Gwenaël Peltier
email: gwenael.peltier@umontpellier.fr

Accelerating invasions along an environmental gradient

Keywords: structured population, nonlocal reaction-diffusion equation, propagation, accelerating fronts AMS Subject Classifications: 35Q92, 45K05, 35B40, 35K57

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Accelerating invasions along an environmental gradient

Introduction

In this paper we study the propagation phenomena of the solution n(t, x, y) to the following nonlocal parabolic Cauchy problem

     ∂ t n -∂ xx n -∂ yy n = r(y -Bx) - R K(t,
x, y, y )n(t, x, y )dy n, t > 0, (x, y) ∈ R 2 , n(0, x, y) = n 0 (x, y), (x, y) ∈ R 2 .

(1)

We shall prove that if the initial data n 0 ≥ 0 has a heavy tail, in a sense to be precised later, then any solution of (1) either goes extinct, or spreads in the favorable direction y -Bx = 0 by accelerating.

Equation [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF] arises in some population dynamics models, see [START_REF] Mirrahimi | Dynamics of sexual populations structured by a space variable and a phenotypical trait[END_REF][START_REF] Prevost | Applications of partial differential equations and their numerical simulations of population dynamics[END_REF]. In this context n(t, x, y) represents a density of population at each time t ≥ 0, structured by a space variable x ∈ R and a phenotypical trait y ∈ R. This population is subject to four biological processes : migration, mutations, growth and competition. The diffusion operators ∂ xx n and ∂ yy n account for migration and mutations respectively. The growth rate of the population is given by r(y -Bx), where r is negative outside a bounded interval. This corresponds to a population facing an environmental gradient : to survive at location x, an individual must have a trait close to the optimal trait y opt = Bx with B > 0. Thus, for invasion to occur, the population has to adapt during migration. As a consequence, it is expected that the population, if it survives, remains confined in a strip around the optimal line y -Bx = 0, where r is typically positive. Finally, we consider a logistic regulation of the population density that is local in the spatial variable and nonlocal in the trait. In other words, we consider that, at each location, there exists an intra-specific competition (for e.g. food) between all individuals, regardless of their traits.

The well-posedness of a Cauchy problem very similar to [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF], but on a bounded domain, has been investigated in [START_REF] Prevost | Applications of partial differential equations and their numerical simulations of population dynamics[END_REF]Theorem I.1]. As mentioned in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF], we believe the arguments could be adapted in our context to show the existence of a global solution on an unbounded domain. A limiting argument would then provide the existence of solutions to [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF] in the whole domain, thanks to the estimates on the tails of the solutions obtained in Lemma 3.4. However, in this article, our main interest lies in the qualitative properties of the solutions.

Survival vs extinction.

As it is well known, the survival or extinction of the population (starting from a localized area) depends on the sign of the generalized principal eigenvalue λ 0 of the elliptic operator -∂ xx n -∂ yy n -r(y -Bx)n. If λ 0 > 0, the population goes extinct exponentially fast in time at rate -λ 0 . If λ 0 < 0, the model, which is of Fisher-KPP type [START_REF] Fisher | The wave of advance of advantageous genes[END_REF][START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF], satisfies the Hair-Trigger Effect : any nonnegative initial data n 0 ≡ 0 leads to the survival of the population and its spreading to the whole space. Since we are concerned with propagation results, we shall assume that λ 0 < 0 and n 0 ≡ 0 in the rest of this introduction.

The local case. When the competition term in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF] is replaced by a local (in x and y) regulation, the equation satisfies the comparison principle and, moreover, one can assume B = 0 without loss of generality (through a rotation of coordinates). This yields to one of the models considered in [START_REF] Berestycki | Traveling fronts guided by the environment for reaction-diffusion equations[END_REF], where the authors show the existence of travelling waves ϕ(x -ct, y) solutions of the equation for speeds c greater than or equal to a critical value c * > 0. Besides, there exists a unique positive stationary solution, which depends only on y ∈ R, denoted here by S(y), and all travelling waves connect the state S(y) when x → -∞ to zero when x → +∞.

One of the main results in [START_REF] Berestycki | Traveling fronts guided by the environment for reaction-diffusion equations[END_REF] concerns the Cauchy problem. When the initial data is compactly supported and satisfies n 0 (x, y) ≤ S(y), the solution n(t, x, y) converges locally uniformly in x towards S(y), while propagating at speed c * in both directions x → ±∞.

