Emeric Bouin
email: bouin@ceremade.dauphine.fr

Christopher Henderson
email: christopher.henderson@ens-lyon.fr

Lenya Ryzhik
email: ryzhik@math.stanford.edu

The Bramson logarithmic delay in the cane toads equations

.

Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the traveling waves grows as (3/(2λ * )) log t. This result relies on a present-time Harnack inequality which allows to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.

Introduction

The cane toads spreading Cane toads were introduced in Queensland, Australia in 1935, to control the native cane beetles in sugar-cane fields. Initially, about one hundred cane toads were released, and by now, their population is estimated to be about two hundred million, leading to disastrous ecological effects. Their invasion has interesting features different from the standard spreading observed in most other species [START_REF] Phillips | Invasion and the evolution of speed in toads[END_REF]. Rather than invade at a constant speed, the annual rate of progress of the toad invasion front has increased by a factor of about five since the toads were first introduced: the toads expanded their range by about 10 km a year during the 1940s to the 1960s, but were invading new areas at a rate of over 50 km a year by 2006. Toads with longer legs move faster and are the first to arrive to new areas, followed later by those with shorter legs. In addition, those at the front have longer legs than toads in the long-established populations -the typical leg length of the advancing population at the front grows in time. The leg length is greatest in the new arrivals and then declines over a sixty year period. The cane toads are just one example of a non-uniform space-trait distributionone other is the expansion of the bush crickets in Britain [START_REF] Thomas | Ecological and evolutionary processes at expanding range margins[END_REF]. There, the difference is between the long-winged and short-winged crickets, with similar conclusions. In all such phenomena, modelling of the spreading rates has to include the trait structure of the population.

The cane toads equation

We consider here a model of the cane toads invasion proposed in [START_REF] Bénichou | Front acceleration by dynamic selection in fisher population waves[END_REF], based on the classical Fisher-KPP equation [START_REF] Fisher | The wave of advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la chaleurde matière et son application à un problème biologique[END_REF]. The population density n(t, x, θ) is structured by a spatial variable x, and a motility variable θ. This population undergoes diffusion in the trait variable θ, with a constant diffusion coefficient, representing mutation, and in the spatial variable, with the diffusion coefficient θ, representing the effect of the trait on the spreading rates of the species. In addition, each toad competes locally in space with all other individuals for resources. If the competition is local in the trait variable, then the corresponding Fisher-KPP model is

u t = θu xx + u θθ + u(1 -u).
(1.1)

It is much more biologically relevant to consider a non-local in trait competition (but still local in space), which leads to

n t = θn xx + n θθ + rn(1 -ρ), (1.2)
where ρ(t, x) = ˆΘ n(t, x, θ)dθ (1.3) is the total population at the position x. Here, Θ is the set of all possible traits. It is either an infinite semi-interval: Θ = [θ, +∞), or an interval Θ = [θ, θ]. For simplicity, we consider the onedimensional case: x ∈ R. Both (1.1) and (1.2) are supplemented by Neumann boundary conditions at θ = θ and θ = θ (in the case when Θ is a finite interval):

n θ (t, x, θ) = n θ (t, x, θ) = 0, t > 0, x ∈ R. (1.4)
The cane toads equation is but one example among other non-local reaction models that have been extensively studied recently [START_REF] Alfaro | Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait[END_REF][START_REF] Berestycki | Propagation in a non local reaction diffusion equation with spatial and genetic trait structure[END_REF][START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF][START_REF] Faye | Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach[END_REF][START_REF] Hamel | On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds[END_REF][START_REF] Nadin | Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation[END_REF][START_REF] Nadin | Wave-like solutions for nonlocal reactiondiffusion equations: a toy model[END_REF]. Mathematically, non-local models are particularly interesting since their solutions do not obey the maximum principle and standard propagation results for the scalar local reaction-diffusion equations do not apply. Rather, on the qualitative level they behave as solutions of systems of reaction-diffusion equations, for which much fewer spreading results are available. The study of the spreading of solutions to the cane toads equations started with a Hamilton-Jacobi framework that was formally developed in [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF], and rigorously justified in [START_REF] Turanova | On a model of a population with variable motility[END_REF] when Θ is a finite interval. Existence of the travelling waves for (1.2) in that case has been proved in [START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF].

As far as unbounded traits are concerned, a formal argument in [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF] using a Hamilton-Jacobi framework predicted front acceleration, observed in the field, and the spreading rate of O(t 3/2 ). A rigorous proof of this spreading rate has been given in [START_REF] Berestycki | Existence of self-accelerating fronts for a non-local reaction-diffusion equations[END_REF][START_REF] Bouin | Super-linear spreading in local and non-local cane toads equations[END_REF].

The main results

In this paper, we consider the spreading rate of the solutions of the non-local cane toads equation (1.2)-(1.3), with x ∈ R and θ ∈ Θ = [θ, θ], and the Neumann boundary conditions (1.4). The initial condition n(0, x, θ) = n 0 (x, θ) ≡ 0 is non-negative and has localized support in a sense to be made precise later. The classical result of [START_REF] Fisher | The wave of advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la chaleurde matière et son application à un problème biologique[END_REF] says that solutions of the scalar KPP equation

v t = v xx + v(1 -v) (1.5)
with a non-negative compactly supported initial condition v 0 (x) = v(0, x) propagate with the speed c * = 2 in the sense that lim t→+∞ v(t, ct) = 0, (1.6) for all c > c * , and lim

t→+∞ v(t, ct) = 1, (1.7)
for all c ∈ [0, c * ]. The corresponding result for the solutions of (1.2) follows from the Hamilton-Jacobi limit in [START_REF] Turanova | On a model of a population with variable motility[END_REF]. The Fisher-KPP result for the solutions of (1.5) has been refined by Bramson in [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]. He has shown the following: for any m ∈ (0, 1), let X m (t) = sup{x : v(t, x) = m}, with s ∈ (0, 1). This level set has the asymptotics X m (t) = 2t -3 2 log t + x m + o(1), as t → +∞.

(1.8)

Here, x m is a constant that depends on m and the initial condition v 0 . Bramson's original proof was probabilistic. A shorter probabilistic proof can be found in a recent paper [START_REF] Roberts | A simple path to asymptotics for the frontier of a branching Brownian motion[END_REF], while the PDE proofs can be found in [START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov[END_REF][START_REF] Uchiyama | The behavior of solutions of some nonlinear diffusion equations for large time[END_REF] and, more recently, in [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]. Various extensions to equations with inhomogeneous coefficients have also been studied in [START_REF] Fang | Branching random walks in time inhomogeneous environments[END_REF][START_REF] Fang | Slowdown for time inhomogeneous branching Brownian motion[END_REF][START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF][START_REF] Maillard | Slowdown in branching Brownian motion with inhomogeneous variance[END_REF][START_REF] Nolen | Power-like delay in time inhomogeneous Fisher-KPP equations[END_REF]. In this paper, we establish a version of (1.8) -but with the weaker O(1) correction rather than o(1) as in (1.8) -for the solutions of the non-local cane toads equation (1.2). We will assume that the initial condition is compactly supported on the right: there exists x 0 such that n 0 (x) = 0 for all x ≥ x 0 . It has been shown in [START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF] that (1.2)-(1.4) admits a travelling wave solution of the form n(t, x, θ) = φ(x -c * t, θ).

It is expected that the function φ(ξ, θ) has the asymptotic decay

φ(ξ, θ) ∼ ξe -λ * ξ Q(θ), (1.9)
with a uniformly positive function Q(θ) > 0. While [START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF] does not show that travelling waves exist for all c > c * , this is expected. This would imply that c * is the minimal speed of propagation for the cane toads equation, in the same sense as c * = 2 is the minimal speed of propagation for the Fisher-KPP equation (see also [START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF]Remark 4]). A precise characterization of the minimal speed c * and the decay rate λ * from [START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF] is recalled in Section 4.1. Here is our main result.

Theorem 1.1. Let n(t, x, θ) satisfy the system (1.2)-(1.4), with the initial condition n 0 (x) ≥ 0 satisfying the assumptions above. There exists m 0 such that for all ε ∈ (0, m 0 ), there is a positive constant C ε such that

lim inf t→∞ inf x≤c * t-3 2λ * log(t)-Cε n(t, x) ≥ m 0 -ε, lim sup t→∞ sup x≥c * t-3 2λ * log(t)+Cε n(t, x) ≤ ε.
The main difficulty in the proof of Theorem 1.1 is the lack of the maximum principle. In order to circumvent this, we obtain a present-time Harnack inequality for n, described below, which is of an independent interest. Using this, we reduce the problem to showing the logarithmic delay for the local Fisher-KPP system (1.1), a much simpler problem, as it obeys the maximum principle. The analysis for the local equation follows the general strategy of [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF], with some nontrivial modifications.

A parabolic Harnack inequality

We will make use of the following version of the Harnack inequality, that is new, to the best of our knowledge. Consider an operator

Lu = ij a ij (x) ∂ 2 u ∂x i ∂x j . (1.10)
Here, A(x) := (a ij (x)) is a Hölder continuous, uniformly elliptic matrix: there exist λ > 0 and Λ > 0 such that ∀x ∈ R n , λI ≤ A(x) ≤ ΛI, in the sense of matrices.

Theorem 1.2. Suppose that u is a positive solution of

u t -Lu = 0, t > 0, x ∈ R n . (1.11)
For any t 0 > 0, R > 0, and p > 1, there exists a constant C such that if t ≥ t 0 and |x -y| ≤ R,

then u(t, x) ≤ C u 0 1-1/p ∞ u(t, y) 1/p . (1.12)
Moreover, C depends only on λ, Λ, n, t 0 , R, and p.

We point out that Theorem 1.2 does not hold with p = 1. Indeed, when n = 1 and (a ij ) = I, the solution u(t, x) = t -1/2 exp -x 2 /4t does not satisfy (1.12).

The paper is organized as follows. First, we prove Theorem 1.2 in Section 2. Then, in Section 3, we use the Harnack inequality to reduce the spreading rate question for the non-local cane toads equation to that for the local problem (1.1). Section 4 contains the proof of the corresponding result for the local equation, with its most technical part presented in Section 5.

Acknowledgement. EB was supported by "INRIA Programme Explorateur". LR was supported by NSF grant DMS-1311903. 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 dAvenir (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 programme (grant agreement No 639638).

A present-time parabolic Harnack inequality

In this section, we prove Theorem 1.2. It is a consequence of a small time heat kernel estimate due to Varadhan [START_REF] Varadhan | On the behavior of the fundamental solution of the heat equation with variable coefficients[END_REF]. Let G(t, x, y) be the fundamental solution to (1.11):

G t = L x G, t > 0, x, y ∈ R n , G(0, •, y) = δ(• -y), (2.1)
so that the solution of

u t -Lu = 0, t > 0, x ∈ R n , u(0, x) = u 0 (x),
can be written, for all t > 0 and x ∈ R n , as

u(t, x) = ˆRn G(t, x, y)u 0 (y)dy.
The notation L x in (2.1) means that the operator L acts on G in the x variable. There are wellknown Gaussian bounds for G (see e.g. [START_REF] Aronson | Non-negative solutions of linear parabolic equations[END_REF][START_REF] Fabes | A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash[END_REF]) of the type

c 1 t n/2 e -c 2 |x-y| 2 t ≤ G(t, x, y) ≤ C 1 t n/2 e -C 2 |x-y| 2 t , for (t, x, y) ∈ R + × R n × R n .
However, these are not precise enough in their dependence on x and y for our purposes, as they do not control the constants c 2 and C 2 very well.

To state Varadhan's estimate, we introduce some notation. Given a matrix A(x) = (a ij (x)), the associated Riemannian metric d A is

d A (x, y) = inf p∈C 1 ([0,1]), p(0)=x,p(1)=y ˆ1 0 ṗ(τ )A -1 (p(τ )) ṗ(τ )dτ.
The ellipticity condition on the matrix A implies that d A and | • | yield equivalent metrics.

Theorem 2.1 (Theorem 2.2 [START_REF] Varadhan | On the behavior of the fundamental solution of the heat equation with variable coefficients[END_REF]). The limit

lim t→0 (-4t log G(t, x, y)) = d A (x, y) 2
holds uniformly for all x and y such that |x -y| is bounded.

