Emeric Bouin
email: bouin@ceremade.dauphine.fr

Christopher Henderson
email: ckhenderson@math.arizona.edu

Lenya Ryzhik
email: ryzhik@math.stanford.edu

The Bramson delay in the non-local Fisher-KPP equation

Keywords: Reaction-diffusion equations, Logarithmic delay, Parabolic Harnack inequality AMS Class. No: 35K57, 35Q92, 45K05, 35C07

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either 2t -(3/2) log t + O(1), as in the local case, or 2t -O(t β ) for some explicit β ∈ (0, 1). Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any β ∈ (0, 1), examples of Fisher-KPP type non-linearities f β such that the front for the local Fisher-KPP equation with reaction term f β is at 2t -O(t β ).

Introduction

The Fisher-KPP equation

u t = u xx + u(1 -u) (1.1)
is one of the simplest models for population spreading, accounting for a competition for resources. However, (1.1) only accounts for a local competition between individuals. When this competition is non-local, one is led to the non-local Fisher-KPP equation

u t -u xx = u(1 -φ ⋆ u), t > 0, x ∈ R, u(0, •) = u 0 . (1.2)
Here, φ is a probability density that represents the strength of the competition between individuals a given distance apart. Equation (1.2) has garnered much interest recently, mostly for two reasons. First, it does not admit a comparison principle, leading to inherent technical difficulties -even proving a uniform upper bound on u is non-trivial [START_REF] Hamel | On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds[END_REF]. Second, unusual behavior may occur, such as the existence of oscillating wave trains behind the front [START_REF] Faye | Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach[END_REF][START_REF] Genieys | Pattern and waves for a model in population dynamics with nonlocal consumption of resources[END_REF][START_REF] Gourley | Travelling front solutions of a nonlocal Fisher equation[END_REF][START_REF] Nadin | Wave-like solutions for nonlocal reactiondiffusion equations: a toy model[END_REF]. Our interest is in the spreading of the solutions of (1.2) when the initial density u 0 is localized. To motivate our work, we recall the known results for the local Fisher-KPP equation (1.1). Going back to the work of Bramson, it is known that if u 0 is compactly supported, the front of u is located at

X(t) = 2t - 3 2 log t + s 0 , (1.3)
where s 0 is a shift depending only on u 0 [START_REF] Bramson | Maximal displacement of branching Brownian motion[END_REF][START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF], with less precise asymptotics obtained earlier by Uchiyama [START_REF] Uchiyama | The behavior of solutions of some nonlinear diffusion equations for large time[END_REF]. These proofs have been simplified in recent years [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Roberts | A simple path to asymptotics for the frontier of a branching Brownian motion[END_REF], with some refinements in [START_REF] Nolen | Refined long time asymptotics for Fisher-KPP fronts[END_REF][START_REF] Nolen | Convergence to a single wave in the Fisher-KPP equation[END_REF], and also extended to the spatially periodic case [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF]. One may think of X(t) = 2t as the position of a traveling wave, and d(t) = (3/2) log t as the delay due to the fact that the initial condition u 0 is compactly supported, so that the solution lags behind the traveling wave.

In the non-local case considered in the present paper, we show that the front position depends on the rate of decay of the kernel φ at infinity. When φ decays fast enough, solutions of (1.2) spread as those of the local equation: the front is at a position as in (1.3), up to a constant order error. However, when φ decays slowly, and the competition at large distances is relatively strong, the delay behind the traveling wave position 2t is not logarithmic but algebraic, of the order O(t β ), with β depending only on the rate of decay of φ.

We now make our assumptions more precise. First, we assume that φ is an even, continuous, and bounded probability density: ˆR φ(x)dx = 1, and φ(x) = φ(-x) for all x ∈ R.

(

In addition, φ has some "mass" near the origin, that is, there exists σ φ > 0 such that φ(x) ≥ σ φ 1 [-σ φ ,σ φ ] .

(1.5)

The behavior of u depends strongly on the tail behavior of φ. Here we make two different assumptions, that are helpful for the upper and lower bounds, respectively. The first assumption, an upper bound on the tail of φ, is that there exists A φ > 0 and r > 1 such that, for all R ≥ 1,

ˆ∞ R φ(x)dx ≤ A φ R -r+1 . (1.6)
Sometimes we will need to complement this with a lower bound on the tail: for all R ≥ 1, we have

ˆ∞ R φ(x)dx ≥ A -1 φ R -r+1 . (1.7)
Roughly, (1.6) and (1.7) mean that φ ∼ x -r for x ≫ 1.

For the initial condition, we assume that u 0 is localized to the left of some point x 0 : 0 ≤ u 0 ≤ 1, ∃x 0 such that u 0 (x) = 0 for all x ≥ x 0 , and lim inf x→-∞ u 0 (x) > 0.

(1.8)

We expect our results to hold when u 0 has "fast" exponential decay, that is, u 0 (x)e (1+ǫ)x → 0 as x → 0 for some ǫ > 0, rather than compactly supported on the right. However, we recall that the front position asymptotics for solutions of (1.1) with u 0 that has a sufficiently slow exponential tail on the right is different from (1.3), see [START_REF] Bramson | Maximal displacement of branching Brownian motion[END_REF][START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF].

The main result of this paper is the following. If r ∈ (1, 3), then the delay is algebraic: there exists C φ > 0, depending only on r, σ φ , and

A φ , such that lim inf t→∞ inf x≤0 u t, 2t -C φ t 3-r 1+r + x > 0, (1.13)
and, if additionally (1.7) holds, then there exists c φ ∈ (0, C φ ), depending only on σ φ , r, and A φ , such that lim

t→∞ sup x≥0 u t, 2t -c φ t 3-r 1+r + x = 0. (1.14)
As we discuss later in greater detail, heuristically, the competition term φ⋆u acts on the scale t γ , with γ = 2/(1 + r). Note that 3 -r

1 + r = 2γ -1, (1.15)
and that, when r > 3, γ < 1/2, which, in turn, suggests that the competition scale is smaller than the diffusive scale √ t. This is one way to see that there is a phase transition at r = 3. As a by-product of our analysis, we also obtain results for the local Fisher-KPP equation

u t = u xx + f (u). (1.16) Let us assume that f is of the KPP class: f (u)/u is decreasing in u near 0, f ∈ C 1 , and f ′ (0) = 1.
A natural question is whether these assumptions are sufficient to ensure that the front location is given by the logarithmic Bramson correction in (1.3). We show, roughly, the following: if

1 - f (u) u ∼ log 1 u 1-r with r > 1,
then the conclusion of Theorem 1.1 holds, with the logarithmic delay for r ≥ 3 and an algebraic delay of the order O(t (3-r)/(1+r) ) for 1 < r < 3. These non-linearities are not purely mathematical curiosities: they are regularly used in biology and are known as Gompertz models, see [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF] and the vast body of literature around it. The statement and proof of this result are contained in Section 6.

Let us mention a few related works. The model (1.2) considered here was first introduced by Britton [START_REF] Britton | Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model[END_REF] and has a quite involved history, see the introduction of [START_REF] Berestycki | The non-local Fisher-KPP equation: travelling waves and steady states[END_REF] for a brief overview. The non-local term φ ⋆ u has different effects depending on whether one is studying the behavior of u behind the front or at the front. Behind the front, there is a possible Turing instability of the steady state of the local Fisher-KPP equation u ≡ 1, which complicates the behavior. For example, wave trains have been constructed by Faye and Holzer [START_REF] Faye | Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach[END_REF] and, in a related setting, in [START_REF] Nadin | Wave-like solutions for nonlocal reactiondiffusion equations: a toy model[END_REF]. Such wave trains have also been observed numerically by Genieys, Volpert, and Auger in [START_REF] Genieys | Pattern and waves for a model in population dynamics with nonlocal consumption of resources[END_REF]. As a result, without finer assumptions on φ, one cannot hope for a stronger result than the lower bounds in Theorem 1.1. As far as the behavior at the front is concerned, the main result in this direction is that traveling waves of speed c = 2 exist [START_REF] Fang | Monotone wavefronts of the nonlocal Fisher-KPP equation[END_REF][START_REF] Gourley | Travelling front solutions of a nonlocal Fisher equation[END_REF] and solutions to the Cauchy problem with compact initial data or which satisfy (1.8) propagate with speed c(t) = 2 + o(1) as t → +∞ [START_REF] Hamel | On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds[END_REF].

As far as algebraic delays are concerned, we point to the work of Fang and Zeitouni [START_REF] Fang | Slowdown for time inhomogeneous branching Brownian motion[END_REF] and Maillard and Zeitouni [START_REF] Maillard | Slowdown in branching Brownian motion with inhomogeneous variance[END_REF], as well as [START_REF] Nolen | Power-like delay in time inhomogeneous Fisher-KPP equations[END_REF] where a Fisher-KPP model with a diffusivity that changes slowly in time was studied, and a delay, roughly, of order t 1/3 was obtained. However, both the set-up and the mechanism for the large delay are quite different in these papers than in the present work. Finally, we also mention the recent paper of Ducrot [START_REF] Ducrot | On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data[END_REF] in which he constructs a class of non-linearities f (x, u), which tend to u(1 -u) as |x| → ∞, such that if the nonlinearity u(1 -u) in (1.1) is replaced by f (x, u), then the front is at 2t -λ log(t) for any λ ≥ 3/2.

While in the final stages of preparing this paper, we learned of a very recent probabilistic study of the delay term by Penington [START_REF] Penington | The spreading speed of solutions of the non-local fisher-kpp equation[END_REF]. In our notation, she obtains the log delay up to an error term O(log log(t)), when r > 3, and an algebraic delay t (3-r)/(1+r)±ǫ for any ǫ > 0 for r ∈ (1, 3). Penington's assumptions on φ are the same as ours when r ≥ 3. However, her assumptions are weaker when r ∈ (1, 3): the R -(r-1) term in (1.6) is replaced by R -(r-1)/2 , at the expense of a slightly less precise power in the correction. The proofs in [START_REF] Penington | The spreading speed of solutions of the non-local fisher-kpp equation[END_REF] are probabilistic, involving the Feynman-Kac formula and an in-depth study of the trajectories of Brownian motion. Overall, our work and [START_REF] Penington | The spreading speed of solutions of the non-local fisher-kpp equation[END_REF] are quite different and reveal different features of the equation.

Heuristics and methods of proof

The upper bound (1.10) is obtained by a rather direct adaptation of the arguments in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]. Let us outline a heuristic argument leading to the upper bound (1.14) for r ∈ [START_REF] Berestycki | The non-local Fisher-KPP equation: travelling waves and steady states[END_REF]3). It also explains how the exponent (3 -r)/(1 + r) comes about. Let the front have a delay d(t) behind 2t, so that inf

x≤2t-d(t) u(t, x) ≥ δ 0 , (1.17)
with some δ 0 > 0. We expect that the solution looks like an exponential to the right of x = 2t -d(t) and until the "front edge" at x = 2t + e(t):

u(t, x) ∼ exp{-(x -2t + d(t))}, for x ∈ (2t -d(t), 2t + e(t)). (1.18)
The diffusive Gaussian decay dominates the exponential "traveling wave" decay for x > 2t + e(t).

Using (1.17) and then (1.7), one may estimate φ ⋆ u(t, x) when x ∈ (2t -d(t), 2t + e(t)) as

φ ⋆ u(t, x) ≥ δ 0 ˆ2t-d(t) -∞ φ(x -y)dy (x -(2t -d(t))) 1-r (e(t) + d(t)) 1-r .
Thus, in order for the exponential in (1.18)

d ′ (t) e(t) 1-r t 1-r 2 d(t) 1-r 2 ,
and thus necessarily d(t) t (3-r)/(1+r) .

We deduce also e(t) t γ , with γ as in (1.15). A way to estimate the solution from below, to get the lower bounds, is to study the linearized Fisher-KPP equation with a Dirichlet boundary condition at 2t + e(t), as in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]. The problem that comes up after removing the exponential factor is

z t = z xx + e ′ (t)(z x -z), t > 0, x > 0, z(t, 0) = 0. (1.23)
Once again, the case r > 3 is treated similarly to [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]. In particular, while the term e ′ (t)z is important and is responsible for the 3/2 pre-factor in the logarithmic correction, the drift e ′ (t)z x is negligible. Roughly, we estimate z(t, x) at x ∼ √ t, and use a "tracing back to a shifted traveling wave" argument, to construct a sub-solution for u.

