stabilityoftravellingfrontsofthefisher-kpp equation in r · vol. 15 (2008) stability of travelling...

Nonlinear differ. equ. appl. 15 (2008), 599–622c© 2008 Birkhauser Verlag Basel/Switzerland1021-9722/040599-24published online 26 November 2008

DOI 10.1007/s00030-008-7041-0

Nonlinear Differential Equationsand Applications NoDEA

Stability of Travelling Fronts of the Fisher-KPPEquation in R

N

Rui Huang

Abstract. Recently, Hamel and Nadirashvili [8] proved that there are infinite-dimensional manifolds of (nonplanar) travelling fronts for the Fisher-KPPequation in R

N . In this paper we study the stability of these travelling frontsof the Fisher-KPP equation in R

N . We prove that all (planar or nonplanar)of the travelling fronts are stable under the condition that the perturbationshave compact supports. We also deal with the case of the perturbations with-out compact supports and prove the stability of the travelling fronts underperturbations that decay exponentially with an appropriate rate.

Mathematics Subject Classification (2000). Primary 35K55, 35K57; Secondary35B35.

Keywords. Stability, travelling fronts, Fisher-KPP equation.

1. Introduction and main results

In this paper we consider the stability of the travelling fronts of the followingsemilinear parabolic equation

ut = Δu + f(u) , 0 < u(x, t) < 1 , x ∈ RN , t ∈ R , (1.1)

where the nonlinearity f is a concave function of class C2 in [0, 1] and satisfies

f(0) = f(1) = 0 , f ′(0) > 0 , f ′(1) < 0 , f(u) > 0 , ∀0 < u < 1 . (1.2)

A typical example of such a function f is the quadratic nonlinearity f(u) = u(1−u)considered by Fisher [4] and Kolmogorov, Petrovsky and Piscunov [10]. Fisher-KPP equation (1.1) arises in various biological models, such as population dynam-ics and gene developments (see Aronson and Weinberger [1], Fife [3] and Mur-ray [12] for details).

By a travelling wave for the equation (1.1), we understand a solution u(x, t)such that

u(x, t + τ) = u(x + c0τν0, t) , ∀(x, t) ∈ RN × R , ∀τ ∈ R , (1.3)

600 R. Huang NoDEA

for some direction ν0 ∈ SN−1 and some speed c0 > 0 (up to a change ν0 → −ν0,one can always assume c0 > 0). Such a wave is propagating in the direction −ν0

with the speed c0. The function u can then be written as

u(x, t) = φ(x + c0tν0) , (1.4)

where φ is uniquely defined by φ(x) = u(x, 0) for all x ∈ RN . The function φ is

such that 0 < φ(x) < 1 for all x ∈ RN and it satisfies the elliptic equation

Δφ − c0ν0∇φ + f(φ) = 0 in RN . (1.5)

Conversely, any solution 0 < φ < 1 of (1.5) gives rise to a travelling wave u(x, t) =φ(x + c0tν0) for (1.1), which propagates in the direction −ν0 with the speed c0. Itis well known that the equation (1.1) has, in dimension N , an N + 1 dimensionalmanifold of planar travelling waves, namely

uν,c,h(x, t) = ϕc(x · ν + ct + h)

where ν belong to SN−1 which is the unit sphere in RN , h ∈ R and c ∈ [c∗,+∞)

withc∗ = 2

√f ′(0) .

Such a front ϕc(x · ν + ct + h) propagates in the direction −ν with the speed c.Especially in space dimension N = 1, there are two dimensional manifolds oftravelling waves:

u+c,h(x, t) = ϕc(x + ct + h) and u−

c,h(x, t) = ϕc(−x + ct + h) .

For any c ≥ c∗, the function ϕc satisfies{

ϕ′′c − cϕ′

c + f(ϕc) = 0 , in R ,

ϕc(−∞) = 0 , ϕc(+∞) = 1 .

Such a solution ϕc is increasing and unique up to translation. Furthermore, up toshifts, the function ϕc satisfies

∀c > c∗ , ϕc(s) ∼ eλcs as s → −∞ ,

where λc = c−√

c2−c∗2

2 > 0. While for the travelling waves with minimal speedc∗ = 2

√f ′(0), it is known that, up to shifts,

ϕc∗(s) ∼ |s|eλ∗s as s → −∞ ,

where λ∗ = λc∗ = c∗

2 (see for instance Bramson [2] and Sattinger [15]).Recently, it was proved by Hamel and Nadirashvili [8, 9] that there exists

an infinite-dimensional manifold of entire solutions (which are defined for all timeand for all point x ∈ R

N ) for the Fisher-KPP equation (1.1). In particular, thereare infinite-dimensional manifolds of (nonplanar) travelling fronts satisfying (1.3).Before introducing these interesting results, we need to give some notations. Let

B(0, c∗) = B(0, 2

√f ′(0)

)= {x ∈ R

N , |x| < c∗}

Vol. 15 (2008) Stability of Travelling Fronts of the Fisher-KPP Equation 601

be the open ball of RN with center 0 and radius c∗. The set of all the unit vectors

in RN is denoted by SN−1. Define the sets

X = SN−1 × [c∗,+∞) , X = SN−1 × (c∗,+∞)

equipped with the topology induced by the Euclidean structure of RN . For any

(ν0, c0) ∈ SN−1 × [c∗,+∞), define

S(ν0,c0) ={(ν, c) ∈ X, c0ν0 · ν = c

}=

{S(c0ν0/2, c0/2) \ B(0, c∗)

},

where S(c0ν0/2, c0/2) is the sphere with center c0ν0/2 and radius c0/2. Let M bethe set of all nonnegative and nonzero Radon-measures μ on X (0 < μ(X) < +∞),such that the restriction μ∗ of μ on the sphere SN−1 × {c∗} can be written as afinite sum of Dirac masses:

μ∗ =∑

1≤i≤k

miδ(νi,c∗)

for some integer k = k(μ) ≥ 0, positive real numbers mi = mi(μ) and somedirections νi = νi(μ) ∈ SN−1. For any μ ∈ M, we denote μ the restriction of μ onthe set X. Then, we have μ = μ∗ + μ. Let MTW be the subset of M defined by

MTW ={μ ∈ M;∃(ν0, c0) ∈ SN−1 × [c∗,+∞), μ is concentrated on S(ν0,c0)

}.

Theorem 1.1 (Hamel and Nadirashvili [8]).

(1) Let u be a travelling wave for (1.1) and assume that u satisfies (1.3), namely,u propagates in the direction −ν0 with speed c0. Then,

(1-a) c0 ≥ c∗;(1-b) the function φ defined by (1.4) is increasing in each direction ν ∈ SN−1

such that ν · ν0 > cos(arcsin

(c∗

c0

)), namely, ν belongs to the open cone

directed by ν0 with angle arcsin(

c∗

c0

). Furthermore, for each such ν, one

has

lims→−∞

φ(a + sν) = 0 , lims→+∞

φ(a + sν) = 1 , ∀a ∈ RN . (1.6)

(1-c) if c0 = c∗, then u is a planar travelling wave with speed c∗, namely,u(x, t) = ϕc∗(x · ν0 + c∗t + h) for some h ∈ R. In other words, if 0 <ϕ < 1 is a solution of (1.5) for c0 = c∗ and for some ν0 ∈ SN−1, thenϕ(x) = ϕc∗(x · ν0 + h) for some h ∈ R.