This agrees with the usual heat kernel when L = ∆ since then A = I and d A (x, y) = |x -y|. We may not use this result as stated as we will require a uniform estimate over all x and y, without a restriction to a compact set. However, it is easy to check that the proof in [START_REF] Varadhan | On the behavior of the fundamental solution of the heat equation with variable coefficients[END_REF], with a few straightforward modifications, implies the following. Theorem 2.2. Given any ε > 0, the following inequalities hold uniformly over all x, y ∈ R n :

lim inf t→0 -4t log G(t, x, y) ≥ (1 -ε)d A (x, y) 2 , (2.2)
lim sup t→0 -4t log G(t, x, y) ≤ (1 + ε)d A (x, y) 2 .
We can now proceed with the proof of Theorem 1.2.

Proof of Theorem 1.2. Without loss of generality we may assume that y = 0 and |x| ≤ R in (1.12). Let us take t 0 > 0 and write, for all t > t 0 and x ∈ R n : u(t, x) = ˆRn G(t 0 , x, y)u(t -t 0 , y)dy.

We have, using the maximum principle, with some s ∈ (0, 1), to be specified later:

u(t, x) = ˆRn G(t 0 , x, z)u(t -t 0 , z)dz ≤ u(t -t 0 , •) 1/q ∞ ˆRn (u(t -t 0 , z)G(t 0 , x, z) sp ) 1/p G(t 0 , x, z) (1-s)q 1/q dz ≤ u 0 1/q ∞ ˆRn u(t -t 0 , z)G(t 0 , x, z) sp dz 1/p ˆRn G(t 0 , x, z) (1-s)q dz 1/q ≤ C u 0 1/q ∞ ˆRn u(t -t 0 , z)G(t 0 , x, z) sp dz 1/p . (2.3)
Here, we have chosen q ∈ (1, ∞) satisfies

1 p + 1 q = 1,
and the constant C > 0 depends on t 0 (in particular, it blows up as t 0 ↓ 0). The last inequality in (2.3) is an application of the bounds in (2.2) since s < 1. Our next step is to show and use the following inequality: there exist a constant C > 0 and s > 1/p that both depend on t 0 , R, and p such that G(t 0 , x, y) sp ≤ CG(t 0 , 0, y), (2.4) for all y ∈ R n and |x| ≤ R. Before proving (2.4), we shall conclude the proof of Theorem 1.2. Using (2.4) in (2.3) gives

u(t, x) ≤ C u 0 1/q ∞ ˆRn u(t -t 0 , y)G(t 0 , 0, y)dy 1/p = C u 0 1/q ∞ u(t, 0) 1/p , (2.5)
which is (1.12) with y = 0. To establish (2.4), we choose s ∈ (0, 1), ε > 0 and θ ∈ (0, 1) such that

sp(1 -ε) > 1 + ε, (2.6)
and

1 -θ = (1 + ε) sp(1 -ε)
.

We may now use Theorem 2.2 to choose t 0 small enough so that

-4t 0 log G(t 0 , x, y) ≥ (1 -ε)d A (x, y) 2 -ε, -4t 0 log G(t 0 , x, y) ≤ (1 + ε)d A (x, y) 2 + ε, (2.7)
for all x, y ∈ R n . Using (2.7) and the triangle inequality

d A (x, y) ≥ |d A (x, 0) -d A (0, y)|, we get log[G(t 0 , x, y) sp ] - spε 4t 0 ≤ -sp(1 -ε) d A (x, y) 2 4t 0 ≤ -sp(1 -ε) d A (x, 0) 2 -2d A (x, 0)d A (y, 0) + d A (y, 0) 2 4t 0 . Young's inequality yields that log[G(t 0 , x, y) sp ] - spε 4t 0 ≤ 1 θ -1 sp(1 -ε)d A (x, 0) 2 4t 0 - sp(1 -ε)(1 -θ)d A (y, 0) 2 4t 0 .
Using the definition of θ and that the Euclidean metric and d A are equivalent, we deduce

log[G(t 0 , x, y) sp ] - spε 4t 0 ≤ CR 2 t 0 - (1 + ε)d A (y, 0) 2 4t 0 ,
with a constant C > 0 that depends on θ, p and ε. Applying the bounds in (2.7) again, we obtain

log[G(t 0 , x, y) sp ] - spε 4t 0 ≤ CR 2 t 0 + log G(t 0 , 0, y) + ε 4t 0 .
Exponentiating, we get (2.4), finishing the proof.

A reduction to the local cane toads problem

In this section, we show how to compare solutions of the non-local cane toads equation to the solutions of a local cane toads problem, of a more general form than (1.1). To do this, we use Theorem 1.2 to eliminate the non-local term in (1.2). This will allow us to find two local cane toads equations to which the solution of (1.2) is a sub-and super-solution, respectively. It has been shown in [START_REF] Turanova | On a model of a population with variable motility[END_REF], that solutions of (

-M |Θ|) ≤ n(1 -ρ) ≤ n,
the comparison principle implies that

e (1-M |Θ|)t h(t, x) ≤ n(t 1 -δ + t, x) ≤ e t h(t, x).
Hence, we may pull the Harnack inequality from h to n: for all (x, θ) ∈ R × Θ and (x , θ

) ∈ R × Θ such that |x -x | + |θ -θ | ≤ R we have n(t 1 , x, θ) ≤ e δ h(δ, x, θ) ≤ Ce δ h(δ, x , θ ) 1/p ≤ Ce δ e (M |Θ|-1)δ n(t 1 , x , θ ) 1/p .
This finishes the proof.

We now construct two local cane toads problems for which n is a sub-and super-solution. We fix p ∈ (1, 3/2) and find C > 0 so that we may apply Proposition 3.1 with t 0 = 1 and R = |Θ|, to obtain (after integration) n(t, x, θ) p C p ≤ ρ(t, x) ≤ Cn(t, x, θ) 1/p . for all t ≥ 1, x ∈ R and θ ∈ Θ. It follows that

n 1 -Cn 1/p ≤ n(1 -ρ) ≤ n 1 - n p C p . (3.3)
This implies that for t ≥ 1 the function n(t, x, θ) is a super-solution to the equation

u t -θu xx -u θθ = u(1 -Cu 1/p ), (3.4)
and a sub-solution to the equation

u t -θu xx -u θθ = u 1 -C -p u p . (3.5)
Here, u and u satisfy the same Neumann boundary conditions (1.4) as n.

We now choose the initial conditions at t = 0: u 0 (x, θ) = u(0, x, θ) and u 0 (x, θ) = u(0, x, θ), so that the ordering

u(t = 1, x, θ) ≤ n(t = 1, x, θ) ≤ u(t = 1, x, θ) (3.6)
holds for all x and θ. This will guarantee that

u(t, x, θ) ≤ n(t, x, θ) ≤ u(t, x, θ) (3.7)
for all t ≥ 1 and all x and θ, because of (3.3). We only describe how u 0 is chosen, but the process is similar for u 0 .

To this end, let h be a solution to the equation

h t -θh xx -h θθ = 0, (3.8)
with the initial condition h 0 (x, θ) = n 0 (x, θ). Define the function h = e (1-M |Θ|)t h, which satisfies

h t = θh xx + h θθ + (1 -M |Θ|) h,
where M is the upper bound for n from (3.1). Notice that n is a super-solution to h. Hence

n(t = 1, x, θ) ≥ h(t = 1, x, θ) = e (1-M |Θ|) h(t = 1, x, θ), (3.9)
for all x and θ. On the other hand, for any a > 0, the function

h = ae t h (3.10)
is a super-solution for the equation for u (3.4). Hence, if u is the solution of (3.4) with the initial condition u 0 = an 0 , then

u(t = 1, x, θ) ≤ aeh(t = 1, x, θ). (3.11)
Putting (3.9) and (3.11) together gives us

u(t = 1, x, θ) ≤ ae M |Θ| n(1, x, θ)
for all x and θ. Thus, if we choose a = exp(-M |Θ|) then the first inequality in (3.6) holds. Similarly, we may choose an initial condition ū0 so that the second inequality in (3.6) holds as well.

The logarithmic correction in the local cane toads fronts

We have shown that there exist functions u and u, satisfying the local cane toads equations (3.4) and (3.5), respectively, such that the solution n of (1.2)-(1.4) satisfies the lower and upper bounds in (3.7). Therefore, Theorem 1.1 is a consequence of the corresponding result for the Fisher-KPP equations. We present the local Fisher-KPP result in a slightly greater generality than what is needed for Theorem 1.1, as the extra generality introduces no extra complications in the proof.

Let D be a uniformly positive and bounded function on a smooth domain Θ ⊂ R d , and let A be a C 1 function on Θ. Let u be the solution to the Fisher-KPP equation

u t -Du xx -∆ θ u + Au x = f (u), (4.1)
with the Neumann boundary conditions:

∂u ∂ν θ (t, x, θ) = 0, (4.2)
and the initial condition u(0, •) = u 0 . Here, ν θ is the normal to ∂Θ. We assume that lim inf

x→-∞ u 0 (x, θ) > 0, (4.3)
uniformly in θ ∈ Θ, that u 0 ≥ 0, and that there is some x 0 such that u 0 (x, θ) = 0 for all x ≥ x 0 . The nonlinearity f is of the Fisher-KPP type: there exist u m > 0, M > 0 and δ > 2/3 such that

f (0) = f (u m ) = 0, f (u) > 0 for all u ∈ [0, u m ], (4.4)
and

u -M δ u 1+δ ≤ f (u) ≤ u, for all u ∈ [0, u m ]. (4.5)
A classical result of Berestycki and Nirenberg [START_REF] Berestycki | Travelling fronts in cylinders[END_REF] shows that (4.1) admits travelling wave solutions of the form u(t, x, θ) = Φ(x -ct, θ), with Φ(x, θ) such that

-cΦ x -DΦ xx -∆ θ Φ + AΦ x = f (Φ), (4.6)
and Φ(-∞, •) = u m , and Φ(+∞, •) = 0. In addition, Φ satisfies the Neumann boundary conditions (4.2), and 0 < Φ(x, θ) < u m for all x and θ. Such travelling waves exist for all c ≥ c * , with the same c * as in Theorem 1.1, and the travelling wave corresponding to the minimal speed has the asymptotics

Φ(ξ, θ) ∼ αξe -λ * ξ Q(θ), as x → +∞,
with the same exponential decay rate λ * and profile Q as in (1.9). Once again, a precise description of c * and λ * in terms of an eigenvalue problem will be given in Section 4.1. What is important for us is that, as far as the function f is concerned, both c * and λ * depend only on f (0) but not, say, on u m or δ.

By translating and scaling and by changing to a constant speed moving reference frame, if necessary, we may assume without loss of generality that u m = 1, f (0) = 1, that the drift A has mean-zero, and, finally, that the initial condition u 0 is not identically equal to zero on the half-cylinder {x > 0, θ ∈ Θ}. Theorem 4.1. Suppose that D and A are as above and f satisfies (4.4)-(4.5). There exist c * > 0 and λ * > 0 that, as far as f is concerned, depend only on f (0), with the following property. Let u satisfy (4.1)-(4.2), with the initial condition u 0 as above (4.3). Then, for any m ∈ (0, u m ), there exist x m > 0 and T m > 0, depending on m, such that if t ≥ T m we have

{x ∈ R : ∃θ ∈ Θ, u(t, x, θ) = m} ⊂ c * t - 3 2λ * log(t) -x m , c * t - 3 2λ * log(t) + x m . (4.7)
Theorem 1.1 follows from Theorem 4.1 and the bounds on n in (3.7), in terms of the solutions of the Fisher-KPP equations (3.4) and (3.5). The reason is that c * and λ * for the two non-linearities in (3.4) and (3.5) coincide, hence the level sets of the corresponding solutions u and u of these two equations stay within O(1) from each other, and (3.7) means that so do the level sets of the solution of (1.2).

The proof of Theorem 4.1 mostly follows the strategy of [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF] where a similar result has been proved in the one-dimensional periodic case. A general multi-dimensional form of the Bramson shift is a delicate problem [33]. However, the particular form of the present problem allows us to streamline many of the details and modifies some of the steps in the proof. Typically, the spreading speed c * of the solutions of the Fisher-KPP type equations can be inferred from the linearized problem, that in the present case takes the form

u t + Au x = Du xx + ∆ θ u + f (0)u. ( 4.8)
The main qualitative difference between the solutions of (4.8) and those of the nonlinear Fisher-KPP problem is that the former grow exponentially in time on any given compact set, while the latter remain bounded. A remedy for that discrepancy is to consider (4.8) in a domain with a moving boundary: x > X(t), with

X(t) = c * t -r(t), (4.9)
with the Dirichlet boundary condition u(t, X(t), θ) = 0. Then the shift r(t) is chosen so that the solutions of the moving boundary problem remain O(1) as t → +∞. It turns out that such "correct" shift is exactly

r(t) = 3 2λ * log t, (4.10)
as in (4.7). This allows to use them as sub-and super-solutions to the nonlinear Fisher-KPP equation, to prove that the front of the solutions to (4.1) is also located at a distance O(1) from X(t) given by (4.9)-(4.10), which is the claim of Theorem 4.1.