The nonlocal but decoupled (B = 0) case. It is worth mentioning that the model (1) when B = 0 has been analyzed in [START_REF] Berestycki | Propagation in a non local reaction diffusion equation with spatial and genetic trait structure[END_REF]. In this context, one can decouple variables x and y, leading to a sequence of scalar Fisher-KPP equations, each obtained by projection on each eigenfunction of the elliptic operator -∂ yy -r(y). This technique [START_REF] Berestycki | Propagation in a non local reaction diffusion equation with spatial and genetic trait structure[END_REF] allows to prove, again, the existence of fronts ϕ(x -ct, y) for speeds c ≥ c * = 2 √ -λ 0 , as well as the survival of the population and its spreading at speed c * for compactly supported initial data.

Akin to the local model in [START_REF] Berestycki | Traveling fronts guided by the environment for reaction-diffusion equations[END_REF], there exists a unique positive stationary state S = S(y) towards which the solution n(t, x, y) converges. When K ≡ 1, the state S is actually the principal eigenfunction associated to λ 0 (with a unique choice of a multiplicative constant for S to be a positive steady state). For example, when r is quadratic, S is gaussian since it satisfies the equation of the harmonic oscillator.

The model [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]. The propagation phenomenon of problem (1) has been investigated in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF][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]. Notice that the authors in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF] also allow the environmental gradient to be shifted (say by Global Warming) at a given forced speed. Since B = 0, the decoupling argument [START_REF] Berestycki | Propagation in a non local reaction diffusion equation with spatial and genetic trait structure[END_REF] cannot be invoked here, which makes the analysis more involved. As far as travelling waves are concerned, problem (1) admits fronts of the form ϕ(x -ct, y -Bx) only for speeds c ≥ c * = 2 -λ0 1+B 2 . However, the construction of those waves relies on a topological degree argument [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] and little is known about their behavior for x → -∞, with y -Bx being constant. This is caused by the presence of a nonlocal competition term in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF], which prevents the equation to enjoy the comparison principle, see also [START_REF] Alfaro | Rapid traveling waves in the nonlocal Fisher equation connect two unstable states[END_REF][START_REF] Berestycki | The non-local Fisher-KPP equation: traveling waves and steady states[END_REF] for similar issues related to the scalar nonlocal Fisher-KPP equation.

The results [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF] for the Cauchy problem (1) are the following : if n 0 is compactly supported, the total population at (t, x), given by N (t, x) = R n(t, x, y)dy, spreads at speed c * . While the convergence of the solution towards a possible steady state remains an open question, it is worth pointing out that the population density remains mainly concentrated around the optimal trait y = Bx.

Accelerating invasions in the one-dimensional case. Before going further, let us here consider the scalar Fisher-KPP equation [START_REF] Fisher | The wave of advance of advantageous genes[END_REF][START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF], say, for simplicity,

∂ t u -∂ xx u = ru(1 -u), t > 0, x ∈ R,
for some r > 0. A result from Hamel and Roques [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] shows that if the initial data displays a heavy tail, i.e. decays more slowly than any exponentially decaying function, then the population invades the whole space by accelerating. This is in contradistinction with the well-studied case of exponentially bounded initial data, where the level sets of the solution spread at finite speed, see [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF][START_REF] Uchiyama | The behavior of solutions of some non-linear diffusion equations for large time[END_REF]. The authors also derive sharp estimates of the location, at large time, of these level sets [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF].

Accelerating invasions in related models. Let us mention that acceleration also occurs for compactly supported initial data if the diffusion term is replaced with a fractional laplacian -(-∂ xx ) α u, 0 < α < 1, see [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF], or with a convolution term J * u -u where the kernel J = J(x) admits a heavy tail in both directions x → ±∞ [START_REF] Garnier | Accelerating solutions in integro-differential equations[END_REF]. The latter case corresponds to models of population dynamics with long-distance dispersal.

The so-called cane-toad equation proposed in [START_REF] Benichou | Front acceleration by dynamic selection in Fisher population waves[END_REF] is another biological invasion model where acceleration may occur. When the trait space is unbounded, propagation of the level sets of order O(t 3/2 ) has been predicted in [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF]. This was then proved rigorously in [START_REF] Berestycki | Existence of self-accelerating fronts for a non-local reactiondiffusion equations[END_REF] with a local competition term, and in [START_REF] Bouin | Super-linear spreading in local and non-local cane toads equations[END_REF] for both local and nonlocal (in trait) competition, using probabilistic and analytic arguments respectively. Notice however that acceleration is not induced here by initial heavy tails, but by a phenotype-dependent term before the spatial diffusion.

