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Asymptotic behavior of solutions of an allen-cahn equation with a nonlocal term
Nonlinear
Analysis
Theory,
Vol. 28, No. 7, pp. 128%1298. 1997 Copyright @ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-%6X/97 $17.00 + 0.00
Methods
& Applicatbm,
Pergamon 0362-546X(95)000217-0
ASYMPTOTIC XINFU
BEHAVIOR OF SOLUTIONS OF AN ALLEN-CAHN EQUATION WITH A NONLOCAL TERM
CHEN, t 1 DANIELLE
HILHORST
+ and ELISABETH
LOGAK
§
t Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.; * Laboratoire d’Analyse Numtrique, CNRS et Universite Paris-Sud, Batiment 425, 91405 Orsay, France; § Departement de Mathematiques et d’Informatique, Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France (Received
16 March
1994; received
in revisedform
Key words andphrases: Allen-Cahn propagation of interfaces,
equation,
27 July 1995; receivedfor
integro-differential
publication
equations,
29 November
singular
and
1995)
limit,
1. INTRODUCTION
In this paper, we study the limiting
behavior, as E I 0, of the solution ~8 of the following problem
s=Au+$f(u..s WE)
au -= 1
u"?x,
J n
U)inQxO%‘,
(1.1)
0 0)
0n =
gE(x)
a!2
x E
x
R+,
cl,
where Sz c IWN (N 2 2) is a smooth bounded domain and n is the unit outward normal to afi. We assume that f(u, v) is smooth and that f(u) = f(u, 0) is the opposite of the derivative of a double equal wells potential taking its global minimum value 0 at u = +-1; more precisely, we assume that f(u) has the properties f(?l)
= 0,
f’(*l)
< 0,
J
f(s) ds = 0
(1.2)
-1
and that there exists a unique value a E (- 1, + 1) such that f(a) = 0
(1.3)
Y(a) > 0.
(1.4)
with
A typical example is f(u) 9 Partially Fellowship.
= u - u3 and f(u, v) = f(u) +
supported by the National Science Part of this work was done during
V/I(U).
Foundation Grant DMS9200459 and the Alfred a visit to the University of Paris-Sud. 1283
P. Sloan
Research
XINFU CHEN etaI.
1284
When f(u, v) is independent of V, i.e. S(u, V) = f(u), equation (l.la) reduces to the AllenCahn equation [l] modelling an antiphase boundary motion and antiphase domain coarsening. The main feature of the solution of the Allen-Cahn equation is that the zero level set of the solution approximates (as E - 0) the motion by mean curvature flow. Formal derivation of this connection was carried out by Allen-Cahn [l], Rubinstein et al. [2], Fife [3] and many others. Rigorous justification was successfully carried out recently by Bronsard and Kohn [4], de Mottoni and Schatzman [5,6], Chen [7], Evans et al. [S], Ilmanen [9] and many others. Problem (PJ is obtained as the limit, as u \. 0 and T \ 0, of the following system of reactiondiffusion equations
t
UC= Au + +u, &2 1 TVI = -Av+uu
ml
-v)
in 0 x R+,
lY -v Y
in fi X Iw+
(1.5)
with appropriate initial and boundary conditions, where E and y are fixed positive constants. (See Section 4 for a rigorous justification.) The system (1.5) is usually referred as a BelousovZhabotinskii model (or FitzHugh-Nagumo model when u = co) which describes, for instance, wave propagation in excitable media, pattern formation in population genetics, propagation of signals along a nerve axon or cardiac tissue, etc. and has been extensively studied; see, for example, Fife [3], Nishiura and Mimura [lo], Ohta and Kawasaki [I 11, Fife and Tyson [12], and the references therein. For fixed positive constants T, 0, and y, the limiting behavior, as E - 0, of solutions of the parabolic system (1.5) was recently studied by X. Chen [13] and X-Y. Chen [14]. It is well-known that the motion by mean curvature flow does not have nontrivial stationary patterns that do not attach aR. In fact, in case R = RN, every initially bounded hypersurface shrinks into points in finite time; see, for example, Evans and Spruck [ 151. However, for equation (1. la), there is a nonlocal term represented by the integral of U, which provides dynamics for the formation of nontrivial stationary patterns. The scaling proposed here leads to a limiting free boundary problem presenting a balance between the local effect of curvature and the effect of the nonlocal term. Hence, problem (PC) provides a richer structure than the Allen-Cahn equation. We now state the main result of this paper. THEOREM 1.1. Assume that S(U, v) E C2([-2, 21 x [-1, 11) and that f(u) := f(u, 0) satisfies (1.2). Let Te CC R be a given smooth closed hypersurface which separates R into two domains Szt and 0,. Let T := ~~~~~~~~~ x {t} be a family of smooth hypersurfaces that moves according to the law
1I-,I~=~= ro, V
= --K+
com2:t
- Isz;I,,
(1.6)
where V is the normal velocity of & (positive if & is expanding), K is the mean curvature of T, (positive if T, is convex), co is a certain constant that depends only on f (see (3.11) in Section 3), Sz; is the region enclosed by T,, R, + = R \ (I’, u a;), and (A] is the measure of the set A. Assume that T, CC 52 for all 0 I t I T*. Then there exists a family of continuous functions {$]~<~~l such that the solution nE of problem (P,) with initial data gE satisfies lpy’(x,t)
= &l
VXER;,
tE
[O,T*].
An Allen-Cahn
equation with a nonlocal term
1285
Actually, a stronger result than theorem 1.1 will be proved. The idea of the proof is to construct sub-super solution pairs and to use a comparison principle. We shall establish the existence, uniqueness, and regularity of the solutions of the (limit) geometric motion problem (1.6) in Section 2 and prove theorem 1.1 in Section 3. Finally, in Section 4 we shall show that the solution of the parabolic system (1.5) with appropriate initial and boundary conditions approaches, as CJ I 0 and T I 0, to the solution of problem (PJ. 2. THE LIMIT
GEOMETRIC
MOTION
PROBLEM
Let IO CC a be a given smooth hypersurface without boundaries. It may consist of several pieces. Let o( be a given constant. Definition. A smooth family of hypersurfaces I = uIEIO,Tl(It x {t]) in fi x [0, T] is called a solution to the geometric motion problem (PO) starting from TOif V = -K + a(lsz:I 1 r, It=O = ro,
- ISZJ,,
t E [O, T],
where V, K, and fit* are the same as stated in theorem 1.1. 2.1. Assume that IO CC Sz c c RN is given and that IO is a smooth hypersurface without boundaries. Then for every o( E 08the motion problem (P) starting from IO admits a unique smooth solution I on [0, T], where T is a positive constant depending only on o(, IO and Sz.
THEOREM
ProojI We shall use a fixed point theorem for contraction a unique solution to problem (P*). Set ~0 = Ifi; 1 - I!& 1, and, for each T > 0, set
mappings to prove the existence of
XT = {cl(t) E C”3([0, Tl) 1 lldt) - CIOIIC~[O,TI) 5 1). Clearly, XT is a closed convex subset of the Banach space C1j3([0, T]). For each p(t) E Xr, consider the motion problem V
= -K + a/J(t),
i r, itzO= ro.
t E [O, T],
(2.1)
By using the local coordinates (d, s) where d = d(x) is the signed distance from x to IO and s = s(x) is the projection of x on IO along the normal of To, one can express I, as a graph over IO represented by a function d = d(s, t), s E IO. t E [0, T], and transfer the motion equation (2.la) into a quasi-linear parabolic equation for d on the manifold I-0 x [O, T]. (See [16] for a detailed calculation.) We remark that the existence of a classical solution of problem (p”oL)also follows from a result of Giga and Goto [17]. By a standard theory for quasi-linear parabolic equations (similar to those used in [16]), one can show that there exists a positive constant TI = F (cc To,Cl) such that (P”) has a unique solution i: E C2+2’3,1+“3 in the time interval [0, Tl ] . In addition, if we define
mt) then
= IQ3 - lfql,
1286
XINFU
CHEN er al.
and IVLO - ~/Jzllc ‘([O,fi]) 5 acI1 - P2IIc’~qo,fiI) vu,,p2 E XT', where C is a constant independent of p, ~1, ~2 E 1,. Consequently, there exists a constant C = C(cx, Is, R) such that, for all t E [0, ri], IITP - P0IIC~~~([O.rl) -< Ct2’3
and lITI
- TL.42IlC’I’([OJ]) 5 Ctz’311P, - c12llc~wo,~l).
These inequalities rely on the fact that, if f belongs to C’ ([O, r]) and if Ilfllc~~~c[o,~I,5 t2/3 (Jf]lc~~to,,~,. Hence, if T is small enough, then ;r’ maps Xr a strict contraction so that by a fixed point theorem for contraction mappings, fixed point P E Xr. Clearly, for this ~1, the solution f of (@) yields a solution The uniqueness of smooth solutions to (P) follows from the contraction whereas the regularity of the solution follows from a bootstrap argument parabolic equations. n
f (0) = 0, then into itself and is f has a unique to (P). property of T, for quasi-linear
We complete this section with a few remarks. Remarks. (1) When o( = 0, (P’)) is called the motion by mean curvature flow and has been extensively studied; see [l&15,19-2 l] and the references therein. (2) Generally, singularities of the solution of problem (P”), such as pinch-off, break-up, etc. arise in finite time, so that classical solutions only exist in finite time. (3) Weak global solutions of geometric motion for a large class of equations have been wellstudied by Chen et al. [18], and Evans and Spruck [15]. Due to the nonlocal behavior of our motion, our problem (Pa) (o( # 0) does not fall into their class. In the sequel we shall fix o( = co where COis a constant which depends only on f and is given in (3.11) in the next section. Also we shall call the solution to the motion problem (P*) with o( = COthe solution to the limit geometric motion problem. 3. CONVERGENCE
OF uE
3.1. Well-posedness of problem (P,) In order to be complete, we first establish a well-posedness result for problem (PE). LEMMA 3.1. Assumethat f(u, v) E Ct([-2,2] x [-1, 11) andthat! := f(u, 0) satisfies (1.2). Then there exists a constant EO> 0 such that, for every E E (0, ~01, if g’ satisfies lgEl < 1 + EOin a, then problem (PJ has a unique solution uE. Proof
Notice that by the properties of j: in (1.2), there exists a constant mo E (0, 1I such that
kf (*(l
+ mo), v) < 0
V v E [ -mi, mfl.
