perturbation of the identity. Let S be its L2 inverse, S = Q-’.From above we have the identity
S(G)= w
+ (Ax2 + B x , fip
$)S(Hl).
Slow Motion Manifolds
1
19
Subsituting in (27) w in terms of E we have
+ f’(Et))?IT- p 1 E
&2EZ,
+
+
p1(Az2 Bz,&L$)S(Hl) - E ~ [ +ABz],, ~ ~ - f’(d)[Az2 B z ] + c$ w,=o, z = O , l ( q H 1 ) = 0. = -plS(E)
+
Taking the inner product with E
+ f ’ ( d ) E- p 1 q m ) 5 Pll(s(q,$l + p11(Az2+ BX,&Ti $)ll(~(m,~)l + 4I ( v 4 1+
(E2E,,
ICII(~~,W)I*
The lefthand side can be estimated from below via Proposition A, part (ii) (since b I H I ) , by (Const.) llE112. On the other hand the right hand side can be estimated from above by 0 ( g ( 0 ) ) llzUll, and so the estimate ll.4l = O ( G ( 0 ) ) (39) follows. In obtaining (39) we made use of (38), and Proposition A part (i). From this we obtain easily the estimate
1141= 0
(m)) 7
and so the proof of Theorem A is complete.
Proof of Theorem B Consider the Manifold M =
{d + vt/tE I }
M
where V = V t is the solution of ( l o ) , ( l l ) ,(12). may be thought as a refinement of the approximate manifold M with elements that now satisfy homogeneous Neumann boundary condition. In a neighborhood of this we introduce the co-ordinate system u =
d + v‘
+ w,(w,$)= 0
w, = 0,
5
= 0,l.
which defines the change of co-ordinates
N . D. Alikakos, P. W. Bates and G. Fusco
20
Derivation of equation in new co-ordinate system Ut
2
= E,,
+
f(U) = E2[&
+ v:, + wx,]+ f(d+ v + w)
On the other hand
and therefore we have
Note also the identity
from which it follows that
By projecting equation (40) on ii$ and using the identity above we obtain the equation
- (E2WZZ
+ f(d+ v +
. I ) -
4 4
(Uzt
@)
4
-
f/(d)V,7&)
which can be written more compactly in the form
By projecting in the complementary directions an equation for w can be obtained, which together with (41) makes up the system that we are required to solve. This is done in [5]. In this paper we assume this important point. Note that O([,O) # 0 in general, and so the
Slow Motion Manifolds
21
size of w is determined by 01([, 0). A computation reveals that the crucial term in the expression for 01([,0) is
Note that for establishing the estimate in Theorem B the estimate on V provided by Theorem A is not sufficient. That estimate shows that this term is 0 ( ( g ( 0 ) ) 2 ) up to algebraic terms in E-', while we dearly need an improved estimate. G. Fusco and J. Hide observed this difficulty long time ago and gave convincing arguments in [6] that V is in fact smaller near the interface +[. In fact it is possible to refine the V-estimate in Theorem A and deduce Theorem B. We describe below one way for carrying this out.
3
The refinement of the V-estimate Consider system (10) ,(11)and (12):
+
&2VXX f'(&)V = cut 4
v,
=
-$, z
= 0,l
(V,+ = 0.
By the variation of parameters formula we have the following repreV:
. sentation for
where for [ # 0 a) zo is the point in ( 0 , l ) at which -iiF3(zo) = 0.
b)
N . D. Alikakos, P. W. Bates and G . Fusco
22
1
t1=+
3
t 2 = -2- t
[Note: { E $ , q } is a fundamental system for the homogeneous equation E’v,, f’(d)v = o ]
+
c) A17 B1 constants.
So far no explicit use of the boundary/othogonality conditions has been made. We will need to utilize these for estimating the constants A17 B1.
