Convergence to a single wave in the Fisher-KPP equation

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1 Convergence to a single wave in the Fisher-KPP equation James Nolen Jean-Michel Roquejoffre Lenya Ryzhik Dedicated to H. Brezis, with admiration and respect Abstract We study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t 3/2 log t + x, the solution of the equation converges as t + to a translate of the traveling wave corresponding to the minimal speed c = 2. The constant x depends on the initial condition u, x. The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple proof based on PDE arguments. 1 Introduction We consider the Fisher-KPP equation: u t u xx = u u 2, t >, x R, 1.1 with an initial condition u, x = u x which is a compact perturbation of a step function, in the sense that there exist x 1 and x 2 so that u x = 1 for all x x 1, and u x = for all x x 2. This equation has a traveling wave solution ut, x = φx 2t, moving with the minimal speed c = 2, connecting the stable equilibrium u 1 to the unstable equilibrium u : φ 2φ = φ φ 2, φ = 1, φ+ =. 1.2 Each solution φξ of 1.2 is a shift of a fixed profile φ ξ: φξ = φ ξ + s, with some fixed s R. The profile φ ξ satisfies the asymptotics φ ξ = ξ + ke ξ + Oe 1+ω ξ, ξ +, 1.3 with two universal constants ω >, k R. The large time behaviour of the solutions of this problem has a long history, starting with a striking paper of Fisher [1], 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: u x = 1 for x, u x = for x, converges to φ in the following sense: there is a function σ t = 2t + ot, 1.4 Department of Mathematics, Duke University, Durham, NC 2778, USA; nolen@math.duke.edu Institut de Mathématiques UMR CNRS 5219, Université Paul Sabatier, 118 route de Narbonne, 3162 Toulouse cedex, France; jean-michel.roquejoffre@math.univ-toulouse.fr Department of Mathematics, Stanford University, Stanford CA, 9435, USA; ryzhik@math.stanford.edu 1

2 such that lim ut, x + σ t = φ x. t + Fisher has already made an informal argument that the ot in 1.4 is of the order Olog t. An important series of papers by Bramson proves the following Theorem 1.1 [5], [6] There is a constant x, depending on u, such that σ t = 2t 3 2 log t x + o1, as t +. Theorem 1.1 was proved through elaborate probabilistic arguments. A generalization is provided by Lau [17], using the decrease of the number of intersection points for any pair of solutions of the parabolic Cauchy problem. A natural question is to prove Theorem 1.1 with purely PDE arguments. In that spirit, a weaker version, precise up to the O1 term, but valid also for a much more difficult case of the periodic in space coefficients, is the main result of [11, 12]: σt = 2t 3 log t + O1 as t 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]. The paper is organized as follows. In Section 2, we shortly describe some connections between the Fisher-KPP equation 1.1 and the branching Brownian motion. In Section 3, we explain, in an informal way, the strategy of the proof of the theorem: in a nutshell, the solution is slaved to the dynamics at x = O t. In Sections 4 and 5, we make the arguments of Section 3 rigorous. Acknowledgment. JN was supported by NSF grant DMS , and LR by NSF grant DMS JMR was supported by the European Union s Seventh Framework Programme FP/ / ERC Grant Agreement n ReaDi - Reaction-Diffusion Equations, Propagation and Modelling, as well as the ANR project NONLOCAL ANR-14-CE LR and JMR thank the Labex CIMI for a PDE-probability quarter in Toulouse, in Winter 214, out of which the idea of this paper grew and which provided a stimulating scientific environment for this project. 2 Probabilistic links and some related models The time delay in models of the Fisher-KPP type has been the subject of various recent investigations, both from the PDE and probabilistic points of view. The Fisher-KPP equation appears in the theory of the branching Brownian motion BBM [19] as follows. Consider a BBM starting at x = at time t =, with binary branching at rate 1. Let X 1 t,..., X Nt t be the descendants of the original particle at time t, arranged in the increasing order: X 1 t X 2 t X Nt t. Then, the probability distribution function of the maximum: vt, x = PX Nt t > x, satisfies the Fisher-KPP equation v t = 1 2 v xx + v v 2, with the initial data v x = 1 x. Therefore, Theorem 1.1 is about the median location of the maximal particle X Nt. Building on the work of Lalley and Sellke [16], recent probabilistic analyses [1, 2, 3, 8, 7] of this particle system have identified a decorated Poisson-type point process which is 2