(3.1)
Let T P 0 be an arbitrary fixed constant and take EO:= min{mo, rni/ ( 1 + mu) Ifi I ] . Then, for every E E (0, EO] and v(t) E XT := 1~ E ~C[O, TI)
( IlNt)Ile([o,~])
5 (1 + m)lfll},
An Allen-Cahn
1287
equation with a nonlocal term
~j.f(t) E [-m$, rni] for all t E [O, T]. It then follows from (3.1) that +f‘(+-( 1 + mo), ED) < 0 and a comparison principle for parabolic equations that problem (PJ with ja u in (1.1 a) replaced by ,u(t ) has a unique solution uP in [O, T] which satisfies -1-Pzo
in 2
X
[O, T].
Define ?’ : XT - @(LO, Tl) by 7’~ = j&,C-, t). Then by the a priori estimate on z+, one can show that 7’ is continuous and compact, and maps XT into itself. Hence by Schauder’s fixed point theorem, 7’ has a fixed point in XT which yields a solution to problem (PE). The uniqueness of a solution of (P,) follows from the Lipschitz continuity of f, the Gronwall’s inequality and a routine calculation; the details are omitted. Since T is arbitrary, the assertion of the lemma follows. n In the sequel, we always use uc to denote the solution of the problem (P,). 3.2. A comparison lemma
Comparison arguments were used for instance by Chen [7,13] and by Ei and Yanagida [22] to show the convergence to geometric motion problems. Here we prove a comparison principle which involves simultaneously a super- and a sub-solution. LEMMA
3.2. Let (u-, u+) be a pair of smooth functions defined on fi u+ (x, t) 5 IL (x,
t)
in 2
x
x
[O, Tl with
[0, T].
(3.2)
Assume that (u-, u+) satisfy the inequalities (u+), - Au, - L max f(u+, ES) 2 0 E2 Jou-sscj* u+
in Q
[O, Tl,
(3.3)
(u-)~ - Au- - .i min f(u-, E2 I* u-9sJn u+
in Sz x [O, Tl,
(3.4)
au-505-an
0) I 0 au+ an
x
on afi x [O, Tl,
u-(x, 0) 5 g’(x) 5 24+(.x,0)
for all x E 0
(3.6)
Then u-(x,
t)
I d(x,
t)
I u+(x,
t)
in s2 x [O, T].
Prooj Set
YT = {p(t) E C?([O, Tl) ( J-u-(., t) I dt) 4 ~u+C.. t) for t E [O, Tl) cl n and, for each p E Y,, let u,, be the solution to uI = Au + (1 /~~)f(u, E/J) with the given boundary and initial conditions in (1.1 b) and (1. lc). Then, by a comparison principle, u- I up I u+
in Q x [O, T].
Define F = T/J = Is2 ZQ,(., t), t E [0, T]. Then one can use Schauder’s fixed point theorem to show that T has a fixed point in YT, which yields a solution ii to problem (PE). Clearly, this
1288
XINFU
CHEN ef al
solution satisfies the assertion of the lemma. Moreover, by uniqueness, li = uE. n In the sequel, we shall call a pair (u-, u+) satisfying (3.2)-(3.5) a sub-super solution pair. 3.3. The travelling wave solutions
For the purpose of constructing subsuper solution pairs, we first introduce travelling solutions of a related one-dimensional parabolic equation. Consider the function
wave
F(u, v, 6) = f(u, v) + 6,
(3.7)
where v and 6 are real parameters. Since FL(‘-c LO, 0) f 0 by the assumptions in (1.2) on f, the implicit function theorem ensures the existence of two positive constants vs and 60 such that for all v E [-vo, vol and 6 E [ -60,60] the equation, for U, F(u, v, 6) = 0, has two solutions h- (v, 6) and h+(v, 6) which are smooth in v and 6, and satisfy h,(O, 0) = kl. In addition, in view of (3.7) and (1.2), they satisfy h,(v, 6) = +-I + O(lvl + ISI), $(“6)
>0
(3.8)
v v E [-vo, vol, s E [--60,601.
(3.9)
We search for a travelling wave solution of the equation ur = u,, + F(u, v, 61,
x E 08,t E Kit,
which converges to h,(v, 6) as x - fo3, i.e. a pair (U(., v,6), W(v,6)) such that u(x,t) = U(x - Wt) satisfies the parabolic equation. Set z = x - Wt. It comes to search for U = U (z, v, 6) and W = W(v, 6) such that U” + W(v, 6)U’ + F(U, v, 6) = 0, ( U(-00, v, 6) = h-(v, S), U(+oo, v, 6) = h+(v, 6).
(3.10)
This problem has a solution (U(z, v, 6), W (v, 6)). The solution is unique up to a translation in z and U is monotonic increasing in the z variable. In view of (3.9) and the monotonic properties of U in z, we can play with this translation invariance so that we can choose a smooth solution U(z, v, 6) satisfying Us(z, v, 6) 2 0. Some properties of this solution are listed in the following lemma. 3.3. There exist small positive constants vo and 60 such that, for all v E [-~a, vol and S E [-ho, 601, there exists a solution (U(z, v, 6), W(v, 6)) to the problem (3.10) which has the following properties: (i) there exist positive constants A and fi such that, for all (v, 6) E [ -VO, VO] X L-60, 601, LEMMA
OcUzrA and OsUglA IU,(+z,v,6)1 + IU,,(fz,v,6)I
VzzER, + IU(~Z,V,S)
-hdv,6)1
I Ae+
VZE 08,;
(ii) the travelling wave velocity W (v, 6) is a smooth function of its arguments and satisfies W(O,O) = 0.
An Allen-Cahn
equation with a nonlocal term
1289
The existence and the properties of U(z, V,b), as well as the smoothness of the function W(v, 6), can be proved by standard methods such as those presented in the Appendix of [3]. See also [23, 241. Finally, we define co := F(O,
0)
3.4. Construction of sub-super solution pairs
We shall now construct sub-super solution pairs to control the solution uE of problem (PE) and show that, in the limit as E - 0, uE tends to a function which takes only two values + 1, and whose boundary of the transition from - 1 to + 1 is the solution of the limit geometric motion problem (PO) where COis defined in (3.11). To construct the sub-super solution pairs, we shall make use of the travelling wave solutions introduced above and a modified distance function that we define now. Let I = U (I, x {t]) be the unique solution of the limit geometric motion problem (PO) rE[o.rl
and let d(x, t) be the signed distance function from x E fi to I,, t E [0, T], that is, 12(x, t) ] = dist(x, I,),
2(x, t) > 0 in Sz:,
and d(x, t) < 0 in Sz,.
Let do be a positive number such that 2(x, t) is smooth in D(2dl)) = ((x, t) E sl x [O, Tl 1 I&x, t)I < 2do) and dist(I,, aR) > 2do Let d(x, t) be a smooth modification
V t E [0, T]
of d” such that
d = Jin DUO) = ((x, t) E Sz x [O, T] 1 Id”(x, t) I < do}, do < Id] I 2d0 and dd” > 0 in D(2do) \ D(do), IdI = 2do and dd” > 0 in fi x [0, T] \ D(2do). Set p(t) = Ifi:
I - I Sz, I. We now prove the following lemma.
3.4. There exist positive constants C* and C** such that if ml 1 1, rn2 1 1, m3 2 1 and m3 1 C*ml, m2 1 C** m3, then, for some positive EO= so(mi, m2, ms) and 0 < E < ~0, the functions u- and u+ defined by
LEMMA
z&(x,t)
= u(
d(x, t) + mlEe’Q’ , Ep(t), +E~WZ~ e’n2’> E
satisfy (3.2)-(3.5); that is, (u-, u+) is a sub-super solution pair. Prooj Since near aR, d = 2do, we have that &&an = 0 on afi x 10, T]. Also, since U, > 0 and U6 2 0, we have that U+ > U- in R x [0, T]. It remains to verify the differential inequalities (3.3) and (3.4). To this end, we first calculate
1290
XINFU
R
CHEN et al.
a,+
fir-
Write Jo2: 24+ ( . , t) as e’narI] + h, (up, J u+ = J [u+ - h+(~p(t), .E~KTQ Q: a,+
E2m3 ern20ISZ: I.