First by evaluating the right hand side of (43)at x = xo we obtain
and therefore by a simple regularity argument utilizing the V-estimate in Theorem A, B1 = O ( G ( O ) / & ) ~ , ( ~ o ) (44) On the other hand the orthogonality condition can be used for relating A1 to B1. Indeed from
we obtain
Noting that $4 is pointwise bounded, and recalling (21) we obtain that the right hand side of (45)is bounded by
I4 (Const .) -. &
It follows, by making use of (21) once more, that
+
[All 5 (Const.)&IB11 (Const.)lcl.
Slow Motion Manifolds
23
Estimates (44), (46) provide the improved estimate on V that we have been seeking. Note that our estimate gets weaker near z = z o (see definition of 4 at z = zo). However this aspect does not present a problem for estimating the term in (42) since near z = z o 4 is exponentially small. By splitting the integral in (42) in two parts we obtain the desired improved estimate. N.A. was partially supported by the National Science Foundation, the Science Alliance, a Program at the University of Tennessee, and the Institute of computational Mathematics in Crete.
B iblio graphy [l] N. D. Alikakos, P. VC': Bates and G. Fusco, Slow Motion f o r the Cahn-Hilliard Equation I n one Space Dimension, J. D. E., to
appear.
[2] P. W. Bates and P. S. Fife, Spectral comparison Principles for the Cahn-Hilliard and Phase-Field Equations, Time Scales for Coarsening, Physica D., to appear.
[3] J. Carr and R. L. Pego, Very slow phase separation in one dimension to appear in Proc. Conf. on Phase Transitions, Nice, M. Rascle, ed. 1987. [4] J. Carr and R. L. Pego, Metastable Patterns in Solutions of ~t = E ~ - fu( ~ )~CPAM , ~ 42 (1989), p. 523-576.
[5] G. Fusco, A Geometric Approach to the Dynamics of ut = E~u,, +f(u)For Small E , Proceeding of the Stuttgart conference in honor of J. K. Hale, ed. K. Kirchgassner. [6] G. Fusco, and J . K. Hale, Slow Motion Manifolds, Dormant Instability and Singular Perturbations, Dynamics and Diff. Equations l (1989), p. 75-94.
24
N . D.Alikakos, P. W . Bates and G. FUSCO
[7] P. Fife, Pattern Dynamics for Parabolic PDE’s, t o appear in “Introduction to Dynamical Systems” Proc. workshop at IMA,
1989, M. Golubitsky et a1 Eds., Springer.
[8] J. Neu, Unpublished lecture Notes.
University of Tennessee
and
University of Crete, Greece
P. W. Bates
Brigham Young University
G. Fusco I1 University of Rome, Italy
Introduction
0
Consider the Cahn-Hilliard equation Ut
as 0
=
(--E2UXX -
f(u)),,
< 2 < 1, subject to boundary conditions u, = u,,, = 0
at
x = 0,l
where F(F' = f) is a double-well function with equal depths. In 111 we rigorously established the existence of some special slow motion manifolds for (1)by carrying out for this equation the general program described in [6] and by taking into account the works of 141 and [ 5 ] . The analysis in [l]is divided naturally into three parts: (i) Spectral analysis of the linearized operator about the elements of an approximate manifold.
N . D. Alikakos, P. W. Bates and G. Fusco
2
(ii) Refinement of the first approximation via linearization. (iii) Existence of the slow motion manifold in a neighborhood of the approximate manifold by a perturbation argument. This method appears to be quite general and applicable to a large class of singular perturbation problems. In the present paper we restrict ourselves to a careful description of (ii). We choose to give a detailed presentation of this part for the bistable equation (a), for 1 layer solutions,
+
= E2UXZ f (.) u, = 0, x = 0 , l . Ut
Though part (ii) is only linear analysis, it provides a very good approximation to the true dynamics on the slow motion manifold (see Theorems A and B below in $2). We note that for (2) the existence of extremely slowly evolving solutions was established rigorously in [4], [6], [ 5 ] , 131 (see also [S] for where the phenomenon was first pointed out by formal analysis). A detailed presentation of part (iii) for (2) is given in [5].We refer the reader to Fife [7] for a large class of singular problems that includes the equations (1) and (2) and for which our methods have relevance.
