Reaction-Diusion Fronts
in Periodically Layered Media
George Papanicolaou and Xue Xin
Courant Institute of Mathematical Sciences
251 Mercer Street, New York, N.Y. 10012
Abstract
We compute the eective wave front speeds of reaction-diusion
equations in periodically layered media with coecents that have
small amplitude oscillations around a uniform mean state. We compare them with the corresponding wave front speeds in the uniform
state. We analyze a one dimensional model where wave propagation
is along the layering direction of the medium and a two dimensional
shear ow model where wave propagation is orthogonal to the layering direction. We nd that the eective wave speed is smaller in the
one dimensional model and is larger in the two dimensional model for
both bistable cubic and quadratic nonlinearities of the KolmogorovPetrovskii-Piskunov form. We derive approximate expressions for the
wave speeds in the bistable case.
1 Introduction
We will consider traveling waves for reaction-diusion equations (R-D)
ut =
ujt
=0
Xn (aij(x)ux )x + f (u)
i
i;j =1
j
= u (x)
(1.1)
0
where a(x) = (aij (x)) is a smooth positive denite matrix, 2-periodic in
each direction in Rn. The nonlinear function f (u) is either the bistable
nonlinearity
f (u) = u(1 , u)(u , ) ; 2 (0; 1=2)
(1.2)
or the quadratic nonlinearity
f (u) = u(1 , u)
(1.3)
of the Kolmogorov-Petrovskii-Piscunov (KPP) form [11]. Travelling wave
solutions of (1.1) have the form u = U (k x , ct; x), where the direction of
propagation k 2 Rn is a unit vector, c = c(k) is the speed and the wave
prole U = U (s; y), s = k x , ct, y = x, satises the equation
(k@s + ry ) (a(y)(k@s + ry )U ) + cUs + f (U ) = 0
U (,1; y) = 0; U (+1; y) = 1;
U (s; ) 2 , periodic :
(1.4)
When a(x) = I , the identity, (1.1) is the usual reaction-diusion equation in
a homogeneous medium, the travelling wave solution has the form U (kx , ct)
and (1.4) is an ordinary dierential equation for U as a function of s. For
a medium with periodic structure it is reasonable to look for wave proles
that have also periodic structure, along with their usual form in the direction
of propagation, as is common in homogenization problems [2]. For bistable
nonlinearities (1.2), traveling waves and their stability are analyzed in [1],
1
[7], in the homogeneous case. For KPP nonlinearities they are analyzed in
[9], [10], [11]. In [15], Xin showed that if a(y) is close to a constant positive
denite matrix, then (1.4) admits a unique solution (U; c) up to a constant
shift in s, with c < 0. This travelling wave solution is stable in one space
dimension. In the multidimensional case it is not known if the travelling
waves constructed in [15] are stable.
We see from (1.4) that the wave speed c and wave prole U are coupled
and must be determined together. However, in the KPP case the asymptotic
speeds can be determined independently of the wave prole and Gartner
and Freidlin [10] obtained the following result: If u is nonnegative(6 0)
and compactly supported, then there exists a number c?(k), the asymptotic
speed in the direction k, such that limt!1 u(t; ctk) = 1, if 0 c < c? (k) and
limt!1 u(t; ctk) = 0, if c > c?(k). To compute c?, let
0
Ly u =
Xn (@x , yj )(aij(y)(@x , yi)u) + fu(0)
i;j =1
j
i
(1.5)
where y 2 Rn is any constant vector and u is any smooth 2-periodic function. Let = (y) be its principle eigenvalue with positive eigenfunction. It
can be shown that is smooth and convex in y. Then
(1.6)
c?(k) = y;kinf> (y;(yk))
In this paper, we calculate approximately the speed of the travelling wave
in a periodically layered medium, uctuating around a constant state with
mean one and small variation. We compare the wave speed in the layered
medium (the eective wave speed) with that in the constant state and see
how the medium aects it. We consider two model problems of propagation
in layered media. In section 2 we analyze traveling waves in a two dimensional shear ow where the ow is layered in the y direction and the wave
is propagating in the y direction, orthogonal to the direction of layering in
the ow. We derive an approximate formula for the speed of the travelling
(
)
0
1
2
2
wave, up to second order in the variations of the shear ow. We nd that
the speed in the two dimensional layered medium is larger than that in the
uniform medium. In section 3 we analyze a one dimensional problem where
the wave propagates along the layering direction of the medium. As in the
2-D shear ow model the qualitative behavior of the wave speed depends
only on the form of the nonlinearity f (u) and not on the detailed properties
of the medium. The approximate calculation of the speed requires the determination of a set of constants which are related to the solution of certain
second order ordinary dierential equations on R . Solvability of the ODE's
is studied in [15]. Here they are solved numerically using a nite dierence
method and the constants are then computed. Our numerical results show
that the wave speed decreases in the layered medium.