(2) (2-a) In dimension N ≥ 2, there exists an infinite-dimensional manifold oftravelling waves for (1.1). Namely, the restriction of the map μ → uμ onMTW ranges in the set of travelling waves for (1.1), and it is one-to-oneon MTW and continuous on MTW ∩ M. If

μ = μ∗ + μ ∈ M

is concentrated on S(ν0,c0) for some (ν0, c0), then uμ is a travelling wavesatisfying (1.4).

602 R. Huang NoDEA

(2-b) In dimension N ≥ 2, for each c0 > c∗ and for each ν0 ∈ SN−1, thereexists an infinite-dimensional manifold of solutions φ(x), 0 < φ < 1, ofthe elliptic equation (1.5).

(2-c) Let u(x, t) be a travelling wave of (1.1) satisfying (1.3). If u is of thetype uμ for some μ ∈ M, then μ is concentrated on S(ν0,c0).

Many works have been devoted to the question of the behavior for large timeand the convergence to travelling waves for the solutions of the Cauchy problemfor the Fisher-KPP equation (1.1), especially in dimension 1. See for exampleKolmogorov, Petrovsky and Piscunov [10], Bramson [2], Sattinger [15, 16] andGallay [5]. For the higher dimensional case, Mallordy and Roquejoffre [11] studiedthis subject in an infinite cylinder. But, as far as we know, there are no stabilityresults about the nonplanar travelling fronts found by Hamel and Nadirashvili(notice that there are infinite-dimensional manifolds of such travelling fronts).That is just what we want to investigate in the present paper.

The first step to attack this problem is to consider the case of perturbationswith compact supports. Physically speaking, it means that the initial value isfixed far away from the origin, and only the tip of the front (the tip of the flame incombustion problems) is perturbed. In this context, the following stability resultis obtained.

Theorem 1.2. For a given measure μ ∈ MTW , let uμ = φc0 be the travelling frontof the equation (1.1). That is, φc0(x) solves the elliptic equation (1.5). Let u(x, t)be the solution of the following Cauchy problem

{ut = Δu + f(u) , x ∈ R

N , t > 0 ,

u(x, 0) = u0(x) , x ∈ RN ,

(1.7)

where 0 ≤ u0(x) ≤ 1 is a uniformly continuous function defined in RN . If u0(x)−

φc0(x) has compact support in RN , then

supx∈RN

|u(x − c0ν0t, t) − φc0(x)| → 0 , as t → +∞ .

We also deal with the case that the perturbations do not have compact sup-port. Before stating the main result in this case, we first reformulate the questionfor the sake of convenience. Without loss of generality, we can restrict ourselvesto the case of ν0 = eN = (0, . . . , 0, 1) up to a rotation of the frame. Then theequation (1.5) reads

Δφ − c0∂φ

∂xN+ f(φ) = 0 . (1.8)

In order to prove that the solution u(x − c0eN t, t) of the Cauchy problem (1.7)tends to the solution φc0(x) of the equation (1.8) as the time t tends to infinity,we can consider the following Cauchy problem

⎧⎨

⎩ut − Δu + c0

∂u

∂xN− f(u) = 0 , x ∈ R

N , t > 0 ,

u(x, 0) = u0(x) , x ∈ RN .

(1.9)


Here, and in what follows, 0 ≤ u0(x) ≤ 1 is always assumed to be a uniformlycontinuous function defined in R

N . Then, we need to prove that the solution u(x, t)of the above Cauchy problem (1.9) tends to the solution φc0(x) of the equation(1.8) as time t tends to infinity.

Lemma 1.3 (Hamel [6]). Let N ≥ 2, 0 < φ(x) < 1 be a solution of (1.8) forsome c ≥ c∗. Then for each λ ∈ (0, 1), the level set {x ∈ R

N , φ(x) = λ} ={x = (x1, . . . , xN−1, xN ) ∈ R

N , xN = gλ(x1, . . . , xN−1)} is well defined and φ(x)satisfies

lim infxN−gλ(x1,...,xN−1)→+∞

φ(x) = 1 , lim supxN−gλ(x1,...,xN−1)→−∞

φ(x) = 0 . (1.10)

Thanks to the above lemma, for each λ ∈ (0, 1) we can define a set

C−l :=

{x = (x1, . . . , xN ) ∈ R

N ;xN ≤ gλ(x1, . . . , xN−1) + l}, (1.11)

which will be used in the proof of the main result in this paper.In the following theorem, we consider initial perturbations which belong to

some functional spaces with exponential weights in the spirit of the pioneeringpapers of Sattinger [15,16].

Theorem 1.4. Assume N ≥ 2, μ = μ∗ + μ ∈ M is concentrated on S(eN ,c0) forsome c0 > c∗, and μ(X) = 0. For the given measure μ, let uμ = φc0 be thetravelling front of the equation (1.1). That is, φc0(x) solves the elliptic equation(1.8). Let u(x, t) be the solution of the Cauchy problem (1.9). Suppose the initialvalue u0(x) ∈ (0, 1) satisfies the following two conditions:(H1) there exists a positive constant C such that

|u0(x) − φc0(x)| ≤ C

∫

S

eλ∗x · νdμ(ν, c) , ∀x ∈ RN ,

where S is a subset of S(eN ,c0) such that S∩{(ν, c) ∈ X; c∗ < c ≤ c∗+ρ} = ∅,for some ρ > 0, μ(S) = 0, and for any given l ∈ R, there exists a positiveconstant Ml < +∞ such that

∫S

eλ∗x · νdμ(ν, c) ≤ Ml for all x ∈ C−l (defined

by (1.11)).(H2) lim inf l→+∞ infx∈C+

lu0(x) = σ > 0, where C+

l = RN \ C−

l .Then there exist constants M > 0 and γ < 0 such that

supx∈RN

|u(x, t) − φc0(x)| ≤ Meλ∗γt , ∀t ≥ 0 .

Remark 1.5. For the assumption (H1) in the above theorem, we mean that forthe given measure μ, there exists a set S satisfying the conditions such that (H1)holds. We can give a simple example to see that S is not an empty set. Forexample, if μ = m1δ(ν1,c1) + m2δ(ν2,c2) with ci = c0νiν0 and ci > c∗, i = 1, 2, thenS = {(ν1, c1), (ν2, c2)} is a subset of S(eN ,c0) satisfying all of the assumptions in(H1). In this case, the assumption (H1) means the existence of a constant C, suchthat

|u0(x) − φc0(x)| ≤ C(m1e

λ∗x · ν1 + m2eλ∗x · ν2

), ∀x ∈ R

N .

604 R. Huang NoDEA

For this example, it is known from [6] that

lim infmax{x · ν1,x · ν2}→+∞

φc0(x) = 1 , lim supmax{x · ν1,x · ν2}→−∞

φc0(x) = 0 .

Then, the assumption (H2) reads

lim infl→+∞

infmax{x · ν1,x · ν2}≥l

u0(x) > 0 .

Physically speaking, this means that the initial value decays to the front fastenough in the region where it approaches 0, and it is not too small in the regionwhere the front approaches 1.