1 Consider for small E
{ where
Ut
> 0 the equation
= E2UXX
u, = 0,
t f(U), 0 < z < 1, t 2 0. x = 0,l
(3)
+
f(u)= -u(u - 1)(u 1).
Let e l ( x ) , e - l ( x ) denote the monotone increasing/decreasing single node, unstable equilibria of (3). By symmetry the node is at z =
i.
Slow Motion Manifolds
1.1
3
The Approximate Manifold
M
We extend el periodically on R and denote by e ( z ) the extended function. Let d ( z ) = e(z - I), o 5 z 5 I. Fix a small number 6 > 0 and consider in L2(0,1) the set -
M =
{d/t€ I
:= (--
1 2
+ 6,-2l - 6)} .
We expect that for small E > 0 M will provide a good approximation to the connections between the equilibria -1, el and + l . Our objective is to establish that indeed this is the case. So we seek the true connection, M , as a graph over M . Note that M contains the equilibria -1, e l , +l.
1.2
The Co-ordinate System
For u in a neighborhood of M we consider the change of co-ordinates
$
in L 2 ( 0 ,1) where V I manifold M in the form:
M =W 1.3
($ := g ). We seek the 1 dimensional
) / tE I > = { 40 = + V ( W E I } .
The Reduced Flow on M
We expect that on M the solution u ( . , t ) to (3) can be represented in the form 44) = u ( t ( t > > Therefore by (3) from ut =
qt
UEi = E2U,,
+
we obtain
f(U).
(4)
N . D. Alikakos, P. W. Bates and G.Fusco
4
The reduced flow on M is given by
where b(J) is a function we seek to determine in terms of f . Hence (4)can be rewritten as
and so using the representation of M b(J)
[$+ vg] = E 2 [&
Recalling that
+ V k ] + f(d+ V ) .
(5)
+ f(d)= 0
E2&
we obtain from ( 5 ) and the condition V 1$ in L2 a system of two equations, b(J)
[$+ vg] v,
+ f(d+ V )- f(d)
= E2Vzz =
-i$
z,
z
(6)
= 0,l.
(V,$) = 0
(7 )
for V(J,z) and b ( [ ) , where (,) denotes the L2 inner product. Note that V(0,z) = 0 and b ( 0 ) = 0. (8)
1.4
The Iteration Scheme
We change independent variables in (6) so that the boundary conditions become homogeneous. For this purpose we set 20
= V - [Ax2
where
A = A ( [ )=
1 2
-
[?&O)
+ Bz] - $(1)]
B = B ( [ ) = -?&(O).
Slow Motion Manifolds
5
(6), (7) and (8) now take the form (A' := b([)
3)
[$+ U J ~+ A'z2 + B'z] = E'w,,
+ f(d+ w + Ax2 + B z ) - f(d)+ E
w,(F,O) =
+
~ ( A Bz),,, ~ ~
%(t,1) = 0,
(w+ Ax2 + B z , $ ) = 0
w(0,z) = Az2
+ Bz,b(O) = 0.
The system above can be written abstractly in the form
where F i ( w , b ) = E'w,,
-+f (J + w + (Ax2+ B z ) ) - f(d)
+ + A'z2 + B'z] + & ( A z 2+ Bz),, F 2 ( ~ , b= ) + Ax2 + B z , 4 ) -b[$
W<
(W
applying the standard Newton scheme to this equation we have (w",Cn)
= (wn-1,cn-I)
We set Wn
wo(t,z)
- [F'(wn-l,c"-l]-l
= Wn-l - h,
= -(Az2 + Bz),
F(wn-1,cn-l).
cn = en-' - I;, CO([)
= 0,
n = 1 , 2 , . ...
More explicitly h = h ( [ , z ) ,k = k([) were determined at each step by
+ f'(d+ tun-' + Ax2 + B z ) h - cn-'---dh = k[$ + w:-' + A'z2 + B'z] + E ~ w , " , ~ +f(d+ wn-l + Ax2 + Bz)- f(d) -cn-'[< + wy-' + A'z2 + B'z] + E'[Az' + Bz],,.