In section 4 we give the corresponding qualitative results for the eective
wave speeds in the case of KPP nonlinearity by using formula (1.6). It
turns out that the same phenomenon occurs: speedup in the two dimensional
shear ow model and slowdown in the one dimensional layered medium.
These results are analogous to what happens to the eective diusivity in
the corresponding linear problems. It increases in the two dimensional shear
ow case and decreases in the one dimensional case.
1
2 Wave Speed in a 2-D Shear Flow Model
The reaction-diusion equation in a two dimensional shear ow is
ut = 4u + W ru + f (u)
W = (0; w(y )); 4 = @y1 y1 + @y2 y2
f (u) = u(1 , u)(u , ); 2 (0; 21 )
2
1
3
2
(2.7)
where w(y ) is a smooth periodic function with period 2. A travelling wave
moving in the y direction has the form
1
2
u = '(y , ct; y ) = '(s; y ); s = y , ct :
2
1
1
2
Substituting the above into (2.7), we see that ' satises
'ss + 'y1 y1 + (c + w(y ))'s + f (') = 0
'(,1; y ) = 0; '(+1; y ) = 1; '(s; ) 2 periodic
1
1
(2.8)
1
and we add a normalization condition
1 '(0; y )dy = 1=2:
2
to make the solution unique. The travelling wave equation (2.8) has also
been studied in [5] for a premixed ame propagation model with Neumann
boundary conditions on y and combustion nonlinearity. For the derivation
of this model and its physical background, we refer to [12] and [14].
We are interested here in the eects of w(y ) on the speed c. If w(y ) is
a small mean zero periodic function, will c be larger than the speed in the
uniform medium or smaller? We expand w(y ) in terms of a small parameter
,
w(y ) = w (y ) + w (y ) + (2.9)
where wi(y )(i = 1; 2; ) have mean zero over [0; 2]. Now expand the
travelling wave and the speed
Z
2
1
0
1
1
1
1
1
1
1
2
1
2
1
1
' = ' + ' + c = c + c + 0
(2.10)
(2.11)
1
0
1
and plug these expansions into (2.8). We obtain the following equations up
to O( )
O(1) ' ;ss + c ' ;s + f (' ) = 0
(2.12)
2
0
0
4
0
0
O() ' ;ss + ' ;y1y1 + c ' ;s + f 0(' )' = ,(c + w (y ))' ;s
(2.13)
O( ) ' ;ss + ' ;y1y1 + c ' ;s + f 0(' )'
= ,c ' ;s , c ' ;s , w ' ;s , w ' ;s , 1=2f 00(' )'
(2.14)
1
0
1
2
1
2
1
1
0
2
0
2
1
0
1
1
2
1
1
2
0
1
0
0
2
0
2
1
From (2.12), we get ' = ' (s), ' (0) = 1=2, and c = c (f ), solution of the
usual travelling wave equation
0
0
0
0
0
'00 + c'0 + f (') = 0
'(,1) = 0; '(0) = 1=2; '(+1) = 1
The solvability condition for (2.13) is
Z (c + w (y ))'
1
1
1
0
dsdy = 0
;s
1
where = ec0s' ;s spans the null space of the adjoint of the linearized equation and the integral is over R [0; 2]. Thus c = , < w (y ) >= 0.