Remark 1.6. If we denote h(x) =∫

Seλ∗x · νdμ(ν, c) and define a Banach space B

by the norm ‖ · ‖B = ‖ ·h−1(x)‖L∞(RN ) + ‖ · ‖L∞(RN ), then, under the same as-sumptions of the above theorem, we can reformulate the result in the followingsense

‖u(x, t) − φc0(x)‖B ≤ Meλ∗γt , ∀t ≥ 0 ,

which can be viewed as a similar result to the one dimensional case given bySattinger [15,16].

We would like to mention here that our results are the first stability resultsabout the nonplanar KPP fronts in the whole space. For the case of perturbationswith compact supports, we employ the maximum principle and the properties ofthe travelling fronts given by Hamel and Nadirashvili [8] to prove the stabilityresult. For the case of the perturbations without compact supports, the method ofthe proof is motivated by Mallordy and Roquejoffre [11]. Although the previouspapers give us a light to deal with the present cases, we can see that the nontrivialtechnical point in our proofs is to prove the uniform convergence in the wholespace.

In the past decades, many papers have been published on the subject of thestability of the travelling waves of the semilinear parabolic equation (1.1) withvarious types of nonlinearities f (see for example [7, 13, 14] and the referencestherein). Especially for the following two cases: one is so-called “ignition tempera-ture” type which can be derived from the combustion theory in the framework ofthe thermo-diffusive approximation (actually, in combustion models, the nonlin-earities f may be less smooth, and even discontinuous). The other one is “bistable”case which arises in genetics or population dynamics. A typical example if such anonlinearity f is cubic nonlinearity f(s) = s(1 − s)(s − θ).

This paper is organized as follows. In Section 2, we consider the case ofperturbations with compact supports and give the proof of the Theorem 1.2. InSection 3, we deal with the case of perturbations without compact supports. Wefirst consider the stability of the travelling waves given by a special kind of mea-sure μ, and give a stability result under certain conditions. Then, following theidea of the special case, we give the proof of our second main result Theorem 1.4.


2. The case of perturbations with compact supports (Theorem 1.2)

In this section, we deal with the case of perturbations with compact supports andgive the proof of the Theorem 1.2 based on the method of maximum principle andthe properties of the travelling waves given by Hamel and Nadirashvili [8].

We first give the following lemma which will be used in the proofs of ourmain theorems.

Lemma 2.1. If f(s) is a concave function of class C2 in [0, 1] and satisfies theassumption (1.2), and f(s) is defined by 0 for all s > 1, then

i) f(s) is a sub-additive function. That is

f(s + t) ≤ f(s) + f(t) , ∀0 ≤ s , t ≤ 1 ;

ii) f(s) ≤ f ′(0)s,∀s ∈ [0, 1];iii) f ′(s) ≤ f ′(0),∀s ∈ [0, 1].

Proof. All of the conclusions in this lemma can be proved by the definition ofconcave functions with f(0) = 0 directly and we omit the details here. �Proof of the Theorem 1.2. We first consider the case c0 > c∗. Since u0(x)−φc0(x)has compact support, we know that there exists a ball BR(0) in R

N with center 0and radius R such that supp (u0(x) − φc0(x)) ⊂ BR(0). Noticing that both u0(x)and φc0(x) take value in (0, 1), we have

φc0(x) − χBR(0)(x) ≤ u0(x) ≤ φc0(x) + χBR(0)(x) , ∀x ∈ RN , (2.1)

where χBR(0)(x) is a function defined by

χBR(0)(x) =

{1 if x ∈ BR(0) ,

0 if x ∈ RN \ BR(0) .

Let v(x, t) solves the following Cauchy problem{

vt = Δv + f(v) , x ∈ RN , t > 0 ,

v(x, 0) = χBR(0)(x) , x ∈ RN .

(2.2)

Noticing the sub-additivity of the function f (see Lemma 2.1), we have∂

∂t

(φc0(x + c0ν0t) + v(x, t)

)− Δ

(φc0(x + c0ν0t) + v(x, t)

)

− f(φc0(x + c0ν0t) + v(x, t)

)

≥ ∂

∂t

(φc0(x + c0ν0t) + v(x, t)

)− Δ

(φc0(x + c0ν0t) + v(x, t)

)

− f(φc0(x + c0ν0t)

)− f

(v(x, t)

)

= 0 ,

which combined with the initial value condition (2.1) and the maximum principlegives that

u(x, t) ≤ φc0(x + c0ν0t) + v(x, t) , ∀x ∈ RN , t ≥ 0 .

606 R. Huang NoDEA

0

x′0

α α = arcsin(c∗/c0)

c0t

c∗t

x′0 + c0ν0t

Figure 1. Bc∗t(x′0 + c0ν0t).

A similar consideration shows that

φc0(x + c0ν0t) − v(x, t) ≤ u(x, t) , ∀x ∈ RN , t ≥ 0 .

If we take a moving frame x′ = x + c0ν0t, then we have for all x′ ∈ RN

φc0(x′) − v(x′ − c0ν0t, t) ≤ u(x′ − c0ν0t, t) ≤ φc0(x

′) + v(x′ − c0ν0t, t) . (2.3)

∀ε > 0, we define a set

Ωε :={x′ ∈ R

N ; 1 − ε ≤ φc0(x′) < 1

}.

It follows from (1.6) and the continuity of the solution φc0 that there exists apoint x′

0 ∈ RN such that φc0(x

′0) = 1 − ε. By Theorem 1.1, we know that φc0 is

increasing in each unit direction ν ∈ SN−1 such that ν · ν0 > cos(arcsin(c∗/c0)).Then we have

φc0(x′) ≥ 1 − ε , ∀x′ ∈ Bc∗t(x′

0 + c0ν0t) ,

where Bc∗t(x′0 + c0ν0t) is a ball in R

N with center x′0 + c0ν0t and radius c∗t (see

Figure 1). This fact implies that

Bc∗t(x′0 + c0ν0t) ⊂ Ωε , ∀t ≥ 0 . (2.4)

Then for any given x′ ∈ RN \ Ωε, we have x′ ∈ R

N \ Bc∗t(x′0 + c0ν0t). That is to

say, x ∈ RN \ Bc∗t(x′

0). Obviously, there exists a positive constant TR such thatBR(0) ⊂ Bc∗t(x′

0) holds for all t ≥ TR. Consider the following Cauchy problem{

wt = Δw + f ′(0)w , x ∈ RN , t > 0 ,

w(x, 0) = χBR(0)(x) x ∈ RN .