E2hz,
(9)
N . D.Alikakos, P. W . Bates and G. Fusco
6
with boundary and initial conditions
and
Note that for n = 1 the equation (9) is elliptic, while for n 2 2 it is parabolic with ( playing the role of the time variable. We expect this scheme to be well defined and convergent (cf. Fusco [ 5 ] ) . In the present paper we restrict our study to the first approximation, n = 1.
2
The lStApproximation
We note that for n = 1 the equation in terms of the original variable V takes the form E2VZX
vx (V,+
+
f’(U<)V = cut -4
(10)
=
-uz, 4 5 = 0,l.
(11)
= 0.
(12)
This equation can be obtained directly by (formally) linearizing (6). We seek to determine V = V([,z), and c = c([) so that (10)) (11) and (12) are satisfied. We will also obtain precise estimates on c ( ( ) and on the difference Ib([) - c(()l. In fact we will show that c([) is the first term in the asymptotic expansion of b ( [ ) . More precisely we will establish the following two theorems:
Theorem A. There is a unique pair (c([))V‘) that solves ( l o ) , (11)) (12), and moreover the following estimates hold for small E :
Slow Motion Manifolds
7
where p1 := p1 is the first eigenvalue of the linearized operator C,,t := f'(d)with homogeneous Neumann conditions, 11 11 the L2-norm.
~~g +
Theorem B (cf [6]) The following estimate holds for small
E:
We note that E $ ( O ) , & ( O ) are transcendentally small in E and so Theorem B above establishes (see part(i) of Proposition A) that the speed b(<) is transcendentally small.
2.1
Preliminaries
Consider the operator
with homogeneous Neumann boundary conditions at x = 0 , l . We will be concerned with properties of the spectrum of C,X. The associated eigenvalue problem is
E~H;,, H;, = 0,
+
f'(d)H; = &"H; z = 0,1, i = 1,2,...
Proposition A For E sufficiently small the spectrum of C,,,,PI'' > pf"
(13)
> . . . satisfies
(ii) There is a positive constant C = C'(S), independent of E and [ such that $," 5 -C < 0. Proposition B Let IT1 be the principal eigenfunction of JC,,~,L2 normalized so that IlHlIl = 1. Then we have
N . D. Alikakos, P. W. Bates and G. Fusco
8
where F ( E )satisfies
and where u = u(n)is the heteroclinic orbit of u
+ f(u)= 0,
connecting -1 to +l. We will assume part (ii) of Proposition A and give proofs of the remaining statements. We refer to [4]for the original proofs of general versions of Propositions A and B and to the appendix in [l] for still another approach. Proof of Proposition B Let 5 = - [ax2 bz] where a = ;[E$,(l) and note that U , satisfies
+
6
+ f/(d)v =
E2EXZ
-
w, = 0,
- eX(0)],b = 6,,(0),
+
+
- [ 2 a ~ ~f ’ ( d ) ( a z 2 bz)]=: Q 2 = 0,l.
(15)
By considering (13) above for i = 1 and subtracting (15) we obtain E2(H1-
E)xz
+ f’(d)(Hl - =) = p1H1 - Q (HI
where
p1 E p;”.
We can express
- 5),= H1
0,
2
= 0,l
(16) (17)
- U , in the form
Hi-E=CiHi+R,
(18)
with R I Hl(in L2 sense) and c1 a constant (depending on E and [) to be determined. Substituting this expression in (16) and (17) we obtain c2Rzz f ‘ ( d ) R= p1(1 - c l ) H ~- Q R, = 0 (R,Hl)= 0
+
From this we easily obtain
9
Slow Motion Manifolds
and from part (ii) of Proposition A it follows that
Going back to (18) and reverting to
~
from which we obtain Hi(1 - ~
1= )
+ R - [ a x 2 4- 6x1.
Therefore by taking L2 norms and using (19) we arrive at I1 - c11 =
IIGII + 0 (Gm)
(Note that by the symmetry o f f a = -b = -&).