(3.13)
Since, by (3.8), h+(~p(t), &2m3em2t) = 1 + O(.EJI~IJ~(~O,~~) + E2mje’“2’), and, by lemma 3.3(i), u+ = h+(~p(t), E2m3emjr) + O(e-@do’2E ) in {x E !A d(x, t) 1 do}, it follows from (3.13) that u+ =
J
0:
J
[
u+ - h+ (up,
E2mjdn2’)]
IxlO
+ E2m3ewt + IQf I + O(EIlPIlCO([O,T,I Here and in the sequel, all the OS are independent sufficiently small. In a similar way, we have u+ =
J
61,
+ e-Bdoi2E
>
_
of ml, m2 and m3 but may require E to be
u+ - h- (zp(t), E2mgP2’ )] L
J {xl-do
- ItI- I + O(~llpll~~~~,~~) + E2rnje’n2r + e-Bdo/2r).
To calculate the integrals on {x E !A I 0 < d(x, t) < do} and on {x E fi I -do < d(x, t) < O}, we make the transformation x E D(do) ++ (s, d), where s(x, t) is the projection of x on G along the normal of I’,, and d(x, t) (= 2(x, t)) is the signed distance function from x f D(do) to I,. Denote by J(s, d) the Jacobi of the transformation and by (., *) the pair (.sp(t), E2rn3en2’ ). Then
do
r + ml Eemzr >-, .) - A+(., *)]~(s, r) dr ds
JJ[ =cJJ[
=
LO
‘(
E
do/E
U(z + ml em21,., .) - A+(-, .)]J(s, EZ) dz ds = O(E),
r, 0
where we have used the change of variables z = I/E in the second equation, and the fact that Ub, -> -) - h+(., -) = O(e+) as z - +a~ in the third equation. Similarly,
1 J
[u+-h-(.:)]~
An Allen-Cahn
equation
with
a nonlocal
1291
term
0 =E
U(Z II
+ ml
em*l,
J
-,
-) -he-(-,
-)]J(s, EZ) ds dzl
G -4/E 0
I CE
JI --m
U(z + ml em”, -, a) -A-(-;)I
dz m,e”‘2’
0
= CE
II --m
I CEml
U(z, ., -)-h-(-AI
dz+C~
J
IU(z;:)-h~(~;)I
dz
0
e'nzf
for some C > 0, since the integral on [w- on the right-hand side of the last equation is convergent (by lemma 3.3(i)), the function IU(z, ., .) -h- (., .) 1 is bounded and ml 2 1. Combining all these estimates, we conclude that there exists a positive constant Cl such that, when E is small enough,
IS
u+(x, t) dx - p(t) I I C,WZ,em*’ E.
n
Similarly,
we can show that
II
U-(X, t) dx - p(t) I I C,WZ~et”*’ E.
n
It then follows that (3.14)
max and that
(3.15) where C2 is a constant independent of ml, rnz and rnj. Now we shall verify the differential inequality (3.3). Set
Then, by (3.14) Lu+
A direct computation
I
(u+)!
-
AU, - iflu+,
E&))
- C2ml ernlr.
gives
(u+)~ = (dl + Em~m2em2’)~
Au, = yUzz
-I- %Ji. &
+ Ep’(t)U,
+ E2m3m2e’“*’ Ua,
1292
XINFU
Moreover, by the definitions
CHEN
et al.
of U and of u+,
-f(u+! W(l)) = m3 e’n*t + Lw-- + W(Ep(t), E2m3 t+‘) U, -. &2
E2
E
E
Hence, f u+ 2 [ dl + Ernlrn2 ern2*-Ad + +[l
- lVd12]$
1-E
W(~p(t), E2rn3eJn*r) U,
E
+ [Ep’(t)C:
+ E2m3m2ern2’ ~61 + [m? - Czrnl] emzr
=: T, + T2 + T3 + T4. First, we estimate TI . Since dl - Ad + CO,U= 0 on r, = {x E R 1 d(x, t) = 0}, it follows from the mean value theorem that Id - Ad + co/.d)l 5 Gld(x,
t)l.
Note that I W(v, 6) - W(O,O) - (aW/lh~)(O, 0)vl = I W(v, 6) - covl = O(I61 + v2), so that ( W(.g.t(t). Zrn, emzf) - q&t)
1 5 C4E2m3 e’n2’
for all f E [O, Tl. Therefore, W(Ep(t), E2rn3en*‘) E
d! + EmI m2 e’n2t - Ad +
= (df - Ad + cop(t)) + ( W(E’(‘)f~2m3e’“2f)
- cop(t)) + Em,m2e1Qf
2 -C3 Idtx, t) I + E(mlm2 - C4m3) e’n2f 2 -4’3 Idk tf + EmI P2’ I + Ee’“*‘(ml (rnz - ~3) - C4m3) 2 -C3ld(x, t) + EmI e’“*’ I if we choose m2 2 C3 + C4m3 2 C**mj. Thus, since Uz > 0, T, ~ -c
Mb, t) + w 3
ernzrI
d(x, t) + EmI em2’ Vi
(
E
E
, w(t),
E2m3
em*’
)
2 -C3 m;x IzU,(z, +, .) [ 2 -CS. As for 7& by the definition
of d, (Vd( = 1 when Id( < do. According to lemma 3.3(i),
IumI‘ (d(x, t) + Em1 e”‘zl , Ep(t), E2m3 efnzf>I
5 A e-BI(dcx,r,+Fm,<“*‘)/El
E
Moreover, for 0 5 t I T and (dl 2 do, we have d(x t) + m E
em*’ l 1 I db, t) I _ m, tin2T 1 6 _ cr 1 &I E E 2E
An Allen-Cahn
1293
equation with a nonlocal term
for E small enough. Therefore, dtx, t) + urnse’nlr
s C6 e-0doj2E.
, Ep(t), E2rn3e’“*’
E
Thus IT21 5 GE- 2 e-Odo/=f
Since, furthermore, h+
if rn3 L C*ml we have
2
T3 2 -c5
-
EM’(~)
U, r -CUE, we deduce that
c6Ep2
e-0do’2E
+ (m3 - C2ml) t+
-CUE
in d X [0, T]
and C* is large enough. Similarly, we can show that by taking m3 and m2 as above, (u-)~ - Au- - 1. E2 lou~~~,ou+ f(u-.
This completes the proof of lemma 3.4. Combining follows.
2 0
ES) I 0
in d X [O, Tl.
n
lemmas 3.2 and 3.4, we can now prove theorem 1.1. A refinement of theorem 1.l
THEOREM 3.5. Assume that, for some positive constants a and b, g’(x) satisfies U(d(X,O)/E
-
for all x E fi and all all E E (0, Eo 1) U((d(X,
t)
-
Em1
E
eMZt)/E,
a, E/J(O),
-bE=) 5 g’(x) I U(d(x,
0)/E
+ a, Ep(O),
E (0, 11. Then there exists a positive constant Eu(t),
-E2W13e’n2’)
5
U’(X,
t)
I
U((d(X,
t) +
~~
EmI
(3.16)
bE=)
E (0, 1) such that for e”*‘)/E,
E/l(t),
E2mspzt)
in Sz x [O, T], where ml = max(1, a), m3 = max(C*mt,
b), m2 = C**ms.
Consequently, IfiIyE(X, t) =
- 1 in Sz; +I inn:
V t E [O, T].
The assertion of the theorem follows directly from lemmas 3.2 and 3.4, and the properties of U in lemma 3.3. Details are omitted. Remarks. (1) From the proof of the theorem, we see that the assertion of theorem 3.5 is also valid if we replace the Neumann boundary condition (1 .l b) by the Cauchy boundary condition d(x, t) = 1 on XI X [0, T]. (2) Our assumption (3.16) on the initial data gc requires that gc is well shaped. For generic initial data, formal analysis and rigorous results (see [7,13,5]) indicate that after a short time (of order O(EJ In ~1)) such a shape will be developed.
1294
XINFU 4. DERIVATION
CHEN OF
ef al.
PROBLEM
(PE)
In this section, we shall show how one can derive problem (P,) from the following system of reaction-diffusion equations ul = Au + ff(~,
Yv)
1
-rvl = -Av+u-”
(PUT)
Wan
Y
PO
av/an = 0 u(x, 0) = uo(x)
in fi X R+, in fi X R+, on aszx R,, on aszx R,, in Q in Q,
v(x, 0) = vo(x) where IY and T are small positive parameters, and E and y are fixed positive constants. More precisely, we shall first show that, as IJ I 0, the solution (ST, vaT) of problem (P”‘) converges to (u’(x, t), v’(r)) which is the unique solution to the following problem I
UT! = AuT + iftuT, T:v~(~)
(PO’)
=
f n
$+(,))
in fi x R+,
uT(x, t) dx - bvT(r)
in 0 x R,, on an x R,, in 0,
&Clan = 0 22(X, 0) = L@(x) vT(0) =
nf
vo(x) dx,
where fn = h so. Then, we shall show that, as T I 0, the solution (Us, v’) of problem (PO’) converges to (u’, ~foz/) where z/ is the unique solution to problem (PJ. For this purpose, we assume that the initial data in problem (PUT) satisfy the hypothesis 240E I-1 -ml), 1 + mol, uo E ff’m, vo E [-y(l + mo), Y(1 + mo)l, i vo E L”(fiz), where mo is a small positive constant satisfying (3.1).
(4.1)
LEMMA 4.1. Assume that f E C’ ([ -2,2] x [ - 1, 11) satisfies (3.1) for some positive constant mo > 0 and that ug and vg satisfy (4.1). Also assume that E and y are fixed positive constants satisfying E]1;2](1 + mo) I rn;.