We use < > to denote the average of the function inside the bracket over
[0; 2]. Equation (2.13) then becomes
0
1
1
1
' ;ss + ' ;y1y1 + c ' ;s + f 0(' )' = ,w (y )' ;s
1
1
0
1
0
1
1
1
0
1
(2.15)
Let (; ) denote the inner product over R [0; 2] divided by 2. From
(2.14) we have
c (' ;s; ) + (w ' ;s; ) + 12 (f 00(' )' ; ) + (w ' ;s; ) = 0
(2.16)
The last term on left hand side is equal to zero since w has mean zero and
so
c (' ;s; ) = , 12 (f 00(' )' ; ) , (w ' ;s; )
(2.17)
Since f 00(' ) = ,6' + 2 + 2,
1
2
1
0
1
0
2
1
2
0
2
2
0
0
0
2
1
1
1
0
c (' ;s; ) = ((3' , , 1)' ; ) , (w ' ;s; )
2
0
2
1
0
5
1
1
with
Z
(' ;s; ) = ' ;sec s' ;sds
R
Z
((3' , , 1)' ; ) = (3' , , 1)' ec s < ' > ds
0
and
2
1
0
R
0
0
1
0
;s
0
1
0
2
1
0
We solve (2.15) by Fourier series. Let
X bmeimy
w (y ) =
1
and
1
m6=0
X am(s)eimy
' (y ; s) =
1
1
1
1
m6=0
where we have used the fact that < >= 0. We substitute these expansions
into (2.15) and collect Fourier coecients to get
1
a00m + c a0m + (f 0(' ) , m )am = ,bm ' ;s
0
2
0
0
(2.18)
Note that ' ;s satises
0
a00m + c a0m + f 0(' )am = 0
0
0
and (2.18) has the unique solution
bm '
am = m
;s
Thus
<' > =
2
1
(2.19)
0
2
X jbmj ' ;s
2
m6=0
m
4
= ' ;s
2
0
where
=
X jbmj :
2
m6=0
6
m
4
2
0
Z
I=
Let
+ 1 )' ec0sds;
(
'
,
;s
3
R1
and
3
0
0
J = ,(w ' ;s; ) :
1
Then
1
c = 3 I + J
2
Z
= '
where
s
By (2.19), we have
X bm eimy
' (s; y ) = ' ;s
1
1
0
and so
ZX
, ( jbm j )'
1
=
m6=0
Z
)=,
J = ,(w ' ;s;
1
2
s m6=0
Let
Then
c0 s ds:
;s e
2
0
m
Z
Z
2
1
;ss '0;se
0
1
c0 sds
X jbmj
2
m6=0
m :
(2.20)
2
J = , 1 ' ;ss' ;sec0sds = , 2
R
= 2 c 1 ' ;sec0sds
R
= 2 c
0
1
< w ' ;s > ds
0
2
=
R1
m
0
Z
R1
(' ;s)sec0sds
2
0
2
0
0
Therefore
c = 3 I + 2 c
2
7
0
(2.21)
Next we compute I = I (). In the case of f (') = '(1 , ')(' , ) with
2 (0; ), we have Huxley's explicit expressions for ' and c
1
2
0
0
' = (1 + e, p2 ),
c = 2p, 1
2
s
1
0
0
Z
I =
Let
1
R
and
1
' ' ;sec0sds
3
0
0
Z
I = +3 1 1 ' ;sec0 sds
R
so that I = I , I . We will show that I = I and hence that I = 0. Since
p
p
' ;s = p1 e,s= (1 + e,s= ),
2
we have
p
p
1
p
I
=
(1 + e,s= ), ( p1 ) exp(,3s= 2)(1 + e,s= ), ec0sds
,1
2
1
p
p
1
p ,1 exp(,3s= 2)(1 + exp(,s= 2)), exp( 2p, 1 s)ds
=
2 2
2
p
1
1
s=
, d
exp
(
,
3
)
exp
((2
,
1)
)(1
+
exp
(
,
))
=
2 ,1
1 ,
y exp , 1
=
y
(1 + y), dy
2
1 1(x , 1) , x, dx
x y
=
2
1 1 1 (1 , 1=x) , dx
=
2
x 1 z (1 , z) , dz
z =x
=
2
z
1
,
=
2 z (1 , z) dz
1 B (3 + 2; 4 , 2)
=
(2.22)
2
2
1
2
1
2
0
Z
1
=
=
2
(
)
3
0
+
Z
Z
Z
Z
Z
Z
Z
2
1
2
+
7
+
2 +3
7
+
3
2
7
1
=1
+
3
4+2
1
4+2
3
2
2
0
1
2
7
+
1
2
3
0
= +1
2
2+2
3
2
0
8
2
6
where B = B (x; y) is the Beta function. Similarly, we have
1 ,
I = 21 +3 1
y (1 + y), dy
= 12 +3 1 z (1 , z) , dz
= 12 +3 1 B (2 + 2; 4 , 2)
Using the identities
x),(y)
B (x; y) = ,(
,(x + y)
,(x + 1) = x,(x)
2
Z
Z
+
3
2
6
0
1
1+2
3
2
0
we obtain
(2.23)
(2.24)
),(4 , 2)
B (3 + 2; 4 , 2) = ,(3 + 2,(7)
),(4 , 2)
= 2 +6 2 ,(2 + 2,(6)
= +3 1 B (2 + 2; 4 , 2)
which implies
I =I
We have shown therefore that I = 0 and hence the speed c() has the expansion
(2.25)
c() = c (1 + 2 + )
where is given by (2.20) and the bm 's are the Fourier coecients of w (y ).