By a well known
-&
= -&~Z(E) we obtain the first estimate and therefore setting in (14). With an extra argument that utilizes (20) we can obtain the second estimate. Proof of part (i) of Proposition A We begin with the identity
1. Set z = &ii$.
By Proposition B established above
N . D. Alikakos, P. W. Bates and G. Fusco
10
On the other hand
-
J’
{&2
1’{
[(H’
- Z)J2
0
+
0
+2Jo =: I
- f’(d)(Hl - 2 ) 2 }
dz
+
- E ~ ( z , ) ~ f’(d).”} da:
1 { E 2 ( H 1 - Z)&
+ 11+ I I I .
-f
‘ ( w f 1
- +} dx
It is not difficult to see that
11 = 0 (by symmetry),
2. We will assume that Therefore
p1
> 0, a fact that
is easy t o establish.
and so P1 [ 1 + 2
J,’ H 1 z d z ] = 0 ( ( & ( 0 ) ) 2 /& )
*
(23)
From (23), using once more (22) we obtain the desired result.
Remarks on the Cahn
- Hilliard Operator
Our approach above for studying the spectrum can be summarized as follows: First part (ii) of Proposition A is established and then information on the size of the small eigenvalue is established, indirectly,
Slow Motion Manifolds
11
via its corresponding eigenfunction whose shape we approximately know (Proposition B). Our strategy in [l]for studying the spectrum of the Cahn-Hilliard operator is similar, thanks to the variational characterization of eigenvalues due to Bates and Fife [a]. We now describe a different approach for estimating the (unstable) eigenvalues for the Cahn-Hilliard problem, which gets directly to the eigenvalue via a reduction t o the bistable case, and which does not presuppose the analog of part (ii) of Proposition A for the Cahn-Hilliard operator. For perturbations with zero average the appropriate eoigenvalue problem for the Cahn-Hilliard equation takes the form - ~ 2 h ; x x x-x ( f ' ( 8 ) h i )
xx
= A:"h;
hi, = hixrx = 0 , x = 0,1, i = 1 , 2 , ... h;dx = 0 ,
Jt
where 8 stands for a multilayer analog of d.The important points about $ are: First that p;'( > 0 and second that information on the smallness of &", the principal eigenvalue of the corresponding eigenvalue problem for the bistable operator, is available. Proposition
o < A;" 5 (Const.) &2 Proof
1. Set
p1
E E := pl' ,h := h l ,
H(x)=
I
X
h(x)dx.
Following Langer we note the identity A1
Jt H 2 d x --_ Jt [E2(hx)2- f ' ( d ) h 2 ]d x . Jth2dx
Jo]
h2dx
(24)
N . D.Alikakos, P. W. Bates and G. Fusco
12
By the hypothesis X I > 0; it follows that E2
I’
1
1
(hx)2dz 5
0
f‘(8)h2dz 1
5 m i h2dx
Where m = m a x t (-f’($(z))),
and therefore
X
L1(h,)2dz 5
&
J’o h2d x .
2. The inequality ( 2 5 ) shows that h is special (as expected) with oscillation that can be measured in terms of The following lemma makes precise this intuition.
5.
Lemma
Consider the Fourier expansion for h, h = C y c k d k . Set B = and let ko be the largest integer k satisfying
2
a k 2 5 2B. Then
1 -lIh1l2
4B
5 IP1I2
7
where llh112 := s,’ h2 dz.
Proof of Lemma Consider the sine/cosine Fourier expansions. By ( 2 5 ) M
and so
M
13
Slow Motion Manifolds
Therefore
Hence ko+l
1
1
On the other hand
and so by the estimate above
2 llh1I2 3. Conclusion. From (24) via the lemma above we obtain ’
X1(Const.)E2
5
-
s,’ [&2(h,)2+ f ’ ( 8 ) h 2 ]d x s,’ h2dx
Ji [&2(h,)2+ f ’ ( 8 ) h 2 ]dx
Jt h2 d x
E9
= Pl . t
1
We note that the estimate given in the proposition above is not included in the general spectral comparison principles in [ 2 ] . We also note that the estimate is not sharp. In fact the optimal estimate is
and though this improvement is not really needed in the study of the 1-dimensional problems, it would be very important for higher dimensional considerations. For the reader who may wonder where we missed we note that our derivation is based on very general considerations and is not taking into account the special shape of the eigenfunc tion.