When r < 3, we choose e(t) = t γ . Since now γ > 1/2, the drift e ′ (t)z x can no longer be neglected, and the choice of the exact exponent γ is necessary to get matching asymptotics. We explicitly construct a sub-solution of u to estimate the solution at the far edge, and then perform a "tracing back" argument with a travelling wave.

Lastly, in the case when r = 3, the diffusive scale and the induced drift have the same order. Here, the balance of these two scales causes a somewhat larger delay.

The local in time Harnack inequality

The main tool that allows us to get "reasonably sharp" asymptotics for the front position is a local-in-time Harnack inequality that is of an independent interest. Proposition 1.2. Suppose that u ∈ L ∞ ([0, T ] × R) is a non-negative function that solves

u t = u xx + c(t, x)u, on [0, T ] × R with c ∈ L ∞ ([0, T ] × R) and T > 0.
Then, for any p ∈ (1, ∞), there exist positive constants α, β, and C, that depend only on c L ∞ ([0,T ]×R) and p, such that, for all x, y ∈ R and t ∈ (0, T ], we have

u(T, x + y) ≤ C u 1-1 p L ∞ ([t,T ])×R u(T, x) 1 p e αt+ βy 2 t . (1.24)
This inequality is an indispensable tool to obtain "reasonably sharp" results for non-local problems. We have used a less precise form of it to obtain the logarithmic delay for solutions of the cane toads equation in [START_REF] Bouin | The Bramson logarithmic delay in the cane toads equations[END_REF], and it has also been used to establish a precise lower bound on the propagation speed of solutions of a Keller-Segel-Fisher system [START_REF] Hamel | Propagation in a Fisher-KPP equation with non-local advection[END_REF]. As far as we know, [START_REF] Bouin | The Bramson logarithmic delay in the cane toads equations[END_REF] is the only other non-local context where a delay asymptotics has been established. It allows us to bound solutions of the non-local Fisher-KPP equation (1.2) in terms of the solutions of a local Fisher-KPP equation with a local time-dependent nonlinearity g(t, u), that is logarithmic in u (Gompertz type). This equation has inherent difficulties coming from the time dependence and the logarithmic behavior near zero, but it is much more tractable because it admits a comparison principle.

The rest of the paper is organized as follows. In Section 2, we present the proofs of the upper bounds (1.10) and (1.14). Section 3 is where the proofs of the lower bounds (1.9), (1.11) and (1.13) are given. In order to complete the proof of the lower bounds, some estimates on linearized problems with moving Dirichlet boundary conditions are obtained in Section 4 and Section 5. In Section 6, we state and prove the result concerning the local KPP equation with logarithmic nonlinearity. The Harnack inequality is proved in Section 7.

The case r > 3 is very close to the local Fisher-KPP equation. The (3/2) log t delay is the best case scenario -in fact, the delay has to be at least that large for any r, so the bound is a quite straightforward application of bounds obtained in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF].

Proof of (1.10). Take t 0 > 0 to be determined later. Working in the moving frame with the logarithmic correction, the function

u mov (t, x) = u t, 2t - 3 2 log 1 + t t 0 + x , satisfies (u mov ) t ≤ 2 - 3 2 1 t + t 0 (u mov ) x + (u mov ) xx + u mov , for all t > 0, x ∈ R, u mov (0, x) = u 0 (x), for all x ∈ R.
We construct a super-solution u as in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]. Let v be the solution to the boundary value problem

v t = 2 - 3 2 1 t + t 0 v x + v xx + v, for all t > 0 and x > 0, v(t, 0) = 0, for all t > 0, v(0, x) = 1 (0,2) (x) for all x > 0.
Then [17, Lemma 2.1] implies that, provided that t 0 is sufficiently large, there exists A 0 ≥ 1 such that for all t ≥ 0, we have v(t, 1) ≥ A -1 0 . We also have the following uniform bound on the solutions to (1.2). Then there exists M > 0 such that, u(t, x) ≤ M for all t > 0 and x ∈ R.

Let us now define ū(t, x) as

u(t, x) = M 1 x≤x 0 + min 1, A 0 v(t, x -x 0 + 1) 1 x≥x 0 ,
where M is as in Lemma 2.1. By construction, u is a super-solution to u mov , and by our assumptions on u 0 (1.8), we also have u(0, x) ≥ u mov (0, x) for all x ∈ R. In addition, [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]Lemma 2.1] implies that there exists T 0 such that, for all z and all t ≥ T 0 ,

v(t, z) ≤ A 0 ze -z .
(2.1)

We are now in a position to conclude the proof. Indeed, as u ≤ u, the upper bound in (2.1) implies lim sup

L→∞ lim sup t→∞ sup x≥L u t, 2t - 3 2 log t + x = lim sup L→∞ lim sup t→∞ sup x≥L u mov (t, x) ≤ lim sup L→∞ lim sup t→∞ sup x≥L u(t, x) ≤ lim L→∞ M A 0 Le -L = 0, (2.2)
which concludes the proof.

The upper bound when r = 3

In this section, we show how to derive the upper bound on the location of the front assuming the lower bound on the location of the front. In other words, we prove (1.12) assuming (1.11), which we prove in the next section.

Proof of (1.12) assuming (1.11). Our proof proceeds similarly as in the previous subsection. Set sφ < S φ to be determined. Using (1.11) and (1.7), we find L > 0 such that, for all x ≥ 0 and t ≥ L,

φ ⋆ u(t, x + 2t -sφ log(t)) ≥ 1 LA φ (x + (S φ -sφ ) log(t) + L) -2 . (2.3)
Next, we use the following result that is proved in Appendix A.

Lemma 2.2. There exists v, sφ > 3/2, and L such that

v t ≥ v xx + v(1 -ν(t, x -(2t -sφ log(t + t 0 ))), t > L, x > 2t -s φ log(t + t 0 ) + L, v(L, x) ≥ u(L, x), x > 2L -sφ log(L + t 0 ) + L, (2.4)
v(t, L + 2t -sφ log(t)) ≥ M + 1 for all t ≥ L, and v(t, x + 2t -sφ log(t)) → 0 as x → ∞ uniformly in t ≥ L.
With Lemma 2.2 in hand, we now conclude. Notice that, (

2.3) implies that v is a super-solution of u in {(t, x) ∈ [L, ∞) × R : x ≥ 2t -sφ log(t) + L}. Let u(t, x) = M + 1, if x ≤ 2t -s φ log(t) + L, min{M + 1, v(t, x)} if x ≥ 2t -s φ log(t) + L.
As in the previous case r > 3, the comparison principle implies that u ≥ u on [L, ∞) × R. The result then follows taking s φ ∈ (s φ , 3/2).

The upper bound when r ∈ (1, 3)

In this section, we show how to derive the upper bound on the location of the front from the lower bound on the location of the front. In other words, we prove (1.14) assuming (1.13), which we prove in the next section.

Proof of the upper bound (1.14) assuming the lower bound (1.13). Note that, by (2.2), we have

lim t→∞ sup x≥2t+t γ u(t, x) = 0.
As a consequence, taking into account (1.15), it suffices to show that

lim t→∞ sup x∈(2t-c φ t 2γ-1 ,2t+t γ ) u(t, x) = 0.
We do this by creating a relevant super-solution to u on the interval (2t-c φ t 2γ-1 , 2t+t γ ). Note that the constant c φ is still to be determined at this stage. Define, for any T > 0 and C φ as in (1.13), the space-time domain (recall that γ > 1/2 for 1 < r < 3):

P T := (t, x) : t ∈ (T, ∞), x ∈ (2t -C φ t 2γ-1 , 2t + t γ ) ,
and, for (t, x) ∈ P T , the function

v(t, x) := B exp -x -2t + 2c φ t 2γ-1 .
On P T , the function v satisfies

v t = v xx + v 1 -2c φ (2γ -1)t γ(1-r) . (2.5)
The rest of the proof is devoted to showing that u is, indeed, a subsolution to (2.5) when the various constants above are suitably chosen: specifically, we show that

u t -u xx -u(1 -2c φ (2γ -1)t γ(1-r) ) ≤ 0 in P T , (2.6)
and

u(t, x) ≤ v(t, x), on ∂P T . (2.7)
First, we show that (2.6) holds. It follows from (1.13) that there exist C φ and δ φ , depending only on φ, and T 0 such that, for all t ≥ T 0 , inf

x≤2t-C φ t 2γ-1 u(t, x) ≥ δ φ .
(2.8)

Using (2.8), we can estimate φ ⋆ u from below, for t ≥ T 0 and x > 2t -C φ t 2γ-1 :

φ ⋆ u(t, x) = ˆR φ(x -y)u(t, y) dy ≥ ˆ2t-C φ t 2γ-1 -∞ φ(x -y)u(t, y) dy ≥ δ φ ˆ2t-C φ t 2γ-1 -∞ φ(x -y) dy = δ φ ˆ+∞ x-2t+C φ t 2γ-1 φ(z) dz ≥ δ φ A -1 φ ˆ+∞ x-2t+C φ t 2γ-1 z -r dz = δ φ A φ (r -1)
x -2t + C φ t 2γ-1 1-r .

(2.9)

Note that, as r > 1, we have

2γ -1 = 3 -r 1 + r = γ + 1 -r 1 + r < γ.
Further increasing T , if necessary, the right-hand side in (2.9) can be estimated, for t ≥ T , as

δ φ A φ (r -1) x -2t + C φ t 2γ-1 1-r ≥ δ φ A φ (r -1) t γ + C φ t 2γ-1 1-r = δ φ A φ (r -1) 1 + C φ t 1-r 1+r 1-r t (1-r)γ ≥ δ φ A φ (r -1) 1 + C φ T 1-r 1+r 1-r t (1-r)γ ≥ 2 1-r δ φ A φ (r -1) t (1-r)γ ≥ 2c φ (2γ -1)t γ(1-r) , (2.10)
as long as c φ is sufficiently small. Now, (2.6) follows from (1.2), (2.9) and (2.10).

To show (2.7), first, we consider the right spatial boundary x = 2t + t γ , t ≥ T . As this point is at the far edge of the front, it is natural to use the linearized problem

u t = u xx + u, t > 0, x ∈ R, u(t = 0, •) = u 0 .
Then, with x 0 as in (1.8), we can write for t ≥ T :

u(t, 2t + t γ ) ≤ u(t, 2t + t γ ) = e t √ 4πt ˆR e -(2t+t γ -y) 2 4t u 0 (y)dy ≤ e t √ 4πt ˆx0 -∞ e -(2t+t γ -y) 2 4t dy = e t √ π ˆ+∞ 2t+t γ -x 0 2 √ t e -y 2 dy ≤ Ce t √ t 2t + t γ -x 0 e -(2t+t γ -x 0 ) 2 4t ≤ C 0 exp -t γ - 1 4 t 2γ-1 ≤ B exp -t γ -2c φ t 2γ-1 = v(t, 2t + t γ ), (2.11)
so long as B ≥ C 0 . Above, we have increased T and decreased c φ if necessary. The constant C 0 depends only on γ and x 0 . Thus, (2.7) holds at x = 2t + t γ for all t ≥ T as long as B ≥ C 0 . At the left boundary

x = 2t -C φ t 2γ-1 , we have v(t, 2t -C φ t 2γ-1 ) = B exp (C φ -2c φ )t 2γ-1 ≥ M ≥ u t, 2t -C φ t 2γ-1 , (2.12)
as long as 2c φ ≤ C φ and B ≥ M . Here, M is the upper bound in Lemma 2.1. Lastly, we check that (2.7) holds at t = T , for 2T -

C φ T 2γ-1 ≤ x ≤ 2T + T γ : v(T, x) = B exp -x -2T + 2c φ T 2γ-1 ≥ B exp -T γ -2c φ T 2γ-1 .
As long as

B ≥ M exp T γ + 2c φ T 2γ-1 , we have that, for all x ∈ [2T -C φ T 2γ-1 , 2T + T γ ] v(T, x) ≥ M ≥ u(T, x), (2.13)
and (2.7) holds on all of ∂P T . It follows from (2.6) and (2.7) that, with T and B sufficently large, and c φ sufficiently small, we have

lim t→∞ sup x≥2t-c φ t 2γ-1 u(t, x) ≤ lim t→∞ sup x≥2t-c φ t 2γ-1 v(t, x) ≤ lim t→∞ B exp -2c φ -c φ t 2γ-1 = 0,
which finishes the proof of the upper bound.