Accelerating invasions in model [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]. Our aim is to study accelerating invasions in model [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF], that is to determine if acceleration occurs if n 0 displays a "heavy tail", a notion that needs to be precised in the two-dimensional framework. First, we prove that if n 0 has a "heavy tail" in the favorable direction y -Bx = 0, then acceleration of the invasion occurs. Second, we derive precise estimates for the large-time location of the level sets of the solution. Finally, we also address the case where the "heavy tail" of n 0 is not positioned along the direction y -Bx = 0: in this case, since ill-directed, the heavy tail does not induce acceleration.

Assumptions and main results

2.1 Functions r, K and n 0 Throughout the paper, we make the following assumption.

Assumption 2.1. The function r(•) ∈ L ∞ loc (R) is confining, in the sense that, for all δ > 0, there exists R > 0 such that r(z) ≤ -δ, for almost all z such that |z| ≥ R.

Additionally, there exists r max > 0 such that r(z) ≤ r max almost everywhere. The function

K ∈ L ∞ ((0, ∞) × R 3 ) satisfies k -≤ K ≤ k + , a.e. on (0, +∞) × R 3 , (3)
for some 0 < k -≤ k + . Moreover, the initial data n 0 is non identically zero, and there exist C 0 > 0 and κ 0 > 0 such that

0 ≤ n 0 (x, y) ≤ C 0 e -κ0|y-Bx| , for almost all (x, y) ∈ R 2 . ( 4
)
An enlightening example of such function r is given by r

(z) = 1 -Az 2 , hence r(y -Bx) = 1 -A(y -Bx) 2 , (5)
for some A > 0. Notice that the width of the strip where r is nonnegative, that is the favorable region, is

2 √ A(1+B 2 )
(see Figure 1). As a result, both parameters A and B > 0 play a critical role to determine whether the population goes extinct or survives. Condition (4) allows us to obtain estimates of the tails of n(t, x, y) in the direction y -Bx → ±∞ as given by Lemma 3.4. Note that condition (4) is not the aforementioned heavy tail condition, for any compactly supported function satisfies it. Before stating our heavy tail condition, we first need to consider some spectral problems.

Some eigenelements

As in [1, Section 4], rather than working in the (x, y) variables, let us write

n(t, x, y) = v(t, X, Y ),
where X (resp. Y ) represents the direction of (resp. the direction orthogonal to) the optimal trait y = Bx, that is

X = x + By √ 1 + B 2 , Y = y -Bx √ 1 + B 2 . (6)
In these new variables, equation ( 1) is recast

∂ t v -∂ XX v -∂ Y Y v = r(Y ) - R K(t, χ, ψ, y )v (t, χ, ψ) dy v, (7)
where we use the shortcuts

r(Y ) := r 1 + B 2 Y , χ = χ(X, Y, y ) := X-BY √ 1+B 2 + By √ 1 + B 2 , ψ = ψ(X, Y, y ) := -B X-BY √ 1+B 2 + y √ 1 + B 2 .
We also note v 0 (X, Y ) = n 0 (x, y) the initial data in the new variables. Next, as recalled in subsection 3.2, we are equipped with a generalized principal eigenvalue λ 0 ∈ R and a generalized principal eigenfunction

Γ 0 ∈ H 2 loc (R) satisfying -∂ Y Y Γ 0 (Y ) -r(Y )Γ 0 (Y ) = λ 0 Γ 0 (Y ) for all Y ∈ R, Γ 0 > 0, ||Γ 0 || L ∞ (R) = 1. (8)
It is worth noting that, in the particular case where r is given by (5), expression (8) corresponds to the harmonic oscillator, for which these eigenelements can be explicitly computed as

λ 0 = A(1 + B 2 ) -1, Γ 0 (Y ) = exp - 1 2 A(1 + B 2 )Y 2 . (9)
Finally, for R > 0, let us consider λ R 0 , Γ R 0 (Y ) the principal eigenelements solving the Dirichlet problem on

(-R, R)          -∂ Y Y Γ R 0 (Y ) -r(Y )Γ R 0 (Y ) = λ R 0 Γ R 0 (Y ) for Y ∈ (-R, R), Γ R 0 (Y ) = 0 for Y = ±R, Γ R 0 (Y ) > 0 for Y ∈ (-R, R), ||Γ R 0 || ∞ = 1. (10)
As recalled in Proposition 3.3, there holds λ R 0 λ 0 as R → +∞. ) is assumed to be greater than u 0 (X), that is a one-dimensional heavy tail in the X direction. The dotted lines delimit the area where r > 0.