3 the limit of the particle distribution seen from the tip : there is a random variable Z > such that the point process defined by the shifted particles {X 1 t ct,..., X Nt t ct}, with ct = 2t 3 log t + log Z, 2 has a well-defined limit process as t. Furthermore, Z is the limit of the martingale Z t = k 2t X k te X kt 2t, and φ x = 1 E [ e Ze x ] for all x R. 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: u t u xx = µxu u where µx is continuous and 1-periodic in R, such that the principal periodic eigenvalue of the operator xx µx is negative. Then there is a minimal speed c > such that for each c c, there is a unique pulsating front U c t, x, up to a time shift [4, 13]. It was shown in [12] that there is s > such that, if ut, x solves 2.1 with a nonnegative, nonzero, compactly supported initial condition u x, and < s s, the s-level set of ut, x satisfies σ s t = c t 3 2λ logt + O1. This implies the convergence of ut, x σ s 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. When µx > everywhere, the solution u of the related model u t u xx = µxu u 2 may be interpreted in terms of the extremal particle in a BBM with a spatially-varying branching rate [12]. 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 dxt = 2κt/T dbt between branching events, rather than following a standard Brownian motion. In terms of PDE, their study corresponds to the model u t = κ 2 t/t u xx + fu, < t < T, x R. 2.2 They proved that if κ is increasing, and and fis of the Fisher-KPP type, the shift is algebraic and not logarithmic in time: there exists C > such that T 1/3 In [2], we proved the asymptotics C XT c eff T CT 1/3, c eff = 2 XT = c eff T νt 1/3 + OlogT, with ν = β 1 1 κsds. στ 1/3 στ 2/3 dτ. 2.3 Here, β < 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 ut to a traveling wave. 3

4 3 Strategy of the proof of Theorem 1.1 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: u t u xx = u u 2, x R, t > 1, 3.1 and proceed with a standard sequence of changes of variables. We first go into the moving frame: x x 2t + 3/2 log t, leading to u t u xx 2 3 2t u x = u u Next, we take out the exponential factor: set ut, x = e x vt, x so that v satisfies v t v xx 3 2t v v x + e x v 2 =, x R, t > Observe that for any shift x R, the function V x = e x φx x is a steady solution of V t V xx + e x V 2 =. We regard 3.3 as a perturbation of this equation, and expect that vt, x e x φx x as t, for some x R. The self-similar variables We note that for x +, the term e x v 2 in 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 z t z xx 3 2t z z x =, x > 3.4 zt, = is a reasonable proxy for 3.3 for x 1, and, as shown in [11, 12], it provides good sub- and super-solutions for vt, x. The main lesson of [11, 12] is that everything relevant to the solutions of 3.4 happens at the spatial scale x t, and their asymptotics may be unraveled by a self-similar change of variables. Here, we will accept the full nonlinear equation 3.3 and perform directly the self-similar change of variables τ = logt, η = x t 3.5 followed by a change of the unknown vτ, η = e τ/2 wτ, η. 4