Lower bounds on the location of the front

The proofs of the lower bounds in Theorem 1.1 are much more involved. They hinge on estimating φ ⋆ u in terms of u in a local way, and then deriving precise heat kernel type estimates on the resulting local equation.

Estimating the non-local term by a local counterpart

To begin, we estimate the convolution term φ⋆u in terms of u under the assumptions of Theorem 1.1. The assumptions of these two theorems differ only in the range of r. In this section, we assume only that r > 1 so our computations apply to all cases. Lemma 3.1. There exists C conv > 0, depending only on φ, such that, for all t ≥ 1 and all x ∈ R,

φ ⋆ u(t, x) ≤ C conv max 1, 1 t log M u(t, x) r-1 2 log M u(t, x) 1-r . (3.1)
Proof. It is here that the local-in-time Harnack inequality is used crucially. Fix any time t ≥ 1 and x, y ∈ R. Proposition 1.2 with p = 2 implies that there exists α > 0 so that

u(t, x + y) ≤ C u(t, x) exp αt ′ + αy 2 t ′ , for all t ′ ∈ (0, t], . (3.2)
Above, we absorbed the uniform bound M of u ∞ given by Lemma 2.1 into the constant C. By increasing M if necessary, we may assume that M ≥ u ∞ + 1, which allows us to simplify notation in the sequel. Using (1.6) and (3.2), we obtain, for R > 0 and t ′ ∈ (0, t] to be determined,

φ ⋆ u(t, x) ≤ ˆR φ(y)u(t, x -y) dy ≤ C ˆBR φ(y) u(t, x)e αt ′ + αR 2 t ′ dy + M ˆBc R φ(y)dy ≤ C u(t, x)R exp αt ′ + αR 2 t ′ + CM R -r+1 . (3.3)
The constant C changes line-by-line for the remainder of the proof and depends only on φ and α.

We now optimize the right-hand side in (3.3) with respect to t ′ ∈ (0, t] and R > 0. If

t ′ = R, then φ ⋆ u(t, x) ≤ C u(t, x)Re 2αR + CM R -r+1 . (3.4)
To roughly balance the two terms in the right side of (3.4), we choose

R = 1 8α log M u(t, x) , (3.5)
the most important point being that R should be of order log u. As we have set t ′ = R in (3.4), and we need to have 0 ≤ t ′ ≤ t, the choice (3.5) is possible only if

t ≥ 1 8α log M u(t, x) . (3.6)
With this, we find, from (3.4):

φ ⋆ u(t, x) ≤ C u(t, x) log M u(t, x) exp - 1 4 log u(t, x) M + C log M u(t, x) 1-r ≤ C u(t, x) 1/4 log M u(t, x) r + 1 log M u(t, x) 1-r ≤ C log M u(t, x) 1-r . (3.7) When (3.6) does not hold, so that t ≤ 1 8α log M u(t, x) . (3.8)
we choose t ′ = t and set

R = t 8α log M u(t, x) 1/2 , in (3.3), leading to φ * u(t, x) ≤ C u(t, x) √ t log 1/2 M u(t, x) exp αt + 1 8 log M u(t, x) + C t log M u(t, x) - (r-1) 2 ≤ C u(t, x) log M u(t, x) exp 1 4 log M u(t, x) + C t log M u(t, x) -(r-1) 2 ≤ Cu(t, x) 1/4 log M u(t, x) + C t log M u(t, x) -(r-1) 2 ≤ C 1 + u(t, x) 1/4 log M u(t, x) t log M u(t, x) r-1 2 t log M u(t, x) -(r-1) 2 ≤ C u(t, x) 1/4 log M u(t, x) r + 1 t log M u(t, x) -(r-1) 2 ≤ C t log M u(t, x) 1-r 2 .
(3.9)

We used (3.8) several times above, as well as the upper bound u(t, x) ≤ M in the last inequality. The combination of (3.7) and (3.9) concludes the proof of the lemma.

A local equation and related bounds

In view of Lemma 3.1, it is natural to introduce the following nonlinearity. Fix r > 1, and for any positive constants θ g and A g , set Θ g := θ g exp{-A

1/(r-1) g } and define g ∈ C 0,1 on (0, Θ g ) as

g(t, u) := A g max 1, t + A 1 r-1 g -1 log θ g u r-1 2 log θ g u 1-r , if u ∈ 0, Θ g . (3.10)
Outside [0, Θ g ] we set g(t, u) = 0 for u < 0 and g(t, u) = 1 for u > Θ g . By construction, g(t, •) is continuous. The "A 1/(r-1) g

" term in the second part of the maximum in the definition of g does not affect the analysis in any way. In fact, any other choice of g that preserves the asymptotics as u and t tend to zero would have the desired properties that we prove in the sequel.

We will make use of the local equation with a moving boundary at the front edge:

w t = w xx + w(1 -g(t, w)), in P g,γ := (t, x) : t > 0, x > 2t + (t + t 0 ) γ -t γ 0 , w(t, 2t + (t + t 0 ) γ -t γ 0 ) = 0, for all t > 0,
w(0, x) = w 0 (x) for all x > 0.

(3.11)

The following proposition contains the crucial lower bounds for the solutions of (3.11) we will need.

Proposition 3.2. Assume that there exists δ w > 0 and x w ∈ R + such that the initial condition w 0 (x) for (3.11) satisfies w 0 (x) ≥ δ w 1 (0,xw) (x).

1. If r > 3, then there exists X w and T 0 such that if x w ≥ X w and t 0 ≥ T 0 then there exists a positive constant B 1 , depending only on x w , δ w , t 0 , γ, and g, such that, for all t sufficiently large, we have

w(t, 2t + t γ + √ t) ≥ B -1 1 t -1 e - √ t-t γ .
2. If r = 3, then set t 0 = 1. There exists N > 0 such that if x w ≥ 1 then there exists a positive constant B 2 , depending only on N , δ w , γ, and g, such that, for all t sufficiently large, we have

w(t, 2t + 2 √ t) ≥ B -1 2 t -1-N e - √ t .
3. If r ∈ (1, 3), then set t 0 = 1. There exists B 3 > 0, depending only on δ w and g, such that if x w ≥ 1 then, for all t ≥ 1, we have

w(t, 2t + t γ + √ t) ≥ B -1 3 e - √ t-t γ -B 3 t 2γ-1 .
We delay the proof of this proposition until Section 4 and now continue the proof of the lower bounds of Theorem 1.1. Having reduced the problem to estimating a delay for a local equation, we now transfer known bounds of Theorem 1.1 on w to bounds on u.

From a bound on w to a bound on u

Let us take θ g = M and A g = C conv in the definition (3.10) of g(t, u) and let the initial condition in (3.11) be w 0 (x) = e -M u 0 (x). A combination of Lemma 3.1 and Proposition 3.2 implies that u is a super-solution for w for t ≥ 1. Further, it follows from considerations as in [4, Section 3], that w(1, x) ≤ u(1, x) for all x ∈ R due to the e -M pre-factor in the definition of w 0 . The maximum principle then implies that w(t, x) ≤ u(t, x) for all t ≥ 1 and all x ∈ R.

Using the assumptions on the initial data (1.8), we can, up to translating u 0 , and thus w 0 as well, assume that w 0 satisfies the hypothesis x w = x 0 ≥ X w in Proposition 3.2. Translating further and using parabolic regularity we may remove the dependence on t 0 . As a direct consequence, we have established: Corollary 3.3. Suppose that u satisfies (1.2) and (1.8) with φ satisfying (1.4), (1.5), and (1.6). Then there exists S 0 , depending only on u 0 and φ, such that:

1. If r > 3, then there exists a positive constant B 1 , depending only on u 0 and φ such that, for all t sufficiently large, we have

u(t, 2t + t γ + √ t -S 0 ) ≥ B -1 1 t -1 e - √ t-t γ .
2. If r = 3, then there exist positive constants N , and B 2 such that, for all t sufficiently large, we have

u(t, 2t + 2 √ t -S 0 ) ≥ B -1 2 t -(1+ N ) e -2 √ t .
3. If r ∈ (1, 3), then there exists a positive constant B 3 , depending only on u 0 and φ, such that, for all t ≥ 1, we have

u(t, 2t + t γ + √ t -S 0 ) ≥ B -1 3 e -t γ -B 3 t 2γ-1 .

From a bound on u on the right to the location of the front

We are now in a position to obtain the lower bounds (1.9), (1.11), and (1.13). Thanks to Corollary 3.3, we fit a suitable translate of a traveling wave solution for (3.11) underneath u, for x ≤ 2t + t γ + √ t.

Proof. We prove the lemma for r > 3, so that γ < 1/2, the proof being the same in the other cases up to situational modifications. We use the parabolic maximum principle. First, we note that, up to increasing s 0 and A V , we may ensure that

v(T 1 , x) ≤ u(T 1 , x) for all x ≤ 2T 1 + √ T 1 + T γ 1 -S 0 .
Second, we claim that, up to increasing s 0 , we have

v(t, 2t + √ t + t γ -S 0 ) ≤ u(t, 2t + √ t + t γ -S 0 ) for all t ≥ T 1 .
Indeed, for t sufficiently large, (3.12) implies, as γ < 1/2:

v(t, 2t + √ t + t γ -S 0 ) = V √ t + t γ + 3 2 log t + s 0 -S 0 (3.14) ≤ 2κ √ t + t γ + 3 2 log t + s 0 -S 0 exp - √ t + t γ + 3 2 log t + s 0 -S 0 ≤ 4κ √ tt -3/2 exp - √ t -t γ -s 0 + S 0 ≤ 4κt -1 exp - √ t -t γ -s 0 + S 0 . It follows that v(t, 2t + √ t + t γ -S 0 ) ≤ 4κe S 0 -s 0 B 1 u(t, 2t + √ t + t γ -S 0 ) ≤ u(t, 2t + √ t + t γ -S 0 ),
for T 1 sufficiently large and all s 0 ≥ S 0 + log(4κB 1 ).

Third, up to increasing A V , the ordering holds true near -∞. Indeed, using Lemma 3.1 and the assumptions (1.8) on u 0 , it is easy to see that there exists δ > 0, depending only on u 0 and φ such that, for any x < 0 with |x| is sufficiently large, the function u(x) = δ cos((x -x)/100) is a sub-solution for u for all t ≥ 1, so that δ = u(x) ≤ u(t, x) for all t ≥ 1. Thus, increasing A V , if necessary, we have that, for all t > 0,

lim x→-∞ v(t, x) < M e -A 1/(r-1) V < δ ≤ inf t≥1 lim inf x→-∞ u(t,
v(t, 2t + √ t + t γ -S 0 ) = V √ t + t γ + 3 2 log(t) + s 0 -S 0 ≥ κ 2 √ t + t γ + 3 2 log t + s 0 -S 0 exp - √ t + t γ + 3 2 log t + s 0 -S 0 ≥ κ 2t 3/2 √ t + t γ + 3 2 log t + s 0 -S 0 exp - √ t -t γ -s 0 + S 0 . (3.15)
Since V is monotonic, γ < 1, and

x ft ≤ 2t ft + t 1 2
ft + t γ ft -S 0 , it follows that up to increasing T 1 , we have that u(t ft , x ft ) = v(t ft , x ft ) ≥ M e -t ft , which, in turn, implies that log M u(t ft , x ft )

-(r-1) ≥ t -r-1 2 ft log M u(t ft , x ft ) -r-1 2 .
In view of the bound on φ ⋆ u obtained in Lemma 3.1, we have that, at (t ft , x ft ),

u t -u xx -u 1 -A V log M u -(r-1) ≥ A V -C conv u log M u 1-r > 0, (3.16)
where we used the fact that A V > C conv in the last inequality. In addition, we note that

v t -v xx -v 1 -A V log M v -(r-1) = 3 2(t + 1) -2 V ′ -V ′′ -V 1 -A V log M V -(r-1) = 3 2(t + 1)
V ′ ≤ 0.