(2.5)


A simple calculation shows that

w(x, t) =ef ′(0)t

(4πt)N/2

∫

RN

e−|x−y|2

4t χBR(0)(y)dy

=ef ′(0)t

(4πt)N/2

∫

BR(0)

e−|x−y|2

4t dy

(2.6)

for all x ∈ RN , t > 0. Notice that f(s) ≤ f ′(0)s,∀s ∈ [0, 1]. Then, using maximum

principle on the problems (2.2) and (2.5) gives that

v(x, t) ≤ w(x, t) , ∀x ∈ RN , t ≥ 0 . (2.7)

Particularly, for all x ∈ RN \ Bc∗t(x′

0) and t > TR, noticing that c∗ = 2√

f ′(0)and BR(0) ⊂ Bc∗t(x′

0), we have

0 ≤ v(x, t) ≤ w(x, t) =ef ′(0)t

(4πt)N/2

∫

BR(0)

e−|x−y|2

4t dy

≤ ef ′(0)t

(4πt)N/2e−

|c∗t−|x′0|−R|2

4t |BR(0)|

=1

(4πt)N/2e−

||x′0|+R|24t +

c∗(|x′0|+R)2 |BR(0)| ,

(2.8)

where |BR(0)| is the Lebesgue measure of the ball BR(0) in RN , and |x′

0| is thenorm of x′

0 in RN . Then, we have

limt→+∞

supx∈RN\Bc∗t(x

′0)

v(x, t) = 0 ,

i.e.lim

t→+∞sup

x′∈RN\Bc∗t(x′0+c0ν0t)

v(x′ − c0ν0t, t) = 0 ,

which combined with (2.3) and (2.4) gives

supx′∈RN\Ωε

|u(x′ − c0ν0t, t) − φc0(x′)| → 0 , as t → +∞ . (2.9)

It remains to prove that the above result holds also for all x′ ∈ Ωε. To this end, wedefine a function u(x′, t) = 1−u(x′− c0ν0t, t),∀x′ ∈ Ωε. Then the function u(x′, t)satisfies the following problem

⎧⎪⎨

⎪⎩

ut = Δu − c0ν0∇u − f(1 − u) , x′ ∈ Ωε , t > 0 ,

u(x′, 0) = 1 − u0(x′) , x′ ∈ Ωε ,

u(x′, t) = 1 − u(x′ − c0ν0t, t) , x′ ∈ ∂Ωε , t ≥ 0 .

(2.10)

By (2.9), we conclude that for any given ε > 0, there exists a positive constant Tε

such that

−ε ≤ u(x′ − c0ν0t, t) − φc0(x′) ≤ ε , ∀x′ ∈ R

N \ Ωε , t ≥ Tε .

608 R. Huang NoDEA

It follows from the continuity of the solutions φc0 and u that the above inequalityholds also for all x′ ∈ ∂Ωε, t ≥ Tε. By the definition of Ωε and the continuity ofthe solution φc0 , we have

1 − ε ≤ φc0(x′) < 1 , ∀x′ ∈ ∂Ωε .

Thus1 − 2ε ≤ u(x′ − c0ν0t, t) < 1 , ∀x′ ∈ ∂Ωε t ≥ Tε . (2.11)

It follows from (2.6) and (2.7) that

0 ≤ v(x, Tε) ≤ w(x, Tε) =ef ′(0)Tε

(4πTε)N/2

∫

BR(0)

e−|x−y|2

4Tε dy

≤ ef ′(0)Tε

(4πTε)N/2e−

|l−R|24Tε |BR(0)|

holds for all x ∈ RN \ Bl(0) with l > R. Then there exists a positive constant

Rε > R such that

0 ≤ v(x, Tε) ≤ ε , ∀x ∈ RN \ BRε

(0) ,

which combined with (2.3) gives that

−ε ≤ u(x′ − c0ν0Tε, Tε) − φc0(x′) ≤ ε , ∀x′ ∈ R

N \ BRε(c0ν0Tε) .

It follows from the definition of the set Ωε gives that

1 − 2ε ≤ u(x′ − c0ν0Tε, Tε) < 1 , ∀x′ ∈ Ωε ∩(R

N \ BRε(c0ν0Tε)

).

By the continuity and positivity of u, we have

δ(ε) = infx′∈BRε (c0ν0Tε)∩Ωε

u(x′ − c0ν0Tε, Tε) > 0 .

Define δ0(ε) = min{δ(ε), 1 − 2ε}. Then we have

0 < δ0(ε) ≤ u(x′ − c0ν0Tε, Tε) < 1 , ∀x′ ∈ Ωε . (2.12)

Notice that u(x′ − c0ν0t, t) satisfies the following equation

ut − Δu + c0ν0∇u = f(u) ≥ 0 ,

which combined with (2.11), (2.12) and the maximum principle implies that

δ0(ε) ≤ u(x′ − c0ν0t, t) < 1 , ∀x′ ∈ Ωε , t ≥ Tε . (2.13)

Define

σ = inf0≤s≤1

f(s)s(1 − s)

> 0 . (2.14)

It follows from (2.10), (2.13) and (2.14) that

ut = Δu − c0ν0∇u − f(u)≤ Δu − c0ν0∇u − σuu ,

≤ Δu − c0ν0∇u − σδ0(ε)u


holds for all x′ ∈ Ωε, t > Tε. Then u(x′, t) satisfies⎧⎪⎨

⎪⎩

ut ≤ Δu − c0ν0∇u − σδ0(ε)u , x′ ∈ Ωε , t > Tε ,

u(x′, Tε) = 1 − u(x′ − c0ν0Tε, Tε) ≤ 1 , x′ ∈ Ωε ,

u(x′, t) = 1 − u(x′ − c0ν0t, t) ≤ 2ε , x′ ∈ ∂Ωε , t ≥ Tε .

(2.15)

We consider a function ψ(t) which solves the following O.D.E{

ψ′(t) + σδ0(ε)ψ(t) = 2σδ0(ε)ε , t > Tε

ψ(Tε) = 1 .(2.16)

Then the maximum principle implies that

u(x′, t) ≤ ψ(t) = 2ε + (1 − 2ε)eσδ0(ε)(Tε−t) , ∀x′ ∈ Ωε , t ≥ Tε .

There exists a constant T ≥ Tε such that

u(x′, t) ≤ ψ(t) ≤ 3ε , ∀x′ ∈ Ωε , t ≥ T .

By the definition of u, we have

1 − 3ε ≤ u(x′ − c0ν0t, t) < 1, ∀x′ ∈ Ωε , t ≥ T .

Then, by the definition of the set Ωε, we conclude

supx′∈Ωε

|u(x′ − c0ν0t, t) − φc0(x′)| ≤ 3ε , ∀t ≥ T ,

which combined with (2.9) implies that

supx′∈RN

|u(x′ − c0ν0t, t) − φc0(x′)| → 0 , as t → +∞ .

Let us now deal with the case c0 = c∗. From the Theorem 1.1, we knowthat φc∗(x + c∗ν0t) is a planar travelling wave with speed c∗ in this case, namely,φc∗(x+ c∗ν0t) = φc∗(x · ν0 + c∗t+h) for some h ∈ R. In fact, we can also prove theresult by the same way. Since u0(x)−φc∗(x · ν0+h) has compact support, we knowthat there exists a constant R ∈ R such that supp (u0(x) − φc∗(x · ν0 + h)) ⊂ {x ∈R

N ;x · ν0 ≤ R}. Noticing that both u0(x) and φc∗(x · ν0 + h) take value in (0, 1),we can conclude that

φc∗(x + h) − χ{x∈RN ;x · ν0≤R}(x) ≤ u0(x)

≤ φc∗(x · ν0 + h) + χ{x∈RN ;x · ν0≤R}(x) , (2.17)

holds for all x ∈ RN , where χ{x∈RN ;x · ν0≤R}(x) is a function defined by

χ{x∈RN ;x · ν0≤R}(x) =

{1 if x ∈ {x ∈ R

N ;x · ν0 ≤ R} ,

0 if x ∈ {x ∈ RN ;x · ν0 > R} .