4.1

The eigenvalue problem defining c * and λ * .

Let us first recall from [START_REF] Berestycki | Travelling fronts in cylinders[END_REF] how c * and λ * are defined in Theorems 1.1 and 4.1. We look for exponential solutions of the linearized cane toads equation (4.8), with f (0) = 1, of the form

u(t, x, θ) = e -λ(x-ct) Q λ (θ). (4.11)
This leads to the following spectral problem on the cross-section Θ for the unique positive eigenfunction Q λ > 0:

     ∆ θ Q λ + (λ 2 D + λA -λc(λ) + 1)Q λ (θ) = 0, in Θ, ∂Q λ ∂ν θ = 0, on ∂Θ.
We will use the normalization

ˆΘ Q λ (θ)dθ = 1. (4.12)
In other words, given λ > 0, we solve the eigenvalue problem

     ∆ θ Q λ + (λ 2 D + λA)Q λ (θ) = µ(λ)Q λ , in Θ, ∂Q λ ∂ν θ = 0, on ∂Θ. (4.13)
It has a unique positive eigenfunction Q λ corresponding to its principal eigenvalue µ(λ) -this is a standard consequence of the Krein-Rutman theorem. The positivity of µ(λ) easily follows by dividing (4.13) by Q λ , integrating, and using the positivity of Q λ and the boundary conditions, along with the normalization ˆΘ A(θ)dθ = 0.

Then, the speed c(λ) is determined by

µ(λ) = λc(λ) -1, (4.14)
that is,

c(λ) = 1 + µ(λ) λ . (4.15)
We will use the notation, well-defined by the following proposition,

c * = min λ>0 c(λ), λ * = argmin λ>0 c(λ), (4.16)
and denote by Q * the corresponding eigenfunction.

c * ˆΘ (Q * ) 2 dθ = ˆΘ[2λ * D(θ) + A(θ)] (Q * ) 2 dθ. (4.17)
Further, we have c (λ * ) > 0.

Proof of Proposition 4.2. Since Q λ ∈ C 2 (Θ) and satisfies Neumann boundary conditions, there exists θ 0 such that ∆Q λ (θ 0 ) = 0. We deduce from (4.12):

c(λ) = 1 λ + A(θ 0 ) + λD(θ 0 ).
As the functions A(θ) and D(θ) are bounded, and

D(θ) is uniformly positive, c(λ) satisfies c(λ) ∼ λ→0 1 λ , λc(λ) = O λ→+∞ (λ 2 ).
The continuity of the function c(λ) implies the existence of a positive minimal speed c * and a smallest positive minimizer λ * . Differentiating (4.12) with respect to λ, we obtain

-λc (λ) -c(λ) + A + 2λD Q λ + λ 2 D + λA -λc(λ) + 1 ∂Q λ ∂λ + ∆ θ ∂Q λ ∂λ = 0.
Let us multiply by Q λ and integrate. We obtain, for all λ > 0,

ˆΘ -λc (λ) -c(λ) + A(θ) + 2λD(θ) Q 2 λ dθ = 0. (4.18)
In particular, for λ = λ * , we have c (λ * ) = 0, and (4.17) follows. Finally, for the last claim, it is easy to see by differentiating twice (4.14) and using c (λ * ) = 0 that

c (λ * ) = µ (λ * ) λ * .
In addition, the variational principle for the principal eigenvalue µ(λ) of (4.13) implies that µ(λ) is a convex function. A straightforward computation shows that actually µ (λ * ) > 0, thus c (λ * ) > 0.

A "heat equation" bound for the local cane toads equation

Motivated by the exponential solutions, we may decompose a general solution u(t, x, θ) of the linearized Fisher-KPP equation (4.8) as

u(t, x, θ) = e -λ * (x-c * t) Q * (θ)p(t, x, θ). (4.19)
The function p(t, x, θ) then satisfies

p t = Dp xx + ∆ θ p -(2λ * D + A) p x + 2 Q * ∇ θ Q * θ • ∇ θ p, (4.20
τ ω(τ ) → ω as τ → +∞, |ω (τ )τ 2 |, (τ + T ) ω(τ ) ≤ C (4.22)
and let p 0 be a non-zero, non-negative function such that that p 0 (x) = 0 for all x > x 0 and such that 1 [0,∞) p 0 is non-zero. Suppose that p satisfies and the initial condition p(0, •) = p 0 . There exists T 0 such that if T ≥ T 0 , then there exist σ > 0 and C > 0 that do not depend on p 0 , and τ 0 > 0 that may depend on p 0 such that

(1 -ω)p τ = Dp xx + ∆ θ p -(2λ * D + A) p x + 2 Q * ∇ θ Q * • ∇ θ p, (4.23
x -c * τ Cτ 3/2 ≤ p(τ, x, θ) ≤ C(x -c * τ ) τ 3/2 , (4.25) for all x ∈ [c * τ, c * τ + σ √ τ ], all θ ∈ Θ and all τ ≥ τ 0 .
As the proof is rather technical, we postpone it for the moment. Its proof is in Section 5.

The upper bound

We will now show how to deduce the statement of Theorem 4.1 from Proposition 4.3, starting with the upper bound. We will thus prove that the delay is at least 3 2λ * log(t) in the following sense:

max{x ∈ R : ∃θ ∈ Θ, u(t, x, θ) = m} ≤ c * t - 3 2λ * log(t) + x m ,
for some constant x m . The idea is to use the linearized problem with a moving Dirichlet boundary condition to create a suitable super-solution. Obviously, the Dirichlet boundary condition prevents the solution of this problem from being directly a super-solution. To overcome this, we show that the solution to the linearized equation is greater than 1 near the moving boundary. Hence, after a suitable cut-off, it will be a true super-solution.

To this end, we consider the solution to the linearized problem with the Dirichlet boundary condition at x = c * t -r log(1 + t/T ), with r and T to be determined:

               z t -Dz xx -∆ θ z + Az x = z, for x > c * t -r log(1 + t/T ), z(t, c * t -r log(1 + t/T ), •) = 0, ∂z ∂ν θ = 0, on ∂Θ, z(0, •) = u 0 . (4.26)
We make a time change

τ = t - r c * log 1 + t T . (4.27)
By fixing T large enough, depending only on r and c * , we may ensure that the function h(τ ) = t is one-to-one, and

1 h (τ ) = 1 - r c * (t + T ) = 1 - r c * (τ + T ) + r log(1 + t/T ) = 1 - r c * (τ + T ) + O(τ -3/2 ). (4.28)
To simplify the notation, we define

ω(τ ) = 1 - 1 h (τ ) , |ω (τ )| = O(τ -2 ). (4.29)
Notice that ω satisfies (4.22). The function z(τ,

•) = z(t, •) satisfies (1 -ω)z τ = Dz xx + ∆ θ z -Az x + z.
Let τ → α(τ ) be a function to be determined later, and decompose z as

z(τ, x, θ) = α(τ )e -λ * (x-c * τ ) Q * (θ)p(τ, x, θ).
The function p satisfies

(1 -ω)p τ = D pxx + ∆ θ p -(A + 2λD)p x + 2 Q * ∇ θ Q * • ∇ θ p + - α h α + rλ * t + T p, (4.30)
and p(τ, c * τ, •) = 0 for all τ . We choose α as the solution of

α α = rλ * t + T h = rλ * τ + T + O 1 (τ + T ) 2 , (4.31)
with the asymptotics:

α(τ ) = exp rλ * log(τ + T ) + O(τ -1 ) = (τ + T ) rλ * (1 + O(τ -1 )). (4.32)
In view of (4.29), we may apply Proposition 4.3 to the solutions of (4.30). This, along with (4.32), implies that if we choose

r = 3 2λ * , (4.33)
then there exist constants σ, C 1 and C 2 and a fixed time τ 0 such that we have

C 1 (x -c * τ )e -λ * (x-c * τ ) ≤ z(τ, x, θ) ≤ C 2 (x -c * τ )e -λ * (x-c * τ ) , (4.34)
for τ ≥ τ 0 and all

x ∈ [c * τ, c * τ + σ √ τ ]. Hence, we may choose M such that M z(τ, c * τ + 1, θ) ≥ 2,
for all τ ≥ τ 0 and θ ∈ Θ.

We may now define a super-solution for the nonlinear Fisher-KPP equation (4.1) as

u(t, x, θ) =    min(1, M z(t, x, θ)), for all x ≥ c * t -r log(1 + t/T ) + 1, 1, for all x ≤ c * t -r log(1 + t/T ) + 1.
Figure 1 depicts a sketch of the solution u of the nonlinear Fisher-KPP problem, and the supersolution u. We also have u(h(τ 0 ), •) ≥ u 0 for a sufficiently large M , since u 0 is compactly supported on the right. Hence, we have

u(t, •) ≤ u(h(τ 0 ) + t, •)
for all t ≥ t 0 .

To conclude, it follows from the form of our super-solution and (4.34) that, given any m ∈ (0, 1), we may choose x m ≥ 1 such that u(t, x, θ) < m for all t ≥ t 0 , all

x ≥ c * t - 3 2λ * log t + x m ,
and all θ ∈ Θ. Thus, for such x we have

u(t, x, θ) ≤ ū(t, x, θ) ≤ m,
for all t ≥ t 0 and θ ∈ Θ. This concludes the proof of the upper bound in Theorem 4.1.

c * t -3 2λ * ln(1 + t T ) u u M z c * t -3 2λ * ln(1 + t T ) + 1

The lower bound

We now prove that the delay is at most 3 2λ * log(t) in the following sense:

min{x ∈ R : ∃θ ∈ Θ, u(t, x, θ) = m} ≥ c * t - 3 2λ * log(t) + x m ,
for some constant C m . The proof of the lower bound requires the same estimates as the upper bound, but the approach is slightly different. Note that the solution to the linearized equation is not a sub-solution to the nonlinear Fisher-KPP equation since f (u) ≤ u. To get around this, we solve the linearized equation with a moving Dirichlet boundary condition at c * t, instead of c * t-(3/2λ * ) log(t), in order to make this solution small. Then, we modify the solution to the linearized equation by an order O(1) multiplicative factor in order to obtain a sub-solution.

The resulting sub-solution will decay in time. Hence, we may not directly conclude a lower bound on the location of the level sets. Instead, we show that this sub-solution is of the correct order e -σ √ t /t at the position c * t + σ √ t. This will allow us to fit a travelling wave underneath the solution u of the Fisher-KPP equation on the half-line x < c * t + σ √ t, and we use this travelling wave to obtain a lower bound on the location of the level sets of u. We will assume without loss of generality that := lim inf

x→-∞ inf θ∈Θ u 0 (x, θ) = 1. (4.35)
It is straightforward to modify the argument below to account for the case < 1. Note that > 0 by assumption (4.3). As a consequence of (4.44) we have that, for all t ≥ 0,

lim inf x→-∞ inf θ∈Θ u(t, x, θ) = 1. ( 4

.36)

A preliminary sub-solution using the linearized system

As outlined above, the first step is to obtain a sub-solution decaying in time. To this end, we look at the solution w to

               w t -Dw xx -∆ θ w + Aw x = w, for x > c * t, w(t, c * t, •) = 0, ∂w ∂ν θ = 0, on ∂Θ, w(0, •) = u 0 . (4.37)
As before, we factor out a decaying exponential, and the eigenfunction Q * :

w(t, x, θ) = e -λ * (x-c * t) Q * (θ)p(t, x, θ). (4.38)
The function p satisfies

p t = Dp xx + ∆ θ p -(2λ * D + A) p x + 2 Q * ∇ θ Q * • ∇ θ p θ , for x > c * t, (4.39)
with the corresponding boundary and initial conditions. Proposition 4.3 with ω = 0 gives an upper bound

|p(t, x + c * t)| ≤ Cx (t + 1) 3/2 ,
that, along with the decomposition (4.38) gives

w(t, •, •) ∞ ≤ C (1 + t) 3/2 (4.40)
This temporal decay allows us to devise a sub-solution of the Fisher-KPP problem, of the form w(t, x, θ) = a(t)w(t, x, θ).