Our result (2.25) says that waves propagating in a particular (a layered)
divergence free, mean zero periodic medium speed up. This phenomenon has
been observed in ame propagation through turbulent media, in [6] and [13],
and formulas for the turbulent ame speed in terms of the laminar ame
speed are similar to (2.25).
1
2
2
0
1
9
1
3 Wave Speed in a 1-D Layered Medium
In this section, we study the speed of the travelling wave solution of the 1-D
bistable reaction-diusion equation
ut = (a(x)ux)x + f (u)
(3.26)
where f (u) given by (1.2) and a(x) = 1 + a (x), with a (x) a smooth, 2periodic, mean zero function and a small parameter. The travelling wave
solution has the form u = u(x , ct; x) = u(s; y), where s = x , ct, y = x.
The wave prole u satises as a function of s and y the equation
1
1
(@s + @y )(a(y)(@s + @y )u) + cus + f (u) = 0
u(,1; y) = 0 ; u(+1; y) = 1
u(s; ) 2 , periodic
< u(0; ) >= 1=2
(3.27)
We write
u(s; y) = u(s; y; ) = ' (s) + ' (s; y) + c = c() = c + c + 2 c + 0
1
2
0
1
2
where ' (s) and c satisfy
0
0
'ss + c 's + f (') = 0
'(,1) = 0; '(0) = 1=2; '(+1) = 1
0
Let
L v = (@s + @y ) v + c vs + fu(' )v :
Expanding in powers of we nd that satises
2
0
0
0
1
L = ,c0 ' ;s , (@s + @y )(a (@s + @y )' )
0
1
1
0
0
The solvability condition for (3.28) is
(,c ' ;s , (@s + @y )(a (@s + @y )' ; ' ;sec0 s) = 0
1
0
1
10
0
0
(3.28)
or
c (' ;s; ' ;sec0 s) = ,((@s + @y )(a (@s + @y )' ); ' ;sec0 s)
= ,(a ' ;ss; ' ;sec0s)
= , < a > (' ;ss; ' ;sec0 s) = 0
1
0
1
0
1
0
0
0
0
1
0
0
Thus c = 0. Expanding to order we nd that satises
2
1
2
L = ,c ' ;s , 2(@s + @y )(a (@s + @y ) ) , fuu(' )( )
0
2
2
1
0
1
0
1
(3.29)
2
The solvability condition for (3.29) is
c (' ;s; ' ;sec0 s) = ,(fuu (' )( ) + 2(@s + @y )(a (@s + @y ) ); ' ;sec0s)
= ,(fuu (' )( ) + 2a (@s + @y ) s; ' ;sec0 s)
2
0
0
1
0
1
0
2
1
2
1
1
1
0
0
where is given by (3.28) which simplies to
1
L = ,a ;y ' ;s , a ' ;ss
0
1
1
1
0
We solve (3.30) by Fourier series. Let
a =
=
1
1
X meimy;
X meimy;
(3.30)
0
X m jmj
= 0;
=0
2
0
< +1
2
0
Then the Fourier coecients of satisfy
1
m;ss + (c + 2mi)m;s + (,m + fu(' ))m
= ,imm ' ;s , m ' ;ss
2
0
0
0
(3.31)
0
Let qm satisfy
qm;ss + (c + 2mi)qm;s + (,m + fu(' ))qm = ,im' ;s , ' ;ss
0
2
0
11
0
0
(3.32)
so that m = m qm . Then the second order correction to the speed has the
form
c (' ;s; ' ;s
2
0
0
X Z jmj jqmj fuu(' )' ;sec sds
s
Z
X
,2
' ec sj j (imq + q )ds
ec0s) = ,
2
2
0
0
0
m;s
m;ss
s
Z
X
= , jmj (fuu(' )' ;sjqmj ec s + 2' ec s(qm;ss + imqm;s))ds
s Z
X
= , jmj ' ;sec s(fuu(' )jqm j + 2(qm;ss + imqm;s))ds
0
2
0
2
2
0
s
0
0
0
;s
0
m
2
0
0
2
0
Since from (3.32) qm = q,m , we get
and hence
qm;ss + imqm;s = q,m;ss , imq,m;s
(3.33)
X jmj Bm
(3.34)
c (' ;s; ' ;sec0 s) = ,2