N . D.Alikakos, P. W. Bates and G. Fusco
14
2.2
Solution of the 1”‘ Approximate Equation [Proof of Theorem A]
Consider (9) for n = 1 and recall the connection with system (lo), (11)) (12). Dropping the superscripts we obtain
{ for
+
+
+
E ~ w , , f’(T8)w = - E ~ [ ABz],, ~ ~ - f’(T8)[Az2 B x ] w, = 0) x = 0 ) l ( w ,&pa$) = -([Ax2 B x ] ,&pa$)
+
C<
+
= F ( E )as defined above;
T$ = -3 was used.
(26)
1. Reduct ion of orthogonality condition
Set
w=w
+ { (Ax2+ B z , &iI 6) + (w,&p 6- HI)} H I
where llHlll = 1. Then (26) takes the form E
~
+ f’(T8)E E ~ =~p ( E - W ) - E ~ [ A+xBzIXz ~ - f‘(d)[Az2 + B z ] + cT$
w, = 0) 2 = 0) 1 (rn,Hl) = 0
(27)
2. The map w +. W
Ti? = w
+ (w,&p$
+
-
+(Ax2 B x , & ~ ~ ) H I =: Q ( w ) ( A z 2 B x ,& p < ) H l .
+
+
The linear map Q is a small perturbation of the identity. In fact
I I Q b ) - wII L IIwIl II&p6
-
HI/I = 41) l l w l l ~
(28)
where at this point we made a very mild use of Proposition B. Therefore I P - wll = ~(1>11~11 +O(~(O>>. (29)
Slow Motion Manifolds
3 . The map c
15
+ w(c)
Now we return to (26) and ignoring for the time being the angle condition (w,&iI$) = - ( [ A x 2 B z ] , & p $ ) , we study the dependence of the solution w on (arbitary) c = c ( 0 . By Proposition A for small E zero is not in the spectrum of L,,J and therefore there is a unique w for any given c. Clearly
+
where
k l , 162
satisfy
E2k2zz
k2,
Let
CT
+
= 0,
fl($)k2
+
= - E ~ [ ABz],, ~ ~ - f'(d)[Az2
2
=0,1.
+ Bz]
(31)
.
+ BE],, - f'($))[Az2 + Bz].
:= -c2[Az2
Estimation of kl 00
kl = ~ ( k l , H i ) H i ,IIHiII = 1, i = 1 , 2 . . . . i=l
Therefore dropping superscripts in the pf", Czl p i H i ( k 1 , H i ) = X z i ( $ , H i ) H i , and SO ( k 1 , H ; ) = ($, H i ) / P i . Hence
Therefore, by Proposition A, part (ii)
N . D. Alikakos, P. W. Bates and G . Fusco
16
Estimation of
k2
Equation (31) can be written in the form
{
c2k2Xx
k2z
+f’(d)k2 =
= 0, x = 0 , l
(0, Hi)Hi
+
0 ’
where gL is the projection of u in the orthogonal complement of the eigenspace generated by H I . Thus P1
where by Proposition A, part (ii)
4. Satisfying the angle condition
Next we show that c can be chosen so that the orthogonality condition (W,H1) = 0 is satisfied. From (27) we see that the requirement on c takes the form 4
C(U[,
H1) = -P1@
-
w,H1)
+ ( c 2 ( A z 2t Bx),,
+ .f’(d)(Ax2 + Bx),Hi).