5 This transforms 3.3 into w τ η 2 w η w ηη w e τ/2 w η + e 3τ/2 ηexpτ/2 w 2 =, η R, τ >. 3.6 This transformation strengthens the reason why the Dirichlet problem 3.4 appears naturally: for η τe τ/2, the last term in the left side of 3.6 becomes exponentially large, which forces w to be almost in this region. On the other hand, for η τe τ/2, 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 τe τ/2. 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, as τ +, we have e τ/2 zτ, η α ηe η2 /4, η >, 3.7 with some α >. Therefore, taking 3.4 as an approximation to 3.3, we should expect that ut, x = e x vt, x e x zt, x e x e τ/2 α ηe η2 /4 = α xe x e x2 /4t, 3.8 at least for x of the order O t. This determines the unique translation: if we accept that u converges to a translate x of φ, then for large x in the moving frame we have Comparing this with 3.8, we infer that ut, x φ x x xe x+x. 3.9 x = logα. The difficulty with this argument, apart from the justification of the approximation ut, x e x zt, x, is that each of the asymptotics 3.8 and 3.9 uses different ranges of x: 3.8 comes from the self-similar variables in the region x O t, while 3.9 assumes x to be large but finite. However, the self-similar analysis does not tell us at this stage what happens on the scale x O1. Indeed, it is clear from 3.6 that the error in the approximation 3.7 is at least of the order Oe τ/2 note that the right side in 3.7 is a solution of 3.6 without the last two terms in the left side. On the other hand, the scale x O1 corresponds to η e τ/2. Thus, the leading order term and the error in 3.7 are of the same size for x O1, which means that we can not extract information directly from 3.7 on that scale. To overcome this issue, we proceed in two steps: first we use the self-similar variables to prove stabilization that is, 3.8 holds at the spatial scales x Ot γ with a small γ >, and not just at the diffusive scale O t. This boils down to showing that wτ, η α ηe η2 /4 for the solution to 3.6, even for η e 1/2 γτ. Next, we show that this stabilization is sufficient to ensure the stabilization on the scale x O1 and convergence to a unique wave. This is the core of the argument: everything happening at x O1 should be governed by the tail of the solution the fronts are pulled. 5

6 4 Convergence to a single wave as a consequence of the diffusive scale convergence The proof of Theorem 1.1 relies on the following two lemmas. The first is a consequence of [11]. Lemma 4.1 The solution of 3.2 with u1, x = u x satisfies lim ut, x = 1, lim x ut, x =, 4.1 x + both uniformly in t > 1. The main new step is to establish the following. Lemma 4.2 There exists a constant α > with the following property. For any γ > and all ε > we can find T ε so that for all t > T ε we have with x γ = t γ. ut, x γ α x γ e xγ e x2 γ/4t εx γ e xγ e x2 γ/6t, 4.2 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 given α >, the behavior of the solutions of u α t 2 u α x t u α x u α + u 2 α =, x x γ t 4.3 u α t, t γ = αt γ e tγ t 2γ 1 /4, for t > T ε, with the initial condition u α T ε, x = ut ε, x. In particular, we will show that u α ±εt, x converge, as t +, to a pair of steady solutions, separated only by an order Oε-translation. Note that the function vt, x = e x u α t, x solves v t v xx + 3 2t v x v + e x v 2 =, x t γ 4.4 vt, t γ = αt γ e t2γ 1 /4. Since we anticipate that the tail is going to dictate the behavior of u α, we choose the translate of the wave that matches exactly the behavior of u α t, x at the boundary x = t γ : set ψt, x = e x φ x + ζt. 4.5 Recall that φ x is the traveling wave profile. We look for a function ζt in 4.5 such that In view of the expansion 1.3, we should have, with some ω > : ψt, t γ = vt, t γ. 4.6 e ζt t γ + ζt + k + Oe ω t γ = αt γ e 1/4t1 2γ, which implies ζt = logα logα kt γ + Ot 2γ, 6