2
where
0
0
Z
2
m>0
Bm = ' ;sec0s(fuu(' )jqmj + 2Re(qm;ss + imqm;s))ds
0
s
2
0
(3.35)
Note that the Bm 's are independent of a (y) and depend only on the
nonlinear function f , since (' ; c ) and qm are determined by solving ODE's
involving f . If qm = pm + irm, then (3.32) becomes
1
0
0
pm;ss + c pm;s , 2mrm;s + (,m + fu (' ))pm = ,' ;ss
rm;ss + c rm;s + 2mpm;s + (,m + fu(' ))rm = ,m' ;s
2
0
2
0
0
0
0
0
(3.36)
and the Bm's can be written as
Bm
Z
=
1
' ec0s(fuu('0 )(p2m + rm2 ) + 2(pm;ss , mrm;s))ds
,1 0;s
+
(3.37)
which after integration by parts gives
Bm
Z
=
1
2('0;sec0s)sspm + 2m('0;sec0s)srm + '0;sec0sfuu('0)(p2m + rm2 )ds
,1
+
(3.38)
12
2
Since of c = c + c + O( ) and c is negative, we see from (3.34) that if
the Bm's are positive then the perturbed speed is larger (in absolute value);
otherwise, if the Bm 's are all negative then the perturbed speed is smaller
(in absolute value). If some Bm's are positve and some are negative, then
the m 's, or a , must be taken into account.
We use a nite dierence method to compute the pm's and rm's from
(3.36) on a large enough nite interval [,N; N ] with zero Dirichlet boundary conditions at the two end points. We use the double precision Linpack
routines to solve the linear equations that result from the discretization of
the ODE's. Then we calculate the Bm 's from (3.38) using a double precision
integration routine from Naglab. We also use Huxley's formulas for (' ; c ).
We veried that the numerical scheme converges as the grid is rened. We
describe here the results of two typical runs. The rst is done on the interval
[,10; 10] with 500 grid points, and = 0:15, where is the middle zero point
of f (u) = u(1 , u)(u , ). We compute B to B and nd that they increase
slowly from ,0:0915 to ,0:08053. They are all negative. The second run is
done on the interval [,20; 20] with 2000 grid points and = 0:30. Now B
to B increase slowly from ,0:05981 to ,0:03660 and again they are all
negative. In all the runs we observed slow convergence of the Bm's to some
negative constant.
These numerical results show that the wave speed is smaller in the layered
medium.
0
2
2
3
0
1
0
1
0
100
1
500
4 Eective Wave Speeds in the KPP case
In this section, we use formula (1.6) to calculate the eective wave speeds
in the model problems of the previous two sections with KPP nonlinearity
(1.3). Since (y) is smooth and convex, the inmum in (1.6) is achieved at
a nite point and thus it is enough to analyze over a compact set of y.
By the Krein-Rutman theorem, is a simple eigenvalue of Ly , so it is stable
13
under perturbation. In our calculations we will drop the term fu(0) in (1.5)
because it only shifts by a constant.