To verify that w is a sub-solution, we note that

w t -Dw xx -∆ θ w + Aw x -f (w) ≤ ȧ(t) a(t) w + w -(w -M δ w 1+δ ),
with δ as in (4.5). Using (4.40), we get

w t -Dw xx -w θθ + Aw x -f (w) ≤ w ȧ(t) a + CM δ (t + 1) 3δ/2 .
We let a(t) be the solution of

- ȧ a = CM δ (t + 1) 3δ/2 . (4.41)
As δ > 2/3, there exists a 0 > 0 so that a(t) > a 0 for all t > 0. Taking a(0) ≤ 1 ensures that

w(0, •) ≤ u 0 (x, •), while (4.41) implies w t -Dw xx -w θθ + Aw x -f (w) ≤ 0.
As a result, the maximum principle implies that w(t, c * t + x, θ) ≤ u(t, c * t + x, θ), for all θ, all t and all x ≥ 0. In particular, the conclusion of Proposition 4.3 implies that there exists σ > 0 and T 0 such that if t ≥ T 0 then

Ca 0 e -σ √ t t ≤ a 0 w(t, c * t + σ √ t, θ) ≤ u(t, c * t + σ √ t, θ). ( 4

.42)

A travelling wave sub-solution

We now use the lower bound (4.42) to fit a travelling wave under u. The sub-solution we will construct is sketched in Figure 2. In order to avoid complications due to boundary conditions at -∞, we fix m to be any constant in (m, 1), and replace the non-linearity f (u) by f (u)(1 -u/m).

Let U be the travelling wave solution to the modified equation moving with speed c * :

-c * U x -DU xx -∆ θ U + AU x -f (U )(1 -U/m) = 0, (4.43)
with the Neumann boundary condition at ∂Θ, and

U (-∞, •) = m, U (+∞, •) = 0. (4.44)
This wave satisfies 0 < U < m, so it sits below u as x tends to -∞: see (4.36). However, it moves too quickly -it does not have the logarithmic delay in time. Instead, we define

U (t, x, θ) = U (x -c * t + s(t), θ). (4.45)
It is easy to check that if ṡ(t) ≥ 0, then U is a sub-solution to (4.43):

U t -DU xx -∆ θ U + AU x -f (U )(1 -U /m) = -(c * -ṡ(t))U x -DU xx -∆ θ U + AU x + f (U )(1 -U/m) = ṡ(t)U x ≤ 0, (4.46)
as U is decreasing in x [START_REF] Berestycki | Travelling fronts in cylinders[END_REF]. Hence, U is a sub-solution. We already know from (4.36) that U sits below u at x = -∞: U (t, x, θ) < u(t, x, θ), for all t > 0 and θ ∈ Θ for all x sufficiently negative. (4.47) Thus, we only need to arrange for U to sit below u at x = c * t + σ √ t, with σ is as in (4.42). The travelling wave has the asymptotics [START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF]

U (x, θ) ∼ xe -λ * x Q * (θ) (4.48)
for large x (uniformly in θ). By translation, we may ensure that

U (x, θ) ≤ εxe -λ * x ,
for all x ≥ 1, with ε > 0 small to be chosen. In view of the definition of U , for t sufficiently large, we have

U (t, c * t + σ √ t, •) ≤ ε(σ √ t + s(t))e -λ * (σ √ t+s(t)) .
Choosing

s(t) = 3 2λ * log(1 + t), (4.49)
using (4.42), and adjusting ε as necessary, we see that

U (t, c * t + σ √ t, •) ≤ Ce -σ √ t a 0 t ≤ u(t, c * t + σ √ t, •). (4.50)
for all t ≥ T 0 . In addition, because of (4.36), it is easy to see that translating U further to the left, we may ensure that U (T 0 , x, θ) ≤ u(T 0 , x, θ), (4.51) for all x ≤ c * T 0 + σ √ T 0 and all θ ∈ Θ. The combination of (4.46), (4.47), (4.50) and (4.51) the inequalities above, along with the maximum principle, implies that

U (t, x, θ) ≤ u(t, x, θ), (4.52)
for all t ≥ T 0 , all x ≤ c * t + σ √ t, and all θ ∈ Θ. To conclude, we need to understand where the level set of height m of U is. We see from (4.45) that there exists L m such that if x < -L m then

U (t, c * t + x -s(t), θ) > m.
Thus, (4.49) and (4.52) mean that

{x ∈ R : ∃θ ∈ Θ, u(t, x, θ) = m} ⊂ c * t - 3 2λ * log(1 + t) -L m , ∞ .
This finishes the proof of the lower bound in Theorem 4.1.

The proof of Proposition 4.3

In this section, we prove Proposition 4.3. The proof of the upper bound in (4.25) is easier than for the lower bound, and this is what we will do first. Essentially, the remainder of the paper will then be devoted to the proof of the lower bound in (4.25).

The self-adjoint form

Our first step is to re-write (4.23) in a self-adjoint form. Let us set

µ = a(Q * ) 2 , a = 1 |Θ| ˆΘ(Q * ) 2 dθ -1
.

(5.1)

Then we have an identity

Dp xx + ∆ θ p + 2 Q * ∇ θ Q * • ∇ θ p = 1 µ (Dµp x ) x + ∇ θ • (µ∇ θ p) . (5.2)
In order to re-write the spatial drift term in the right side of (4.23), we look for a corrector β that satisfies with some r ∈ R. The solvability condition for (5.3) is

∆ θ β = 2λ * D + A -r in Θ, (5.3) ∂β ∂ν θ = 0 on ∂Θ, c * t -3 2λ * ln(t) c * t c * t + σ √ t U ∼ O e -σ √
r = ˆΘ[2λ * D(θ) + A(θ)]µ(θ) dθ |Θ| = a ˆΘ[2λ * D(θ) + A(θ)](Q * (θ)) 2 dθ |Θ| = c * . (5.4)
We used (4.17 Note that the average of the advection term in x in (5.6) equals to c * . We now state a lemma regarding almost-linear solutions to (5.5) and its adjoint. The latter will be crucial in the proof of the upper bound for p. The former will be required later. We denote by L * the formal adjoint of the operator L with respect to the Lebesgue measure, and set

C τ = [c * τ, +∞) × Θ, Lemma 5.1. There exist functions ζ and f solving          µ∂ τ ζ = Lζ, on C τ , ∂ζ ∂ν θ = 0, on ∂Θ, ζ(τ, c * τ, •) = 0,
and

µ∂ τ f = -L * f, on C τ , ∂f ∂ν θ = 0 on ∂Θ, f (τ, c * τ, •) = 0, (5.7)
such that f τ , ζ τ ≤ 0. Moreover, there exists a constant C > 0 such that all x ≥ c * τ ,

C -1 (x -c * τ ) ≤ ζ(t, x, θ), f (t, x, θ) ≤ C (x -c * τ ) ,
and

|∂ τ f |, |∂ τ ζ| ≤ C.
We omit the proof as it is very close to [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF].

The proof of the upper bound

We now prove the upper bound in (4.25), namely, there exists a positive constant such that

p(τ, x, θ) ≤ C 0 (x -c * τ ) (τ + 1) 3/2 , (5.8)
for all τ > 0, x > c * t and θ ∈ Θ. We use a standard strategy: a Nash-type inequality is used to obtain the L 2 decay in terms of the L 1 norm, and then the uniform decay follows by a duality argument.

We first derive an L 1 -L 2 bound. Using (5.5)-(5.6), integrating by parts gives that for any τ > 0, we have

1 -ω 2 d dτ ˆCτ µ(θ)p(τ, x, θ) 2 dxdθ = - ˆCτ µ(θ) D(θ)p x (τ, x, θ) 2 + |∇ θ p(τ, x, θ)| 2 dxdθ.
(5.9)

The dissipation in the right side may be estimated using a Nash type inequality for half-cylinders of the form Ω = [0, ∞) × Θ, with Θ ⊂ R d , for functions such that φ(0, •) ≡ 0:

∇φ 2 2 ≥ C 1 + φ 2 xφ 1 10d 3(3+d) -1 φ 2 2 φ 2 xφ 1 4 3
.

(5.10)

The proof of the one-dimensional version of (5.10) can be found in [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF]. We describe the required modifications for d > 1 in Section 5.8. This gives:

ˆCτ µ(θ) D(θ)p x (τ, x, θ) 2 + |∇ θ p(τ, x, θ)| 2 dxdθ ≥ CI 2 (τ ) 1 + I 2 (τ ) 1/2 I 1 (τ ) 10d 3(3+d) -1 I 2 (τ ) 1/2 I 1 (τ ) 4 3
.

(5.11) Here, we have defined

I 1 (τ ) := ˆCτ µ(θ)(x -c * τ )p(τ,
x, θ)dxdθ, and,

I 2 (τ ) := ˆCτ µ(θ)p(τ, x, θ) 2 dxdθ.
We point out that we used in (5.11) that µ is bounded uniformly away from 0 and ∞.

Next, we look at

I(τ ) : = ˆCτ µ(θ)f (τ, x, θ)p(τ, x, θ)dxdθ,
with f as in (5.7). If ω ≡ 0, then I(τ ) is a conserved quantity. In general, following the proof of [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF]Lemma 5.4], one can show that there exists a constant C > 0 such that

C -1 I(0) ≤ I(τ ) ≤ C ˆC0 p 0 dxdθ + I(0) .
(5.12)

Using Lemma 5.1, we see that I(τ ) and I 1 (τ ) are comparable:

1 C ˆCτ µ(θ)f (τ, x, θ)p(τ, x, θ)dxdθ ≤ I 1 (τ ) ≤ C ˆCτ µ(θ)f (τ, x, θ)p(τ, x, θ)dxdθ.
As a consequence, we have

C -2 I 1 (0) ≤ C -1 I 1 (τ ) ≤ N := ˆC0 p 0 dxdθ + I 1 (0) . (5.13)
Using (5.13) together with (5.9) and (5.11), we obtain

I 1 (τ ) 4/3 I 2 (τ ) 5/3 + I 1 (τ ) (4-2d)/(3+d) I 2 (τ ) 5/(3+d) I 2 (τ ) ≤ - 1 C(1 -ω(τ ))
.

(5.14)

An elementary argument, starting with this differential inequality, using the decay assumptions on ω and (5.13), gives an upper bound

I 2 (τ ) ≤ CN 2 (τ + 1) 3/2 ,
(5.15) regardless of the cross-section dimension d ≥ 1. In other words, we have the bound

p(τ, •) L 2 (Cτ ) ≤ C (τ + 1) 3/4 ˆC0 (1 + x)|p 0 (x, θ)|dxdθ.
(5.16)

We may now apply the standard duality argument. Let S τ be the solution operator mapping p 0 to p(τ, •). The bound (5.15) applies that S * τ satisfies

|S * τ p 0 | ≤ C(1 + x -c * τ ) (τ + 1) 3/4 p 0 L 2 (C 0 ) .
(5.17) However, S * τ is the solution operator for a parabolic equation of the same type, except for the reverse drift direction, thus it also obeys the bound (5.16), and hence S τ itself obeys (5.17) as well. Decomposing S τ = S τ /2 • S τ /2 and applying the bounds (5.16) and (5.17) separately, we get

|p(τ, x, θ)| ≤ C(1 + x -c * τ ) (τ + 1) 3/2

ˆC0

(1 + x)p 0 (x, θ)dxdθ. (5.18) This proves (5.8) for x > c * τ + 1. However, as p(τ, c * τ, •) = 0, using the parabolic regularity for x ∈ (c * τ, c * τ + 1), we obtain the upper bound (5.8) for all x > c * τ .

The lower bound for p

We now prove the lower bound on p in Proposition 4.3, namely, there exists a positive constant such that

p(τ, x, θ) ≥ (x -c * τ ) C 0 (τ + 1) 3/2 , (5.19)
for all τ > 0, x > c * t and θ ∈ Θ.

Approximate solutions

For the proof of Proposition 4.3 will make use of approximate solutions of our problem that satisfy the bounds claimed in this Proposition. Let Q λ be the eigenfunction in (4.12), and set

χ = - 1 Q λ ∂Q λ ∂λ λ=λ * , (5.20)
and

D : = ˆΘ (D + c * χ -2λ * Dχ -Aχ) (Q * ) 2 dθ ˆΘ(Q * ) 2 dθ -1
.