(3.17)

Hence, setting ψ = u -v, (3.16) and (3.17), imply that

ψ t -ψ xx -cψ > 0,
where we define

c := u 1 -A V log M u -(r-1) -v 1 -A V log M v -(r-1) u -v .
Notice that, due to the Lipschitz continuity of w → w 1 -

A V log M w -(r-1)
on compact subsets of [0, M ), c ∈ L ∞ . On the other hand, using that t ft is the first time that ψ touches zero and x ft is the location of a minimum of ψ, we have that

ψ t -ψ xx -cψ ≤ 0.
This yields a contradiction, finishing the proof.

The lower bounds now follow easily.

Proof of (1.9), (1.11), and (1.13). We conclude the proof by noticing that, for all t ≥ T 1 ,

inf x≤2t-(3/2) log t u(t, x) = inf x≤0 u t, x + 2t - 3 2 log t ≥ inf x≤0 v t, x + 2t - 3 2 log t = inf x≤0 V x + s 0 = V (s 0 ), (3.18)
which means that (1.9) holds. The proofs of (1.13) and (1.11) are similar and, thus, omitted.

Proof of Proposition 3.2

To obtain estimates on the solution of (3.11), we consider the corresponding linearized problem with the Dirichlet boundary condition: ṽt = ṽxx + ṽ, on {(t, x) : t > 0, x > 2t + (t + t 0 ) γ -t γ 0 }, ṽ(t, 2t + (t + t 0 ) γ -t γ 0 ) = 0, for all t > 0, ṽ(0, x) = w 0 (x), for all x > 0, (

where w 0 is as in Proposition 3.2.

The case r > 3

The following key lemma about solutions to (4.1) allows us to prove Proposition 3.2 when r > 3.

We prove this lemma in Section 5.1.

Lemma 4.1. Assume r > 3. If t 0 and x w are sufficiently large, depending only on γ, there exist positive constants T and B, depending only on w 0 and t 0 , such that, for all t ≥ T , we have ṽ(t, •) ∞ ≤ Be -t γ and

ṽ(t, 2t + t γ + √ t) ≥ B -1 t -1 exp - √ t -t γ .
We now finish the proof of Proposition 3.2. Let ṽ be as in Lemma 4.1. We may assume, without loss of generality, that T ≥ 1, and set

δ = min B -1 , B -1 θ g e -A 1/(r-1) g , e -T .
We also take a continuous function a(t) ≤ 1 for all t ≥ 0, to be determined, and set v(t, x) = δa(t)ṽ(t, x).

(4.2)

Using (4.1), we obtain

v t -v xx -v(1 -g(t, v
)) = δa ′ ṽ + δaṽ t -δaṽ xx -δaṽ + δaṽg(t, δaṽ) = δṽ a ′ + ag(t, δaṽ) . (4.3)

Thus, v is a sub-solution of w for t ≥ T as long as a ′ + ag(t, δaṽ) ≤ 0.

Using the upper bound on ṽ along with the definition of δ, we see that this inequality would hold if

a ′ + aA g max 1, t + A 1 r-1 g -1 log 1 ae -t γ r-1 2 log 1 ae -t γ 1-r ≤ 0. ( 4

.4)

A lengthy but straightforward computation using, in particular, that A g ≥ 1, shows that (4.4) is satisfied if we take

a(t) = exp β (t + 1) 2γ-1 -1 , with a suitable β > 0.
Hence v is a sub-solution of w. Further, arguing as in [4, Section 3] and using the choice of δ and a, we have that v(T, x) ≤ w(T, x) for all x ≥ 2T + (T + t 0 ) γ -t γ 0 . The maximum principle then implies that v(t, x) ≤ w(t, x) for all t > T and x > 2t + (t + t 0 ) γ -t γ 0 . The conclusion of the proposition follows immediately from Lemma 4.1 since 2t + t γ ≥ 2t + (t + t 0 ) γ -t γ 0 .

The case r = 3

We follow here the same strategy as for r > 3, but the estimates on ṽ are obtained differently.

Lemma 4.2. For r = 3 and t sufficiently large, there exist λ and B > 0 such that

ṽ(t, •) ∞ ≤ Be - √ t ,
and

ṽ(t, 2t + 2 √ t) ≥ B -1 t -1-λ exp -2 √ t .
With this lemma, proved in Section 5.2, one may repeat the argument for r > 3, building a sub-solution v(t, x) as in (4.2), with δ > 0 sufficiently small, and a(t) such that

a ′ + aA g max 1, t + √ A g -1 log e √ t Cδa log e √ t Cδa -2 ≤ 0.
The above inequality is satisfied with a(t) = (t + √ A g ) -N for all t ≥ 1 so long as δ is chosen small enough and N is chosen large enough, depending only on A g and C.

The estimate when r ∈ (1, 3)

Here we directly construct a sub-solution of w. We seek a sub-solution ṽ solving ṽt ≤ ṽxx + ṽ, for t > 0, x > 2t + (t + 1) γ -1, ṽ(t, 2t + (t + 1) γ -1) = 0, for t > 0. (

Recall that t 0 = 1 in parts 2 and 3 of Proposition 3.2. Given a > 0, set

v(t, x) = x (1 + t) 3 exp -x- γ 2 x 1+t γ-1 -1+t γ - γ 2 4(2γ -1) +a (1+t) 2γ-1 - x 2 2(1 + t) . (4.6)
Here, the key computation is the following:

Lemma 4.3. There exists a 0 > 0 such that if a ≥ a 0 then ṽ(t, x) = v(t, x -(2t + (t + 1) γ -1)) solves (4.5).
We delay the proof of Lemma 4.3 until Section 5.3 and proceed with the proof of Proposition 3.2.

A bound for small times. Unfortunately, v is not compactly supported at t = 0, so we need to "fit it under" w at a later time. To do this, we first obtain a preliminary lower bound on w at time 1 by using the infinite speed of propagation of the heat equation. Recall that w 0 ≥ δ w 1 (-∞,xw) and 1 -g(t, w) ≥ 0. Hence, we have w t -w xx ≥ 0, so that w is a super-solution to the heat equation with a Dirichlet boundary condition fixed at

x 0 := 2 • 2 + (2 + t 0 ) γ -t γ 0 = 3 γ + 3,
≥ δ w e -x 2 /8-(xw-x 0 ) 2 /8 √ 8π 2 x cosh x(x w -x 0 ) 4 -1 ≥ x C δ w e -x 2 /8 , (4.7)
for some C independent of all parameters, as long as x w ≥ x 0 + 1. We used here that cosh(x) -1 ≥ x 2 /C for some universal C > 0. On the other hand, from the explicit expression (4.6) for v, we get

v(2, x -x0 ) ≤ C(x -x 0 ) exp -x -x γ 2 3 γ-1 - x 2 6 + xx 0 3 .
Thus, there exists ǫ > 0 such that

ǫṽ(2, x) = ǫv(2, x -x0 ) ≤ w(2, x) for x ≥ x0 = 3 γ + 3.
The subsolution. We now follow the same strategy as before, constructing a sub-solution of the form v(t, x) = δa(t)ṽ(t, x) on

P := {(t, x) : t ≥ 1, x > 2t + (1 + t) γ -1}.
Another lengthy but straightforward computation shows that v(t, x) is a sub-solution for w on P if we choose a(t) = exp -βt 2γ-1 for a suitable β and δ sufficiently small. Note also that ṽ and w satisfy the same boundary conditions at x = 2t + (1 + t) γ -1. Finally, choosing δ ≤ ǫ and using the computation (4.7) and the discussion following it, we see that v(2, x) ≤ w(2, x) for all x > 3 + 3 γ .

The conclusion of the proposition when r ∈ (1, 3) follows by simply using the explicit form of v(t, x).

Estimates on the linearized KPP equation

In this section, we adopt the convention that any constant denoted C may chance line-by-line but depends only on φ and u 0 .

5.1 The case r > 3: the proof of Lemma 4.1

The key observation is that γ < 1/2 when r > 3. Thus, the t γ term is of a lower order than the diffusive scale √ t. This allows us to use the strategy in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF], obtaining energy estimates in self-similar variables. Since the present proof is similar to that in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF], we provide a rather brief treatment.

Proof of Lemma 4.1. We begin by removing an exponential factor from ṽ and changing to the moving frame: let z(t, x) := e x ṽ(t, 2t + (t + t 0 ) γ -t γ 0 + x), x > 0. This function satisfies

z t = z xx + γ(t + t 0 ) γ-1 z x -z ,
for t > 0, x > 0, z(t, 0) = 0, for t > 0, z(0, x) = e x w 0 (x), for x > 0.

(

We now turn to self-similar variables, which are natural for the diffusive process. Let

τ = log 1 + t t 0 , y = (t + t 0 ) -1/2 x,
and ζ(τ, y) = z t 0 (e τ -1), t 1/2 0 e τ /2 y . Then ζ satisfies the equation

ζ τ = ζ yy + y 2 ζ y + 1 + γ(t 0 e τ ) γ-1/2 ζ y -1 + γ(t 0 e τ ) γ ζ.
We remove the integrating factor above, setting

ζ(τ, y) = e -(τ +t γ 0 (e γτ -1)) ζ(τ, y), so that ζ satisfies ζτ = L ζ + γt γ-1/2 0 e (γ-1/2)τ ζy , (5.2)
with

L := ∂ 2 y + y 2 ∂ y + 1. (5.3)
It is now heuristically clear that the last term in (5.2) should be not important due to the e (γ-1/2)τ term and the fact that γ < 1/2. The following lemma is proved in Appendix A.

Lemma 5.1. Let ζ solve ζτ = L ζ + εe (γ-1/2)τ ζy ,
with initial data ζ(τ = 0, •) = ζ0 . There exists ε 0 > 0 such that for all compact subsets K ⊂ R + there exists C K > 0 such that for all ε < ε 0 ,

ζ(τ, y) = y e -y 2 /4 2 √ π ˆ∞ 0 ξ ζ0 (ξ)dξ + O(ε) + e (γ-1/2)τ h(τ, y) ,
for all y > 0, τ > 0, and such that | h(τ, y)| ≤ C K for all τ > 0 and y ∈ K.

Undoing the various changes of variable, we get

ṽ(t, 2t + (t + t 0 ) γ -t γ 0 + x) = e -x z(t, x) = e -x ζ log 1 + t t 0 , x (t + t 0 ) 1/2 (5.4) = xe -x t 0 e -((t+t 0 ) γ -t γ 0 ) (t + t 0 ) 3/2 e -x 2 4(t+t 0 ) 2 √ π ˆ∞ 0 ξe ξ w 0 ( √ t 0 ξ)dξ + O t γ-1 2 0 + 1 + t t 0 γ-1 2 h(t, x) ,
where h(t, x) = h log 1 + t t 0 , (t + t 0 ) -1 2 x . First, notice that the L ∞ bound on ṽ in Lemma 4.1 follows immediately from the expression above on sets of the form [2t + t γ , 2t + t γ + σ √ t]. To obtain bounds on sets of the form [2t + t γ + σ √ t, ∞), we simply use that e -t ṽ is a sub-solution to the heat equation on R (see, e.g., (1.20)). Hence, we obtain that, for x ≥ 0, ṽ(t, 2t + t γ + σ

√ t + x) ≤ Ce t √ t exp - (2t + t γ + σ √ t + x) 2 4t ≤ Ce - √ t-t γ , (5.5)
where C is some constant depending only on the initial data and γ. Second, we have

ˆ∞ 0 ξe ξ w 0 (t 1/2 0 ξ)dξ + O(t γ-1 2 0 ) ≥ δ w ˆxw/t 1/2 0 0 ξe ξ dξ + O(t γ-1 2 0
).