(4.2)
Then, for all positive constants cr and T, the problem (PUT) has a unique solution (P, vgT). Moreover, for every T > 0, there exists a constant C = C(T), which is independent of u and T, such that the following estimates hold IP(X,
t)I I 1 + mo
IVUT(X,t)I 5 Y(1 + mo)
V(x, t) E fi x [O, co),
V(x, t) E sz x [O,oo),
(4.3) (4.4)
An Allen-Cahn
equation with a nonlocal term
II~aTIIL”(O,T;H’(n,,+ lI~uTtIILqRX(O,T))5 c,
1295
(4.5)
T lVvuT1*
I
(4.6)
Ca,
f.J
OSa T
II
p7-2
C rs76
v
6
E
(0.
T),
(4.7)
60
v t E (0, Tl,
(4.8)
v t E (0, T].
(4.9)
R d 1dt I n
vUT(., t) 1 I $
ProoJ: By a method of invariant regions [25, Chapter 14, Section B] and the conditions (3.1) and (4.2), one can show that the set { (u, v) ] ]U( I 1 + mo, ] v ] I y( 1 + mo) ) is positive invariant for the parabolic system (POT). Therefore, by a classical theory for parabolic systems [25, Chapter 141, the problem (P"') has a unique global solution (P, vaT) and the solution satisfies the estimates (4.3) and (4.4). The estimates (4.5)-(4.9) follow from the standard energy estimates. Here we briefly give the procedures. To obtain (4.5) we multiply the equation for zP by zPTI, integrate by parts on fi x (0, T) and use the uniform boundedness of zPT and PT. The estimates (4.6) follow by multiplying the equation for vUT by vaT and integrating by parts on !A x (0, T) . The estimates (4.7) and (4.8) follow by multiplying the equation for vVT by tvuTl and integrating by parts on R x (0, T), whereas (4.9) follows directly from integrating the equation for vaT on 0 x {t } . n
In the sequel, T > 0 is an arbitrary fixed constant. With the basic estimates in lemma 4.1, we can now prove the following result. Let (uuT, vuT ) be the unique solution of the problem (P”‘) given in lemma 4.1. Then there exist functions u~(x, t) E L”(0, T;H’(fl)) n H’(fl x (0, T)) and vT(x, t) E L*(O, T;H’(fl)) ~7H’(R x (6, T)) (for all 6 E (0, T)), and a sequence {a,,}~=, such that as m - 00, o;, I 0 and the following limits exist: (i) lim zPT = uT weakly in H’(fi x (0, T)), strongly in C([O, T];L*(fl)) and a.e. in 0 X m-+m (0, T); weakly in H’(R x (6, T)), strongly in (ii) t?Z-+CC lim vanIT = vT weakly in L*(O, T;H’(Cl)), C([6, T];L*(s2)) and a.e. in fi x (6, T), for all 6 E (0, T); (iii) vT = v’(t) E C?,‘([O, T]) and $-oPnT(x, t) dx - vT in C?**([O, T]) for all o( E (0, 1). LEMMA 4.2.
Proof: The assertions (i) and (ii) are immediate compactness of the set {u E L”(to, T;H’(SZ)),
consequences of lemma 4.1 and the relative ul E L2(to, T;L*(sZ))}
in C(to, T; L*(0)), with to E [0, T). It remains to prove (iii). From the lower semi-continuity the functional M; - ]o w* and the estimate (4.8) it follows that
of
1296
XINFU CHENetd T
T
JJ 60
IVrT(x, t)12 dx 5 liminf m-m
JJ 60
(VPnT(x, t)j2 = 0
V 6 E (0, T],
so that VrT = 0 a.e. in Sz x (0, T), and therefore vT = vT (t). The convergence result in (iii) and the regularity of vT then follow from the second assertion of this lemma and the estimate (4.9) n in lemma 4.1. The proof is completed. As we shall show below, the limit (Us, vT) obtained in lemma 4.2 is a solution to the problem (PO’). Since one can directly prove that the problem (PO’) has a unique solution, we can strengthen the conclusion of lemma 4.2 as follows. THEOREM 4.3. Let (u’(x, t), ~~0)) be the unique solution to the problem
be the unique solution to the problem (I’“‘).
(PO’) and (zPT, vu’)
Then, as u I 0,
(fP, vUT) - (Us, v’) weakly in H’(a
x (0, T)) x L2(0, T;@(R)).
PruojI First we assume that (Us, vT) is the pair of functions stated in lemma 4.2. Since the function zPT satisfies
for all 4 E C” (fi x [O, T]) satisfying $I ( *, T) = 0, sending m - W, we deduce from Lebesgue’s dominated convergence theorem and lemma 4.2 that ur satisfies the parabolic equation for uT in the problem (pT), as well as the homogeneous Neumann boundary condition and initial condition. Since vUmTsatisfies the equation Tf”n’“T(l) - Tfvo = j fuUtflT - ; /fPT R n 00 OCI sending m - 0 we deduce that vT satisfies the ordinary differential equation and the initial condition in the problem (I”“). Therefore, (Us, rT) is a solution to the problem (I’(“). Since the solution (u’, vT) to the problem (I““) is unique, we conclude that the whole sequence n { (UUT, VT I 0~~~1 converges to (Us, vT) as o I 0. This completes the proof of the theorem. Finally, we derive the problem (PE) from (PO’). THEOREM 4.4.
For each T E (0, 11, let (Us, v’) be the unique solution to problem (PO’). Then, as T I 0, (Us, v7) converges to (uE, Y&G), where uE is the unique solution to problem (P,). ProoJ From the uniform
lIuTII~m I 1 + Multiplying integrating
estimates in lemma 4.1 we can derive that n-20,
llvTIILm
5 .dl
+ mO),
li~T'l~kz(~~(O.T))
the ordinary differential equation for vT in the problem the resulting equation over [0, T], we obtain
s
c
(P”‘) by f&r
- ivT and
An Allen-Cahn
where C is a constant independent
equation with a nonlocal term
1297
of T. It then follows that
llvT - YfUrllL2((o,T)) - O n
as T - O.
Consequently,
as T - 0. It then follows that uT - z.8as T - 0, where uE is the unique solution to the problem # (P,). This completes the proof of the theorem. Acknowledgements-The
authors are very grateful to Professor Y. Nishiura for many stimulating discussions. REFERENCES
1. ALLEN S. & CAHN J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27, 10841095 (1979). 2. RUBINSTEIN J., STERNBERG P, & KELLER J. B., Fast reaction, slow diffusion and curve shorting, SIAM .I appl. Math. 49, 116-133 (1989). 3. FIFE P. C., Dynamicsfor Internal Layers and Diffitsive Interfaces, CCMS-NSF Regional Conf. Ser. in Appl. Math. SIAM, Philadelphia (1988). 4. BRONSARD L. & KOHN R., Motion by mean curvature as the singular limit of Ginzburg-Landau model, J dt@ Eqns 90, 211-237 (1991). 5. DE MOTTONI F! & SCHATZMAN M., Development of interfaces in OB”, Proc. R. Sot. Edinb. 116A, 207-220 (1990).
6. DE MOTTONI I? & SCHATZMAN M., Geometrical evolution of developed interfaces, Trans. Am. math. Sot. (to appear). 7. CHEN X., Generation and propagation of interfaces for reactiondiffusion equations, 1 difl Eqns 96, 116-141 (1992). 8. EVANS L. C., SONER H. M. & SOUGANIDIS P. E., Phase transitions and generalized motion by mean curvature, Communs pure appl. Math. (to appear). 9. ILMANEN T., Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, preprint. 10. NISHIURA Y. & MIMURA M., Layer oscillations in reactiondiffusion systems, SIAM 1 appl. Math. 49, 481514 (1989). 11. OHTA T. & KAWASAKI K., Equilibrium morphology of block copolymer melts, Macromolecules 19, 2621-2632 (1986). 12. FIFE P C. & TYSON J., Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J them. Phys. 73, 2224-2237 (1980). 13. CHEN X., Generation and propagation of interfaces for reaction-diffusion systems, Trans. Am. math. SOC. 334, 877-913 (1992).
1298
XINFU
CHEN
et al.
15. EVANS L. C. & SPRUCK J., Motion of level sets by mean curvature, J d@ Geom. 33, 635-681 (1991). 16. CHEN X. & REITICH F., Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J math. Analysis Applic. 162, 350-362 (1992). 17. GIGA Y. & GOT0 S., Geometric evolution of phase-boundaries, in On The Evolution of Phase Boundaries (Edited by M. E. GURTIN and G. B. MCFADDEN). Springer, Berlin (1992). 18. CHEN Y. G., GIGA Y. & GOT0 S., Uniqueness and existence of viscosity solutions of generalized mean curvature Bow equations, 1 d18 Geom. 33, 749-786 (1991). 19. GRAYSON M., The heat equation shrinks plane curves to points, J d@ Geom. 26, 283-314 (1987). 20. HAMILTON R., Three manifolds with positive Ricci curvature, 1 d13 Geom. 17, 2555306 (1982). 21. HUISKEN G., Flow by mean curvature of convex surfaces into spheres, J. diff: Geom. 20, 237-266 (1984). 22. EI S. I. & YANAGIDA E., Dynamics of interfaces in competition-diffusion systems, SIAM .I appl. Mach. 54, 1355-1373 (1994). 23. ARONSON D. G. & WEINBERGER H. E, Nonlinear diffusion in population genetics, combustion and nerve propagation, in Partial D@rential Equations and Related Topics, Lect. Notes in Math., Vol. 446, pp. 549. Springer, New York (1975). 24. FIFE P. C. & HSIAO L., The generation and propagation of internal layers, Nonlinear Analysis 12, 1941 (1988). 25. SMOLLER J., Shock Waves and Reacfion-D@sion Equations. Springer, New York (1983).