(5.21)

To see that D > 0, we differentiate (4.18) in λ to obtain

ˆΘ 2Q ∂Q ∂λ c (λ)λ + c(λ) -2λD -A + Q 2 c (λ)λ + 2c (λ) -2D -A dθ = 0. (5.22)
Evaluating (5.22) at λ = λ * , we obtain, as c (λ * ) = 0:

0 = ˆΘ -2(Q * ) 2 χ (c * -2λ * D -A) + (Q * ) 2 c (λ)λ -2D -A dθ.
Now, (5.21) and (4.18) show that this is

c (λ * )λ * ˆΘ(Q * ) 2 dθ = 2D ˆΘ(Q * ) 2 dθ.
Since c (λ * ) > 0 by Proposition 4.2, we conclude that D > 0. The approximate solutions are described by the following analogue of [21, Proposition 5.2].

Proposition 5.2. Let χ ∈ R, then there is a function S(τ, x, θ) such that, for any σ > 0,

(1 -ω) ∂S ∂τ -DS xx -∆ θ S + (2λD + A) S x - 2 Q * ∇ θ Q * • ∇ θ S = O(τ -3 ) (5.23)
and

S(τ, x, •) - x -c * τ + χ + χ τ 3/2 e -(x-c * τ ) 2 4Dτ ≤ Cτ -3/2 x -c * τ √ τ 2 + O(τ -2 ), (5.24
)

for all x ∈ [c * τ, c * τ + σ √ τ ].
The constant C depends on σ.

The approximate solutions do approximate true solutions on [c * τ, c * τ + σ √ τ ], as seen from the following.

Proposition 5.3. Fix σ > 0, and let S be as in Proposition 5.2. Suppose that ξ satisfies for τ > 0,

       (1 -ω)ξ τ = Dξ xx + ∆ θ ξ -(2λ * D + A) ξ x + 2 Q * ∇Q * • ∇ξ, x ∈ [c * τ, c * τ + σ √ τ ], ξ(τ, c * τ, •) = S(τ, c * τ, •), ξ(τ, c * τ + σ √ τ , •) = S(τ, c * τ + σ √ τ , •).
(5.25)

Then there is a positive constant

τ 0 such that, if τ ≥ τ 0 and x -c * τ ∈ (0, σ √ τ ), then | (ξ -S) (τ, x, •)| ≤ C τ 3/2 .
The proof of Proposition 5.3 is a relatively straightforward energy estimate of the difference ξ-S that can be obtained almost exactly as in [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF]Proposition 5.3].

The size of the solution at distance O( √ τ )

Another key step is to establish the magnitude of p at distances of the order O( √ τ ) from x = c * τ . With the following proposition, we control p at the endpoints of the interval [c * τ, c * τ +σ √ τ ]. Then, the previous propositions allow us to control p in the remainder of the interval as S approximates p. Proposition 5.4. Let p be as in Proposition 4.3. There are constants σ > 0 and C 0 > 0 so that

1 C 0 τ ≤ p(τ, c * τ + σ √ τ ) ≤ C 0 τ (5.26) whenever τ ≥ 1.

Sketch of the proof of Proposition 4.3

We now outline how to combine Propositions 5.2 to 5.4 to obtain the lower bound in Proposition 4.3. Proposition 5.4 controls p at the point c * τ +σ √ τ in a way consistent with (4.25). On the other hand, by choosing χ = -(1+ χ ∞ ) in Proposition 5.2, the combination of Propositions 5.2 and 5.3 allows us to build a sub-solution ξ -to p. Then, re-applying Proposition 5.3, we see that ξ -satisfies the bounds in (4.25) except on a finite interval [c * τ, c * τ +x 0 ], for some x 0 . By the comparison principle, we may then transfer these bounds to p and use parabolic regularity to remove the condition on x 0 , finishing the proof of the claim. Thus, it remains to prove Propositions 5.2 and 5.4, which is done in the rest of this paper.

The proof of Proposition 5.2

Our strategy is the same as in [21, Proposition 5.2], though the details are different, so we include a sketch of the proof for reader's convenience. We begin with the multi-scale expansion

S(τ, x, θ) = 1 τ S 0 (z) + S 1 (z, θ) √ τ + S 2 (z, θ) τ + S 3 (z, θ) τ 3/2 , z = x -c * τ √ τ .
Plugging this into the left hand side of (5.23), we obtain the equation

(1 -ω) τ - S 0 τ - 3S 1 2τ 3/2 - 2S 2 τ 2 - 5S 3 2τ 5/2 + (1 -ω) τ -c * S 0 z τ 1/2 -c * S 1 z τ -c * S 2 z τ 3/2 -c * S 3 z τ 2 + (1 -ω) τ - z 2 S 0 z τ - z 2
S 1 z τ 3/2 - z 2 S 2 z τ 2 - z 2 S 3 z τ 5/2 + D τ - S 0 zz τ - S 1 zz τ 3/2 - S 2 zz τ 2 - S 3 zz τ 5/2 + 1 τ LS 1 τ 1/2 + LS 2 τ + LS 3 τ 3/2 + (2λ * D + A) τ S 0 z √ τ + S 1 z τ + S 2 z τ 3/2 + S 3 z τ 2 = 0. (5.27)
Here, we have defined the operator

L = ∆ θ + 2 Q * ∇ θ Q * • ∇ θ .
Grouping the terms of order τ -3/2 in (5.27), we obtain

LS 1 = (c * -2λ * D -A) S 0 z . (5.28)
It is easy to verify that (5.28) has a solution of the form

S 1 = χ 0 S 0 z + φ 1 , χ 0 = χ + χ, (5.29)
where φ 1 only depends on z. The terms of order τ -2 in (5.27) give

-S 0 -c * S 1 z - z 2 S 0 z -DS 0 zz + LS 2 + 2λDS 1 z + AS 1 z = 0.
(5.30)

Using expression (5.29) for S 1 , multiplying (5.30) by (Q * ) 2 and integrating in θ, we obtain

S 0 + z 2 S 0 z + DS 0 zz = 0 (5.31)
with D as in (5.21), so that

S 0 (z) = z exp - z 2 4D . (5.32)
With this in hand, we return to (5.30) that we write as

LS 2 = (c * -2λD + A)(S 1 ) z + (D + c * χ 0 -2λ * Dχ 0 + Aχ 0 -D)S 0 zz . (5.33)
One solution of (5.33) is

S 2 (z, θ) = χ 0 (θ)(φ 1 ) z (z) + Ŝ2 (θ)S 0 zz (z)
, where Ŝ2 (θ) is any solution to

L Ŝ2 = D + c * χ 0 -2λ * Dχ 0 + Aχ 0 -D, (5.34)
with the Neumann boundary conditions. The definition of D ensures that solution of (5.34) exists. Continuing, we examine the terms of order τ -5/2 to obtain

- 3 2 S 1 + ωc * S 0 z -c * S 2 z - z 2 S 1 z -DS 1 zz + 2λDS 2 z + LS 3 + AS 2 z = 0. (5.35)
Here, we replaced ω by ω/τ at the expense of lower order terms which we may absorb into the O(τ -3 ) term in (5.23). Multiplying by (Q * ) 2 and integrating over θ yields the solvability condition for S 3 :

- 3 2 φ 1 - z 2 (φ 1 ) z -D(φ 1 ) zz = (3β 1 -ωc * ) S 0 z + zβ 1 S 0 zz + β 2 S 0 zzz . (5.36)
Here, we have defined

β 1 : = 1 2 ´Θ χ 0 (Q * ) 2 dθ ´Θ(Q * ) 2 dθ , β 2 : = ´Θ(c * Ŝ2 -2λD Ŝ2 -A Ŝ2 + D)(Q * ) 2 dθ ´Θ(Q * ) 2 dθ .
We may now choose φ 1 to be the unique solution to (5.36) with φ 1 (0) = 0 and (φ 1 ) z (0) = 0. Since S 0 and its derivatives are bounded, there exists a constant C such that |φ 1 (z)| ≤ Cz 2 for all |z| ≤ σ. For the sake of clarity, we write φ 1 = z 2 φ, with a bounded function φ.

Finally, grouping the τ -3 terms together and setting them to zero, we get an equation for S 3 . It follows from the elliptic regularity theory that, for z ≤ σ, S 3 is uniformly bounded. To summarize, we have found an approximate solution, in the sense that (5.23) holds, of the form

S = x -c * τ τ 3/2 e -(x-c * τ ) 2 4Dτ + χ 0 1 -z 2 4Dτ τ 3/2 e -(x-c * τ ) 2 4Dτ + x -c * τ τ 5/2 φ(z) + S 2 τ 2 + S 3 τ 5/2 .
(5.37)

It also clearly satisfies the condition (5.24). This concludes the proof. The lower bound in (5.26) is a consequence of an integral bound.

Lemma 5.5. There exists a time T 0 > 0 and constants c 0 , B, and N , depending only on the initial data, such that for any τ > T 0 there exists a set

I τ ⊂ [c * τ + N -1 √ τ , c * t + N √ τ ] with |I τ | ≥ B √ τ and with 1 c 0 τ
≤ ˆΘ p(τ, x, η)dη.

(5.38) Proposition 5.4 follows from Lemma 5.5 and a standard heat kernel bound. Indeed, let us assume that ω = 0, as we may otherwise apply the time change

dτ = dτ 1 -ω(τ )
.

Let Γ be the heat kernel for (5.5)-(5.6) with the Dirichlet boundary condition at x = c * t. That is, the solution of

µψ τ = (Dµψ x ) x + ∇ θ • (µ∇ θ ψ) -(∆ θ β + c * ) ψ x , τ > s, x > c * τ, θ ∈ Θ, ∂ψ ∂ν θ = 0, on ∂Θ, (5.39) ψ(τ, c * τ, •) = 0, ψ(s, •) = ψ,
can be written as ψ(τ, x, θ) = ˆCs Γ(τ, x, θ, s, y, η) ψ(y, η)µ(η)dydη.

(5.40)

As in [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF], one can show the following, starting with the standard heat kernel bound in a cylinder. Set Φ(s) = s for s ∈ [0, 1] and Φ(s) = √ s for s > 1, then for all δ > 0, there exists a constant K such that

Γ(τ, x, θ, s, y, η) ≥ 1 KΦ(τ -s) exp -K |x -y| 2 + |θ -η| 2 Φ(τ -s) (5.41)
whenever R > 0, τ ∈ (s, s + R 2 ], and x, y ∈ (c * τ + ξ -δR, c * τ + ξ + δR). A straightforward computation using (5.40) going from the time s = τ /2 to τ shows that the integral bound (5.38), combined with the pointwise lower bound (5.41) on the heat kernel, lead to a pointwise lower bound on p in Proposition 5.4.

Proof of Lemma 5.5

An exponentially weighted estimate

As in [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF], one may show that for all α > 0, there exists a function η α that satisfies

µ∂ τ η α = -L * (µη α ) + ℵ(α)µη α , on C τ , (5.42) ∂η α ∂ν θ = 0 on ∂Θ, η α (τ, c * τ, •) = 0.
as well as the exponential bounds e α(x-c * τ ) -e -α(x-c * τ ) Cα ≤ η α (t, x, θ) ≤ C e α(x-c * τ ) -e -α(x-c * τ ) α .

(5.43)

The eigenvalue ℵ(α) in (5.42) behaves as

ℵ(α) = ℵ 0 α 2 + O(α 3 ), (5.44)
as α tends to zero, with some ℵ 0 > 0. Moreover, we have

|∂ τ η α | ≤ C for all x ∈ [c * τ, c * τ + α -1 ],
and

|∂ τ η α | ≤ Cαη α for all x ≥ c * τ + α -1 . (5.45)
With this in hand, we define

V α (τ ) = (1 -ω(τ ))
ˆCτ µ(θ) η 2α (τ, x, θ) p(τ, x, θ) q(τ, x, θ) dxdθ.

(5.46)

Here, we write p = qζ, (5.47) and ζ is as in Lemma 5.1. Lemma 5.5 is a consequence of the following estimate.