7 and thus for γ, 1/3, we have The equation for the function ψ is ζt C t 1+γ. ψ t ψ xx + 3 2t ψ x ψ + e x ψ 2 = ζψ + ζψ x + 3 2t ψ x ψ = O x t = Ot 1+γ, x < t γ. In addition, the left side above is exponentially small for x < t γ because of the exponential factor in 4.5. Hence, the difference st, x = vt, x ψt, x satisfies s t s xx + 3 2t s x s + e x v + ψs = Ot 1+γ, x t γ 4.7 st, t γ = Oe tγ, st, t γ =. Proposition 4.3 We have lim sup st, x =. 4.8 t + x t γ Proof. The 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 t γ, t γ is π 2 /t 2γ, investigating 4.7 is, heuristically, equivalent to solving the ODE f t + 1 2γt 2γ f = t1 γ The coefficient 1 2γ is chosen simply for convenience and can be replaced by another constant. The solution of 4.9 is ft = f1e t 2γ t 1 s γ 1 e t 2γ+1 +s 2γ+1 ds. Note that ft tends to as t + a little faster than t 3γ 1 as soon as γ < 1/3, 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 supersolution to 4.7, in the form x st, x = t λ cos t γ+ε, 4.1 where λ, γ and ε will be chosen to be small enough. We now set T ε = 1 for convenience. We have, for x t γ : st, x t λ, s xx = t 2γ+2ε st, x, 4.11 s t = λ C x s + gt, x, gt, x t t λ+γ+ε+1 C st, x, t1+ε and 3 2t s x st, x Ct 1 st, x Gathering 4.11 and 4.12 we infer the existence of q > such that, for t large enough: t xx 3 2t x 1 st, x qt 2γ+2ε st, x q 1 2 t 2γ+2ε+λ O, t1 2γ as soon as γ, ε and λ are small enough. Because the right 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. 7

8 Proof of Theorem 1.1 We are now ready to prove the theorem. Given ε >, take T ε as in Lemma 4.2. Let u α t, x be the solution of 4.3 for t > T ε, and the initial condition u α T ε, x = ut ε, x. Here, ut, x is the solution of the original problem 3.2. It follows from Lemma 4.2 that for any t T ε, we have for all x t γ. From Proposition 4.3, we have uniformly in x t γ, t γ, with Because ε > is arbitrary, we have u α εt, x ut, x u α +εt, x, e x[ u α ±εt, x φ x + ζ ± t ] = o1, as t +, 4.13 ζ ± t = 1 t γ logα ± ε + Ot 2γ. lim t + ut, x φ x + x =, with x = logα, uniformly on compact sets. Together with Lemma 4.1, this concludes the proof of Theorem The diffusive scale x O t and the proof of Lemma 4.2 Our analysis starts with 3.6, which we write as w τ + Lw e τ/2 w η + e 3τ/2 ηexpτ/2 w 2 =, η R, τ >. 5.1 Here, the operator L is defined as Lv = v ηη η 2 v η v. 5.2 Its principal eigenfunction on the half-line η > with the Dirichlet boundary condition at η = is φ η = η 2 e η2 /4, as Lφ =. The operator L has a discrete spectrum in L 2 R +, weighted by e η2 /8, its non-zero eigenvalues are λ k = k 1, and the corresponding eigenfunctions are related via φ k+1 = φ k. The principal eigenfunction of the adjoint operator L ψ = ψ ηη ηηψ ψ is ψ η = η. Thus, the solution of the unperturbed version of 5.1 on a half-line satisfies pτ, η = η e η2 /4 2 π p τ + Lp =, η >, pτ, =, ξv ξdξ + Oe τ e η2 /6, as τ +, 5.4 and our task is to generalize this asymptotics to the full problem 5.1 on the whole line. weight e η2 /6 in 5.4 is, of course, by no means optimal. We will prove the following: 8 The