Analysis
Theory,
Vol. 28, No. 7, pp. 128%1298. 1997 Copyright @ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-%6X/97 $17.00 + 0.00
Methods
& Applicatbm,
Pergamon 0362-546X(95)000217-0
ASYMPTOTIC XINFU
BEHAVIOR OF SOLUTIONS OF AN ALLEN-CAHN EQUATION WITH A NONLOCAL TERM
CHEN, t 1 DANIELLE
HILHORST
+ and ELISABETH
LOGAK
§
t Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.; * Laboratoire d’Analyse Numtrique, CNRS et Universite Paris-Sud, Batiment 425, 91405 Orsay, France; § Departement de Mathematiques et d’Informatique, Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France (Received
16 March
1994; received
in revisedform
Key words andphrases: Allen-Cahn propagation of interfaces,
equation,
27 July 1995; receivedfor
integro-differential
publication
equations,
29 November
singular
and
1995)
limit,
1. INTRODUCTION
In this paper, we study the limiting
behavior, as E I 0, of the solution ~8 of the following problem
s=Au+$f(u..s WE)
au -= 1
u"?x,
J n
U)inQxO%‘,
(1.1)
0 0)
0n =
gE(x)
a!2
x E
x
R+,
cl,
where Sz c IWN (N 2 2) is a smooth bounded domain and n is the unit outward normal to afi. We assume that f(u, v) is smooth and that f(u) = f(u, 0) is the opposite of the derivative of a double equal wells potential taking its global minimum value 0 at u = +-1; more precisely, we assume that f(u) has the properties f(?l)
= 0,
f’(*l)
< 0,
J
f(s) ds = 0
(1.2)
-1
and that there exists a unique value a E (- 1, + 1) such that f(a) = 0
(1.3)
Y(a) > 0.
(1.4)
with
A typical example is f(u) 9 Partially Fellowship.
= u - u3 and f(u, v) = f(u) +
supported by the National Science Part of this work was done during
V/I(U).
Foundation Grant DMS9200459 and the Alfred a visit to the University of Paris-Sud. 1283
P. Sloan
Research
XINFU CHEN etaI.
1284
When f(u, v) is independent of V, i.e. S(u, V) = f(u), equation (l.la) reduces to the AllenCahn equation [l] modelling an antiphase boundary motion and antiphase domain coarsening. The main feature of the solution of the Allen-Cahn equation is that the zero level set of the solution approximates (as E - 0) the motion by mean curvature flow. Formal derivation of this connection was carried out by Allen-Cahn [l], Rubinstein et al. [2], Fife [3] and many others. Rigorous justification was successfully carried out recently by Bronsard and Kohn [4], de Mottoni and Schatzman [5,6], Chen [7], Evans et al. [S], Ilmanen [9] and many others. Problem (PJ is obtained as the limit, as u \. 0 and T \ 0, of the following system of reactiondiffusion equations
t
UC= Au + +u, &2 1 TVI = -Av+uu
ml
-v)
in 0 x R+,
lY -v Y
in fi X Iw+
(1.5)
with appropriate initial and boundary conditions, where E and y are fixed positive constants. (See Section 4 for a rigorous justification.) The system (1.5) is usually referred as a BelousovZhabotinskii model (or FitzHugh-Nagumo model when u = co) which describes, for instance, wave propagation in excitable media, pattern formation in population genetics, propagation of signals along a nerve axon or cardiac tissue, etc. and has been extensively studied; see, for example, Fife [3], Nishiura and Mimura [lo], Ohta and Kawasaki [I 11, Fife and Tyson [12], and the references therein. For fixed positive constants T, 0, and y, the limiting behavior, as E - 0, of solutions of the parabolic system (1.5) was recently studied by X. Chen [13] and X-Y. Chen [14]. It is well-known that the motion by mean curvature flow does not have nontrivial stationary patterns that do not attach aR. In fact, in case R = RN, every initially bounded hypersurface shrinks into points in finite time; see, for example, Evans and Spruck [ 151. However, for equation (1. la), there is a nonlocal term represented by the integral of U, which provides dynamics for the formation of nontrivial stationary patterns. The scaling proposed here leads to a limiting free boundary problem presenting a balance between the local effect of curvature and the effect of the nonlocal term. Hence, problem (PC) provides a richer structure than the Allen-Cahn equation. We now state the main result of this paper. THEOREM 1.1. Assume that S(U, v) E C2([-2, 21 x [-1, 11) and that f(u) := f(u, 0) satisfies (1.2). Let Te CC R be a given smooth closed hypersurface which separates R into two domains Szt and 0,. Let T := ~~~~~~~~~ x {t} be a family of smooth hypersurfaces that moves according to the law
1I-,I~=~= ro, V
= --K+
com2:t
- Isz;I,,
(1.6)
where V is the normal velocity of & (positive if & is expanding), K is the mean curvature of T, (positive if T, is convex), co is a certain constant that depends only on f (see (3.11) in Section 3), Sz; is the region enclosed by T,, R, + = R \ (I’, u a;), and (A] is the measure of the set A. Assume that T, CC 52 for all 0 I t I T*. Then there exists a family of continuous functions {$]~<~~l such that the solution nE of problem (P,) with initial data gE satisfies lpy’(x,t)
= &l
VXER;,
tE
[O,T*].
An Allen-Cahn
equation with a nonlocal term
1285
Actually, a stronger result than theorem 1.1 will be proved. The idea of the proof is to construct sub-super solution pairs and to use a comparison principle. We shall establish the existence, uniqueness, and regularity of the solutions of the (limit) geometric motion problem (1.6) in Section 2 and prove theorem 1.1 in Section 3. Finally, in Section 4 we shall show that the solution of the parabolic system (1.5) with appropriate initial and boundary conditions approaches, as CJ I 0 and T I 0, to the solution of problem (PJ. 2. THE LIMIT
GEOMETRIC
MOTION
PROBLEM
Let IO CC a be a given smooth hypersurface without boundaries. It may consist of several pieces. Let o( be a given constant. Definition. A smooth family of hypersurfaces I = uIEIO,Tl(It x {t]) in fi x [0, T] is called a solution to the geometric motion problem (PO) starting from TOif V = -K + a(lsz:I 1 r, It=O = ro,
- ISZJ,,
t E [O, T],
where V, K, and fit* are the same as stated in theorem 1.1. 2.1. Assume that IO CC Sz c c RN is given and that IO is a smooth hypersurface without boundaries. Then for every o( E 08the motion problem (P) starting from IO admits a unique smooth solution I on [0, T], where T is a positive constant depending only on o(, IO and Sz.
THEOREM
ProojI We shall use a fixed point theorem for contraction a unique solution to problem (P*). Set ~0 = Ifi; 1 - I!& 1, and, for each T > 0, set
mappings to prove the existence of
XT = {cl(t) E C”3([0, Tl) 1 lldt) - CIOIIC~[O,TI) 5 1). Clearly, XT is a closed convex subset of the Banach space C1j3([0, T]). For each p(t) E Xr, consider the motion problem V
= -K + a/J(t),
i r, itzO= ro.
t E [O, T],
(2.1)
By using the local coordinates (d, s) where d = d(x) is the signed distance from x to IO and s = s(x) is the projection of x on IO along the normal of To, one can express I, as a graph over IO represented by a function d = d(s, t), s E IO. t E [0, T], and transfer the motion equation (2.la) into a quasi-linear parabolic equation for d on the manifold I-0 x [O, T]. (See [16] for a detailed calculation.) We remark that the existence of a classical solution of problem (p”oL)also follows from a result of Giga and Goto [17]. By a standard theory for quasi-linear parabolic equations (similar to those used in [16]), one can show that there exists a positive constant TI = F (cc To,Cl) such that (P”) has a unique solution i: E C2+2’3,1+“3 in the time interval [0, Tl ] . In addition, if we define
mt) then
= IQ3 - lfql,
1286
XINFU
CHEN er al.
and IVLO - ~/Jzllc ‘([O,fi]) 5 acI1 - P2IIc’~qo,fiI) vu,,p2 E XT', where C is a constant independent of p, ~1, ~2 E 1,. Consequently, there exists a constant C = C(cx, Is, R) such that, for all t E [0, ri], IITP - P0IIC~~~([O.rl) -< Ct2’3
and lITI
- TL.42IlC’I’([OJ]) 5 Ctz’311P, - c12llc~wo,~l).
These inequalities rely on the fact that, if f belongs to C’ ([O, r]) and if Ilfllc~~~c[o,~I,5 t2/3 (Jf]lc~~to,,~,. Hence, if T is small enough, then ;r’ maps Xr a strict contraction so that by a fixed point theorem for contraction mappings, fixed point P E Xr. Clearly, for this ~1, the solution f of (@) yields a solution The uniqueness of smooth solutions to (P) follows from the contraction whereas the regularity of the solution follows from a bootstrap argument parabolic equations. n
f (0) = 0, then into itself and is f has a unique to (P). property of T, for quasi-linear
We complete this section with a few remarks. Remarks. (1) When o( = 0, (P’)) is called the motion by mean curvature flow and has been extensively studied; see [l&15,19-2 l] and the references therein. (2) Generally, singularities of the solution of problem (P”), such as pinch-off, break-up, etc. arise in finite time, so that classical solutions only exist in finite time. (3) Weak global solutions of geometric motion for a large class of equations have been wellstudied by Chen et al. [18], and Evans and Spruck [15]. Due to the nonlocal behavior of our motion, our problem (Pa) (o( # 0) does not fall into their class. In the sequel we shall fix o( = co where COis a constant which depends only on f and is given in (3.11) in the next section. Also we shall call the solution to the motion problem (P*) with o( = COthe solution to the limit geometric motion problem. 3. CONVERGENCE
OF uE
3.1. Well-posedness of problem (P,) In order to be complete, we first establish a well-posedness result for problem (PE). LEMMA 3.1. Assumethat f(u, v) E Ct([-2,2] x [-1, 11) andthat! := f(u, 0) satisfies (1.2). Then there exists a constant EO> 0 such that, for every E E (0, ~01, if g’ satisfies lgEl < 1 + EOin a, then problem (PJ has a unique solution uE. Proof
Notice that by the properties of j: in (1.2), there exists a constant mo E (0, 1I such that
kf (*(l
+ mo), v) < 0
V v E [ -mi, mfl.
