Convergence to a Single Wave in the Fisher-KPP Equation∗

James NOLEN Jean-Michel ROQUEJOFFRE Lenya RYZHIK

(Dedicated to Haim Brezis,with admiration and respect)

1 Introduction

We consider the Fisher-KPP equation

with an initial condition u(0,x)=u0(x)which is a compact perturbation of a step function,in the sense that there exist x1and x2,so that u0(x)=1 for all x≤x1,and u0(x)=0 for all x≥x2.

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Each solution φ(ξ)of(1.2)is a shift of a fixed profile φ∗(ξ): φ(ξ)= φ∗(ξ+s),with some fixed s ∈ R.The profile φ∗(ξ)satisfies the asymptotics

with two universal constantsω0>0,k∈ R.

The large time behaviour of the solutions of this problem has a long history,starting with a striking paper of Fisher[10],which identifies the spreading velocity c∗=2 via numerical computations and other arguments.In the same year,the pioneering KPP paper[15]proved that the solution of(1.1),starting from a step function:u0(x)=1 for x≤0,u0(x)=0 for x>0,converges toφ∗in the following sense:There is a function

such that

Theorem 1.1(see[5–6])There is a constant x∞,depending on u0,such that

Models with temporal variation in the branching process have also been considered.In[9],Fang and Zeitouni studied the extremal particle of such a spatially homogeneous BBM where the branching particles satisfy

A natural question is to prove Theorem 1.1 with purely PDE arguments.In that spirit,a weaker version,precise up to the O(1)term(but valid also for a much more difficult case of the periodic in space coefficients),is the main result of[11–12],

Here,we will give a simple and robust proof of Theorem 1.1.These ideas are further developed to study the refined asymptotics of the solutions in[21].

2 Probabilistic Links and Some Related M odels

We regard(3.3)as a perturbation of this equation,and expect that v(t,x)→ exφ(x−x∞)as t→ ∞ for some x∞∈R.

Lemma 4.2There exists a constantα∞>0 with the following property.For anyγ>0 and allε>0,we can find Tε,so that for all t>Tεwe have

with the initial dataTherefore,Theorem 1.1 is about the median location of the maximal particle XNt.Building on the work of Lalley and Sellke[16],recent probabilistic analyses(see[1–3,7–8])of this particle system have identified a decorated Poissontype point process which is the limit of the particle distribution “seen from the tip”:There is a random variable Z>0,such that the point process defined by the shifted particleswith

has a well-defined limit process as t→∞.Furthermore,Z is the limit of the martingale

and

As we have mentioned,the logarithmic term in Theorem 1.1 arises also in inhomogeneous variants of this model.For example,consider the Fisher-KPP equation in a periodic medium

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where λ∗>0 is the rate of exponential decay(as x → ∞)of the minimal front Uc∗,which depends on μ(x)but not on s or on u0.This implies the convergence of u(t,x− σs(t))to a closed subset of the family of minimal fronts.It is an open problem to determine whether convergence to a single front holds,not to mention the rate of this convergence.Whenμ(x)>0 everywhere,the solution u of the related model may be interpreted in terms of the extremal particle in a BBM with a spatially-varying branching rate(see[12]).

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Theorem 1.1 was proved through elaborate probabilistic arguments.Bramson also gave necessary and sufficient conditions on the decay of the initial data to zero(as x→ +∞)in order that the solution converges to φ∗(x)in some moving frame.Lau[17]also proved those necessary and sufficient conditions(for a more general nonlinear term)using a PDE approach based on the decrease in the number of the intersection points for a pair of solutions of the parabolic Cauchy problem.The asymptotics ofσ∞(t)were not identified by that approach.

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They proved that ifκis increasing,and f is of the Fisher-KPP type,the shift is algebraic and not logarithmic in time:There exists C>0,such that

In[20],we proved the asymptotics

Here,β<0 is the first zero of the Airy function.Maillard and Zeitouni[18]refined the asymptotics further,proving a logarithmic correction to(2.3),and convergence of u(T)to a traveling wave.