9 Lemma 5.1 Let wτ, η be the solution of 3.6 on R, with the initial condition w, η = w η such that w η = for all η > M, with some M >, and w η = Oe η for η <. There exists α > and a function hτ such that lim hτ =, and such that we have, for any τ + γ, 1/2: wτ, η = α + hτη + e η2 /4 + Rτ, ηe η2 /6, η R, 5.5 with and where η + = max, η. Rτ, η C γ e 1/2 γ τ, Once again, the weight e η2 /6 is not optimal. Lemma 4.2 is an immediate consequence of this result. Indeed, ut, x = e x x twlog t,, t hence Lemma 5.1 implies, with x γ = t γ, e xγ ut, x γ α x γ e x2γ/4t = tw log t, x γ α x γ e x2 γ/4t t = hlog tx γ e x2γ/4t + tr log t, x γ e x2γ/6t. t 5.6 We now take T ε so that hlog t < ε/3 for all t > T ε. For the second term in the right side of 5.6 we write x γ R log t, te x 2 γ /6t Ct γ e x2 γ /6t εx γ e x2 γ /6t 5.7 t for t > T ε sufficiently large, as soon as γ < 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. The approximate Dirichlet boundary condition Let us come back to why the solution of 5.1 must approximately satisfy the Dirichlet boundary condition at η =. Recall that w is related to the solution of the original KPP problem via wτ, η = ue τ, ηe τ/2 e τ/2+ηeτ/2. The trivial a priori bound < ut, x < 1 implies that we have < wτ, η < e τ/2+ηeτ/2, η <, 5.8 and, in particular, we have We also have < wτ, e 1/2 γτ e eγτ. 5.9 w τ τ, η = u t e τ, ηe τ/2 e τ/2+ηeτ/2 + η 2 u xe τ, ηe τ/2 e ηeτ/2 + η 2 eτ/2 1 2 ueτ, ηe τ/2 e τ/2+ηeτ/2, 9

10 so that w τ τ, e 1/2 γτ = u t e τ, e 1/2 γτ e τ/2 eγτ 1 2 e 1/2 γτ u x e τ, e 1/2 γτ e eγτ 1 2 eγτ + 1ue τ, e γτ e τ/2 eγτ = Oe γeγτ, 5.1 for γ > sufficiently small. Thus, the solution of 5.1 satisfies < wτ, e 1/2 γτ e eγτ, w τ τ, e 1/2 γτ Ce γeγτ, which we will use as an approximate Dirichlet boundary condition at η =. An upper barrier Consider the solution of 5.11 w τ + Lw e τ/2 w η =, τ >, η > e 1/2 γτ, 5.12 wτ, e 1/2 γτ = e eγτ, with a compactly supported initial condition w η = w, η chosen so that w η u 1 ηe η. Here, γ, 1/2 should be thought of as a small parameter. It follows from 5.11 that wτ, η is an upper barrier for wτ, η. That is, we have It is convenient to make a change of variables wτ, η wτ, η, for all τ > and η > e 1/2 γτ. wτ, η = pτ, η + e 1/2 γτ + e eγτ gη + e 1/2 γτ, 5.13 where gη is a smooth monotonic function such that gη = 1 for η < 1 and gη = for η > 2. The function p satisfies p τ + L p + γe 1/2 γτ e τ/2 p η = Gτ, ηe eγτ, η >, pτ, =, 5.14 for τ >, with a smooth function Gτ, η supported in η 2, and the initial condition p η = w η 1 e 1 gη, 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 side and the last term in the left side of 5.14 as a small perturbation. A variant of Lemma 2.2 from [11] implies that pt, ηe η2 /6 L 2 R + for all T >, as well as the following estimate: Lemma 5.2 Consider ω, 1/2 and Gτ, η smooth, bounded, and compactly supported in R +. Let pτ, η solve p τ + Lp εe ωτ p η + p + Gτ, η, τ >, η >, pτ, = with the initial condition p η such that p ηe η2 /6 L 2 R +. There exists ε > and C > depending on p such that, for all < ε < ε, we have e η 2 /4 + pτ, η = η 2 ξp ξdξ+εr 1 τ, η +εe ωτ R 2 τ, ηe η2 /6 +e τ R 3 τ, ηe η2 /6, 5.16 π where R 1,2,3 τ, C 3 C for all τ >. 1