The heavy tail condition

We can now turn to the two-dimensional heavy tail condition. First, we recall or introduce some definitions for one-dimensional functions. A function w : R → R is said to be asymptotically front-like if

lim inf -∞ w > 0, w > 0, lim +∞ w = 0. (11)
A positive function w : R → R is said to have a heavy tail in +∞ if lim x→+∞ w(x)e εx = +∞, ∀ε > 0, (1D Heavy Tail).

Typical examples are "lighter heavy tails" [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF], algebraic tails [START_REF] Bouin | Super-linear spreading in local and non-local cane toads equations[END_REF], and "very heavy tails" [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF], that is w(x) ∼ Ce -bx a , as x → +∞, with C, b > 0 and a ∈ (0, 1),

w(x) ∼ Cx -a , as x → +∞, with C, a > 0, (13)
w(x) ∼ C(ln x) -a , as x → +∞, with C, a > 0.

We now state our two-dimensional heavy tail condition for equation [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]. Note that this condition is expressed in the new variables, thus it applies to v 0 . Assumption 2.2 (2D heavy tail condition). Let us consider the new coordinates (X, Y ) given by [START_REF] Berestycki | Traveling fronts guided by the environment for reaction-diffusion equations[END_REF]. The initial data v 0 (X, Y ) = n 0 (x, y) is such that there exists u 0 ∈ L ∞ (R) satisfying (11)- [START_REF] Berestycki | Existence of self-accelerating fronts for a non-local reactiondiffusion equations[END_REF], so that

v 0 (X, Y ) ≥ u 0 (X)1 [σ-,σ+] (Y ), (2D Heavy Tail). ( 16
)
for some reals σ -< σ + .

Let us emphasize that we do not assume that 0 ∈ (σ -, σ + ), meaning the initial data may not overlap the optimal trait line (see Figure 1). Moreover, the interval (σ -, σ + ) may not only be arbitrarily far from zero, but also arbitrarily small. The key assumption is the correct orientation of the heavy tail, that is in the direction X → +∞, as highlighted by subsection 2.6. In the survival case λ 0 < 0 and under Assumption 2.2, we shall prove that the solution of (1) is accelerating.

We now aim at providing a precise estimate of the location of the level sets at large times. To do so, we first introduce the following definition.

Definition 2.3. A function w is said to satisfy the condition (Q) if          w ∈ L ∞ (R) and is uniformly continuous, lim inf -∞ w > 0, w > 0, lim +∞ w = 0, ∃ξ 0 ∈ R such that w is C 2 and nonincreasing on [ξ 0 , +∞), w (x) = o(w(x)) as x → +∞, (Q)
where o denotes the Landau symbol "little-o".

Notice that any function w satisfying (Q) also satisfies w (x) = o(w(x)) as x → +∞, and thus displays a one-dimensional heavy tail in +∞, see [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF]. For the scalar Fisher-KPP equation, when the initial data satisfies (Q), the authors in [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] derived precise estimates on the location of the level sets of the solution. In our context, we make the following assumption on the initial data.

Assumption 2.4 ((Q)-Initial bounds). Let us consider the new coordinates (X, Y ) given by [START_REF] Berestycki | Traveling fronts guided by the environment for reaction-diffusion equations[END_REF]. The initial data v 0 (X, Y ) = n 0 (x, y) is such that there exist functions u 0 , u 0 satisfying (Q) so that

u 0 (X)1 [σ-,σ+] (Y ) ≤ v 0 (X, Y ) ≤ u 0 (X)Γ 0 (Y ), (17)
for some reals σ -< σ + .

In particular, if the initial data satisfies Assumption 2.4, then it satisfies Assumption 2.2. As far as the Y direction is concerned, when r is of the form (5), the eigenfunction Γ 0 is given by ( 9), so that [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] amounts to a gaussian control on the initial data. In the general case of a confining growth function [START_REF] Alfaro | Rapid traveling waves in the nonlocal Fisher equation connect two unstable states[END_REF], one can prove that Γ 0 (Y ) decays at least exponentially when |Y | → +∞, see subsection 3.2. Under Assumption 2.4, we shall derive some precise estimates on the large-time position of the level sets, see Theorem 2.7.

The extinction case

As we shall see, under Assumption 2.1, the population either goes extinct or survives depending on the sign of the principal eigenvalue λ 0 . In this short section we simply expose the result of [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], which covers the case λ 0 > 0.