(3.1)
Let T P 0 be an arbitrary fixed constant and take EO:= min{mo, rni/ ( 1 + mu) Ifi I ] . Then, for every E E (0, EO] and v(t) E XT := 1~ E ~C[O, TI)
( IlNt)Ile([o,~])
5 (1 + m)lfll},
An Allen-Cahn
1287
equation with a nonlocal term
~j.f(t) E [-m$, rni] for all t E [O, T]. It then follows from (3.1) that +f‘(+-( 1 + mo), ED) < 0 and a comparison principle for parabolic equations that problem (PJ with ja u in (1.1 a) replaced by ,u(t ) has a unique solution uP in [O, T] which satisfies -1-Pzo
in 2
X
[O, T].
Define ?’ : XT - @(LO, Tl) by 7’~ = j&,C-, t). Then by the a priori estimate on z+, one can show that 7’ is continuous and compact, and maps XT into itself. Hence by Schauder’s fixed point theorem, 7’ has a fixed point in XT which yields a solution to problem (PE). The uniqueness of a solution of (P,) follows from the Lipschitz continuity of f, the Gronwall’s inequality and a routine calculation; the details are omitted. Since T is arbitrary, the assertion of the lemma follows. n In the sequel, we always use uc to denote the solution of the problem (P,). 3.2. A comparison lemma
Comparison arguments were used for instance by Chen [7,13] and by Ei and Yanagida [22] to show the convergence to geometric motion problems. Here we prove a comparison principle which involves simultaneously a super- and a sub-solution. LEMMA
3.2. Let (u-, u+) be a pair of smooth functions defined on fi u+ (x, t) 5 IL (x,
t)
in 2
x
x
[O, Tl with
[0, T].
(3.2)
Assume that (u-, u+) satisfy the inequalities (u+), - Au, - L max f(u+, ES) 2 0 E2 Jou-sscj* u+
in Q
[O, Tl,
(3.3)
(u-)~ - Au- - .i min f(u-, E2 I* u-9sJn u+
in Sz x [O, Tl,
(3.4)
au-505-an
0) I 0 au+ an
x
on afi x [O, Tl,
u-(x, 0) 5 g’(x) 5 24+(.x,0)
for all x E 0
(3.6)
Then u-(x,
t)
I d(x,
t)
I u+(x,
t)
in s2 x [O, T].
Prooj Set
YT = {p(t) E C?([O, Tl) ( J-u-(., t) I dt) 4 ~u+C.. t) for t E [O, Tl) cl n and, for each p E Y,, let u,, be the solution to uI = Au + (1 /~~)f(u, E/J) with the given boundary and initial conditions in (1.1 b) and (1. lc). Then, by a comparison principle, u- I up I u+
in Q x [O, T].
Define F = T/J = Is2 ZQ,(., t), t E [0, T]. Then one can use Schauder’s fixed point theorem to show that T has a fixed point in YT, which yields a solution ii to problem (PE). Clearly, this
1288
XINFU
CHEN ef al
solution satisfies the assertion of the lemma. Moreover, by uniqueness, li = uE. n In the sequel, we shall call a pair (u-, u+) satisfying (3.2)-(3.5) a sub-super solution pair. 3.3. The travelling wave solutions
For the purpose of constructing subsuper solution pairs, we first introduce travelling solutions of a related one-dimensional parabolic equation. Consider the function
wave
F(u, v, 6) = f(u, v) + 6,
(3.7)
where v and 6 are real parameters. Since FL(‘-c LO, 0) f 0 by the assumptions in (1.2) on f, the implicit function theorem ensures the existence of two positive constants vs and 60 such that for all v E [-vo, vol and 6 E [ -60,60] the equation, for U, F(u, v, 6) = 0, has two solutions h- (v, 6) and h+(v, 6) which are smooth in v and 6, and satisfy h,(O, 0) = kl. In addition, in view of (3.7) and (1.2), they satisfy h,(v, 6) = +-I + O(lvl + ISI), $(“6)
>0
(3.8)
v v E [-vo, vol, s E [--60,601.
(3.9)
We search for a travelling wave solution of the equation ur = u,, + F(u, v, 61,
x E 08,t E Kit,
which converges to h,(v, 6) as x - fo3, i.e. a pair (U(., v,6), W(v,6)) such that u(x,t) = U(x - Wt) satisfies the parabolic equation. Set z = x - Wt. It comes to search for U = U (z, v, 6) and W = W(v, 6) such that U” + W(v, 6)U’ + F(U, v, 6) = 0, ( U(-00, v, 6) = h-(v, S), U(+oo, v, 6) = h+(v, 6).
(3.10)
This problem has a solution (U(z, v, 6), W (v, 6)). The solution is unique up to a translation in z and U is monotonic increasing in the z variable. In view of (3.9) and the monotonic properties of U in z, we can play with this translation invariance so that we can choose a smooth solution U(z, v, 6) satisfying Us(z, v, 6) 2 0. Some properties of this solution are listed in the following lemma. 3.3. There exist small positive constants vo and 60 such that, for all v E [-~a, vol and S E [-ho, 601, there exists a solution (U(z, v, 6), W(v, 6)) to the problem (3.10) which has the following properties: (i) there exist positive constants A and fi such that, for all (v, 6) E [ -VO, VO] X L-60, 601, LEMMA
OcUzrA and OsUglA IU,(+z,v,6)1 + IU,,(fz,v,6)I
VzzER, + IU(~Z,V,S)
-hdv,6)1
I Ae+
VZE 08,;
(ii) the travelling wave velocity W (v, 6) is a smooth function of its arguments and satisfies W(O,O) = 0.
An Allen-Cahn
equation with a nonlocal term
1289
The existence and the properties of U(z, V,b), as well as the smoothness of the function W(v, 6), can be proved by standard methods such as those presented in the Appendix of [3]. See also [23, 241. Finally, we define co := F(O,
0)
3.4. Construction of sub-super solution pairs
We shall now construct sub-super solution pairs to control the solution uE of problem (PE) and show that, in the limit as E - 0, uE tends to a function which takes only two values + 1, and whose boundary of the transition from - 1 to + 1 is the solution of the limit geometric motion problem (PO) where COis defined in (3.11). To construct the sub-super solution pairs, we shall make use of the travelling wave solutions introduced above and a modified distance function that we define now. Let I = U (I, x {t]) be the unique solution of the limit geometric motion problem (PO) rE[o.rl
and let d(x, t) be the signed distance function from x E fi to I,, t E [0, T], that is, 12(x, t) ] = dist(x, I,),
2(x, t) > 0 in Sz:,
and d(x, t) < 0 in Sz,.
Let do be a positive number such that 2(x, t) is smooth in D(2dl)) = ((x, t) E sl x [O, Tl 1 I&x, t)I < 2do) and dist(I,, aR) > 2do Let d(x, t) be a smooth modification
V t E [0, T]
of d” such that
d = Jin DUO) = ((x, t) E Sz x [O, T] 1 Id”(x, t) I < do}, do < Id] I 2d0 and dd” > 0 in D(2do) \ D(do), IdI = 2do and dd” > 0 in fi x [0, T] \ D(2do). Set p(t) = Ifi:
I - I Sz, I. We now prove the following lemma.
3.4. There exist positive constants C* and C** such that if ml 1 1, rn2 1 1, m3 2 1 and m3 1 C*ml, m2 1 C** m3, then, for some positive EO= so(mi, m2, ms) and 0 < E < ~0, the functions u- and u+ defined by
LEMMA
z&(x,t)
= u(
d(x, t) + mlEe’Q’ , Ep(t), +E~WZ~ e’n2’> E
satisfy (3.2)-(3.5); that is, (u-, u+) is a sub-super solution pair. Prooj Since near aR, d = 2do, we have that &&an = 0 on afi x 10, T]. Also, since U, > 0 and U6 2 0, we have that U+ > U- in R x [0, T]. It remains to verify the differential inequalities (3.3) and (3.4). To this end, we first calculate
1290
XINFU
R
CHEN et al.
a,+
fir-
Write Jo2: 24+ ( . , t) as e’narI] + h, (up, J u+ = J [u+ - h+(~p(t), .E~KTQ Q: a,+
E2m3 ern20ISZ: I.