Lemma 5.6. There is a constant C 0 depending on p 0 such that

V τ -1/2 (τ ) ≤ C 0 τ -3/2 . (5.48)
We first show how to conclude the proof of Lemma 5.5 from Lemma 5.6. Note that (5.48) implies

ˆ∞ 0 ˆΘ e 2x/ √ τ -e -2x/ √ τ x p(τ, c * τ + x, θ) 2 dxdθ 1/2 ≤ C 0 τ , (5.49)
Fix N > 0 to be determined later, then (5.49) gives, in particular:

ˆ∞ N √ τ ˆΘ xp(τ, c * τ + x, θ)dxdθ = ˆ∞ N √ τ ˆΘ e x/ √ τ √ x p(τ, c * τ + x, θ)e -x/ √ τ x 3/2 dxdθ ≤ ˆ∞ N √ τ ˆΘ e 2x/ √ τ x p(τ, c * τ + x, θ) 2 dxdθ 1/2 ˆ∞ N √ τ ˆΘ e -2x/ √ τ x 3 dxdθ 1/2 ≤ C τ ˆ∞ N √ t ˆΘ e -2x/ √ τ x 3 dxdθ 1/2 ≤ C 0 N 3 e -N/2 . ( 5
I τ : = x ∈ [c * τ + √ τ /N, c * τ + N √ τ ] : ˆΘ p(τ, x, η)dη ≥ C 0 4N 2 τ .
Then, (5.51) implies

3C 0 4 ≤ ˆIτ ˆΘ(x -c * τ )p(τ, x, θ)dxdθ ≤ ˆIτ ˆΘ C 0 (x -c * τ ) 2 τ 3/2 dxdθ ≤ |I τ | C 0 N 2 τ 1/2 ,
and the proof of Lemma 5.5 is complete.

5.7

The proof of Lemma 5.6

Throughout this section we use the assumption that τ ≤ α -2 . The proof relies on two observations. First, we have the following energy-dissipation inequality for V α :

V α (τ ) ≤ ℵ(2α) - ω (τ ) 1 -ω(τ ) + Cαω(τ ) V α (τ ) -2D α (τ ) + C 0 (τ + 1) 5/2 ,
(5.52)

with the dissipation

D α = ˆCτ µ(θ)η 2α (τ, x, θ)ζ(τ, x, θ) D(θ)|q x (τ, x, θ)| 2 + |q θ (τ, x, θ)| 2 dxdθ.
(5.53)

Recall that the function ζ is defined in Lemma 5.1, and q is as in (5.47). Since this computation is quite involved, we delay it for the moment.

The second observation is that the dissipation D α may be related to V α by the inequality

D α ≥ 1 C 0 V 5/3 α (5.54)
where C 0 is a constant depending only on p 0 and τ ∈ [0, T ]. We also delay the proof of (5.54). The combination of (5.52) and (5.54) yields the differential inequality

V α ≤ ℵ(2α) - ω 1 -ω + Cα τ + 1 V α - 1 C 0 V 5/3 α + C 0 (τ + 1) 5/2 .
(5.55)

Let us define

Z(τ ) = (τ + 1) 3/2 V α exp(-Φ(τ )), Φ(τ ) = ℵ(2α)τ + log(1 -ω(τ )) + Cα log(τ + 1).
Note that, as τ ≤ α -2 , we know, due to the asymptotics (5.44) for ℵ(2α), that

|Φ(s)| ≤ C for all 0 ≤ s ≤ τ , (5.56)
with a constant C > 0 that is independent of α > 0 sufficiently small. Thus, (5.48) would follow if we show that that Z is uniformly bounded above. However, it follows from (5.55) and (5.56) that Z satisfies

Z ≤ C Z τ + 1 + C 0 τ + 1 - 1 C 0 (τ + 1) Z 5/3 .
This implies

Z 5/3 ≤ max C (Z + C 0 ) , Z(0) 5/3 .
Hence, Z is bounded uniformly above. Thus, to finish the proof of Lemma 5.6, it only remains to show (5.52) and (5.54).

Proof of the differential inequality (5.52) for V α Differentiating V α , we obtain

V α = - ω 1 -ω V α + (1 -ω) ˆCτ µ [(∂ τ η 2α ) pq + η 2α p τ q + η 2α pq τ ] dxdθ.
(5.57)

Let us re-write the integral in (5.57). By the definition of η 2α , we have

(1 -ω) ˆCτ µ (∂ τ η 2α ) pq dxdθ = (1 -ω) ˆCτ [-L * (µη 2α ) + ℵ(2α)µη α ] pq dxdθ, = -(1 -ω) ˆCτ µη 2α L(pq) dxdθ + ℵ(2α)V α .
Using equation (4.23) for p, we deduce

V α = ℵ(2α) - ω 1 -ω V α + ˆCτ µη 2α [-(1 -ω(τ )) L(pq) + L(p)q + (1 -ω)pq τ ] dxdθ.
(5.58)

The last integral requires a bit of work. Note that L(pq) = pL(q) + qL(p) + 2Dp x q x + 2p θ q θ , and

(1 -ω)q τ = L(q) + q ζ (L (ζ) -(1 -ω) ζ τ ) + 2D ζ x ζ q x + 2 ζ θ ζ q θ , = L(q) + q ζ (L (ζ) -ζ τ ) + ω ζ τ ζ q + 2D ζ x ζ q x + 2 ζ θ ζ q θ .
Thus, we may re-write (5.58) as

V α = ℵ(2α) - ω 1 -ω V α + ω ˆCτ µη 2α L(pq)dxdθ + ω ˆCτ µη 2α pq ζ τ ζ dxdθ -2 ˆCτ µη 2α (Dp x q x + p θ q θ )dxdθ + 2 ˆCτ µη 2α p D ζ x ζ q x + ζ θ ζ q θ dxdθ.
The last two terms in the right side can be combined as

-2 ˆCτ µη 2α (Dp x q x + αp θ q θ ) dxdθ + 2 ˆCτ µη 2α D ζ x ζ q x + ζ θ ζ q θ p dxdθ = -2 ˆCτ µη 2α D p x - ζ x ζ p q x + p θ - ζ θ ζ p q θ dxdθ = -2 ˆCτ µη 2α ζ D|q x | 2 + |q θ | 2 dxdθ = -2D α , hence V α = ℵ(2α) - ω 1 -ω V α -2D α + ω ˆCτ µη 2α L(pq) + p ζ τ ζ q dxdθ.
(5.59)

The estimate (5.52) will be complete after estimating the last term in the right side. We write

ˆCτ µη 2α L(pq) + p ζ τ ζ q dxdθ = ˆCτ pqL * (µη 2α ) + µη 2α ζ τ ζ pq dxdθ ≤ ˆCτ µ[∂ τ η 2α ]pq dxdθ, (5.60)
as ζ τ ≤ 0. We use (5.45) to obtain

ˆCτ µ∂ τ η 2α pqdxdθ ≤ ˆc * τ +α -1 c * τ ˆΘ µ∂ τ η 2α pqdxdθ + ˆ∞ c * τ +α -1 ˆΘ µ∂ τ η 2α pqdxdθ ≤ ˆc * τ +α -1 c * τ ˆΘ µpqdxdθ + Cα ˆ∞ c * τ +α -1 ˆΘ µη 2α pqdxdθ.
The second term above is CαV α , as desired. For the first term, we apply the upper bound (5.8) for p and the asymptotics for ζ in Lemma 5.1 to obtain ˆc * τ +α -1

c * τ ˆΘ µpqdxdθ ≤ C 0 (τ + 1) 3/2 ˆc * τ +α -1 c * τ ˆΘ pdxdθ.
Integrating (5.5), we see that ´Cτ µpdxdθ is non-increasing in time. Hence, we obtain that

ω ˆCτ µη 2α L(pq) + p ζ τ ζ q dxdθ ≤ C 0 (τ + 1) 5/2 + C α τ + 1 V α .
Returning to (5.59), we obtain the desired differential inequality

V α = ℵ(2α) - ω (τ ) -ω(τ ) + C α τ + 1 V α -2D α + C 0 (τ + 1) 5/2 .
Proof of the inequality (5.54) relating V α and D α It is helpful to define ϕ(τ, z, θ) = e αz 3 q(τ, c * τ + |z|, θ), with (z 1 , z 2 , z 3 ) = z ∈ R 3 , and consider the following quantities

Îα := 1 2π ˆR3 ×Θ ϕ(τ, z, θ)dzdθ = ˆCτ e α(x-c * τ ) -e -α(x-c * τ ) α (x -c * τ )q(τ, x, θ)µ dzdθ, Vα := 1 2π ˆR3 ×Θ ϕ(τ, z, θ) 2 dzdθ = ˆCτ e 2α(x-c * τ ) -e -2α(x-c * τ ) α (x -c * τ )q 2 (τ, x, θ)µ dzdθ, Dα := 1 2π ˆR3 ×Θ |∇ϕ(τ, z, θ)| 2 dzdθ = ˆCτ e 2α(x-c * τ ) -e -2α(x-c * τ ) 2α (x -c * τ )(|∇q(τ, x, θ)| 2 -α 2 q(τ, x, θ) 2 )µ dzdθ.
(5.61)

They can be related by the following Nash-type inequality.

∇φ 2 2 ≥ C 1 + φ 2 φ 1 2d(k+2) k(k+d) -1 φ 2 2 φ 2 φ 1 4 k .
(5.62) Inequality (5.62) is a multi-dimensional version of a Nash-type inequality in [START_REF] Fannjiang | Quenching of reaction by cellular flows[END_REF], while the one-dimensional version of (5.10) is in [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF]. Its proof is in Section 5.8.

We may apply Proposition 5.7 to φ in the cylinder R

3 × Θ to obtain Dα ≥ 1 C V 5/3 α Î-4/3 α 1 + V 5d/(3d+9) α Î-10d/(3d+9) α (5.63)
Using the bounds for ζ in Lemma 5.1 and the exponential bounds (5.43) for η α , we see that

C -1 Vα ≤ V α ≤ C Vα and Dα ≤ CD α . (5.64) We claim that 1 C 0 ≤ Îα ≤ C 0 and V α ≤ C 0 , (5.65)
so that (5.63) implies

D α ≥ 1 C 0 V 5/3 α ,
which is (5.54).

To finish, we need to show that (5.65) holds. We begin with the inequality for Îα in (5.65). Let us introduce

I α = (1 -ω) ˆCτ µ(θ)η (τ, x, θ)p(τ, x, θ)dxdθ. (5.66)
We note that C -1 I α ≤ Îα ≤ CI α , by (5.43). Hence, we need only show that I α is bounded away from infinity and zero uniformly in τ and α for all τ ≤ α -2 . Let us differentiate I α :

I α = - ω 1 -ω I α + (1 -ω) ˆCτ µ [p∂ τ η α + η α p τ ] dxdθ.
Using (4.23) and (5.42) allows us to rewrite the integral involving p τ :

I α (τ ) = ℵ(α) - ω (τ ) 1 -ω(τ ) I α (τ ) -ω(τ ) ˆCτ µp∂ τ η α dxdθ.
(5.67)

The last term may be estimated as

ω ˆCτ µp∂ τ η α dxdθ ≤ Cω ˆc * τ +α -1 c * τ ˆΘ pdxdθ + Cωα ˆ∞ c * τ +α -1
ˆΘ pη α dxdθ.

(5.68)

The second term in (5.68) is bounded by C(τ + 1) -1 αI α . The first requires a bit more work. First, split the integral as ˆc * τ +α -1

c * τ ˆΘ pdxdθ = ˆc * τ +min{τ 2/3 ,α -1 } c * τ ˆΘ pdxdθ + ˆc * τ +α -1 c * τ +min{τ 2/3 ,α -1 } ˆΘ pdxdθ. (5.69)
The first term is estimated using (5.8) to obtain

ˆc * τ +min{τ 2/3 ,α -1 } c * τ ˆΘ pdxdθ ≤ ˆmin{τ 2/3 ,α -1 } 0 ˆΘ C 0 x (τ + 1) 3/2 dxdθ ≤ C 0 (τ + 1) 1/6 .
Arguing as in [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF]Lemma 5.4], we may bound p by the solution to (4.23) in the whole cylinder R × Θ. Thus, the heat kernel bounds of, e.g. [START_REF] Norris | Long-time behaviour of heat flow: global estimates and exact asymptotics[END_REF], imply that p(τ, x + c * τ, •) ≤ C 0 e (5.72)

Using that τ ≤ α -2 and that ℵ(α) ∼ α 2 , by (5.44), we have that τ Cα e ℵ(α)τ ≤ C. Using this and choosing T at least as large as (2C 0 /I α (0)) 12 in (5.72) finishes the proof of the first estimate in (5.65). We note that, for all α, we have I α (0) ≥ ˆCτ xp 0 (x, θ)dxdθ, so that our condition on T can be made uniform in α. Now we consider V α . Fix N to be determined later and assume that τ 2/3 < N α -1 , the other case being treated via a very similar computation. We decompose the integral as For the first integral, using Lemma 5.1 and the definition of q, we may apply (5.8) to bound p and q as C(x -c * τ )(τ + 1) -3/2 and C(τ + 1) -3/2 , respectively. Using also (5.43) For the last integral in (5.73), we use the bound q and p as in the last step and bound η 2α by Ce 2αx /α. This yields

ˆ∞ c * τ +N α -1 ˆΘ η 2α pqdxdθ ≤ C (τ + 1) 2 ˆ∞ N α -1 e 2αx-x 2
C(τ +1) α dx.