(34)
Note that the right hand side of (34) depends (linearly) on c. We need to estimate the coefficient of c on that side and show it is negligible with respect to ( $ , H I ) . From (29), (30) we have IIW -
41 = o ( ~ > I I ~ 1 I I I+c lo ( W 2 l l + Wm))
where in fact the o(1) term is O(~x(0)/& (see (28) and recall estimate (14)). Therefore the first term on the right of (34) can be estimated as follows:
Slow Motion Manifolds
17
which by (32) (and also (33)) equals
From this computation it is clear that (34) is solvable for c. Notice also that in obtaining this conclusion we used only the very weak estimate (consequence of (14))
5 . Calculating c
Consider equation (34). We begin with the coefficient of c:
-($,fh) = ($+l) = (&
H1 - & P ( 4
6) + &P(E)(&&
Next we examine the second term on the right of (34), (.,HI) (see definition of CT below (31)): Integrating by parts we obtain
+
(.,HI) = - P I ( I I I , A z ~ B z ) - (2A t B)H1(1) t BHl(0) and therefore
(.,Hi) = $(1)H1(1)
-
$(O)
- p1(H1,Az2t
Bz).
(36)
Note that (a,H1) It=o= 0. We will show that the term involving is negligible. For this purpose we need to calculate further. First by the symmetry o f f , ?$(l)= -?$(O), & ( O ) = -&(1). Next from
E2Hlzz t f’(+)Hl = PlH1 H I , = 0,
z =
0,l
N. D . Alikakos, P. W. Bates and G. Fusco
18
and
E2(g)xx + f’(d)$= 0
we obtain by simple manipulations the identity
Utilizing this in (36) we arrive at (031)
= - p16(o)
[J,’ HIT& dx - gx(0) J, H1(x2 - x ) dx] . 1
&X(O)
Finally, employing estimate (14) we obtain
Now combining (37) with the estimate on ( $ , H I ) at the beginning of step 5 and with estimate ( 3 5 ) we obtain that
6. The V-estimate
Consider (27) for the special value of c for which the orthogonality condition holds. Recall also that
where
Q(w)= w
+ ( w ,f i F $
- H1)Hi.
By estimate (as),Q is a small perturbation of the identity. Let S be its L2 inverse, S = Q-’.From above we have the identity
S(G)= w
+ (Ax2 + B x , fip
$)S(Hl).
Slow Motion Manifolds
1
19
Subsituting in (27) w in terms of E we have
+ f’(Et))?IT- p 1 E
&2EZ,
+
+
p1(Az2 Bz,&L$)S(Hl) - E ~ [ +ABz],, ~ ~ - f’(d)[Az2 B z ] + c$ w,=o, z = O , l ( q H 1 ) = 0. = -plS(E)
+
Taking the inner product with E
+ f ’ ( d ) E- p 1 q m ) 5 Pll(s(q,$l + p11(Az2+ BX,&Ti $)ll(~(m,~)l + 4I ( v 4 1+
(E2E,,
ICII(~~,W)I*
The lefthand side can be estimated from below via Proposition A, part (ii) (since b I H I ) , by (Const.) llE112. On the other hand the right hand side can be estimated from above by 0 ( g ( 0 ) ) llzUll, and so the estimate ll.4l = O ( G ( 0 ) ) (39) follows. In obtaining (39) we made use of (38), and Proposition A part (i). From this we obtain easily the estimate
1141= 0
(m)) 7
and so the proof of Theorem A is complete.
Proof of Theorem B Consider the Manifold M =
{d + vt/tE I }
M
where V = V t is the solution of ( l o ) , ( l l ) ,(12). may be thought as a refinement of the approximate manifold M with elements that now satisfy homogeneous Neumann boundary condition. In a neighborhood of this we introduce the co-ordinate system u =
d + v‘
+ w,(w,$)= 0
w, = 0,
5
= 0,l.