11 For any ε >, we may choose T sufficiently large, and ω, 1/2 γ so that p τ + L p εe ωτ T p η + Gτ, η, τ > T, η >, pτ, = This follows from Then, applying Lemma 5.2 for τ > T, we have e η 2 /4 + pτ, η = η 2 ξ pt, ξdξ+εr 1 τ, η +εe ωτ T R 2 τ, ηe η2 /6 +e τ T R 3 τ, ηe η2 /6. π 5.18 We claim that with a suitable choice of w, the integral term in 5.18 is bounded from below: η pτ, ηdη 1, for all τ > Indeed, multiplying 5.14 by η and integrating gives d dτ η pτ, ηdη = γe 1/2 γτ e τ/2 pτ, ηdη + e eγτ Gτ, ηηdη. 5.2 The function Gτ, η need not have a sign, hence a priori we do not know that pτ, η is positive everywhere. However, it follows from 5.14 that the negative part of p is bounded as pτ, ηdη C, for all τ >, with the constant C which does not depend on w η on the interval [2,. Thus, we deduce from 5.2 that for all τ > we have η pτ, ηdη η w ηdη C, 5.21 with, once again, C independent of w. Therefore, after possibly increasing w we may ensure that 5.19 holds. It follows from 5.19 and 5.18 that there exists a sequence τ n +, C > and a function W η such that C 1 ηe η2 /4 W η Cηe η2 /4, 5.22 and lim /8 n + eη2 pτ n, η W η =, 5.23 uniformly in η on the half-line η. The same bound for the function wτ, η itself follows: also uniformly in η on the half-line η. A lower barrier lim /8 n + eη2 wτ n, η W η =, 5.24 A lower barrier for wτ, η is devised as follows. First, note that the upper barrier for wτ, η we have constructed above implies that e 3τ/2 ηexpτ/2 wτ, η C γ e expγτ/2, 11

12 as soon as η e 1/2 γτ, with γ, 1/2, and C γ > is chosen sufficiently large. defined as the solution of Thus, a lower barrier wτ, η can be w τ + Lw e τ/2 w η + C γ e expγτ/2 w =, wτ, e 1/2 γτ =, η > e 1/2 γτ, 5.25 and with an initial condition w η w η. This time it is convenient to make the change of variables wτ, η = zτ, η e 1/2 γτ so that z τ + Lz + γe 1/2 γτ e τ/2 z η + C γ e expγτ/2 z =, η >, zτ, =, 5.26 We could now try to use an abstract stable manifold theorem to prove that Iτ := ηzτ, ηdη c >, for all τ > That is, Iτ remains uniformly bounded away from. However, to keep this paper self-contained, we give a direct proof of We look for a sub-solution to 5.26 in the form pτ, η = ζτφ η qτηe η2 /8 e F τ, 5.28 where F τ = τ and with the functions ζτ and qτ satisfying C γ e expγs/2 ds, ζτ ζ >, ζτ <, qτ >, qτ = Oe τ/ In other words, we wish to devise pτ, η as in such that p, η z, η = w η + 1, 5.3 and with Lτp, 5.31 Lτp = p τ + Lp + γe 1/2 γτ e τ/2 p η. Notice that the choice of F τ in 5.28 has eliminated a low order term involving C γ e expγτ/2. For convenience, let us define hτ = γe 1/2 γτ e τ/2, which appears in Because Lφ = and because Lηe η2 /8 = ηle η2 /8 = η ηe η2 /8, 12

13 we find that Lτp = ζφ + ζhτφ Let us write this as η 1 e η2 /8 Lτp = ζη 1 φ e η2 /8 + η 1 hτ q + η ηe 4 q η2 /8 + q η2 /8 4 e η2 hτ qe η2 /8 hτ. η ζe η2 /8 φ 2 + q 4 1 q + η q Our goal is to choose ζτ and qτ such that 5.29 holds and the right side of 5.32 is non-positive after a certain time τ, possibly quite large. However, and this is an important point, this time τ will not depend on the initial condition w η. Let us restrict the small parameter γ to the interval, 1/4. Observe that if τ > is sufficiently large, then hτ < and hτ e τ/4 for all τ τ. As φ η = ηe η2 /4, note that in 5.32 both φ ηeη2 /8 and φ ηe η2 /8 are bounded functions. In particular, if τ is large enough then for all τ τ, η. Note also that for all η η 1 = 28 we have φ e η2 /8 hτ e τ/4 η and η Therefore, on the interval η [η 1, and for τ τ, 5.32 is bounded by η 1 e η2 /8 Lτp η 1 hτζe η2 /8 φ q + q ζτe τ/4 q + q, assuming qτ > and ζ <. Hence, if qτ and ζτ are chosen to satisfy the differential inequality q + q e τ/4 ζ, τ τ, 5.34 then we will have Lτp for τ τ and η η 1, 5.35 provided that ζ, as presumed in Still assuming ζ on τ, +, a sufficient condition for 5.34 to be satisfied is: q + q e τ/4 ζτ, τ τ. Hence, we choose qτ = e τ τ e τ/4 ζτ Note that qτ satisfies the assumptions on q in Let us now deal with the range η [, η 1 ]. The function η 1 φ η is bounded on R and it is bounded away from on [, η 1 ]. Define ε 1 = min η [,η η 1 φ ηe η2 /8 >. γ] As hτ < for τ τ, on the interval [, η 1 ], we can bound 5.32 by η 1 e η2 /8 Lτp ε 1 ζτ + η 1 hτ ζe η2 /8 φ q q 3 4 q