Let v(x · ν0, t) solves the Cauchy problem (2.2) with initial value v(x · ν0, 0) =χ{x∈RN ;x · ν0≤R}(x),∀x ∈ R

N . Then by the same arguments as in the previous casewe have

φc∗(x · ν0 + c0t + h) − v(x · ν0, t) ≤ u(x, t) ≤ φc∗(x · ν0 + c0t + h) + v(x · ν0, t)

610 R. Huang NoDEA

holds for all x ∈ RN , t ≥ 0. If we take a moving frame x′ = x + c∗ν0t, then

x′ · ν0 = x · ν0 + c∗t. In this moving frame, the above inequalities read as

φc∗(x′ · ν0 + h) − v(x′ − c0ν0t, t) ≤ u(x′ − c0ν0t, t)

≤ φc∗(x′ · ν0 + h) + v(x′ − c0ν0t, t) . (2.18)

∀ε > 0, we define a set

Ωε :={x′ ∈ R

N ; 1 − ε ≤ φc∗(x′ · ν0 + h) < 1}

.

It follows from (1.6) and the continuity of the solution φc∗ that there exists apoint x′

0 ∈ RN such that φc0(x

′0 · ν0 + h) = 1 − ε. By Theorem 1.1, we know that

φc∗(x′0 · ν0 + h) is increasing in each unit direction ν ∈ SN−1 such that ν · ν0 > 0.

Then we have

φc0(x′0 · ν0 + h) ≥ 1 − ε , ∀x′ ∈ {x′ ∈ R

N ;x′ · ν0 ≥ x′0 · ν0} .

This fact implies that

{x′ ∈ RN ;x′ · ν0 ≥ x′

0 · ν0} ⊂ Ωε , ∀t ≥ 0 . (2.19)

Then for any given x′ ∈ RN\Ωε, we have x′ ∈ {x′ ∈ R

N ;x′ · ν0 < x′0 · ν0}. That is to

say, x ∈ {x ∈ RN ;x · ν0 ≤ x′

0 · ν0 − c∗t}. Obviously, there exists a positive constantTR such that x · ν0 ≤ x′

0 · ν0 − c∗t ≤ −R holds for all t ≥ TR. Let w(x · ν0, t) solvesthe Cauchy problem (2.5) with initial value w(x · ν0, 0) = χ{x∈RN ;x · ν0≤R}(x), ∀x ∈R

N . Then, by a similar arguments as in the previous case we can inform that

0 ≤ v(x · ν0, t) ≤ w(x · ν0, t) =ef ′(0)t

√4πt

∫ +∞

−∞e−

|x · ν0−y|24t χ{y∈R;|y|≤R}(y)dy

=ef ′(0)t

√4πt

∫ R

−R

e−|x · ν0−y|2

4t dy

≤ 2Ref ′(0)t

√4πt

e−|x′

0 · ν0−c∗t−R|24t

= 2R1√4πt

e−|x′

0 · ν0−R|24t +

c∗(x′0 · ν0−R)

2 .

holds for all x ∈ {x ∈ RN ;x · ν0 ≤ x′

0 · ν0 − c∗t} and t ≥ TR. Then, we have

limt→+∞

supx∈{x∈RN ;x · ν0≤x′

0 · ν0−c∗t}v(x · ν0, t) = 0 ,

i.e.lim

t→+∞sup

x′∈{x′∈RN ;x′ · ν0≤x′0 · ν0}

v(x′ − c0ν0t, t) = 0 ,

which combined with (2.18) and (2.19) gives

supx′∈RN\Ωε

|u(x′ − c0ν0t, t) − φc∗(x′ · ν0 + h)| → 0 , as t → +∞ . (2.20)

It remains to prove that the above result holds also for all x′ ∈ Ωε. To this end,we define a function u(x′, t) = 1 − u(x′ − c0ν0t, t),∀x′ ∈ Ωε. Then the function


u(x′, t) satisfies the problem (2.10). By a complete similar arguments with thecase c0 > c∗, we conclude that (2.13) still holds in the present case. Then u(x′, t)satisfies the problem (2.15). Then, by the same way as in the case c0 > c∗ we canprove that

supx′∈Ωε

|u(x′ − c0ν0t, t) − φc∗(x′ · ν0 + h)| ≤ 3ε , ∀t ≥ T ,


supx′∈RN

|u(x′ − c0ν0t, t) − φc∗(x′ · ν0 + h)| → 0 , as t → +∞ .

The proof is completed. �

3. The case of perturbations without compact supports(Theorem 1.4)

In this section, we consider the stability of the travelling waves when the pertur-bations do not have compact supports. We first focus ourselves on the travellingwaves given by a special kind of measure μ, that is, μ = μ∗ + μ ∈ MTW , whereμ∗ =

∑1≤i≤k miδ(νi,c∗), μ = m1δ(ν1,c1) + m2δ(ν2,c2) + μ ∈ MTW and ci > c∗,

i = 1, 2.

Lemma 3.1. Consider two solutions u and v of the equation

∂tu = Δu − c0∂u

∂xN+ f(u) , x ∈ R

N , t > 0 ,

and suppose that the initial data u0(x) and v0(x) take values in (0, 1) and satisfythe following two conditions:

i) u0(x) ≤ v0(x) in RN ;

ii) |u0(x) − v0(x)| = O(eλ∗x · ν1 + eλ∗x · ν2), as x ∈ C−l and l → −∞, where C−

l

(see Figure 2) is defined by

C−l = {x ∈ R

N ;x · νi ≤ l, i = 1, 2} , (3.1)

ν1 and ν2 are the unit vectors in SN−1 such that there exist c1 and c2 withci > c∗, i = 1, 2 and (ν1, c1) and (ν2, c2) are in the support of μ.

Then for all t ≥ 0 and x ∈ RN , we have

0 ≤ v(x, t) − u(x, t) ≤ Ch(x, t) , (3.2)

where

h(x, t) = eλ∗(x · ν1−(c1−c∗)t) + eλ∗(x · ν2−(c2−c∗)t) , (3.3)

λ∗ = c∗/2, and C is a positive constant.

612 R. Huang NoDEA

0

xN

x1, · · · , xN−1

C+l

C−l

lν2lν1

Figure 2. C−l and C+

l .

Proof. The left hand side of the inequality (3.2) follows from the maximum princi-ple. Now we are in the position to prove the right hand side of the inequality (3.2).By the Lemma 2.1, we inform that there exists a function w such that

(∂t − Δ + c0

∂

∂xN

)(v − u) = f(v) − f(u) = f ′(w)(v − u) ≤ f ′(0)(v − u) .

λ∗ = c∗/2 solves the following equation

−(λ∗)2 + c∗λ∗ − f ′(0) = 0 .

Then, we have

−λ∗(ci − c∗) = (λ∗)2 − λ∗c1 + f ′(0) , i = 1, 2 .

Thus∂

∂th(x, t) = −λ∗(c1 − c∗)eλ∗(x · ν1−(c1−c∗)t) − λ∗(c2 − c∗)eλ∗(x · ν2−(c2−c∗)t)

=((λ∗)2 − λ∗c1 + f ′(0)

)eλ∗(x · ν1−(c1−c∗)t)

+((λ∗)2 − λ∗c2 + f ′(0)

)eλ∗(x · ν2−(c2−c∗)t)

=((λ∗)2 − λ∗c0eN · ν1 + f ′(0)

)eλ∗(x · ν1−(c1−c∗)t)

+((λ∗)2 − λ∗c0eN · ν2 + f ′(0)

)eλ∗(x · ν2−(c2−c∗)t)

=(Δ − c0ν0∇ + f ′(0)

)h(x, t) .