First we treat the shear ow model. The eigenvalue problem for is
Ly u = (@x1 , z ) u + (@x2 , z ) u + w(y )(@x2 , y )u = u
u = u(x; ) = 1 + u (x) + u (x) + = (y; ) = (y) + (y) + (y) + (4.39)
1
2
2
1
2
1
0
2
2
2
2
1
2
where (y) = y + y . Denoting by ' dierentiation with respect to , we
get from (4.39)
2
1
0
2
2
(@x1 , y ) u0 + (@x2 , y ) u0 + w(x )(@x2 , y )u + w(x )(@x2 , y )u0
= 0 u + u0
(4.40)
2
1
2
2
1
2
1
2
and
(@x1 , y ) u00 + (@x2 , y ) u00 + 2w(x )(@x2 , y )u0 + w(x )(@x2 , y )u00
= 00u + 20u0 + u00
(4.41)
2
1
2
2
1
2
1
2
Letting = 0 in (4.40) gives
(@x1 , y ) u0 + (@x2 , y ) u0 , w(x )y = 0 + (y + y )u0
1
2
2
2
1
2
1
2
or
2
2
u0x1x1 + u0x2x2 , 2y u0x1 , 2y u0x2 , w(x )y = 0
Averaging over x and using < w >= 0, we get
1
2
1
2
(4.42)
(4.43)
1
0 = (y) = 0
1
and hence
4xu0 , 2y rxu0 = w(x )y
1
2
(4.44)
The only solution of this equation is u0 = u0(x ) with
1
u0x1x1 , 2y u0x1 = w(x )y
1
1
14
2
(4.45)
Letting = 0 in (4.41) gives
(@x1 , y ) u00 + (@x2 , y ) u00 + 2w(x )(@x2 , y )u0 = 00 + u00
2
1
or
2
2
1
2
0
4xu00 , 2y rxu00 + 2w(x )(u0x , y u0) = 00
1
2
2
(4.46)
(4.47)
and averaging over x gives
00 = 2 (y) = 2 < w(x )(u0x2 , y u0) >= (,2y ) < w(x )u0 >
2
1
2
2
1
(4.48)
Let us solve (4.45) by Fourier series
u0 =
w (x ) =
1
X eimx u0m
m6
X eimx m
1
=0
1
m6=0
Substituting into (4.45) gives
u0m = ,m y,2mmy i
2
2
and so
1
X jmj y
,m , 2my i
m6
X jmj Ref 1 g
2y
,m , 2my i
m>
X jmj ,m
2y
< w(x )u0 > =
2
2
1
2
=0
=
2
2
2
0
=
2
Therefore by (4.48)
00 = 4y
2
2
or
1
2
X jmj
m>0
m + 4m y
4
m>0
2
2
1
m + 4y > 0
2
(y; ) (y)
0
15
1
2
2
1
2
1
(4.49)
(4.50)
which implies that
(y; ) inf (y) = c?(0) > 0
c?() = yinf
(4.51)
y2 > y
2>
y
This means that the eective wave speed is larger than the corresponding
speed in the uniform medium in the shear ow model.
Let us now turn to the 1-D model. The eigenvalue problem for is
0
0
Ly u
u
<a >
1
=
=
=
=
0
2
2
(@x , y)((1 + a )(@x , y)u) = u
1 + u + u + y + (y) + (y) + 0
1
2
1
2
2
2
1
(4.52)
2
Dierentiating (4.52) twice with respect to , we get
(@x , y)(a (@x , y)u) + (@x , y)((1 + a )(@x , y)u0) = 0 u + u0
1
1
(4.53)
and
2(@x , y)(a (@x , y)u0) + (@x , y)((1 + a )(@x , y)u00) = 00 u + 20u0 + u00
(4.54)
1
1
Letting = 0 in (4.53), we have
,(@x , y)(a y) + (@x , y) u0 = 0 + y u0
(4.55)
,y(a ;x , ya ) + u0xx , 2yu0x = 0
(4.56)
2
1
or
1
1
Averaging over x, and using < a >= 0, we get
1
0 = (y) = 0
1
16
2
We then have
u0xx , 2yu0x = y(a ;x , ya )
Letting = 0 in (4.54), we get
1
(4.57)
1
2(@x , y)(a (@x , y)u0) + (@x , y) u00 = 00 + y u00
2
1
2
(4.58)
Averaging over x gives
00 = (,2y) < a (u0x , yu0) >
(4.59)
1
Let us solve (4.57) by Fourier series
X eimxm
m6
X eimxu0m
a =
1
=0
u0 =
m6=0
Substituting into (4.57) gives
m (im , y )
u0m = y
,m , 2ymi
(4.60)
2
and so
00 = (,2y) < a (u0x , yu0) >
= (,2y ) jm j (im , y)
,m , 2ymi
X
m6
X
2ymi g
(,4y ) jm j Ref y ,,mm,,2ymi
m>
X
)(,m + 2ymi) g
(,4y ) jm j Ref (y , m ,m2ymi
+ 4y m
m>
X
(,4y ) jm j m + 3y < 0
(4.61)
1
2
2
2
2
=0
=
=
=
which implies
2
2
2
2
2
0
2
2
2
2
4
0
2
2
m>0
2
2
2
m + 4y
2
2
(y; ) (y; 0)
17
2
2
or
c? () c? (0)
This means that the wave speed in the 1-D layered medium is less than that
in the corresponding uniform medium. We conclude that the qualitative
behavior of eective wave speeds in the KPP case is the same as that in the
bistable case.
Acknowledgement
We would like to thank Marco Avellaneda for many helpful discussions. The
work was supported by the National Science Foundation under grant DMS9003227.
Keywords
Reaction-Diusion Equations, Homogenization, Travelling Waves.
18
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