14 For η [1, η 1 ], where η 1 < 1, we have η 1 e η2 /8 Lτp ε 1 ζτ + e τ/4 ζ + q To make this non-positive, we choose ζ to satisfy q 3 4 q ε 1 ζτ q 3 4 q e τ/4 ζ + q = e τ/4 ζτ 7 4 qτ e τ/4 ζτ + qτ, 5.39 where the last equalilty comes from Assuming ζ <, we have ζτ < ζτ, so a sufficient condition for 5.39 to hold when τ τ is simply ε 1 ζτ 3qτ. 5.4 For η near, the dominant term in 5.37 is η 1 hτ ζe η2 /8 φ q. Define ε 2 = min η [,1] φ ηe η2 /8 >. Therefore, if we can arrange that ζτ > qτ/ε 2, then for η [, 1], we have ζe η2 /8 φ q, so η 1 hτ ζe η2 /8 φ q. In this case, η 1 e η2 /8 Lτp ε 1 ζτ q 3 4 q which is non-positive for τ τ, due to In summary, we will have Lτp in the interval η [, η 1 ] and τ τ if ζ satisfies 5.4 and ζτ > qτ/ε 2 for τ τ. In view of this, we let ζτ have the form ζτ = a 2 + a 3 e τ τ/4. Thus, 5.4 holds if ε 1a 3 4 e τ τ /4 3q = 3e τ τ 4e τ/4 a 2 + a 3, τ τ. Hence it suffices that ε 1 a e τ/4 a 2 + a 3 4 holds; this may be achieved with a 2, a 3 > if τ is large enough. Then we may take a 2 large enough so that ζτ > qτ/ε 2 also holds for τ τ ; this condition translates to: a 2 + a 3 e τ τ/4 1 e τ τ + 4 ε 2 3 e τ/4 a 2 + a 3, τ τ. This also is attainable with a 2 > 1 ε 2 and a 3 > if τ is chosen large enough. This completes the construction of the subsolution pτ, η in Let us come back to our subsolution zτ, η. From the strong maximum principle, we know that zτ, η > and η zτ, >. Hence, there is λ > such that wτ, η λ pτ, η, 14