3 Strategy of the Proof of Theorem 1.1

3.1 Why converge to a traveling wave?

We first provide an informal argument for the convergence of the solution of the initial value problem to a traveling wave.Consider the Cauchy problem(1.1),starting at t=1 for the convenience of the notation

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and proceed with a standard sequence of changes of variables.We first go into the moving frame

leading to

Next,we take out the exponential factor:Set

so that v satisfies

We postpone the proof of this lemma for the moment,and show how it is used.A consequence of Lemma 4.2 is that the problem for the moment is to understand,for a givenα>0,the behavior of the solutions of

Observe that for any shift x∞∈ R,the function V(x)=exφ(x−x∞)is a steady solution of

followed by a change of the unknown

3.2 The self-similar variables

We note that for x→ +∞,the term e−xv2in(3.3)is negligible,while for x→ −∞ the same term will create a large absorption and force the solution to be close to zero.For this reason,the linear Dirichlet problem

This transformation strengthens the reason why the Dirichlet problem(3.4)appears naturally:For

the last term in the left-hand side of(3.6)becomes exponentially large,which forces w to be almost 0 in this region.On the other hand,for

this term is very small,so it should not play any role in the dynamics of w in that region.The transition region has width of the order

3.3 The choice of the shift

Also,through this change of variables,we can see how a particular translation of the wave will be chosen.Considering(3.4)in the self-similar variables,one can show(see[11,14])that,asτ→+∞,we have

with someα∞>0.Therefore,taking(3.4)as an approximation to(3.3),we should expect that

Comparing this with(3.8),we infer that

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We conclude this section with some remarks about the generality of the argument.Although we assume,for simplicity,that the reaction term in(1.1)is quadratic,our proof also works for a more general reaction term.Specifically,the function u−u2in(1.1)may be replaced by a C2function f:[0,1]→R satisfying f(0)=0=f(1),f?(0)>0,f?(1)<0,and f?(s)≤ f?(0)for all s∈[0,1].In particular,these assumptions imply that there is C>0,such that 0≤f?(0)s−f(s)≤Cs2for all s∈[0,1].Without loss of generality,we may suppose that f?(0)=1.Then,if g(u)=u−f(u),the equation(3.3)for v becomes

and the equation(3.6)for w becomes

where 0≤g(s)≤Cs2and g?(s)≥0.Then all of the arguments below(and in[11])work in this more general setting.Finally,the arguments also apply to fronts arising from compactly supported initial data u0≥0(not just perturbations of the step-function).In that case,one obtains two fronts propagating in opposite directions.Combined with[11],our arguments here imply that Theorem 1.1 holds for both fronts.That is,the fronts moving to±∞are at positionswith

where the shiftsandmay diff er and depend on the initial data.

Recall thatφ∗(x)is the traveling wave profile.We look for a function ζ(t)in(4.5)such that

4 Convergence to a Single Wave as a Consequence of the Diff usive Scale Convergence

Lemma 4.1The solution of(3.2)with u(1,x)=u0(x)satisfies

both uniformly in t>1.

Proposition 4.1Forwe have

The main new step is to establish the following.

satisfies the Fisher-KPP equation

with xγ=tγ.

between branching events,rather than following a standard Brownian motion.In terms of PDE,their study corresponds to the model

for t>Tε,with the initial conditionIn particular,we will show thatconverge,as t→ +∞,to a pair of steady solutions,separated only by an order O(ε)-translation.Note that the function v(t,x)=exuα(t,x)solves

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In view of the expansion(1.3),we should have,with someω0>0,

which implies,for

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and thus

The equation for the functionψis

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ProofThe issue is whether the Dirichlet boundary conditions would be stronger than the force in the right side of(4.7).Since the principal Dirichlet eigenvalue for the Laplacian in,investigating(4.7)is,heuristically,equivalent to solving the ODE

The coefficient(1−2γ)is chosen simply for convenience and can bereplaced by another constant.

The solution of(4.9)is

Note that f(t)tends to 0 as t→ +∞ a little faster than t3γ−1as soon as,so the analog of(4.8)holds for the solutions of(4.9).With this idea in mind,we are going to look for a super-solution of(4.7),in the form

and

Gathering(4.11)and(4.12),we infer the existence of q>0,such that,for t large enough,

as soon asεandλare small enough,sinceBecause the right-hand side of(4.7)does not depend on s,the inequality extends to all t≥1 by replacing s by As,with A large enough,and(4.8)follows.

Let us note that the term e−x(v+ ψ)in(4.7),which results from the quadratic structure of the nonlinearity,is positive.For a more general nonlinearity f(u)replacing u−u2,the monotonicity of g(u)=uf?(0)−f(u)may be used in an analogous way.

4.1 Proof of Theorem 1.1

for all x ≤ tγ.From Proposition 4.1,we have

uniformly in x ∈ (−tγ,tγ)with

Becauseε>0 is arbitrary,we have

with x∞=−logα∞,uniformly on compact sets.Together with Lemma 4.1,this concludes the proof of Theorem 1.1.