From the initial value condition ii), we can choose a constant C > 0 such that

v0(x) − u0(x) ≤ Ch(x, 0) , ∀x ∈ RN .


The right hand side of the inequality (3.2) then follows from the maximum prin-ciple. The proof of this lemma is completed. �

Theorem 3.2. Assume N ≥ 2, μ = μ∗ + μ ∈ MTW , where μ = m1δ(ν1,c1) +m2δ(ν2,c2) + μ ∈ MTW and ci > c∗, i = 1, 2. Denote the solution of the ellipticequation (1.8) by φμ = φc0 and let u(x, t) be the solution of the following Cauchyproblem

⎧⎨

⎩ut − Δu + c0

∂u

∂xN− f(u) = 0 , x ∈ R

N , t > 0 ,

u(x, 0) = u0(x) , x ∈ RN .

(3.4)

Suppose the initial value u0 ∈ (0, 1) satisfies the following two conditions:

i) |u0(x) − φc0(x)| = O(eλ∗x · ν1 + eλ∗x · ν2) as x ∈ C−l and l → −∞, where C−

l

is defined by (3.1);ii) there exists a positive constant r ∈ R such that

φc0(x − reN ) ≤ u0(x) , ∀x ∈ RN .

Then there exist constants C, l > 0 and γ < 0 such that

supx∈RN

|u(x, t) − φc0(x)| ≤ Ceλ∗(l+γt) , ∀t ≥ 0 .

Proof. Define u0(x) = min {u0(x), φc0(x)}. It follows from the first initial valuecondition and the boundedness of u0(x) and φc0(x) that there exists a constant Csuch that

0 ≤ φc0(x) − u0(x) ≤ |φc0(x) − u0(x)| ≤ Ch(x, 0) , ∀x ∈ RN ,

where h(x, t) is defined by (3.3). We denote by u(x, t) the solution of the Cauchyproblem (3.4) with initial value u0(x), then from the Lemma 3.1, we have

0 ≤ φc0(x) − u(x, t) ≤ Ch(x, t) , ∀t ≥ 0 , x ∈ RN .

Similarly, we define u0(x) = max {u0(x), φc0(x)}. Then there exists a constant Csuch that

0 ≤ u0(x) − φc0(x) ≤ |φc0(x) − u0(x)| ≤ Ch(x, 0) , ∀x ∈ RN .

We denote by u(x, t) the solution of the Cauchy problem (3.4) with initial valueu0(x), then from the Lemma 3.1, we have

0 ≤ u(x, t) − φc0(x) ≤ Ch(x, t) , ∀t ≥ 0 , x ∈ RN .

It follows from the maximum principle that

φc0(x) − Ch(x, t) ≤ u(x, t) ≤ u(x, t) ≤ u(x, t) ≤ φc0(x) + Ch(x, t)

holds for all t ≥ 0 and x ∈ RN . That is

|u(x, t) − φc0(x)| ≤ Ch(x, t) , ∀t ≥ 0 , x ∈ RN .

614 R. Huang NoDEA

From the above inequality, we can inform that for any given constant l

supx∈C−

l

|u(x, t) − φc0(x)| ≤ C supx∈C−

l

h(x, t)

= C supx∈C−

l

∑

i=1,2

eλ∗(x · νi−(ci−c∗)t)

≤ C∑

i=1,2

eλ∗(l−(ci−c∗)t) .

(3.5)

Then it remains to show that there exists a real constant l such that u(x, t) tendsto φc0(x) uniformly with respect to x ∈ C+

l = RN \ C−

l as t tends to infinity. First,we claim that

lim infl→+∞

infx∈C+

l

φc0(x) = 1 . (3.6)

By (3.5) in [8], we know the following lower bound of the solution uμ of theequation (1.1)

uμ(x, t) ≥ max

(

max1≤i≤k

φc∗(x · νi + c∗t + c∗ ln mi),

∫

X

φc(x · ν + ct + c ln M)1M

dμ

)

.

≥∫

X


dμ ,

where M = μ(X). By the notation of (1.4), we have

φc0(x + c0tν0) = uμ(x, t) ≥∫

X


dμ

Recalling (ν, c) ∈ S(ν0,c0), we have c0ν0 · ν = c. Then from the above inequalityand the fact that μ = m1δ(ν1,c1) + m2δ(ν2,c2) + μ, we have

φc0(x) ≥∫

X

φc(x · ν + c ln M)1M

dμ

≥ m1

Mφc1(x · ν + c1 ln M) +

m2

Mφc2(x · ν + c2 ln M) .

Then we have

lim infl→+∞

infx∈C+

l

φc0(x) ≥ min(

m1

M,m2

M

)> 0 . (3.7)

Assume now that there exists ε > 0 and a sequence xn ∈ C+ln

such that

φc0(xn) ≤ 1 − ε , ln → +∞ .

Up to extraction of some subsequence, the functions φn(x) = φc0(x+xn) convergein C2

loc(RN ) to a solution φ∞ of the equation (1.5) such that φ∞(0) ≤ 1 − ε. On


the other hand, the inequality (3.7) and the fact that xn ∈ C+ln

and ln → +∞imply that

m∞ := infRN

φ∞ ≥ min(

m1

M,m2

M

)> 0 ,

whence

min(

m1

M,m2

M

)≤ m∞ ≤ 1 − ε < 1 .

Let x′n ∈ R

N be a sequence such that u∞(x′n) → m∞ as n → +∞. Up to extraction

of some subsequence, the functions ϕn(x) = φ∞(x + x′n) converge in C2

loc(RN ) to

a solution ϕ∞ of the equation (1.5) such that

m∞ = ϕ∞(0) = minRN

ϕ∞ ,

which combined with the equation (1.5) gives that f(m∞) ≤ 0, which contradictsthe positivity of f on

[min

(m1

M, m2

M

), 1

). Thus the claim (3.6) holds. It follows

from the second initial condition that there exists a positive constant r ∈ R suchthat

φc0(x − reN ) ≤ u0(x) , ∀x ∈ RN .

Then the maximum principle gives that

φc0(x − reN ) ≤ u(x, t) , ∀t ≥ 0 , x ∈ RN .

By the claim (3.6), we can choose l large enough such that

1 − ε ≤ φc0(x − reN ) ≤ u(x, t) ≤ 1 , ∀t ≥ 0 , x ∈ C+l ,

Obviously, we also have

1 − ε ≤ φc0(x) ≤ 1 , ∀x ∈ C+l .

For all t ≥ 0, x ∈ C+l , v(x, t) = u(x, t) − φc0(x) satisfies the following equation

vt − Δv + c0ν0∇v − f(v + φc0) + f(φc0) = vt − Δv + c0ν0∇v − f ′(w)v= 0 ,

where w ∈ (1 − ε, 1). We define a parabolic operator as follows

L =∂

∂t− Δ + c0

∂

∂xN− f ′(w) .