Choosing first x w ≥ √ t 0 and t 0 ≫ 1 so that the first two terms in the parentheses in (5.4) are positive and then choosing T 0 large depending on t 0 and α, we have that, for all 0 ≤ x ≤ √ t + t 0 and t ≥ T 0 , ṽ(t, 2t + (t

+ t 0 ) γ -t γ 0 + x) ≥ x C e -x-((t+t 0 ) γ -t γ 0 ) (t + t 0 ) 3/2 .
The lower bound on ṽ(t, 2t + t γ + √ t) is immediate after evaluating at x = t γ -(t + t 0 ) γ -t γ 0 + √ t. This concludes the proof.

5.2

The case r = 3: the proof of Lemma 4.2 Note that in this case γ = 1/2. As a consequence, the drift induced by the moving boundary has the same order as the diffusion. It is thus useful to modify the t γ term in the moving boundary by a small multiplicative factor.

Proof of Lemma 4.2. To begin, fix ǫ > 0. Work in the moving frame 2t + [(t + 1) 1/2 -1] and remove an exponential factor, as previously:

z(t, x) := e x ṽ(t, x + 2t + [(1 + t) 1/2 -1]).
Passing then to self-similar coordinates τ = log(t + 1) and y = (t + 1) -1/2 x, so that ζ(τ, y) := z e τ -1, e τ /2 y , we see that ζ satisfies

ζ τ = Lζ + 1 2 ζ y -1 + 1 2 e τ /
We finish using the following lemma, proved in Appendix A. This result falls outside of [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF] and Lemma 5.1 because the ζy term in (5.6) is no longer a remainder term. Here, µ > λ > 0, ψ 2 = 1, and ψ and h(τ, y) are bounded on all compact subsets of [0, ∞). Also, for every compact set K ⊂ [0, ∞), there exists C K > 0 such that C -1 K y ≤ ψ(y) ≤ C K y. for all y ∈ K.

We are now able to establish the upper bound on ṽ for all x ∈ [2t + ((1 + t) 1/2 -1), 2t + 2 √ t]. Indeed, returning to the original variables in (5.7), we find

|ṽ(t, x)| = e -(1+t) 1/2 y ζ(τ, y) = e -(1+t) 1/2 y-τ -(e τ /2 -1) ζ(τ, y) ≤ Ce -(1+t) 1/2 y-τ -(e τ /2 -1)-y 2 8 -y 4 ψ(y)e -λτ + h(τ, y)e -µτ ≤ Ce - √ t .
(5.8)

In fact, the estimate (5.8) holds for all x ≥ 2t + [(1 + t) 1/2 -1] since, as above, ṽ may be estimated for x ∈ [2t + 2 √ t, ∞) using the same approach as in (5.5) (see also (1.20)). Indeed,

ṽ(t, x) ≤ C √ t x e t-x 2 2t ≤ C √ t e t-(2t+2 √ t) 2 4t = C √ t e -2 √ t-1 ≤ Ce - √ t .
We now establish the lower bound from (5.7). Taking t sufficiently large and evaluating at x = 2t + 2 √ t, we see that ṽ

(t, 2t + 2 √ t) ≥ α t 1+λ e -2 √ t ,
for some α depending only on u 0 . This concludes the proof.

5.3

The case r ∈ (1, 3): the proof of Lemma 4.3 To motivate some of the steps in the following proof, we briefly discuss a heuristic. In the stationary frame, as we did in (1.20), we may always estimate ṽ above by ignoring the Dirichlet boundary condition and using the fact that e -t ṽ solves the heat equation. Thus, ṽ(t, x+2t+t γ ) t -1/2 exp t-

(x + 2t + t γ ) 2 4t = t -1/2 exp -x- x 2 4t - x √ t t γ-1/2 2 -t γ - t 2(γ-1/2) 4 .
Recalling that γ > 1/2, we see that on the diffusive scale x ∼ √ t, the Gaussian term x 2 /4t and the t -1/2 in front are (much) lower order and, thus, negligible, but all other terms are large. Hence, our sub-solution should contain all such terms to be reasonably sharp. In particular, while the xt γ-1 term appears small at first glance since γ < 1, it is not negligible in the diffusive scale x ∼ √ t. While the terms depending only on t show up as obvious integrating factors, this term will not. Hence, the key to the proof below is in carefully taking account of this term. Note that here we see the effect of γ > 1/2.

Proof of Lemma 4.3. We show how to "guess" the form of the sub-solution v. We begin by removing an exponential from ṽ and changing to the moving frame. Define, for x ∈ R + , z(t, x) := e x ṽ(t, 2t + (t + 1) γ -1 + x), so that (4.5) becomes z t ≤ z xx + γ(t + 1) γ-1 (z x -z), t > 0, x > 0, z(t, 0) = 0, z(0, x) = e x w 0 (x).

(5.9)

Turning to self-similar variables, τ = log(1 + t), y = (t + 1) -1/2 x, and ζ(τ, y) = z(e τ -1, e τ /2 y),

we wish to construct ζ that satisfies the inequality

ζ τ ≤ ζ yy + y 2 ζ y + γe (γ-1/2)τ ζ y -γe γτ ζ. (5.10)
As γ > 1/2, the drift in (5.10) is not a perturbation anymore. The heuristic discussion preceding this proof indicates that we should consider ζ(τ, y) = e -αye (γ-1/2)τ ψ(τ, y), with α ∈ R to be determined. Then we require .11) with L as in (5.3). To remove the drift term, we set α = γ/2. Then (5.11) becomes

ψ τ ≤ Lψ + (γ -2α)e (γ-1/2)τ ψ y -1 + α(γ -α)e (2γ-1)τ + γe τ γ ψ -α(1 -γ)ye (γ-1/2)τ ψ. ( 5
ψ τ -Lψ + 1 + γ 2 4 e (2γ-1)τ + γe τ γ ψ + γ 2 (1 -γ)ye (γ-1/2)τ ψ ≤ 0.
Further, writing ψ(τ, y) = exp -τ -e γτγ 2 4(2γ -1) e (2γ-1)τ Ψ(τ, y),

we arrive at Ψ τ -LΨ + 1 2 γ(1 -γ)ye (γ-1/2)τ Ψ ≤ 0.
(5.12)

To deal with the last term in (5.12), let a, a ′ , b > 0 be constants to be determined and define

Ψ(τ, y) = y exp -ae τ (2γ-1) -a ′ τ - y 2 b .
By a direct computation, we see that

Ψ τ -LΨ + γ 2 (1 -γ)ye (γ-1/2)τ Ψ = -a ′ -a(2γ -1)e τ (2γ-1) - y 2 b 4 b -1 + 6 b - 3 2 + γ(1 -γ) 2 ye (γ-1/2)τ Ψ.
(5.13)

It is clear that to have (5.12), we must choose b < 4. For simplicity, we take b = 2, and

a ′ = 6 b - 3 2 = 3 2 , so that (5.13) becomes Ψ τ -LΨ + γ 2 (1 -γ)ye (γ-1/2)τ Ψ = -a(2γ -1)e τ (2γ-1) + γ(1 -γ) 2 ye (γ-1/2)τ - y 2 2 Ψ.
The choice

a ≥ γ 2 (1 -γ) 2 8(2γ -1)
ensures that (5.12) holds. Returning to our original variables, we see that

v(t, x) = ṽ(t, 2t + (t + 1) γ -1 + x) = e -x ζ(log(1 + t), (t + 1) -1/2 x) = e -x e -α(t+1) -1/2 x(1+t) γ-1/2 ψ(log(1 + t), (t + 1) -1/2 x) = 1 1 + t exp -x - γ 2 x 1 + t γ-1 -(1 + t) γ - γ 2 (1 + t) 2γ-1 4(2γ -1) Ψ(log(1 + t), (t + 1) -1/2 x) = x (1 + t) 3 exp -x - γ 2 x(1 + t) γ-1 -(1 + t) γ - γ 2 4(2γ -1) + a (1 + t) 2γ-1 - x 2 2(1 + t) .
This concludes the proof.

The Fisher-KPP equation with a Gompertz non-linearity

A side effect of our analysis gives the asymptotics for a related local equation:

u t -∆u = f r (u). (6.1)
Here, we assume that f r ∈ C 1 , r ∈ (1, ∞), and there exist positive constants θ f , δ f , and A f such that

f r (0) = 0, f r (u) > 0 for all u ∈ (0, θ f ), f r (θ f ) = 0, f r (u) = 0 for all u ≥ θ f , (6.2)
and

u 1 -A f log 1 u 1-r ≤ f r (u) ≤ u 1 -A -1 f log 1 u 1-r , (6.3)
for u ∈ (0, δ f ).

Theorem 6.1. Suppose that the initial condition u 0 (x) for (6.

sup x≤0 u t, 2t -C f t 3-r 1+r + x -θ f = 0. ( 6.9)
The proof of (6.4) follows directly from Section 3. The proofs of (6.5), (6.6), and (6.9) follow from what was done in Section 4, combined with a standard argument saying that the convergence is necessarily to the steady state θ f (see, e.g., [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]). The bounds (6.7) and (6.8) need additional ingredients; the work is similar so we only detail the computations for (6.8). Indeed, since our non-linearity is local, we cannot "pull" information from the front as we did above when we used the value of u at the front to bound φ ⋆ u far ahead of the front. In order to get around this, we state a weak lower bound on u. Lemma 6.2. Let the hypotheses of Theorem 6.1 be satisfied. Then there exists δ f > 0, depending only on f , such that u(t, x) ≥ exp{-δ f t γ } for all t sufficiently large and all x ≤ 2t + t γ , where we again define γ = 2/(1 + r).

Such a bound follows from the analysis of the lower bound in part (3) of Proposition 3.2 and requires no new ideas. As such, we omit the proof.

The main point in the proof of Theorem 6.1 is to use the lower bound in Lemma 6.2 on u along with the form of the non-linearity to replace the estimate of φ ⋆ u that we used in the proof of the upper bound in Theorem 1.1 when r ∈ (1, 3).

Proof of (6.8) assuming Lemma 6.2. We use a super-solution v(t, x) := B exp -x -2t + 2c f t 2γ-1 , with c f > 0 to be determined. Then v satisfies

v t = v xx + v 1 -2c f (2γ -1)t 2γ-2 .
On the other hand, using the bound on f (6.3) along with Lemma 6.2, we have that, for all t sufficiently large and x ≤ 2t + t γ ,

u t -u xx = f r (u) ≤ u 1 -A f log 1 u(t, x) 1-r ≤ u 1 -A f δ 1-r f t γ(1-r) .
Recalling 2γ -2 = γ(1 -r), and choosing c f such that A f δ r-1 f ≥ 2c f (2γ -1), we see that v is a super-solution for u.

7 The local-in-time Harnack inequality: Proposition 1.2

Proof of Proposition 1.2. Up to a shift in time, we may assume that t = 0. We may also assume that c ≡ 0. Indeed, let u ± (t, x) = e ±t c L ∞ ([0,T ]×R) w(t, x),

where w solves the heat equation w t = w xx , with the initial condition w(t = 0, x) = u(t = 0, x). Then u + is a super-solution to u while u -is a sub-solution to u. Hence, we have In view of this inequality, it is enough to prove the claim for w, that is, solutions to the heat equation.

Let G be the one-dimensional heat kernel G(t, x) = (4πt) -1/2 e -x 2 /(4t) . Fix s = (p + 1)/2p, notice that s ∈ (0, 1) and sp > 1, and let q be the dual exponent of p. Then we have w(T, x + y) = ˆR w(0, z)G(T, x + y -z)dz ≤ w 1-1/p ∞ ˆR w(0, z) 1/p G(T, x + y -z) s G(T, x + y -z) 1-s dz ≤ w 1-1/p ∞ ˆR w(0, z)G(T, x + y -z) sp dz e -q(1-s) x 2 4T dx 1 q = (4πT ) -1 2 (1-s)

4T π q(1 -s)

1 2 1 q = C p T 1 2q -1 2 (1-s) = C p T s 2 -1 2p .
We now seek a bound on G(T, x+y -z) sp in terms of G(T, x-z). To this end, we recall that sp > 1, let x ′ = x -z and we compute G(T, x ′ + y) sp G(T, x ′ ) = (4πT ) In the second line we used the explicit choice of s to simplify the exponent of T . This concludes the proof.