5 The Diffusive Scaleand the Proof of Lemma 4.2

Our analysis starts with(3.6),which we write as

Here,the operator L is defined as

Its principal eigenfunction on the half-lineη>0 with the Dirichlet boundary condition atη=0 is

as Lφ0=0.The operator L has a discrete spectrum inweighted by,its non-zero eigenvalues areλk=k≥1,and the corresponding eigenfunctions are related via

The principal eigenfunction of the adjoint operator

isThus,the solution of the unperturbed version of(5.1)on a half-line

satisfies

and our task is to generalize this asymptotics to the full problem(5.1)on the whole line.The weightin(5.4)is,of course,by no means optimal.We will prove the following.

Lemma 5.1Let w(τ,η)be the solution of(3.6)on R,with the initial condition w(0,η)=w0(η)such that w0(η)=0 for allη >M,with some M>0,and w0(η)=O(eη)for η<0.Thereand a function h(τ)such thatand such that we have,for any

with

where η+=max(0,η).

Once again,the weightis not optimal.Lemma 4.2 is an immediate consequence of this result.Indeed,

hence Lemma 5.1 implies,with xγ=tγ,

We now take Tεso thatfor all t>Tε.For the second term in the right-hand side of(5.6),we write

for t>Tεsufficiently large,as soon asγ?< γ.This proves(4.2).Thus,the proof of Lemma 4.2 reduces to proving Lemma 5.1.We will prove the latter by a construction of an upper and lower barrier for w with the correct behaviors.

5.1 The approximate Dirichlet boundary condition

Let us come back to why the solution of(5.1)must approximately satisfy the Dirichlet boundary condition atη=0.Recall that w is related to the solution of the original KPP problem via

The trivial a priori bound 0

and,in particular,we have

We also have

so that

forγ>0 sufficiently small.Thus,the solution of(5.1)satisfies

which we will use as an approximate Dirichlet boundary condition atη=0.

5.2 An upper barrier

Consider the solution of

with a compactly supported initial conditionchosen so thatHere,??should be thought of as a small parameter.

It follows from(5.11)that w(τ,η)is an upper barrier for w(τ,η).That is,we have

It is convenient to make a change of variables

where g(η)is a smooth monotonic function such that g(η)=1 for 0 ≤ η<1 and g(η)=0 for η>2.The functionsatisfies

forτ>0,with a smooth function G(τ,η)supported in 0≤ η ≤ 2,and the initial condition

which also is compactly supported.

We will allow(5.14)to run for a large time T,after which time we can treat the right-hand side and the last term in the left-hand side of(5.14)as a small perturbation.A variant of Lemma 2.2 from[11]implies thatfor all T>0,as well as the following estimate.

Lemma 5.2Considerand G(τ,η)smooth,bounded,and compactly supported in R+.Let p(τ,η)solve

with the initial condition p0(η)such thatThere existsε0>0 and C>0(depending on p0),such that,for all 0< ε< ε0,we have

wherefor allτ>0.

For anyε>0,we may choose T sufficiently large,andso that

This follows from(5.14).Then,applying Lemma 5.2 forτ>T,we have

We claim that with a suitable choice ofthe integral term in(5.18)is bounded from below:

Indeed,multiplying(5.14)byηand integrating gives

The function G(τ,η)need not have a sign,hence we do not know thatis positive everywhere.However,it follows from(5.14)that the negative part ofis bounded as

for all τ>0,with the constant C0which does not depend on w0(η)on the interval[2,∞).Thus,we deduce from(5.20)that for allτ>0,we have

with,onceagain,independent ofTherefore,after possibly increasingwe may ensure that(5.19)holds.

It follows from(5.18)–(5.19)that there exists a sequenceand a functionsuch that

and

uniformly in η on the half-line η ≥ 0.The same bound for the function w(τ,η)itself follows

also uniformly inηon the half-lineη≥0.

5.3 A lower barrier

A lower barrier for w(τ,η)is devised as follows.First,note that the upper barrier for w(τ,η)we have constructed above implies that

as soon as

withand Cγ>0 is chosen sufficiently large.Thus,a lower barriercan be defined as the solution of

with an initial conditionThis time it is convenient to make the change of variables

so that

We could now try to use an abstract stable manifold theorem to prove that

That is,remains uniformly bounded away from 0.However,to keep this paper self contained,we give a direct proof of(5.27).We look for a sub-solution to(5.26)in the form

where

with the functions ζ(τ)and q(τ)satisfying

In other words,we wish to deviseas in(5.28)–(5.29),such that

and

with

Notice that the choice of F(τ)in(5.28)has eliminated a low order term involvingFor convenience,let us define

which appears in(5.26).Because Lφ0=0 and

we find that

Let us write this as

Our goal is to choose ζ(τ)and q(τ),such that(5.29)holds and the right-hand side of(5.32)is non-positive after a certain timeτ0,possibly quite large.However,and this is an important point,this time τ0will not depend on the initial condition w0(η).

for allτ≥ τ0,η ≥ 0.