Then we have Lv = 0. Define a function

v(x, t) := Ceλ∗(l+γt) , ∀t ≥ 0 , x ∈ RN ,

where γ = max {−mini=1,2(ci − c∗), f ′(1 − ε)/λ∗} < 0, C = max{ε/eλ∗l, 2C

}

and C is the constant defined in Lemma 3.1. For all x ∈ C+l and t ≥ 0, we have

Lv(x, t) =(λ∗γ − f ′(w)

)v(x, t)

≥(f ′(1 − ε) − f ′(w)

)v(x, t)

≥ 0 = Lv .

616 R. Huang NoDEA

We also have

v(x, 0) = u(x, 0) − φc0(x) ≤ ε ≤ Ceλ∗l = v(x, 0) , ∀x ∈ C+l .

It follows from (3.5) that

v(x, t) ≤ Ch(x, t)

≤ Ceλ∗(l−(c1−c∗)t) + Ceλ∗(l−(c2−c∗)t)

≤ 2Ceλ∗(l−γt) ≤ v(x, t)

holds for all x ∈ ∂C+l = ∂C−

l and t ≥ 0. Then the maximum principle gives that

v(x, t) ≤ v(x, t) = Ceλ∗(l+γt) , ∀t ≥ 0 , x ∈ C+l . (3.8)

By a similar consideration, we have

v(x, t) ≥ v(x, t) := −Ceλ∗(l+γt) , ∀t ≥ 0 , x ∈ C+l .

Then we have

supx∈C+

l

|u(x, t) − φc0(x)| ≤ Ceλ∗(l+γt) , ∀t ≥ 0 ,


supx∈RN

|u(x, t) − φc0(x)| ≤ Ceλ∗(l+γt) , ∀t ≥ 0 .

The proof of the theorem is completed. �Now we are in the position to give the proof of the Theorem 1.4.

Proof of the Theorem 1.4. We first define a function

h(x, t) =∫

S

eλ∗(x · ν−(c−c∗)t)dμ(ν, c) . (3.9)

Let u0(x) = min {u0(x), φc0(x)}. Then by the assumption (H1), there exists aconstant C such that

0 ≤ φc0(x) − u0(x) ≤ |φc0(x) − u0(x)| ≤ Ch(x, 0) , ∀x ∈ RN .

We denote by u(x, t) the solution of the Cauchy problem (1.9) with initial valueu0(x). It follows from the maximum principle that

φc0(x) − u(x, t) ≥ 0 , ∀t ≥ 0 , x ∈ RN .

Since φc0(x) − u(x, t) ≥ 0 and f is concave, there exists a function w such that(

∂t − Δ + c0∂

∂xN

) (φc0(x) − u(x, t)

)= f

(φc0(x)

)− f

(u(x, t)

)

= f ′(w)(φc0(x) − u(x, t)

)

≤ f ′(0)(φc0(x) − u(x, t)

).

Since λ∗ satisfies the equation (λ∗)2 − c∗λ∗ + f ′(0) = 0, one has

−λ∗(c − c∗) = (λ∗)2 − λ∗c + f ′(0) , ∀c > c∗ .


Thus

ht = −∫

S

λ∗(c − c∗)eλ∗(x · ν−(c−c∗)t)dμ(ν, c)

=∫

S(ν0,c0)

((λ∗)2 − λ∗c + f ′(0)

)eλ∗(x · ν−(c−c∗)t)dμ(ν, c)

=∫

S(ν0,c0)

((λ∗)2 − λ∗c0eN · ν + f ′(0)

)eλ∗(x · ν−(c−c∗)t)dμ(ν, c)

= Δh − c0∂

∂xNh + f ′(0)h .

Then the maximum principle gives that

φc0(x) − u(x, t) ≤ Ch(x, t) , ∀t ≥ 0 , x ∈ RN .

Similarly, let u0(x) = max {u0(x), φc0(x)}. Then there exists a constant C suchthat

0 ≤ u0(x) − φc0(x) ≤ |φc0(x) − u0(x)| ≤ Ch(x, 0) , ∀x ∈ RN .

We denote by u(x, t) the solution of the Cauchy problem (1.9) with initial valueu0(x). Then, we have

0 ≤ u(x, t) − φc0(x) ≤ Ch(x, t) , ∀t ≥ 0 , x ∈ RN .

The maximum principle gives that

φc0(x) − Ch(x, t) ≤ u(x, t) ≤ u(x, t) ≤ u(x, t) ≤ φc0(x) + Ch(x, t) ,

for all t ≥ 0 and x ∈ RN . That is

|u(x, t) − φc0(x)| ≤ Ch(x, t) , ∀t ≥ 0 , x ∈ RN . (3.10)

Then for any given real constant l, we have

supx∈C−

l

|u(x, t) − φc0(x)| ≤ C supx∈C−

l

h(x, t)

= C supx∈C−

l

∫

S

eλ∗(x · ν−(c−c∗)t)dμ(ν, c)

≤ Ce−λ∗ρt supx∈C−

l

∫

S

eλ∗x · νdμ(ν, c)

≤ CMle−λ∗ρt .

(3.11)

In what follows, we investigate the long time behavior of the solutions in thedomain C+

l = RN \C−

l . By the assumption (H2), there exist a constant l0 > 0 suchthat u0(x) ≥ σ/2, for all x ∈ C+

l0−1. It follows from the Lemma 1.3 that for anygiven ε > 0, there exists a constant lε > l0 such that

1 − ε

2≤ φc0(x) < 1 , ∀x ∈ C+

lε. (3.12)

618 R. Huang NoDEA

By the inequality (3.11), we know that there exists a positive constant Tε suchthat

|u(x, t) − φc0(x)| ≤ ε

2, ∀t ≥ Tε , x ∈ C−

lε. (3.13)

It follows from (3.12) and (3.13) that

1 − ε ≤ u(x, t) < 1 , ∀t ≥ Tε , x ∈ ∂C+lε

= ∂C−lε

. (3.14)

Since lε > l0, we also have

u0(x) ≥ σ

2, ∀x ∈ C−

lε−1 . (3.15)

For any given x0 ∈ ∂C+lε

, there exists a positive constant rx0 such that B(x0, rx0) ⊂C+

lε−1, where B(x0, rx0) is a ball with center x0 and radius rx0 . Let v(x, t) solvethe following initial boundary value problem

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

vt = Δv − c0∂v

∂xN+ f(v) , 0 < t < Tε , x ∈ B(x0, rx0) ,

v(x, t) = 0 , 0 ≤ t ≤ Tε , x ∈ ∂B(x0, rx0) ,

v(x, 0) = v0(x) , x ∈ B(x0, rx0) ,

where the initial value v0(x) satisfies 0 ≤ v0(x) ≤ σ/2 for all x ∈ B(x0, rx0) andv0(x) = 0 for all x ∈ ∂B(x0, rx0). Then we have

u0(x) ≥ v0(x) , ∀x ∈ B(x0, rx0) .

It follows from (3.15) and the positivity of u0(x) for all x ∈ RN that

u(x, t) ≥ v(x, t) = 0 , ∀x ∈ ∂B(x0, rx0) , 0 ≤ t ≤ Tε .