15 where p is given by 5.28 with ζ and q defined above, and we have for τ τ : wτ, η λ pτ, η. This, by 5.29, bounds the quantity Iτ uniformly from below, so that 5.29 holds with a constant c > that depends on the initial condition w. Therefore, just as in the study of the upper barrier, we obtain the uniform convergence of possibly a subsequence of wτ n, on the half-line η e 1/2 γτ to a function W η which satisfies C 1 ηe η2 /4 W η Cηe η2 /4, 5.42 and such that Convergence of wτ, η: proof of Lemma 5.1 lim /8 n + eη2 wτ n, η W η =, η > Let X be the space of bounded uniformly continuous functions uη such that e η2 /8 uη is bounded and uniformly continuous on R +. We deduce from the convergence of the upper and lower barriers for wτ, η and ensuing uniform bounds for w that there exists a sequence τ n + such that wτ n, itself converges to a limit W X, such that W on R, and W η > for all η >. Our next step is to bootstrap the convergence along a sub-sequence, and show that the limit of wτ, η as τ + exists in the space X. First, observe that the above convergence implies that the shifted functions w n τ, η = wτ + τ n, η converge in X, uniformly on compact time intervals, as n + to the solution w τ, η of the linear problem τ + Lw =, η >, 5.44 w τ, =, w, η = W η. In addition, there exists α > such that w τ, η converges to ψη = α ηe η2 /4, in the topology of X as τ +. Thus, for any ε > we may choose T ε large enough so that w τ, η α ηe η2 /4 εηe η2 /8 for all τ > T ε, and η > Given T ε we can find N ε sufficiently large so that In particular, we have wt ε + τ n, η + e 1/2 γtε w T ε, η εηe η2 /8, for all n > N ε α ηe η2 /4 2εηe η2 /8 wτ Nε + T ε, η + e 1/2 γtε α ηe η2 /4 + 2εηe η2 / We may now construct the upper and lower barriers for the function wτ +τ Nε +T ε, η +e 1/2 γtε, exactly as we have done before. It follows, once again from Lemma 5.2 applied to these barriers that any limit point φ of wτ, in X as τ + satisfies α Cεηe η2 /4 φ η α + Cεηe η2 / As ε > is arbitrary, we conclude that wτ, η converges in X as τ + to ψη = α ηe η2 /4. Taking into account Lemma 5.2 once again, applied to the upper and lower barriers for wτ, η constructed starting from any time τ >, we have proved Lemma 5.1, which implies Lemma

16 References [1] E. Aïdékon, J. Berestycki, É. Brunet, Z. Shi, Branching Brownian motion seen from its tip, Probab. Theory Relat. Fields , pp [2] L.-P. Arguin, A. Bovier, and N. Kistler, Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab , pp [3] L.-P. Arguin, A. Bovier, and N. Kistler, The extremal process of branching Brownian motion. Probab. Theory Relat. Fields pp [4] H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math , [5] M.D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31, 1978, [6] M.D. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44, [7] E. Brunet and B. Derrida. A branching random walk seen from the tip, Journal of Statistical Physics , pp [8] E. Brunet and B. Derrida. Statistics at the tip of a branching random walk and the delay of traveling waves. Eur. Phys. Lett. 87, [9] M. Fang and O. Zeitouni, Slowdown for time inhomogeneous branching Brownian motion, J. Stat. Phys. 149, 212, 1 9. [1] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 1937, [11] F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Het. Media 8, 213, [12] F. Hamel, J. Nolen, J.-M. Roquejoffre, and L. Ryzhik, The logarithmic time delay of KPP fronts in a periodic medium, J. Europ. Math. Soc. 18, 216, [13] F. Hamel, L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Europ. Math. Soc. 13, 211, [14] C. Henderson, Population stabilization in branching Brownian motion with absorption, to appear in CMS, 215. [15] A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov, Étude de l équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Inter. A 1, 1937, [16] S.P. Lalley and T. Sellke, A conditional limit theorem for the frontier of a branching Brownian motion. Annals of Probability, 15, 1987, [17] K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovskii and Piskunov, J. Diff. Eqs. 59, 1985, [18] P. Maillard, O. Zeitouni, Slowdown in branching Brownian motion with inhomogeneous variance, to appear in Ann. IHP, Prob. Stat. 16

17 [19] H.P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii- Piskunov, Comm. Pure Appl. Math , [2] J. Nolen, J.-M. Roquejoffre and L. Ryzhik, Power-like delay in time inhomogeneous Fisher-KPP equations, Comm. Partial Diff. Equations, 4, 215, [21] J. Nolen, J.-M. Roquejoffre and L. Ryzhik, Sharp large-time asymptotics in the Fisher-KPP equation, forthcoming. [22] M. Roberts, A simple path to asymptotics for the frontier of a branching Brownian motion, Ann. Prob. 41, 213, [23] J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 1997,