which defines the change of co-ordinates
N . D. Alikakos, P. W. Bates and G. Fusco
20
Derivation of equation in new co-ordinate system Ut
2
= E,,
+
f(U) = E2[&
+ v:, + wx,]+ f(d+ v + w)
On the other hand
and therefore we have
Note also the identity
from which it follows that
By projecting equation (40) on ii$ and using the identity above we obtain the equation
- (E2WZZ
+ f(d+ v +
. I ) -
4 4
(Uzt
@)
4
-
f/(d)V,7&)
which can be written more compactly in the form
By projecting in the complementary directions an equation for w can be obtained, which together with (41) makes up the system that we are required to solve. This is done in [5]. In this paper we assume this important point. Note that O([,O) # 0 in general, and so the
Slow Motion Manifolds
21
size of w is determined by 01([, 0). A computation reveals that the crucial term in the expression for 01([,0) is
Note that for establishing the estimate in Theorem B the estimate on V provided by Theorem A is not sufficient. That estimate shows that this term is 0 ( ( g ( 0 ) ) 2 ) up to algebraic terms in E-', while we dearly need an improved estimate. G. Fusco and J. Hide observed this difficulty long time ago and gave convincing arguments in [6] that V is in fact smaller near the interface +[. In fact it is possible to refine the V-estimate in Theorem A and deduce Theorem B. We describe below one way for carrying this out.
3
The refinement of the V-estimate Consider system (10) ,(11)and (12):
+
&2VXX f'(&)V = cut 4
v,
=
-$, z
= 0,l
(V,+ = 0.
By the variation of parameters formula we have the following repreV:
. sentation for
where for [ # 0 a) zo is the point in ( 0 , l ) at which -iiF3(zo) = 0.
b)
N . D. Alikakos, P. W. Bates and G . Fusco
22
1
t1=+
3
t 2 = -2- t
[Note: { E $ , q } is a fundamental system for the homogeneous equation E’v,, f’(d)v = o ]
+
c) A17 B1 constants.
So far no explicit use of the boundary/othogonality conditions has been made. We will need to utilize these for estimating the constants A17 B1.
First by evaluating the right hand side of (43)at x = xo we obtain
and therefore by a simple regularity argument utilizing the V-estimate in Theorem A, B1 = O ( G ( O ) / & ) ~ , ( ~ o ) (44) On the other hand the orthogonality condition can be used for relating A1 to B1. Indeed from
we obtain
Noting that $4 is pointwise bounded, and recalling (21) we obtain that the right hand side of (45)is bounded by
I4 (Const .) -. &
It follows, by making use of (21) once more, that
+
[All 5 (Const.)&IB11 (Const.)lcl.
Slow Motion Manifolds
23
Estimates (44), (46) provide the improved estimate on V that we have been seeking. Note that our estimate gets weaker near z = z o (see definition of 4 at z = zo). However this aspect does not present a problem for estimating the term in (42) since near z = z o 4 is exponentially small. By splitting the integral in (42) in two parts we obtain the desired improved estimate. N.A. was partially supported by the National Science Foundation, the Science Alliance, a Program at the University of Tennessee, and the Institute of computational Mathematics in Crete.
B iblio graphy [l] N. D. Alikakos, P. VC': Bates and G. Fusco, Slow Motion f o r the Cahn-Hilliard Equation I n one Space Dimension, J. D. E., to
appear.
[2] P. W. Bates and P. S. Fife, Spectral comparison Principles for the Cahn-Hilliard and Phase-Field Equations, Time Scales for Coarsening, Physica D., to appear.
[3] J. Carr and R. L. Pego, Very slow phase separation in one dimension to appear in Proc. Conf. on Phase Transitions, Nice, M. Rascle, ed. 1987. [4] J. Carr and R. L. Pego, Metastable Patterns in Solutions of ~t = E ~ - fu( ~ )~CPAM , ~ 42 (1989), p. 523-576.
[5] G. Fusco, A Geometric Approach to the Dynamics of ut = E~u,, +f(u)For Small E , Proceeding of the Stuttgart conference in honor of J. K. Hale, ed. K. Kirchgassner. [6] G. Fusco, and J . K. Hale, Slow Motion Manifolds, Dormant Instability and Singular Perturbations, Dynamics and Diff. Equations l (1989), p. 75-94.
24
N . D.Alikakos, P. W . Bates and G. FUSCO
[7] P. Fife, Pattern Dynamics for Parabolic PDE’s, t o appear in “Introduction to Dynamical Systems” Proc. workshop at IMA,
1989, M. Golubitsky et a1 Eds., Springer.
[8] J. Neu, Unpublished lecture Notes.