Since τ ≤ α -2 , we may choose N such that N Cα(τ + 1))

-2α ≥ 1 α(τ + 1) for all α sufficiently small. Hence, we have

C (τ + 1) 2 ˆ∞ N α -1 e 2αx-x 2
C(τ +1) α dx ≤ C (τ + 1) 2 ˆ∞ N α -1 e 2αx-N x Cα(τ +1) α dx ≤ C (τ + 1) 2 ˆ∞ N α -1 e -x α(τ +1) α dx ≤ C (τ + 1)
.

Combining this bound with (5.75) and (5.76), we have that, for all τ ≤ α -2 , V α (τ ) ≤ C(τ + 1) -1 , which, in particular, implies the upper bound on V α in (5.65).

5.8

The proof of Proposition 5.7

Here we prove the Nash-type inequality on cylinders that we use above. We point out that when the L 2 norm is small relative to the L 1 norm, this yields the same inequality as in R k . The main point here is that using this inequality we see that solutions to the heat equation on R k × Θ decay at the same rate as solutions to the heat equation in R k . Our approach is similar to the one used in [START_REF] Fannjiang | Quenching of reaction by cellular flows[END_REF]. However, some computational challenges arise since we lack an explicit formula for the solutions of k + 1 order polynomials. We note that, by extending φ if necessary and scaling, we may assume without loss of generality that Θ = [0, 1] d .

First, we represent φ in terms of its Fourier series in the θ variable, and its Fourier transform in the x variable. This yields φ(x, θ) = (5.80)

The second and third terms in (5.79) may be estimated in the same way so we show only the second term. It can be bounded as: In the interest of legibility, we define the following constants

I = φ 2 1 , J = ∇φ 2 2 , and
K = φ 2 2 ,
and we re-write the above inequality as

1 C K ≤ ρ k (ρ d + 1)I + 1 ρ 2 J.
Define X to be the quantity in order to optimize this inequality. Hence, the above inequality becomes 1 ≤ X k+d + αX k ,

X def = J 1 k+2 CI 2-d k+2

Proposition 4 . 2 .

 42 The function λ → c(λ) has a minimum c * , and

  ) for τ > 0, x > c * τ , and θ ∈ Θ, with the Neumann boundary condition (4.21), the Dirichlet boundary condition for τ > 0, p(τ, c * τ, •) = 0, (4.24)

Figure 1 :

 1 Figure 1: A sketch of the solution u and the super-solution u.

Figure 2 :

 2 Figure 2: A sketch of the sub-solution U , the solution u of the Fisher-KPP problem, and of the solution of the linearized problem with the Dirichlet boundary condition at x = c * t.

  ) and (5.1) in the last step above. Thus, (4.23) can be recast as (1 -ω)µp τ = Lp, (5.5) with the operator L Lp = (Dµp x ) x + ∇ θ • (µ∇ θ p) -(∆ θ β + c * ) p x . (5.6)

5. 5

 5 Understanding p at x -c * τ ∼ O( √ τ ): the proof of Proposition 5.4

  .50) On the other hand, we also have ˆ√τ/N 0 ˆΘ xp(τ, c * τ + x, θ)dxdθ ≤ C 0 N -3 . Hence, choosing N sufficiently large, depending only on the initial data of p and not on time, we have ˆN√ τ √ τ /N ˆΘ xp(τ, c * τ + x, θ)dxdθ ≥ C 0 . (5.51) Let us set

Proposition 5 . 7 .

 57 Let Θ ⊂ R d be a smooth, bounded domain, and Ω = R k × Θ. There exists a constant C, depending only on d, k, and |Θ| such that if φ is any function in L 1 (Ω) ∩ H 1 (Ω) satisfying Neumann boundary conditions on the boundary ∂Θ, then

-x 2 C 1 c * τ +min{τ 2 / 3 ,α - 1 } 1 min{τ 2 / 3 ,

 21231123 where C 0 is a constant depending on p 0 . Hence, the second integral in (5.69) yieldsˆc * τ +α -ˆΘ pdxdθ ≤ C ˆαdxdθ ≤ C 0 e -(τ +1) 1/3 /C .From (4.22), we see that |ω| ≤ C(τ + T ) -1 , with T to be chosen. This, along with the previous two inequalities and (5.68), implies thatω ˆCτ µp∂ τ η α dxdθ ≤ C 0 (τ + T )(τ + 1) 1/6 ≤ C 0 T 1/12 (τ + 1) 13/12 .We used above the first assumption on ω in(4.22). Hence, we obtainI α -ℵ(α) -ω 1 -ω + O(α/(τ + 1)) I α ≤ C 0 T 1/12 (τ + 1) 13/12 .(5.71) Integrating (5.71), using, once again, (4.22), yields the inequalityC(τ + 1) Cα e ℵ(α)τ I α (0) -C 0 T 1/12 ≤ I α (τ ) ≤ C(τ + 1) Cα e ℵ(α)τ I α (0) + C 0 T 1/12 .

V 1 c * τ +τ 2/ 3 ˆΘ

 13 α (τ ) = ˆc * τ +τ 2/3 c * τ ˆΘ η 2α pqdxdθ + ˆc * τ +N αη 2α pqdxdθ + ˆ∞ c * τ +N α -1ˆΘ η 2α pqdxdθ. (5.73)

1 c * τ +τ 2/ 3 ˆΘ η 2α pqdxdθ ≤ Ce 2αN (τ + 1) 2 ˆNα - 1 τ 2 / 3 xe -x 2 C

 1321232 to bound η 2α by a linear function:η 2α (τ, x, θ) ≤ Ce 2αN (x -c * τ ) on [c * τ, c * τ + N α -1 ], (5.74) we get ˆc * τ +τ 2/3 c * τ ˆΘ η 2α pqdxdθ ≤ Ce 2αN (τ + 1) 3 ˆc * τ +τ 2/3 c * τ (x -c * τ ) 2 dx ≤ Ce 2αN (τ + 1) . (5.75) For the second integral in (5.73), we use the same bounds for q and η 2α but we bound p with the Gaussian bound (5.70). This yields ˆc * τ +N α -(τ +1) dx ≤ Ce 2αN (τ + 1) e -(τ +1) 1/3 /C . (5.76)

2 , 2 n ˆRk |ξ| 2 + n 2 | 2 . 2 = 2 1

 2222222 n∈Z d ˆRk φn (ξ)e iξ•x cos (πnθ) dξ (2π) k+d where φn (ξ) := ˆΘ ˆRk φ(x, θ)e -iξ•x cos (πnθ) dxdθ (2π) k+dBefore we continue, we note two things. First, we have that| φn (ξ)| ≤ φ 1 , φn (ξ)| 2 dξ (2π) k+d (5.78)Fix a constant ρ to be determined later. We now decompose φ 2 into outer and inner parts asφ 2 |n|≤ρ ˆBρ(0) | φn (ξ)| 2 dξ + |n|>ρ ˆRd | φn (ξ)| 2 dξ + n ˆBρ(0) c | φn (ξ)| 2 dξ.(5.79)The first term in (5.79) may be bounded as|n|≤ρ ˆBρ(0) | φn (ξ)| 2 dξ ≤ |n|≤ρ ˆBρ(0) φ n dξ ≤ Cρ k (ρ + 1) d φ n 2 1

ρ 2 |ξ| 2 + n 2 |

 22 φn (ξ)| 2 dξ ≤

  Proof of Proposition 3.1. The proof is by comparing n to a solution to an associated linear heat equation. Take t 1 ≥ t 0 and let h be the solution toh t = θh xx + h θθ ,

	1.2) satisfy a uniform bound
	n(t, x, θ) ≤ M	(3.1)
	for all (t, x, θ) ∈ [0, ∞) × R × Θ with a constant M depending only on θ and θ. With this in hand,
	we first show that we may bootstrap Theorem 1.2 to hold for n as well.
	Proposition 3.1. For any t 0 > 0, R > 0, and p > 1, there is a constant C > 0 such that if t ≥ t 0
	and |θ -θ | + |x -x | ≤ R, and n is a solution of (1.2)-(1.4), then
	n(t, x, θ) ≤ Cn(t, x , θ ) 1/p .	(3.2)
	with the Neumann boundary conditions
	h θ (t, x, θ) = h θ (t, x, θ) = 0,
	and the initial condition
	h(0, x, θ) = n(t 1 -δ, x, θ),
	with δ = min{1, t 0 /2}. Theorem 1.2 implies 1 that there is a constant C depending only on M , δ, R
	and p such that, for any |x -x | ≤ R and θ ∈ [θ, θ], we have
	h(t, x, θ) ≤ Ch(t, x , θ) 1/p ,
	for all t ≥ δ.
	On the other hand, as
	n(1

Strictly speaking, to apply Theorem 1.2, we need n to be defined on R

, not on R × Θ. This obstacle, however, may be avoided considering a periodic extension of n to R 2 ; see [35, Section 2.1] for more details.

where we define

It is straight-forward to verify that this polynomial has exactly one positive root which must be at least as large as (2(1 + α)) -1/k . Hence, it follows that

Returning to our earlier notation, we obtain

, Re-arranging this inequality and substituting in for I, J, and K concludes the proof.


Clifford W Fong Eigenenergy
email: cwfong@internode.on.net

Keywords: trimethylmethylcyclopentadienylplatinum, dissociative electron attachment, FEBID, transition states, molecular orbital computations

may come

The molecular mechanism of dissociative electron attachment of trimethylmethylcyclopentadienylplatinum in focused electron beam induced deposition

Introduction

Trimethylmethylcyclopentadienylplatinum(IV) (TMCP) is a precursor used in focused electron beam induced deposition (FEBID) which is a single-step, direct-write nanofabrication technique used to produce nanoscale three-dimensional metal-containing structures on surfaces. It is known that metal purity due to incomplete precursor decomposition is one factor that limits FEBID resolution. Low-energy (<100 eV) secondary electrons generated by interactions of the primary beam with the substrate are involved in the decomposition reaction of the precursor molecules. Thorman et al 2015 [START_REF] Thorman | The role of low-energy electrons in focused electron beam induced deposition: four case studies of representative precursors[END_REF] has extensively reviewed and summarised the different low-energy electron-induced fragmentation processes that can be initiated by the secondary electrons generated in FEBID: dissociative electron attachment, dissociative ionization, neutral dissociation, and dipolar dissociation. Comparing gas phase and surface (physisorption) precursor processes allows insight into the primary deposition mechanisms, which could be used to design future FEBID precursors and optimize deposition conditions. This study examines the dissociative electron attachment (DEA) of TMCP. Shen et al 2012 [START_REF] Shen | Physisorption of an organometallic platinum complex on silica: an ab initio study[END_REF] have extensively examined the physisorption of TMCP on silica using ab initio DFT molecular orbital methods. It was found that the physisorption of TMCP is dependent on both (i) the orientation of TMCP relative to the surface, and (ii) the adsorption site on the substrate. The most stable configuration was found with the methylcyclopentadienyl and the methyl groups of the molecule oriented towards the surface. (Figure 1) Van der Waals corrections to the DFT calculations were crucial for the stabilization of the molecule on the surface, indicative of Van der Waals forces being crucial in the physisorption process.