Then the maximum principle implies that

u(x, t) ≥ v(x, t) , ∀0 ≤ t ≤ Tε , x ∈ B(x0, rx0) . (3.16)

On the other hand, since x0 is an interior point of B(x0, rx0), the strong maximumprinciple implies that there exist a positive constant τx0 such that v0(x0, t) ≥ τx0

holds for all 0 ≤ t ≤ Tε. Then by (3.16) and the arbitrariness of the x0, we concludethat there exists a positive constant τ such that

u(x, t) ≥ τ , ∀0 ≤ t ≤ Tε , x ∈ ∂C−lε

.

Obviously, there exists a concave function fε(x) (see Figure 3) such that

fε(x) ≤ f(x) , ∀0 ≤ x ≤ 1 , and

fε(0) = fε(1 − ε) = 0 , f ′ε(1 − ε) ≤ 0 ≤ f ′

ε(0) .

We consider the following O.D.E{

ζ ′(t) = fε(ζ(t)) , t > 0 ,

ζ(0) = λ ,

where λ = min{σ/2, τ/ef ′ε(0)Tε}. It follows from the assumptions of the function

fε thatζ ′(t) = fε

(ζ(t)

)≤ f ′

ε(0)ζ(t) .


1−ε 10 x

f(x)

fε(x)

Figure 3. f and fε.

Then we have

ζ(t) ≤ λef ′ε(0)Tε ≤ τ ≤ u(x, t) , 0 ≤ t ≤ Tε , x ∈ ∂C+

lε. (3.17)

Obviously, the solution ζ(t) of the above O.D.E is increasing and satisfies

limt→+∞

ζ(t) = 1 − ε , (3.18)


ζ(t) ≤ 1 − ε ≤ u(x, t) , t ≥ Tε , x ∈ ∂C+lε

.

The above inequality combined with (3.17) gives that

ζ(t) ≤ u(x, t) , t ≥ 0 , x ∈ ∂C+lε

. (3.19)

By (3.15) we also haveζ(0) ≤ u(x, 0) , ∀x ∈ C+

lε. (3.20)

Then it follows from the maximum principle that

u(x, t) ≥ ζ(t) , ∀t ≥ 0 , x ∈ C+lε

.

By (3.18), we conclude that there exists a positive constant T > Tε such that

u(x, t) ≥ 1 − 2ε , ∀t ≥ T , x ∈ C+lε

,


|u(x, t) − φc0(x)| ≤ ε , ∀t ≥ T , x ∈ C+lε

. (3.21)

For all t ≥ T , x ∈ C+lε

, v(x, t) = u(x, t) − φc0(x) satisfies the following equation

Lv = vt − Δv + c0∂v

∂xN− f(v + φc0) + f(φc0)

= vt − Δv + c0∂v

∂xN− f ′(w)v = 0 ,

620 R. Huang NoDEA

where w ∈ (1 − ε, 1). Define a function

v(x, t) := CMlεeλ∗γt , ∀t ≥ T , x ∈ R

N ,

where γ = max {−ρ, f ′(1 − ε)/λ∗} is negative and C = max{

εMlεeλ∗γT , C

}, C is

the constant in (3.11). For all x ∈ C+lε

and t ≥ 0, we have

Lv =(λ∗γ − f ′(w)

)v ≥

(f ′(1 − ε) − f ′(w)

)v ≥ 0 = Lv .

From (3.21), we have

v(x, t)∣∣t=T

= u(x, t)∣∣t=T

− φc0(x) ≤ ε ≤ CMlεeλ∗γT = v(x, t)

∣∣t=T

.

From (3.11), we have

v(x, t) ≤ CMlεe−λ∗ρt ≤ CMlεe

λ∗γt ≤ v(x, t)

holds for all x ∈ ∂C+lε

= ∂C−lε

and t ≥ T . It follows from the maximum principlethat

v(x, t) ≤ v(x, t) = CMlεeλ∗γt , ∀t ≥ T , x ∈ C+

lε. (3.22)

By a similar consideration, we have

v(x, t) ≥ v(x, t) := −CMlεeλ∗γt , ∀t ≥ T , x ∈ C+

lε.

Then we have

supx∈C+

l

|u(x, t) − φc0(x)| ≤ supx∈C+

l

|v(x, t)| ≤ CMlεeλ∗γt , ∀t ≥ T ,


supx∈RN

|u(x, t) − φc0(x)| ≤ CMlεeλ∗γt , ∀t ≥ T .

Since u and φc0 are globally bounded, we conclude that there exists a constant Msuch that

supx∈RN

|u(x, t) − φc0(x)| ≤ Meλ∗γt , ∀t ≥ 0 .

The proof of this theorem is completed. �

Acknowledgements

The author would like to thank Professors F. Hamel and J.-M. Roquejoffre fortheir helpful support and suggestions in this work.


References

[1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arisingin population genetics, Adv. Math., 30 (1978), 33–76.

[2] M. Bramson, Convergence of solutions of the Kolmogorov equation to travellingwaves, Memoirs Amer. Math. Soc., 44 (1983).

[3] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notesin Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.

[4] R. A. Fisher, The advance of advantageous genes. Ann. Eugenics, 7 (1937), 335–369.

[5] Th. Gallay, Local stability of critical fronts in nonlinear parabolic partial differentialequations, Nonlinearity, 7 (1994), 741–764.

[6] F. Hamel, Conical fronts and more general curved fronts for homoneneous equa-tions in RN , Notes de cours en Ecole d’Ete de l’Institut Fourier, 2005, http://www-fourier.ujf-grenoble.fr/

[7] F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of conical fronts in a com-bustion model, Ann. Sci. Ecole Normale Superieure, 37 (2004), 469–506.

[8] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in R

N , Arch. Ration. Mech. Anal., 157 (2001), 91–163.

[9] F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. PureAppl. Math., 52 (1999), 1255–1276.

[10] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l’ equation de ladiffusion avec croissance de la quantite de matiere et son application a un problemebiologique, Bulletin Universite d’Etat a Moscou (Bjul. Moskowskogo Gos. Univ.),Serie Internationale, Section A, 1 (1937), 1–26.

[11] J. F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higherdemensions, SIAM J. Math. Anal., 26 (1995), 1–20.

[12] J. D. Murray, Mathematical Biology, Springer-Verlag, 1989.

[13] J.-M. Roquejoffre, Stability of travelling fronts in a curved flame model, Part II:Non-linear orbital stability, Arch. Rat. Mech. Anal., 117 (1992), 119–153.

[14] J.-M. Roquejoffre, Convergence to travelling waves for solutions of a class of semi-linear parabolic equation, J. Diff. Equations, 108 (1994), 262–295.

[15] D. H. Sattinger, Stability of waves of nonlinear parabolic systems, Adv. Math., 22(1976), 312–355.

[16] D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Equa-tions, 25 (1977), 130–144.

622 R. Huang NoDEA

Rui HuangUniversite Aix-Marseille IIILATPFaculte des Sciences et TechniquesAvenue Escadrille Normandie-Niemen13397 Marseille Cedex 20FranceandDepartment of MathematicsJilin UniversityChangchun 130012P.R. Chinae-mail: [email protected]

Received: 10 September 2007.

Revised: 21 March 2008.

Accepted: 25 April 2008.

stabilityoftravellingfrontsofthefisher-kpp equation in r · vol. 15 (2008) stability of travelling...

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