(3.13)
Since, by (3.8), h+(~p(t), &2m3em2t) = 1 + O(.EJI~IJ~(~O,~~) + E2mje’“2’), and, by lemma 3.3(i), u+ = h+(~p(t), E2m3emjr) + O(e-@do’2E ) in {x E !A d(x, t) 1 do}, it follows from (3.13) that u+ =
J
0:
J
[
u+ - h+ (up,
E2mjdn2’)]
IxlO
+ E2m3ewt + IQf I + O(EIlPIlCO([O,T,I Here and in the sequel, all the OS are independent sufficiently small. In a similar way, we have u+ =
J
61,
+ e-Bdoi2E
>
_
of ml, m2 and m3 but may require E to be
u+ - h- (zp(t), E2mgP2’ )] L
J {xl-do
- ItI- I + O(~llpll~~~~,~~) + E2rnje’n2r + e-Bdo/2r).
To calculate the integrals on {x E !A I 0 < d(x, t) < do} and on {x E fi I -do < d(x, t) < O}, we make the transformation x E D(do) ++ (s, d), where s(x, t) is the projection of x on G along the normal of I’,, and d(x, t) (= 2(x, t)) is the signed distance function from x f D(do) to I,. Denote by J(s, d) the Jacobi of the transformation and by (., *) the pair (.sp(t), E2rn3en2’ ). Then
do
r + ml Eemzr >-, .) - A+(., *)]~(s, r) dr ds
JJ[ =cJJ[
=
LO
‘(
E
do/E
U(z + ml em21,., .) - A+(-, .)]J(s, EZ) dz ds = O(E),
r, 0
where we have used the change of variables z = I/E in the second equation, and the fact that Ub, -> -) - h+(., -) = O(e+) as z - +a~ in the third equation. Similarly,
1 J
[u+-h-(.:)]~
An Allen-Cahn
equation
with
a nonlocal
1291
term
0 =E
U(Z II
+ ml
em*l,
J
-,
-) -he-(-,
-)]J(s, EZ) ds dzl
G -4/E 0
I CE
JI --m
U(z + ml em”, -, a) -A-(-;)I
dz m,e”‘2’
0
= CE
II --m
I CEml
U(z, ., -)-h-(-AI
dz+C~
J
IU(z;:)-h~(~;)I
dz
0
e'nzf
for some C > 0, since the integral on [w- on the right-hand side of the last equation is convergent (by lemma 3.3(i)), the function IU(z, ., .) -h- (., .) 1 is bounded and ml 2 1. Combining all these estimates, we conclude that there exists a positive constant Cl such that, when E is small enough,
IS
u+(x, t) dx - p(t) I I C,WZ,em*’ E.
n
Similarly,
we can show that
II
U-(X, t) dx - p(t) I I C,WZ~et”*’ E.
n
It then follows that (3.14)
max and that
(3.15) where C2 is a constant independent of ml, rnz and rnj. Now we shall verify the differential inequality (3.3). Set
Then, by (3.14) Lu+
A direct computation
I
(u+)!
-
AU, - iflu+,
E&))
- C2ml ernlr.
gives
(u+)~ = (dl + Em~m2em2’)~
Au, = yUzz
-I- %Ji. &
+ Ep’(t)U,
+ E2m3m2e’“*’ Ua,
1292
XINFU
Moreover, by the definitions
CHEN
et al.
of U and of u+,
-f(u+! W(l)) = m3 e’n*t + Lw-- + W(Ep(t), E2m3 t+‘) U, -. &2
E2
E
E
Hence, f u+ 2 [ dl + Ernlrn2 ern2*-Ad + +[l
- lVd12]$
1-E
W(~p(t), E2rn3eJn*r) U,
E
+ [Ep’(t)C:
+ E2m3m2ern2’ ~61 + [m? - Czrnl] emzr
=: T, + T2 + T3 + T4. First, we estimate TI . Since dl - Ad + CO,U= 0 on r, = {x E R 1 d(x, t) = 0}, it follows from the mean value theorem that Id - Ad + co/.d)l 5 Gld(x,
t)l.
Note that I W(v, 6) - W(O,O) - (aW/lh~)(O, 0)vl = I W(v, 6) - covl = O(I61 + v2), so that ( W(.g.t(t). Zrn, emzf) - q&t)
1 5 C4E2m3 e’n2’
for all f E [O, Tl. Therefore, W(Ep(t), E2rn3en*‘) E
d! + EmI m2 e’n2t - Ad +
= (df - Ad + cop(t)) + ( W(E’(‘)f~2m3e’“2f)
- cop(t)) + Em,m2e1Qf
2 -C3 Idtx, t) I + E(mlm2 - C4m3) e’n2f 2 -4’3 Idk tf + EmI P2’ I + Ee’“*‘(ml (rnz - ~3) - C4m3) 2 -C3ld(x, t) + EmI e’“*’ I if we choose m2 2 C3 + C4m3 2 C**mj. Thus, since Uz > 0, T, ~ -c
Mb, t) + w 3
ernzrI
d(x, t) + EmI em2’ Vi
(
E
E
, w(t),
E2m3
em*’
)
2 -C3 m;x IzU,(z, +, .) [ 2 -CS. As for 7& by the definition
of d, (Vd( = 1 when Id( < do. According to lemma 3.3(i),
IumI‘ (d(x, t) + Em1 e”‘zl , Ep(t), E2m3 efnzf>I
5 A e-BI(dcx,r,+Fm,<“*‘)/El
E
Moreover, for 0 5 t I T and (dl 2 do, we have d(x t) + m E
em*’ l 1 I db, t) I _ m, tin2T 1 6 _ cr 1 &I E E 2E
An Allen-Cahn
1293
equation with a nonlocal term
for E small enough. Therefore, dtx, t) + urnse’nlr
s C6 e-0doj2E.
, Ep(t), E2rn3e’“*’
E
Thus IT21 5 GE- 2 e-Odo/=f
Since, furthermore, h+
if rn3 L C*ml we have
2
T3 2 -c5
-
EM’(~)
U, r -CUE, we deduce that
c6Ep2
e-0do’2E
+ (m3 - C2ml) t+
-CUE
in d X [0, T]
and C* is large enough. Similarly, we can show that by taking m3 and m2 as above, (u-)~ - Au- - 1. E2 lou~~~,ou+ f(u-.
This completes the proof of lemma 3.4. Combining follows.
2 0
ES) I 0
in d X [O, Tl.
n
lemmas 3.2 and 3.4, we can now prove theorem 1.1. A refinement of theorem 1.l
THEOREM 3.5. Assume that, for some positive constants a and b, g’(x) satisfies U(d(X,O)/E
-
for all x E fi and all all E E (0, Eo 1) U((d(X,
t)
-
Em1
E
eMZt)/E,
a, E/J(O),
-bE=) 5 g’(x) I U(d(x,
0)/E
+ a, Ep(O),
E (0, 11. Then there exists a positive constant Eu(t),
-E2W13e’n2’)
5
U’(X,
t)
I
U((d(X,
t) +
~~
EmI
(3.16)
bE=)
E (0, 1) such that for e”*‘)/E,
E/l(t),
E2mspzt)
in Sz x [O, T], where ml = max(1, a), m3 = max(C*mt,
b), m2 = C**ms.
Consequently, IfiIyE(X, t) =
- 1 in Sz; +I inn:
V t E [O, T].
The assertion of the theorem follows directly from lemmas 3.2 and 3.4, and the properties of U in lemma 3.3. Details are omitted. Remarks. (1) From the proof of the theorem, we see that the assertion of theorem 3.5 is also valid if we replace the Neumann boundary condition (1 .l b) by the Cauchy boundary condition d(x, t) = 1 on XI X [0, T]. (2) Our assumption (3.16) on the initial data gc requires that gc is well shaped. For generic initial data, formal analysis and rigorous results (see [7,13,5]) indicate that after a short time (of order O(EJ In ~1)) such a shape will be developed.
1294
XINFU 4. DERIVATION
CHEN OF
ef al.
PROBLEM
(PE)
In this section, we shall show how one can derive problem (P,) from the following system of reaction-diffusion equations ul = Au + ff(~,
Yv)
1
-rvl = -Av+u-”
(PUT)
Wan
Y
PO
av/an = 0 u(x, 0) = uo(x)
in fi X R+, in fi X R+, on aszx R,, on aszx R,, in Q in Q,
v(x, 0) = vo(x) where IY and T are small positive parameters, and E and y are fixed positive constants. More precisely, we shall first show that, as IJ I 0, the solution (ST, vaT) of problem (P”‘) converges to (u’(x, t), v’(r)) which is the unique solution to the following problem I
UT! = AuT + iftuT, T:v~(~)
(PO’)
=
f n
$+(,))
in fi x R+,
uT(x, t) dx - bvT(r)
in 0 x R,, on an x R,, in 0,
&Clan = 0 22(X, 0) = L@(x) vT(0) =
nf
vo(x) dx,
where fn = h so. Then, we shall show that, as T I 0, the solution (Us, v’) of problem (PO’) converges to (u’, ~foz/) where z/ is the unique solution to problem (PJ. For this purpose, we assume that the initial data in problem (PUT) satisfy the hypothesis 240E I-1 -ml), 1 + mol, uo E ff’m, vo E [-y(l + mo), Y(1 + mo)l, i vo E L”(fiz), where mo is a small positive constant satisfying (3.1).
(4.1)
LEMMA 4.1. Assume that f E C’ ([ -2,2] x [ - 1, 11) satisfies (3.1) for some positive constant mo > 0 and that ug and vg satisfy (4.1). Also assume that E and y are fixed positive constants satisfying E]1;2](1 + mo) I rn;.