Results

Three possible reaction schemes have been identified for the addition of an electron to TMCP either in the free state in vacuo or as the physisorbed on a surface. It has been generally assumed that processes involved in electron attachment or DEA are not materially affected by whether the TMCP is in the gas phase or physically absorbed on a surface. [START_REF] Thorman | The role of low-energy electrons in focused electron beam induced deposition: four case studies of representative precursors[END_REF][START_REF] Shen | Physisorption of an organometallic platinum complex on silica: an ab initio study[END_REF] TMCP + e TMCP• TMCP• TS Scheme 1

TMCP + 2e TMCP•• TMCP•• TS1 TMCP•• TS2 Scheme 2 TMCP• + e TMCP•• TMCP•• TS1 TMCP•• TS2 Scheme 3
The calculated structure of TMCP is shown in Table 1 and Figure 1 and it is almost identical with the known x-ray structures of trimethyl-h5-methylcyclopentadienylplatinum(IV) and of trimethyl-h5-cyclopentadienylplatinum(IV). [START_REF] Xue | Characterization of (methylcyclopentadienyl)trimethylplatinum and low-temperature organometallic chemical vapor deposition of platinum metal[END_REF][START_REF] Adamson | Crystal and molecular structure of cyclopentadienyl(trimethyl)platinum(IV), (π-C 5 H 5 )PtMe 3[END_REF] Scheme 1 shows the formation of the radical species TMCP• which has an adiabatic electron affinity (AEA) of 0.3 eV, which is in agreement with the findings of Engman 2012, 2013 [START_REF] Engmann | Gas phase low energy electron induced decomposition of the focused electron beam induced deposition (FEBID) precursor trimethyl (methylcyclopentadienyl) platinum(IV)[END_REF][START_REF] Engmann | Low Energy Electron Interactions with Precursor Molecules Relevant to Focused Electron Beam Induced Deposition[END_REF] who found that main DEA pathway for low energy electron is the loss of a methyl group from TMCP which occurs starting from 0 eV reaching a maximum at 0.5 eV. The free energy and configurational entropy of electron attachment at 298.15K to TMCP (ΔG eattach and TΔS eattach ) are -12.7 and 2.1 kcal/mol respectively. The transition state TS for the loss of a methyl group from the TMCP• radical species has been located by identifying the Pt---CH 3 stretching frequency vibration, and has an activation free energy ΔG ‡ TS and an activation configurational entropy TΔS ‡ TS of 16.2 and 2.7 kcal/mol. See Figure 2 and Table 1.

Figure 2. Schematic representation dissociative electron attachment of trimethylmethylcyclopentadienylplatinum(IV) on a surface

Scheme 2 and 3 show the addition of two electrons to either TMCP or an additional electron to the TMCP• species. Both TS have been identified, with structures shown in Figure 2 and Table 1. In both cases, the TS correspond to the loss of a methyl group by cleavage of the Pt---CH 3 bond. The ΔG ‡ TS are 27.7 and 17.4 kcal/mol for TS1 and TS2.

The notable feature of the TMCP• radical and the various TSs is the change from a h5-methylcyclopentadienylPt structure to a structure close to a quasi h1-methylcyclopentadienylPt structure (quasi-h1-MCP), as well as the lengthening of one Pt---CH 3 bond. Figure 2 shows the Pt-C bond to the near C atom of the quasi-h1-MCP ring and the Pt-C bond to the far C atom of the quasi-h1-MCP ring, with values given in Table 1.

Natural population analysis NPA charges as shown in Table 1 illustrate the changes in electron density associated with electron attachment, particularly to the Pt centre and the nearest C atom of the quasi h1-MCP ring, and the methyl group which eventually is lost when the Pt---CH 3 bond is ruptured.

Discussion

Assuming the results discussed above are representative of both gas phase and surface physisorption, then the data is consistent with the previous finding related to DEA of TMCP where loss of a methyl group is the principal mechanism. [START_REF] Thorman | The role of low-energy electrons in focused electron beam induced deposition: four case studies of representative precursors[END_REF][START_REF] Engmann | Gas phase low energy electron induced decomposition of the focused electron beam induced deposition (FEBID) precursor trimethyl (methylcyclopentadienyl) platinum(IV)[END_REF][START_REF] Engmann | Low Energy Electron Interactions with Precursor Molecules Relevant to Focused Electron Beam Induced Deposition[END_REF][START_REF] Wnuk | [END_REF] Shen 2012 [START_REF] Shen | Physisorption of an organometallic platinum complex on silica: an ab initio study[END_REF] has shown that TMCP adsorbed on a silica surface does not undergo any structural deformation. The important finding in this study is that radical species formed from TMCP have a Pt-quasi h1-MCP structure, as shown in Figure 2. This structure is far less stable than a Pt-h5-MCP structure for the intermediate species formed in DEA processes, as the quasi h1-mCP ring is notionally a one electron ligand compared to the 5 electrons of a h5-MCP ligand. The notional oxidation state changes from Pt(IV) in TMCP to a state somewhere around Pt (III) or lower. It is likely that the highly reactive Pt-quasi h1-MCP structure would undergo radical surface polymerization reactions associated with the h1-MCP ring, as the ring now has a reactive diene-like structure with a quasi σ Pt-C near bond instead of a h5 stabilized structure with a π Pt-C 5 bonding structure.

Engmann [START_REF] Engmann | Gas phase low energy electron induced decomposition of the focused electron beam induced deposition (FEBID) precursor trimethyl (methylcyclopentadienyl) platinum(IV)[END_REF] has shown that the onset of the [MeCpPtMe2] -formation in the DEA ion yield curves is to 0 eV, and the peak intensity of this fragment is close to 0.5 eV. Methane is the main gaseous product of low electron irradiation of TMCP. [START_REF] Engmann | Gas phase low energy electron induced decomposition of the focused electron beam induced deposition (FEBID) precursor trimethyl (methylcyclopentadienyl) platinum(IV)[END_REF][START_REF] Wnuk | [END_REF] TMCP was found to desorb from a surface when heated to room temperature prior to electron irradiation, but no compound was found to desorb from the surface after irradiation. Hence electron irradiation induces chemical changes to the physisorbed TMCP which produces a chemically bound deposit containing Pt and carbonaceous material. [START_REF] Wnuk | [END_REF] This observation is consistent with a quasi (h1-MCP)Pt(CH 3 ) 2 • intermediate radical species, or similar, being formed on the surface after electron irradiation.

It has been previously suggested that it is difficult to see how the platinum can contact the surface in the ground state of the molecule without some dislocation of the h5-MeCp ring to lower hapticity. [3] Shen [START_REF] Shen | Physisorption of an organometallic platinum complex on silica: an ab initio study[END_REF] has found that the most stable orientation has a Pt to silica surface distance of 2.87 Å, with no structural deformation of the absorbed TMCP. The calculated adsorption energy was -0.669 eV (or -15.4 kcal/mol), and orientation of the adsorbed TMCP was more important than the adorption site.

Thorman [START_REF] Thorman | The role of low-energy electrons in focused electron beam induced deposition: four case studies of representative precursors[END_REF] has discussed the importance of secondary electrons produced within the substrate at or close to the surface where they can induce fragmentation of the adsorbed precursor. Species such as TMCP•• can arise either after initial formation of TMCP• via secondary electrons or directly from the precursor TMCP.

The structures of the identified radical intermediates indicate that these species can undergo DEA processes while the precursor is still physically adsorbed on a surface, since the formation of a Pt-quasi h1-MCP structure and Pt---CH 3 bond elongation are both possible as these processes move the Pt and one methyl away from the surface, as shown in Figure 2.

The adsorption energy of -15.4 kcal/mol for TMCP is comparable to the calculated TS energies for the TMCP• TS, TMCP•• TS1 and TMCP•• TS2 of 16.2, 27.7 and 17.4 kcal/mol respectively, suggesting that these reactive species can form when adsorbed on a surface. Low energy secondary electrons can easily induce the formation of these species.

For FEBID purposes, the DEA precursor processes in reaction schemes 1, 2 or 3 can occur at very low incident electron energies (less than 1.2 eV).

Conclusions

The structures of the radical anion of trimethylmethylcyclopentadienylplatinum(IV), and the transition states involved in the cleavage of the Pt-CH 3 bond in the radical anions from one and two electron dissociative electron attachment have been identified. The radical anions and the various transition states have a Pt-quasi h1-methylcyclopentadienyl structure quite different from the h5-methylcyclopentadienyl structure of the starting precursor. Using the known most stable orientation of trimethylmethylcyclopentadienylplatinum(IV) adsorbed on a surface, it can be concluded that the processes involved in surface dissociative electron attachment can occur in the adsorbed precursor. For FEBID purposes, the DEA precursor processes involved in the identified reaction schemes can occur at very low incident electron energies (less than 1.2 eV).

Experimental Methods

All calculations were carried out using the Gaussian 09 package on optimised structures. Atomic charges were calculated from natural population analysis as atomic charges produced by NPA are not dependant on basis set selection. Calculations were at the WB97XD/6-311 ++ G** level of theory for all atoms except for Pt where the relativistic ECP SDD Stuttgart-Dresden basis set for transition metals was used. Comparison calculations were also conducted using the LanL2DZ and B3LYP functionals with the 6-311 ++ G** basis sets, and the wB97XD/aug-cc-cPDV level with similar results. It was found that the wB97XD/6-311 ++ G** and wB97XD/aug-cc-pVDZ combinations gave results for TMCP very close to the x-ray structures [START_REF] Xue | Characterization of (methylcyclopentadienyl)trimethylplatinum and low-temperature organometallic chemical vapor deposition of platinum metal[END_REF][START_REF] Adamson | Crystal and molecular structure of cyclopentadienyl(trimethyl)platinum(IV), (π-C 5 H 5 )PtMe 3[END_REF] and gave almost identical AEA values for the TMCP• species. The wB97XD functional is a long range corrected hybrid with damped dispersion correction which gives good results with non-covalent and covalent systems. [START_REF] Minenkov | The accuracy of DFT-optimized geometries of functional transition metal compounds: a validation study of catalysts for olefin metathesis and other reactions in the homogeneous phase[END_REF][START_REF] Peverati | The quest for a universal density functional: the accuracy of density functionals across a broad spectrum of databases in chemistry and physics[END_REF] Free energy ΔG and configurational entropy TΔS values (at 298.15K) were derived from vibrational thermochemistry analysis. Transition states were confirmed by identifying the singular negative vibration that related to stretching of Pt-CH 3 or Pt-h1-C near (MCP) bond. Adiabatic electron affinity in eV was calculated from the SCF difference method AEA = E(M)-E(M -) at the optimised geometry of M -in the gas phase. The AEA was confirmed as an optimised energy minima as there were no negative frequencies.

Figure 1 .

 1 Figure 1. Schematic representation of physisorption of the most stable orientation of trimethyl-h5-methylcyclopentadienylplatinum(IV) TMCP on a surface. The circles represent methyl groups, and the methylcyclopentadienyl ring is penta-hapto coordinated to the Pt atom.

Table 1 . Molecular properties of the various species identified in schemes 1-3.

 1 = trimethylmethylcyclopentadienylplatinum(IV), TMCP• = TMCP radical, TS = transition state, MCP = methylcyclopentadienyl group, pentahapto = h5, Pt-h5 C(near) = platinum to nearest C on methylcyclopentadienyl group (see Figure2), shortest distance or bond from Pt to nearest C of MCP = near, longest distance or bond from Pt to far C of MCP = far, free energy of electron attachment = ΔG eattach , configurational entropy of electron attachment at 298.15K = TΔS eattach , transition state free energy = ΔG ‡ TS , NPA charge = charge, adiabatic electron affinity = AEA

				TMCP	TMCP•	TMCP• TS	TMCP•• TS1 TMCP•• TS2
	h5 MCP C-C Å	1.41-1.43
	Pt-MCP Å	2.35-2.37 2.32 near	2.32 near	2.54 near	2.36 near
					3.75 far	3.51 far	4.22 far	3.92 far
	Pt-CH 3 Å	2.06	2.10, 2.07,	3.0, 2.05,	2.15, 2.04,	2.85, 2.05,
					2.09	2.02	2.12	2.01
	Pt charge	0.284	0.414	0.277	0.150	0.190
	Pt-CH 3 charge	-0.646	-0.798,	-0.590,	-0.991,	-1.132,
					-0.813,	-0.835,	-0.799,	-0.856,
					-0.804	-0.716	-0.958	-0.829
	Pt-h5 C(near)	-0.305	-0.470	-0.460 near	-0.419 near	-0.457 near
	charge		near
	AEA eV		0.3
	ΔG eattach kcal/mol		-12.7
	TΔS eattach kcal/mol		2.1
	ΔG ‡	TS kcal/mol			16.2	27.7	17.4
	TΔS ‡	TS kcal/mol			2.7	2.2	0.6

TMCP