(
A Proofs of Lemmas 2.2, 5.1 and 5.2.

Proof of Lemma 5.1. The proof of this lemma is similar to that of a corresponding estimate in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]. However, the proof there only deals with moving boundary conditions of the form 2t + r log(t). Hence, for completeness, we provide a streamlined proof. Recall that ζ solves ζτ = L ζ + εe (γ-1/2)τ ζy .

To rectify the fact that the operator L is not self-adjoint, we remove a Gaussian term. Let

Lemma 2 . 1 .

 21 [START_REF] Hamel | On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds[END_REF] Theorem 1.2] Suppose that u satisfies (1.2) with initial data u 0 satisfying (1.8).

  on the time interval [0, 2]. It follows that w(2, x + x 0 ) ≥ 1 √ 8π ˆ∞ 0 w 0 (y+x 0 ) e -|x-y| 2 /8 -e -|x+y| 2 /8 dy ≥ δ w e -x 2 /8 √ 8π ˆxw-x0 0 e -y 2 /8 e xy/4 -e -xy/4 dy

  2 ζ, with L as in (5.3). Finally, pulling out the zeroth order factor ζ(τ, y) = e -τ -(e τ /2 -1) ζ(τ, y), we see that ζ solves ζτ = Lζ + 1 2 ζ y .

Lemma 5 . 2 .

 52 Let ζ solve (5.6), then it can be represented as ζ(τ, y) = exp -y) dy ψ(y)e -λτ + y h(τ, y)e -µτ .(5.7)

L

  ∞ u(T, x) 1/p ≤ u + (T, x + y) u - 1-1/p L ∞ u -(T, x) 1/p ≤ e 2 c L ∞ T w(T, x + y) w 1-1/p L ∞ w(T, x) 1/p.

∞

  ˆR w(0, z)G(T, x + y -z) sp dz

  ˆRn

=

  Ce βy 2 /T w 1-1/p ∞ w(t, x) 1/p .

  ζ(τ, y) = e -y 2 /8 ζ * (τ, y), then ζ * satisfies ζ * τ + M ζ * = εe (γ-1/2)τ ζ * y -

  Theorem 1.1. Suppose that u satisfies (1.2) and (1.8) with φ satisfying (1.4), (1.5), and (1.6). If r > 3, then the solution u propagates with a logarithmic delay: If r = 3, then the solution u propagates with a larger logarithmic delay: there exists S φ > s φ > 3/2

		lim inf t→∞	inf x≤0	u t, 2t -	3 2	log t + x > 0,	(1.9)
	and	lim L→∞	lim sup t→∞	sup x≥L	u t, 2t -	3 2	log t + x = 0.	(1.10)
	such that
		lim inf t→∞	inf x≤0	u t, 2t -S φ log t + x > 0,	(1.11)
	and
		lim

t→∞ sup x≥0 u t, 2t -s φ log t + x = 0.

(1.12)

  We also need the exponential to be above u(t, x) at the front edge. To control u there, we use that, letting h = e -t u, h is a sub-solution to the heat equation. In other words,

		h t ≤ h xx
	and, hence, for all x ≫ 1, e -t u(t, x) = h(t, x) ≤ ˆe-|x-y| 2 4t	u 0 (y)dy	√ t x	e -x 2 4t .	(1.20)
	Thus, for u to sit below the exponential super-solution at x = 2t + e(t), we require
	exp t -	(2t + e(t)) 2 4t		exp{-(e(t) + d(t))},
	that is,
		e(t) 2 ≥ 4td(t).	(1.21)
	Since e(t) should be o(t), we get
		lim t→+∞	d(t) e(t)	= 0.	(1.22)
	Combining (1.19), (1.21) and (1.22) gives, for t large,
		to be a super-solution to (1.2) inside (2t-d(t), 2t+e(t)),
	we need
		(e(t) + d(t)) 1-r d ′ (t).	(1.19)

  x). Now, assume for the sake of a contradiction that there exists a first touching time (t ft , x ft ) such that t ft ≥ T 1 , x ft ≤ 2t ft + √ t ft + t γ ft -S 0 , and u(t ft , x

ft ) = v(t ft , x ft ), and u(t, x) > v(t, x) for all t ∈ [T 1 , t ft ) and x < 2t + √ t + t γ -S 0 . Our goal is to obtain a contradiction by estimating φ ⋆ u and looking at the equation satisfied by u -v.

First, we estimate φ ⋆ u(t ft , x ft ) using Lemma 3.1. By increasing s 0 if necessary, we obtain

  1) is as in(1.8). If r > 3, then the solution u(t, x) propagates with a logarithmic delay:

		lim L→∞	lim sup t→∞	sup x≥L	u t, 2t -	3 2	log t + x = 0,	(6.4)
	and	lim L→∞	lim sup t→∞	sup x≤-L	u t, 2t -	3 2	log t + x -θ f = 0.	(6.5)
			lim t→∞	sup x≥0	u t, 2t -c f t	3-r 1+r + x = 0,	(6.8)
	and
		lim t→∞

If r = 3, u(t, x) propagates with a larger logarithmic delay: there exists

S A > s A > 3/2 such that lim inf t→∞ sup x≤0 u t, 2t -S A log t + x -θ f = 0,

(6.6)

and lim t→∞ sup x≤0 u t, 2t -s A log t + x = 0, (

6.7)

If r ∈ (1, 3), then the delay is algebraic: there exist C f > c f > 0, depending only on f r , such that

Upper bounds on the location of the frontIn this section, we prove the upper bounds (1.10) and(1.14) in Theorem 1.1.

Acknowledgements

The authors thank Nicolas Champagnat for the reference [8]. EB was supported by "INRIA Programme Explorateur". LR was supported by NSF grants DMS-1311903 and DMS-1613603. Part of this work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). In addition, CH has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No 639638) and was partially supported by the NSF RTG grant DMS-1246999 and DMS-1907853.

Fix any A V > C conv and let V be a traveling wave solution of

and V (+∞) = 0.

The existence, uniqueness up to translation, and monotonicity of V is given by, for example, [START_REF] Berestycki | Travelling fronts in cylinders[END_REF]. We also recall (see [START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF]3]) the fact that there exists β r,Av > 0, depending only on r and A V , and κ > 0 such that, as ξ → ∞

where the shift s 0 is to be determined below and λ is as in Proposition 3.2 and Corollary 3.3.

Lemma 3.4. There exists T 1 > 0 and s 0 such that if s 0 ≥ s 0 , then v(t, x) ≤ u(t, x) if r = 3, t ≥ T 1 , and x ≤ 2t + t γ + √ t -S 0 where S 0 is the shift given in Corollary 3.3.

The principle eigenvalue of M is associated to the eigenfunction

Define the non-negative quadratic form

Multiplying (A.1) by ζ * and integrating, we obtain

Integrating this inequality in τ and using the L 2 bound above, we obtain

We now show that the component of ζ * that is orthogonal to ψ decays in time. Let

Using (A.1), we obtain

from which we deduce that ζ⊥ (τ,

Gathering all estimates concludes the proof.

Proof of Lemma 5.2. Recall that ζ solves

To pass to a self-adjoint form, write

This operator is now self-adjoint with a compact resolvent. Let ψ and λ be the principal eigenfunction and eigenvalue of the operator above satisfying the boundary condition ψ(0) = 0 and the normalisation ψ

and thus λ > 0. Write

where µ is the second eigenvalue of M . After a time differentiation we have

and as a consequence of (A.4), that

Then, locally we have ζ * ⊥ ∞ (τ ) e -µτ by parabolic regularity. This yields

where h is bounded in τ , locally in y. To finish, we simply note that, by elliptic regularity theory, for any compact set K ⊂ [0, ∞), there exists

since ψ(0) = 0. This concludes the proof.

Proof of Lemma 2.2. As before, we pass to the moving frame and remove an exponential: let

Then, we want

Changing now to self-similar variables and recalling the definition

and N (τ, y) := t 0 e τ ν t 0 (e τ -1), t

The important point here is that N (∞, •) is neither infinity nor zero as it would be for another choice of r. This is what induces the larger delay.

Then we must find ζ that satisfies:

We have changed the order to emphasize that the right hand side is a small error. We now define ζ. As above M +N (0, •) is self-adjoint with a compact resolvent. Let ζ0 and λ 0 be its principle eigenelements. Using the Rayleigh quotient and testing with ye -y 2 /8 , we see immediately that λ 0 > 0. We need only verify that ζ satisfies (A.6); indeed, setting sφ = 3/2 + λ 0 /2, The first, third, and fourth terms are all positive. A simple maximum principle argument yields that -ζy is positive for all y ≥ y 0 for some y 0 > 0. The Hopf maximum principle implies (λ 0 /2) ζsφ εe -τ /2 ζy > 0 for all y > e -τ /2 /C for some C > 0. Thus, ζ satisfies (A.6) on [e -τ /2 /C, ∞), which, after translating back to physical variables, concludes the proof.


Carlos M Mateo

Juan Antonio Corrales

Youcef Mezouar

A Manipulation Control Strategy for Granular Materials based on a Gaussian Mixture Model

Keywords: robot control, image processing, manipulation, granular material, optimization, non-rigid registration

In the context of metal additive manufacturing, one of the most attractive tasks to be robotized is the cleaning process of metal powder after the printing operations. This task presents a challenging scenario for most of robot manipulation approaches in the literature. In this paper we present an approach, marker-less and real time affordable, which address the cleaning problem like a shape manipulation control problem. This control strategy is designed as an optimization problem. The error function is written as a lagrangian function using an objective function based on Gaussian Mixture Model (GMM). The local optimization is performed by a gradient descent and a global optimization process is used to avoid local minima.

Introduction

Robotize cleaning process in metal additive manufacturing nowadays is a hot topic because this process is still handmade, regardless, the manipulated material is an unhealthy substance for people. Metal powder can be easily inhaled and absorbed by cardiovascular system, producing serious health disorders. In addition, this material also presents explosive behavior in contact with oxygen, making the manipulation of this material unsafe [START_REF] Gibson | Additive Manufacturing Technologies[END_REF]. Moreover, during the printing process of pieces, it is wasted more than half of the source product since just a few parts of the powder material is melted. Therefore, we make safer and ecological additive industry by robotizing this task since we avoid involve human operators in risky situation and the rates of re-usability and recycling are increased.

We define the cleaning task as a close-loop form control strategy to drive the robot to the parts to be cleaned while avoiding obstacles. Specially, we consider the operation of removing powder as a soft manipulation task where the powder bed plus the printed piece plus the powder is a whole object (Fig. 1). So, here the goal is to transform the current object shape (cube of powder with the object inside) into a desired one (the object without any powder). During last years in the literature, we find a few approaches aimed to resolve concrete problems in object (granular material) manipulation tasks, such as [START_REF] Balatti | Towards Robot Interaction Autonomy : Explore , Identify , and Interact[END_REF]. Others like, the work [START_REF] Sanchez | Robotic manipulation and Sensing of Deformable Objects in Domestic and Industrial Applications: A Survey[END_REF] by Sanchez et al. makes a review of advances in robotic manipulation task of non-rigid objects. Works [START_REF] Mateo | Computation of curvature skeleton to measure deformations in surfaces[END_REF][START_REF] Gil | Visual/Tactile sensing to monitor grasps with robot-hand for planar elastic objects[END_REF] by Mateo et al. present a method for understanding how the surface changes during manipulation task, implementing a Dijkstra-based method to model the deformation of the object. The authors also present a method in [START_REF] Mateo | 3D visual data-driven spatiotemporal deformations for non-rigid object grasping using robot hands[END_REF] to predict when the object surface changes drastically during manipulation tasks to achieve dexterous manipulation and prevent damages in objects. The previous cited works are focused mainly in the perception of the objects, but works more related to the control of robots for the manipulation of elastic objects are [START_REF] Navarro-Alarcon | A Depth-Based Algorithm for Manipulating Deformable Objects Using Smooth Parametric Surfaces and Energy Minimisation[END_REF][START_REF] Navarro-Alarcon | Visual Shape Servoing of Deformable Objects : The Fundamentals[END_REF] by Navarro-Alarcon, in which an strategy to control the shape of objects with robots, using Fourier transform as features, is presented. In this line, works like [START_REF] Cherubini | Towards vision-based manipulation of plastic materials[END_REF][START_REF] Schenck | Learning Robotic Manipulation of Granular Media[END_REF] realize similar tasks, where Cherubini in [START_REF] Cherubini | Towards vision-based manipulation of plastic materials[END_REF] presents a study to understand how to address the dexterous manipulation of granular materials using robot hands without a previous planning. At the same time Schenck et al. in [START_REF] Schenck | Learning Robotic Manipulation of Granular Media[END_REF] present a deep learning strategy to teach robots to address the problem presented by Cherubini.