Therefore,on the intervalη ∈ [η1,∞)and for τ≥ τ0,(5.32)is bounded by

assuming q(τ)>0 andHence,if q(τ)andζ(τ)are chosen to satisfy the differential inequality

then we will have

provided thatas presumed in(5.29).Still assuminga sufficient condition for(5.34)to be satisfied is

Hence,we choose

Note that q(τ)satisfies the assumptions on q in(5.29).

Let us now deal with the range η∈ [0,η1].The functionis bounded on R and it is bounded away from 0 on[0,η1].Define

As h(τ)<0 for τ≥ τ0,on the interval[0,η1],we can bound(5.32)by

Forη ∈ [1,η1],whereη−1<1,we have

To make this non-positive,we chooseζto satisfy

where the last equalilty comes from(5.36).Assumingwe have ζ(τ)<ζ(τ0),so a sufficient condition for(5.39)to hold when τ≥τ0is simply

For η near 0,the dominant term in(5.37)isDefine

Therefore,if we can arrange that,then for η∈ [0,1],we have,so

In this case,

which is non-positive forτ≥ τ0,due to(5.39).In summary,we will have L(τ)p ≤ 0 in the intervalη ∈ [0,η1]and τ≥ τ0ifζsatisfies(5.40)andfor τ≥ τ0.In view of this,we let ζ(τ)have the form

Thus,(5.40)holds if

Hence it suffices that

holds;this may be achieved with a2,a3>0 ifτ0is large enough.Then we may take a2large enough,so thatholds forτ≥ τ0;this condition translates to

This also is attainable withand a3>0 if τ0is chosen large enough.This completes the construction of the subsolutionin(5.28).

Let us come back to our subsolutionFrom the strong maximum principle,we know thatandHence,there is λ0>0,such that

where p is given by(5.28)withζand q defined above,and we have forτ≥τ0,

This,by(5.29),bounds the quantity I(τ)uniformly from below,so that(5.29)holds with a constant c0>0 that depends on the initial condition w0.

Therefore,just as in the study of the upper barrier,we obtain the uniform convergence of(possibly a subsequence of)on the half-lineto a functionwhich satisfies

and such that

5.4 Convergence of w(τ,η):Proof of Lemma 5.1

Let X be the space of bounded uniformly continuous functions u(η),such thatis bounded and uniformly continuous on R+.We deduce from the convergence of the upper and lower barriers for w(τ,η)(and ensuing uniform bounds for w)that there exists a sequencesuch that w(τn,·)itself converges to a limit W∞∈ X,such that W∞≡ 0 on R−,and W∞(η)>0 for allη>0.Our next step is to bootstrap the convergence along a sub-sequence,and show that the limit of w(τ,η)asτ→ +∞ exists in the space X.First,observe that the above convergence implies that the shifted functions wn(τ,η)=w(τ+ τn,η)converge in X,uniformly on compact time intervals,as n → +∞ to the solution w∞(τ,η)of the linear problem

In addition,there exists α∞>0,such that w∞(τ,η)converges toin the topology of X asτ→+∞.Thus,for anyε>0,we may choose Tεlarge enough,so that

Given Tε,we can find Nεsufficiently large so that

In particular,we have

We may now construct the upper and lower barriers for the function w(τ+ τNε+Tε,η +exactly as we have done before.It follows,once again from Lemma 5.2 applied to these barriers that any limit pointφ∞of w(τ,·)in X asτ→ +∞ satisfies

Asε>0 is arbitrary,we conclude that w(τ,η)convergesin X asTaking into account Lemma 5.2 once again,applied to the upper and lower barriers for w(τ,η)constructed starting from any timeτ>0,we have proved Lemma 5.1,which implies Lemma 4.2.

AcknowledgementsLenya Ryzhik and Jean-Michel Roquejoffre thank the Labex CIMI for a PDE-probability quarter in Toulouse,in Winter 2014,out of which the idea of this paper grew and which provided a stimulating scientific environment for this project.

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