(4.2)
Then, for all positive constants cr and T, the problem (PUT) has a unique solution (P, vgT). Moreover, for every T > 0, there exists a constant C = C(T), which is independent of u and T, such that the following estimates hold IP(X,
t)I I 1 + mo
IVUT(X,t)I 5 Y(1 + mo)
V(x, t) E fi x [O, co),
V(x, t) E sz x [O,oo),
(4.3) (4.4)
An Allen-Cahn
equation with a nonlocal term
II~aTIIL”(O,T;H’(n,,+ lI~uTtIILqRX(O,T))5 c,
1295
(4.5)
T lVvuT1*
I
(4.6)
Ca,
f.J
OSa T
II
p7-2
C rs76
v
6
E
(0.
T),
(4.7)
60
v t E (0, Tl,
(4.8)
v t E (0, T].
(4.9)
R d 1dt I n
vUT(., t) 1 I $
ProoJ: By a method of invariant regions [25, Chapter 14, Section B] and the conditions (3.1) and (4.2), one can show that the set { (u, v) ] ]U( I 1 + mo, ] v ] I y( 1 + mo) ) is positive invariant for the parabolic system (POT). Therefore, by a classical theory for parabolic systems [25, Chapter 141, the problem (P"') has a unique global solution (P, vaT) and the solution satisfies the estimates (4.3) and (4.4). The estimates (4.5)-(4.9) follow from the standard energy estimates. Here we briefly give the procedures. To obtain (4.5) we multiply the equation for zP by zPTI, integrate by parts on fi x (0, T) and use the uniform boundedness of zPT and PT. The estimates (4.6) follow by multiplying the equation for vUT by vaT and integrating by parts on !A x (0, T) . The estimates (4.7) and (4.8) follow by multiplying the equation for vVT by tvuTl and integrating by parts on R x (0, T), whereas (4.9) follows directly from integrating the equation for vaT on 0 x {t } . n
In the sequel, T > 0 is an arbitrary fixed constant. With the basic estimates in lemma 4.1, we can now prove the following result. Let (uuT, vuT ) be the unique solution of the problem (P”‘) given in lemma 4.1. Then there exist functions u~(x, t) E L”(0, T;H’(fl)) n H’(fl x (0, T)) and vT(x, t) E L*(O, T;H’(fl)) ~7H’(R x (6, T)) (for all 6 E (0, T)), and a sequence {a,,}~=, such that as m - 00, o;, I 0 and the following limits exist: (i) lim zPT = uT weakly in H’(fi x (0, T)), strongly in C([O, T];L*(fl)) and a.e. in 0 X m-+m (0, T); weakly in H’(R x (6, T)), strongly in (ii) t?Z-+CC lim vanIT = vT weakly in L*(O, T;H’(Cl)), C([6, T];L*(s2)) and a.e. in fi x (6, T), for all 6 E (0, T); (iii) vT = v’(t) E C?,‘([O, T]) and $-oPnT(x, t) dx - vT in C?**([O, T]) for all o( E (0, 1). LEMMA 4.2.
Proof: The assertions (i) and (ii) are immediate compactness of the set {u E L”(to, T;H’(SZ)),
consequences of lemma 4.1 and the relative ul E L2(to, T;L*(sZ))}
in C(to, T; L*(0)), with to E [0, T). It remains to prove (iii). From the lower semi-continuity the functional M; - ]o w* and the estimate (4.8) it follows that
of
1296
XINFU CHENetd T
T
JJ 60
IVrT(x, t)12 dx 5 liminf m-m
JJ 60
(VPnT(x, t)j2 = 0
V 6 E (0, T],
so that VrT = 0 a.e. in Sz x (0, T), and therefore vT = vT (t). The convergence result in (iii) and the regularity of vT then follow from the second assertion of this lemma and the estimate (4.9) n in lemma 4.1. The proof is completed. As we shall show below, the limit (Us, vT) obtained in lemma 4.2 is a solution to the problem (PO’). Since one can directly prove that the problem (PO’) has a unique solution, we can strengthen the conclusion of lemma 4.2 as follows. THEOREM 4.3. Let (u’(x, t), ~~0)) be the unique solution to the problem
be the unique solution to the problem (I’“‘).
(PO’) and (zPT, vu’)
Then, as u I 0,
(fP, vUT) - (Us, v’) weakly in H’(a
x (0, T)) x L2(0, T;@(R)).
PruojI First we assume that (Us, vT) is the pair of functions stated in lemma 4.2. Since the function zPT satisfies
for all 4 E C” (fi x [O, T]) satisfying $I ( *, T) = 0, sending m - W, we deduce from Lebesgue’s dominated convergence theorem and lemma 4.2 that ur satisfies the parabolic equation for uT in the problem (pT), as well as the homogeneous Neumann boundary condition and initial condition. Since vUmTsatisfies the equation Tf”n’“T(l) - Tfvo = j fuUtflT - ; /fPT R n 00 OCI sending m - 0 we deduce that vT satisfies the ordinary differential equation and the initial condition in the problem (I”“). Therefore, (Us, rT) is a solution to the problem (I’(“). Since the solution (u’, vT) to the problem (I““) is unique, we conclude that the whole sequence n { (UUT, VT I 0~~~1 converges to (Us, vT) as o I 0. This completes the proof of the theorem. Finally, we derive the problem (PE) from (PO’). THEOREM 4.4.
For each T E (0, 11, let (Us, v’) be the unique solution to problem (PO’). Then, as T I 0, (Us, v7) converges to (uE, Y&G), where uE is the unique solution to problem (P,). ProoJ From the uniform
lIuTII~m I 1 + Multiplying integrating
estimates in lemma 4.1 we can derive that n-20,
llvTIILm
5 .dl
+ mO),
li~T'l~kz(~~(O.T))
the ordinary differential equation for vT in the problem the resulting equation over [0, T], we obtain
s
c
(P”‘) by f&r
- ivT and
An Allen-Cahn
where C is a constant independent
equation with a nonlocal term
1297
of T. It then follows that
llvT - YfUrllL2((o,T)) - O n
as T - O.
Consequently,
as T - 0. It then follows that uT - z.8as T - 0, where uE is the unique solution to the problem # (P,). This completes the proof of the theorem. Acknowledgements-The
authors are very grateful to Professor Y. Nishiura for many stimulating discussions. REFERENCES
1. ALLEN S. & CAHN J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27, 10841095 (1979). 2. RUBINSTEIN J., STERNBERG P, & KELLER J. B., Fast reaction, slow diffusion and curve shorting, SIAM .I appl. Math. 49, 116-133 (1989). 3. FIFE P. C., Dynamicsfor Internal Layers and Diffitsive Interfaces, CCMS-NSF Regional Conf. Ser. in Appl. Math. SIAM, Philadelphia (1988). 4. BRONSARD L. & KOHN R., Motion by mean curvature as the singular limit of Ginzburg-Landau model, J dt@ Eqns 90, 211-237 (1991). 5. DE MOTTONI F! & SCHATZMAN M., Development of interfaces in OB”, Proc. R. Sot. Edinb. 116A, 207-220 (1990).
6. DE MOTTONI I? & SCHATZMAN M., Geometrical evolution of developed interfaces, Trans. Am. math. Sot. (to appear). 7. CHEN X., Generation and propagation of interfaces for reactiondiffusion equations, 1 difl Eqns 96, 116-141 (1992). 8. EVANS L. C., SONER H. M. & SOUGANIDIS P. E., Phase transitions and generalized motion by mean curvature, Communs pure appl. Math. (to appear). 9. ILMANEN T., Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, preprint. 10. NISHIURA Y. & MIMURA M., Layer oscillations in reactiondiffusion systems, SIAM 1 appl. Math. 49, 481514 (1989). 11. OHTA T. & KAWASAKI K., Equilibrium morphology of block copolymer melts, Macromolecules 19, 2621-2632 (1986). 12. FIFE P C. & TYSON J., Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J them. Phys. 73, 2224-2237 (1980). 13. CHEN X., Generation and propagation of interfaces for reaction-diffusion systems, Trans. Am. math. SOC. 334, 877-913 (1992).
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15. EVANS L. C. & SPRUCK J., Motion of level sets by mean curvature, J d@ Geom. 33, 635-681 (1991). 16. CHEN X. & REITICH F., Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J math. Analysis Applic. 162, 350-362 (1992). 17. GIGA Y. & GOT0 S., Geometric evolution of phase-boundaries, in On The Evolution of Phase Boundaries (Edited by M. E. GURTIN and G. B. MCFADDEN). Springer, Berlin (1992). 18. CHEN Y. G., GIGA Y. & GOT0 S., Uniqueness and existence of viscosity solutions of generalized mean curvature Bow equations, 1 d18 Geom. 33, 749-786 (1991). 19. GRAYSON M., The heat equation shrinks plane curves to points, J d@ Geom. 26, 283-314 (1987). 20. HAMILTON R., Three manifolds with positive Ricci curvature, 1 d13 Geom. 17, 2555306 (1982). 21. HUISKEN G., Flow by mean curvature of convex surfaces into spheres, J. diff: Geom. 20, 237-266 (1984). 22. EI S. I. & YANAGIDA E., Dynamics of interfaces in competition-diffusion systems, SIAM .I appl. Mach. 54, 1355-1373 (1994). 23. ARONSON D. G. & WEINBERGER H. E, Nonlinear diffusion in population genetics, combustion and nerve propagation, in Partial D@rential Equations and Related Topics, Lect. Notes in Math., Vol. 446, pp. 549. Springer, New York (1975). 24. FIFE P. C. & HSIAO L., The generation and propagation of internal layers, Nonlinear Analysis 12, 1941 (1988). 25. SMOLLER J., Shock Waves and Reacfion-D@sion Equations. Springer, New York (1983).