The contribution of this work lies in the proposal methodology (strategy) to resolve the problem of powder removal in metal additive manufacturing. Being the goal of this strategy the transformation of the current shape of the scenario to the desired one. The methodology combines visual perception techniques, optimization methods and robot control schemes. The robot should be moved to minimize the disparity between the current shape and the target shape. We decided to use a strategy to drive the robot to those surface points with maximum disparity and at the same time closest to the current robot position. Here, the disparity is computed using a registration point set method. In the literature there are works which address the problem of registering two different nonrigid surfaces: Coherent Point Drift (CPD) [START_REF] Myronenko | Non-rigid point set registration: Coherent Point Drift[END_REF]; Volume Deformation [START_REF] Innmann | Volumedeform: Real-time volumetric non-rigid reconstruction[END_REF]; or Killing Fusion [START_REF] Slavcheva | KillingFusion: Non-rigid 3D Reconstruction without Correspondences[END_REF]. All these works follow the same key concept, retrieve the transformation (deformation) which map one surface to another as rigidly as possible. The control of the robot is formalized using a gradient descent algorithm to optimize the robot velocity [START_REF] Corke | Robotics, vision and control: fundamental algorithms In MATLAB R[END_REF][START_REF] Lynch | Modern robotics : mechanics, planning, and control[END_REF], expressed in R 3 manifold.

Strategy Definition

This approach is devoted to defining the control policy for robotic manipulation tasks in cleaning operations. This method continuously controls a robot until the task is done, without stops for sensing neither re-planning. We consider finished the task when the current scenario shape matches with a reference (or target) scenario. The diagram of the method is pictured in Fig. 2. As is illustrated in the flowchart, the framework splits the approach into three different blocks: perception; and control; level planning. We use the visual perception part to generate the control references (attractors points) used in the robot control. These emerge from the disparity map W, computed in the block (e) in Fig. 2, by using the reference D * and current D depth images of the scenario.

We use a 3D model to define the target scenario, i.e. a mesh composed by the platform (powder bed) where the printed piece will be built and the piece itself (Fig. 1 (b) flat surface and cylinder). Using the 3D model, we create a virtual depth image D * . By a ray tracing technique using a virtual camera placed in the same pose than the real one, a top-view configuration. Where the camera z-axis (optical view axis) is aligned with the z-axis of the world reference frame (Fig. 1 (c)). We use as world frame the bottom-right corner of the scenario (Fig. 1 (b)).

Once we have the set of projected points x ∈ D * and y ∈ D in image space, we re-project this set into the euclidean space as the set of N points X ∈ R N ×3 and Y ∈ R N ×3 . This is done by the well-known re-projection equation X = P + x where the projection matrix P = K [I 3×3 | 0 3×1 ] is composed by the extended intrinsic camera parameters matrix K 4×3 , and a matrix 3 × 4, extrinsic camera parameters, where the first block is an identity matrix I 3×3 and the last column 0 3×1 (a zero column). The Moore-Penrose pseudo-inverse projection matrix is computed as P + = (P T P) -1 P T . Equally for the current projected points y ∈ D to obtain Y. To both point sets (point clouds) X and Y, we execute a passthrough filter f p : R N ×3 → R M ×3 , where M < N , (Fig. 2 (a) and (b)) to select those points which lie in the current depth layer l i (Fig. 1 (c)), according with, where

f p (X, z low , z up ) = {X | z X > z low ∧ z X < z up } , (1)
fp fc fc fp fm x X y Y X Y t ∂x ∂ J ẋ V IK + W e q - r e < ε3 ∆z yes (a) (c) (b) (d) (e) (f) (g) (h) (i) (j)
z X is the z component of any point in {X 1 , X 2 , • • • , X M } ∈ X.
We use the Canny edge filter f c [START_REF] Canny | A Computational Approach to Edge Detection[END_REF] to extract the contours of the silhouette, in the image projection of the survivor points X and Y in equation 1. Function f c (Fig. 2 (c) and (d)) works in image space, thus x and y are projected before to apply this filter. We use x = PX (and y = PY ) relation to project back the 3D points into image plane. Thus, final X and Y point sets are composed by the point re-projection of the sets f c (x ) and f c (y ). We compute the error between the two final point sets X and Y using Coherent Point Drift (CPD) operator [START_REF] Myronenko | Point set registration: Coherent point drifts[END_REF][START_REF] Myronenko | Non-rigid point set registration: Coherent Point Drift[END_REF] (Fig. 2). CPD is a 3D registration algorithm aimed to resolve the problem of 3D non-rigid alignment. This operation returns a set of transformations t ∈ R 3×N , that represent the translations between each pair of correlated points.

Map Function f m

Disparity field W is used in this work to balance the importance of each boundary point Y according to the robot pose r and the visual feedback t. So, we purpose the following mapping function f m : R N ×3 → R N subject to the robot pose r and t to get W (Fig. 2 block (e)). Thus, the fundamental idea of this function is to define a reachability rank of points. That is, weight with the maximum likelihood the point with better ratio between: distance to r; and maximum disparity between current Y ∈ R 3 and the desired X ∈ R 3 position. Note that here, we keep using the correlation space between X and Y obtained after applying CPD algorithm beside robot position r.

Mathematically, we can express the mapping function f m (Y | r, t) as a multiplication of two different weights terms W = W t W d . Where W d is the vector of weights according to the robot pose r (i.e. distance between point in the current surface and the robot) and W t with the visual feedback (i.e. distance between current and desired surfaces). Then, W t is formally expressed like the normalized vector according with infinity norm, i.e. the set of t is weighted with respect to the maximum t. This is because we emphasize those points with more transformation, formally this is as follows,

t = t 2 , W t = t t ∞ ( 2
)
where t is a vector in R N stacking all euclidean norm of all elements in t. On the other hand, W d is the normalized distance vector d = Y -r according with the negative infinity norm, i.e. the set d is balanced w.r.t. the minimum d. This is because we look for the closest point,

d = d 2 , W d = d d -∞ . ( 3
)
Values in Equation 2 and 3 are bounded in the range (0, 1], thus W is also bounded in the same range. Where the most relevant point Y i ∈ Y is valued with a 1.

Objective Function Definition

At this point all filter, registration and mapping functions used in the perception block are presented. This subsection is devoted to present the objective function used in the control law which is shown in Fig. 2.

Control here is formalized as an optimization problem where the parameters that minimize the objective function should be found. In our case, the parameter that we want to optimize is the robot velocity. The goal (or objective) is to move the robot r to the maximum-interesting points W, discussed above. Our objective function is constrained to the fact that the robot does not have to cross over the desired shape X. This constraint is imposed because X is the contour of the solid 3D printed object plus the machine support (powder bed Fig. 1), therefore crossing over this area means a collision between the robot and the scene.

To convert a constrained into an unconstrained problem, we use the Lagrange multipliers method [START_REF] Nocedal | Numerical Optimization[END_REF]. Using this function, we can find the local minimal of our objective function subject to our constraint,

L (r, w) = f (r) -wg (r | X) (4)
where, f (r) is the objective function, g (r | X) is the equality constraint and w (free parameter, in our experimentation w = 1 2 ) is a weight to balance the relevance of g. Note that in L is still differentiable.

Because the intrinsically noisy nature of image sensors, we define the objective function using a Mixture Model (MM) function. Concretely the MM used here is the Gaussian Mixture Model GMM,

f (r) = -log N i α i N (r | Y i , σ), (5)
where

N (r | Y i , σ
) is a Gaussian distribution function centered in the point y i with a spread factor of σ and a peak value equal to α i . Note that there is a Gaussian for each point in the current contour. Gaussian's peak in GMM is treated as the prior-knowledge (in our case W) of the i-th distribution and is bounded between [0, 1]. In the other hand, constraint equation g is defined in terms of an exponential artificial field,

g (r | X) ≡ E (r | X) . (6)
Note that we do not use the Gaussian Mixture Model to represent the position of the robot but to relate the current robot pose with respect to the points in the current surface shape Y. Also, MM is used to deal with the intrinsic uncertainty of visual information.

Following section shows more detailed develop about objective function f , constraint g and lagrangian L.

Task-Space Robotic Control

The most straightforward approach to move a robot is to design a velocity controller which relates the error function variations during time ė with the robot velocity V = [υ, ω], Chaummette and Hutchinson in [START_REF] Chaumette | Visual servo control. I. Basic approaches[END_REF]. Formally, this is written as ė = ∇eV, where ∇e is the gradient of the error function e, υ ∈ R 3 is the linear velocity and ω ∈ R 3 the angular velocity, thus the velocity vector is in V ∈ R 6 .

In this work the error function is equivalent to the Lagrangian equation 4, e ≡ L (r, w). Considering that the hidden idea is to minimize the error ensuring that the decreasing follows a monotonic exponential function with respect to time ė = -λ e e, being the controller gain λ e an scalar positive constant, the obtained expression is as follow, V = -λ e ∇e + e.

(

) 7
where ∇e + is the Moore-Penrose pseudo-inverse gradient matrix. We extend the control law (equation 7) to optimize the spread σ of N while the robot velocity V. Therefore, our final velocity vector V ≡ [V, σ] ∈ R 7 is composed by V and the gaussian's spread time variation σ. Consequently, the gradient ∇e is differentiated also with respect to σ, not just w.r.t. the robot pose. This is because by controlling the spread of GMM variance, we are controlling the velocity of the robot. The idea is to situate the robot pose in the slop of the GMM where it is the maximum value variation of the function and therefore the gradient have the maximum value.

Error definition

As it has been introduced before, removing particle media from a scene can be considered as a surface shape control problem. Then, controlling the difference between desired X and current Y shapes can be expressed as the optimization of an error function e. Moreover, the error function e, as we discussed above, is equivalent to the lagrangian operator L, defined in equation 4.

Objective function f in L is defined using a GMM in equation 5, where the i-th gaussian distribution

N (r | Y i , σ) is defined as, N (r| Y i , σ) = exp - 1 2 r -Y i 2 σ 2 , ( 8
)
and the prior-knowledge (guassian's peak) of the i-th gaussian is equivalent to α i ≡ W i . The potential field term E is designed using a sigmoid function and is defined as follows,

E (r | X) = - 1 1 + exp (-r -X -∞ ) (9)
As the equation shows this potential barrier is based on the minimum distance between any point of the desired surface X and current robot pose of r. Therefore, the potential barrier is going to be governed just by the closest desired contour point X i to r. Then, by injecting equation 8 and 9 in 4 we obtain our error function as,

e = -log M i W i exp - 1 2 r -Y i 2 σ 2 + w 1 1 + exp (-r -X -∞ ) ( 10
)
Equation 10 represents the error function of our method. This formulation can be see has a problem of regularization where we have two different forces: attraction (first term); and repulsive (second term). As it is expressed in the first part of the equation, as smaller is the difference between robot pose with the points in current surface shape r -Y i 2 , smaller is the energy of the error function. In contrast the second term as smaller is the distance between the robot pose and the point in desired surface shape -r -X -∞ , bigger is the energy of the error function.

Global robot pose Optimization & Level planning

We use the Deterministic annealing algorithm [START_REF] Ueda | Deterministic annealing EM algorithm[END_REF] to govern the global behavior of our control scheme, while the control scheme is devoted to walk towards the most interesting points in the current surface according with W. The global optimization strategy prevents that the system falls in stationary state. These stationary states are reached when the robot position is too far from the influence of f or too close to the GMM's peak. This global optimization is as follows,

σ t+1 = λ σ σ t , e < ε 1 λ -1 σ σ t , e > ε 2 (11) (a) (b) (c)
Fig. 3: Experiments over 3 different scenarios. Green dot represents robot pose, path is represented in white color, current surface is the red surface and purple is the desired surface. Note that the purple area is a restricted area for the robot.

where λ σ is the global gain of the system and ε 1 and ε 2 are the thresholds used for activating the global optimization.

In the other hand, the global evolution of the cleaning task l 1 → l 2 → • • • → l i evolves by removing layers of powder (Fig. 1). The method flowchart as a level planning block (Fig. 2 block (i)) aimed to move the robot in z-direction in the cleaning task. Then, the depth range are z low = z low + ∆z and z up = z up + ∆z (this is a range area surrounding a level l i ) if the differences between current and target surfaces is lower than a desired value X -Y 2 < ε 3 .

Experiments

Here, a set of 3 different simulations are designed (Fig 3), assuming the robot should clean in z direction, i.e. layer by layer, where each test represents a depth map in a specific layer. This strategy is chosen because it reduces drastically the point set density (because only those points in the silhouette of the shape are considered) in comparison to take all the points in the 3D surface, hence the computational time is reduced proportionally. We first evaluate the characteristics of the proposed method using simulations. And secondly, we have implemented this strategy in a real robot to prove the viability. This approach has 2 different gains: one for the local optimization λ e ; and the other for the global optimization λ σ . Moreover, this scheme has three bound parameters ε 1 , Fig. 4: The disparity between shapes evolution over time. x-axis represents the iteration ε 2 and ε 3 , two first to keep e in a desired level and the last to know when a depth level is cleaned (or deformed). Due the fact that this strategy comprised a global optimization step to bound the error function inside an error levels, this method cannot be evaluated directly on the study of the error evolution. Therefore, the convergence is analyzed based on the shape similarities evolution between current and target shapes.

We found that the behavior of system is suitable for each test, doing just a single set up of these parameters. The gains for error minimization and variance optimization are λ e = 0.03 and λ σ = 200. The limits for error function in this experimentation are set in the range ε 1 = 0.1 and ε 2 = 0.15. Additionally, we have initialized the value of GMM spread to σ = 0.005. One of the advantages of this method is the smooth transformation that generates. This ability can be appreciated in Fig. 3(a) where the shape is transformed from a square to a circle. This kind of behavior is progressively doing the task while avoiding problems of local minimum. In this work, we consider that the robot is in a local minimum when this is stop and still remind powder to remove. Although it is not common, we still have problem with local minima in concave shapes. In other hand, the Fig. 3(b) presents a challenging situation where three unconnected powder areas should be removed. This test shows how the system deals with local tasks to get a global solution. The experiment 3(c)) was designed to demonstrate how the controller manages to convert a simple shape into a much more complex shape (concave shape).

It is demonstrated for all experiments that the system presents a local and global behavior. All experiments complete their task after approximately 2500 iterations. The time to process a single iteration depends on the amount of points in the silhouette. More concretely the relation between the number of points and the time to perform the task is linear. In these experiments, the Finally, we implement this approach over a real platform to demonstrate its applicability in real scenarios (Fig. 6). We use the collaborative robot Baxter equipped with a Kinect camera in the end effector of the right arm and an AR10 humanoid hand in the left arm. We decided to use a hard actuation (touching) over the material (kinetic sand) to avoid uncertainties produced by soft actuation (vacuum aspiration). Fig. 6 presents a series of key frames of the manipulation task. This experiment is following the described trajectory in the third simulated test (Fig. 3(c)).

Conclusions

This work presents a novel approach to manipulate granular materials using a robotic manipulator. Here, it is defined the cleaning tasks problems as a shape deformation problem. The goal is reached using our proposed control scheme based on a Gaussian Mixture Model. One of the major contributions of this work is the novel control scheme which combines local and global optimization by means of a gradient descent and a Deterministic Annealing method. Another important contribution is that here the task control is not defined in terms of error evolution. The error evolution here is just used to control the robot velocity. In contrast here, task evolution is defined in terms of surface shape disparity (Fig. 4). The experiments show that this method reach solutions in real time when the point density of silhouette contour is low. But this approach cannot guaranty an optimal solution when desired shape has complex topology (many concave shapes). In this work we initialize the gaussian distribution's spread although this can be discovered by the global optimization algorithm. We do this because as near these values are to the optimal ones, the convergence is faster. This fact opens the possibility to extend this method to predict optimal values of σ by prior clues using learning strategies. Similarly, the definitions of boundaries in the energy function can be improved by using another source of information which predicts best boundaries for each problem.

Fig. 1 :

 1 Fig. 1: Platform setup where (a) shows the real robot in front of a table with one object occluded inside the powder. Image (b) represents the virtual scenario with object, solid cylinder, which is covered by the powder, transparent cube. (c) Front view of the virtual scenario.

Fig. 2 :

 2 Fig. 2: Method flowchart. The perception task follows an open-loop scheme where is being compared current and target scenario. Control part is designed as a close-loop for minimizing the error between current and target robot position.

Fig. 5 :

 5 Fig. 5: Evolution over the time (x-axis) of: (a) GMM σ, y-axis; and (b) error e function, y-axis

Fig. 6 :

 6 Fig. 6: Real robot experiment. Hard actuation over kinetic sand.

Acknowledgements Research supported by project CoMManDIA (SOE2/ P1/ F0638) which is cofinanced by Interreg Sudoe Programme (European Regional Development Fun), and by the Chair Industrial of SIGMA Clermont Chaire SIG-AM "Fabrication Additive Multisectorielle".


Maé Guinet

Bernard Nicolardot

Anne-Sophie Voisin
email: anne-sophie.voisin@inra.fr

Multifunctional analysis of ecosystem services relative to the nitrogen fluxes provided by ten legume crops

Introduction

In the context of agroecological transition, ecosystem services should be maximized to ensure agriculture production [START_REF] Tibi | Evaluation des services écosystémiques rendus par les écosystèmes agricoles. une contribution au programme EFESE[END_REF]. The provision of ecosystem services relies in particular on the reintroduction of spatial and temporal biodiversity [START_REF] Isbell | High plant diversity is needed to maintain ecosystem services[END_REF]. Indeed, species diversity provides a variety of functions supporting ecosystem services. Legumes deliver unique and/or complementary functions to those of other groups of species. Specifically, they should play a key role in the provision of ecosystem services relative to nitrogen (N) fluxes when reintroduced in cropping systems mainly based on cereals. Indeed, legumes enable the production of protein-rich seeds. They also have the unique ability to establish a symbiosis with specific soil Rhizobium bacteria that leads to N accumulation during their growth through the symbiotic N2 fixation process [START_REF] Guinet | Comparative effect of inorganic N on plant growth and N 2 fixation of ten legume crops: towards a better understanding of the differential response among species[END_REF]. Legumes also supply N to the following crops through the mineralization of their N-rich residues [START_REF] Angus | Break crops and rotations for wheat[END_REF]. Nevertheless, some negative impacts can result from the introduction of legumes in cropping systems. Hence, their lower ability to retrieve soil inorganic N compared to cereals [START_REF] Hauggaard-Nielsen | Temporal and spatial distribution of roots and competition for nitrogen in peabarley intercrops-a field study employing 32P technique[END_REF] and the desynchronization between N supply by legume residues and the N demand of the following crop can lead to N losses by leaching during legumes growth cycle and/or after their harvest. As a large diversity of grain legumes exist, an important issue is to characterize and distinguish them based on their ability to deliver functions supporting ecosystem services relative to nitrogen in order to assist the choice of legume species according to the expected objectives. However, the choice of grain legumes according to the provision of those ecosystem services remains difficult due to a lack of references for a diversity of species. The objectives of our study was i) to quantify N fluxes induced by legumes and to identify several explanatory plant traits of these N fluxes in order to establish the functional profiles of ten grain legume crops and ii) to better understand the synergy and trade-off between the different N fluxes and hence the explanatory plant traits.

Materials and Methods

Two field experiments lasting two years were carried at the INRA experimental site of Bretenière (Dijon, France) in 2014[START_REF] Angus | Break crops and rotations for wheat[END_REF]and 2016[START_REF] Tibi | Evaluation des services écosystémiques rendus par les écosystèmes agricoles. une contribution au programme EFESE[END_REF]. The first year (2014and 2016)), nine legumes were cultivated in absence of N fertilization: chickpea, common bean, common vetch, faba bean, lentil, lupin, Narbonne vetch, pea and soybean. In 2016, fenugreek was also cultivated. Seeds of the ten legumes were inoculated at sowing with species-specific strains of N2-fixing bacteria to ensure symbiotic N2 fixation. Two cereals were also sown as pre-crops for comparison with legumes. Legume and cereal seeds were harvested and residues were chopped and incorporated into the soil. No seeds were harvested for Narbonne vetch in 2014 and 2016 and for chickpea in 2016, due to climatic conditions unsuitable for seed production for these two species. The second year, winter wheat was sown in October as a following crop for each of the legume and cereal pre-crops, and was not supplied with N fertilization. Wheat was then harvested in July 2015 and 2017, respectively. During the two year legumewheat succession, five N fluxes were measured or estimated using the STICS model: 1) amount of N in harvested legume seeds, 2) amount of N derived from the air in legume shoots through the symbiotic N2 fixation process, 3) amount of N derived from the soil in legume shoots through soil inorganic N uptake by the roots, 4) amount of N leached between legume harvest and wheat harvest, and 5) amount of N in the shoots of the following wheat. Legume plant traits considered as explanatory for these N fluxes were measured on legumes during the two field experiments or during a greenhouse experiment where the same ten legume species were cultivated in rhizotrons. For the ten legumes species, a redundancy analysis was performed using the five N fluxes as response variables and the plant traits as explanatory variables in order to study the link between the N fluxes and identify the most explanatory plant traits.

Results

Our results indicate that the amount of N in legume seeds and the amount of N derived from the air were positively correlated with four plant traits: legume shoot biomass, nitrogen harvest index (amount of seed N / total amount of shoot N), mean seed weight and seed N concentration. The two previous N flux were uncorrelated with the amount of N in the following wheat, which was negatively correlated with the legume residue C:N ratio. A clear antagonism between the amount of N leached after legume harvest and the amount of soil inorganic N retrieved by legumes was found. The latter N flux was positively associated with the root lateral expansion rate of legumes and the belowground nodule mass fraction (nodule biomass / belowground biomass). Those profiles enabled to distinguish legumes with high ability to retrieve soil inorganic N during their growth cycle and with low N losses by leaching after their harvest (chickpea, common bean, and soybean) against species with low ability to retrieve soil inorganic N and with higher risks of N leaching (common vetch, faba bean, fenugreek, lentil, pea and Narbonne vetch). Yet, the latter legume species tended to induce higher amounts of N in the shoots of the following wheat in comparison to chickpea, common bean, and soybean. Finally, faba bean in 2016 and soybean in 2014 had the highest amounts of N derived from the air as well as the highest amounts of N in seeds compared to the other species.

Discussion and Conclusions

Based on the simultaneous study of five N fluxes and several plant traits considered as determinant for the provision of the N fluxes, ten grain legume species were distinguished according to their functional profile. Hence, the characterization of grain legume species according to a combination of plant traits enable to evaluate their potential abilities to deliver N functions and the resultant ecosystem services. Yet, negative impacts such as N leaching could be compensated by adequate agricultural practices such as legume-cereal mixture or the establishment of cover-crop during the fallow period. Lastly, variation within each legume species could be characterised in the future by measuring plant traits that were identify in this study, to select the right genotype in order to improve N transfers.

Figure 1 .

 1 Figure 1. Redundancy analysis (RDA) biplot of N fluxes and legume pre-crop treatments for the 2014-2015 experiment () and the 2016-2017 experiment (). Ellipses show the 95% confidence interval associated with each legume pre-crop treatments in both experiments. LegG_N: amount of N in harvested legume seeds; Ndfa: amount of N derived from the air; Ndfsoil: amount of N derived from the soil; Nleached: amount of N leached between legume harvest and wheat harvest; Wheat_N: amount of N in the shoots of the following wheat.