Proposition 2.5 (Extinction case [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]). Assume λ 0 > 0. Let r, K, n 0 satisfy Assumption 2.1. Suppose that there is k > 0 such that

n 0 (x, y) ≤ kΓ 0 y -Bx √ 1 + B 2 .
Then any global nonnegative solution of (1) satisfies

n(t, x, y) ≤ kΓ 0 y -Bx √ 1 + B 2 e -λ0t , (18)
which implies ||n(t, •, •)|| L ∞ (R 2 ) = O(e -λ0t
), that is an exponentially fast extinction.

The proof of Proposition 2.5 is elementary as n(t, x, y) and the right-hand side of ( 18) are respectively subsolution and supersolution of the parabolic operator

∂ t n -∂ xx n -∂ yy n -r(y -Bx)n.
The maximum principle yields the result.

Main result : acceleration in the invasion case

We now investigate the case where the principal eigenvalue λ 0 is negative. In order to capture the spreading speed of the population in the space variable, we look at the evolution of the total population in (t, x), regardless of their trait. Thus, for any µ > 0, we define the level set of n by

E n µ (t) = x ∈ R R n(t, x, y)dy = µ .
Let us emphasize again that, because of the nonlocal competition term, problem (1) does not enjoy the comparison principle. In such situation, and as mentioned in the introduction, the behavior "behind the front" is typically out of reach, see [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF][START_REF] Alfaro | Rapid traveling waves in the nonlocal Fisher equation connect two unstable states[END_REF][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 | The non-local Fisher-KPP equation: traveling waves and steady states[END_REF][START_REF] Faye | Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach[END_REF]. For such a reason, we are mainly interested in the spreading properties of E n µ (t) for small values µ.

Let us recall that under Assumption 2.1, if λ 0 < 0 and if n 0 ≡ 0 has compact support, then the population survives and the solution propagates at speed c * = 2 -λ0 1+B 2 , see [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]Theorem 4.2]. Our first result shows that there is acceleration when, instead of being compactly supported, the initial data admits a heavy tail in the X direction, in the sense given by Assumption 2.2.

Theorem 2.6 (2D initial heavy tail implies acceleration). Assume λ 0 < 0. Let r, K, n 0 satisfy Assumptions 2.1 and 2.2. Let n be any global nonnegative solution of (1). Then there exists β > 0 such that for any µ ∈ (0, β), there holds

1 t min E n µ (t) → +∞, as t → +∞.
In other words, if the initial data is greater than or equal to a front-like function with a heavy tail in the direction X → +∞, the solution is accelerating. We will only give a sketch of the proof in subsection 4.4, as it is similar to the proof of Theorem 2.7 below.

We now state our main result, namely Theorem 2.7, which is an accurate estimate of the position of the accelerating level sets under Assumption 2.4. In the rest of this article, for any function f : R → R, we denote

f -1 (a) the set {x ∈ R | f (x) = a}.
Theorem 2.7 (Asymptotic position of the accelerating level sets). Assume λ 0 < 0. Let r, K, n 0 satisfy Assumptions 2.1 and 2.4. Let R > 0 be large enough such that λ R 0 < 0 (see subsection 2.2). Let n be any global nonnegative solution of (1).

Then there exists β > 0 so that for any µ ∈ (0, β), ε ∈ (0, -λ R 0 ), Γ > 0 and γ > 0, there exists T * = T * µ,ε,γ,Γ,R ≥ 0 such that for all t ≥ T * , the set E n µ (t) is nonempty, compact, and satisfies

E n µ (t) ⊂ 1 √ 1 + B 2 min u -1 0 Γe -(-λ R 0 -ε)t , max u -1 0 γe -(-λ0+ε)t . (19)
Let us make some comments on this theorem. Observe first that for t large enough there holds

Γe -(-λ R 0 -ε)t ∈ 0, lim inf -∞ u 0 , γe -(-λ0+ε)t ∈ 0, lim inf -∞ u 0 , thus the sets u -1 0 Γe -(-λ R 0 -ε)t and u -1 0 γe -(-λ0+ε
)t are non-empty and bounded. Additionally, Assumption 2.4 implies that u 0 ≤ Cu 0 with C = min (σ-,σ+) Γ 0 > 0. In conjunction with λ 0 < λ R 0 , it follows that for t possibly even larger there holds

min u -1 0 Γe -(-λ R 0 -ε)t < max u -1 0 γe -(-λ0+ε)t ,
giving a meaning to [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF]. Next, notice that, given any two values µ and µ in (0, β), both level sets E n µ (t) and E n µ (t) are included in the same interval given by expression [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF]. As a consequence, Theorem 2.7 implies that for any ε ∈ (0, -λ R 0 ) and positive real numbers γ and Γ, there holds

lim inf t→+∞ inf x≤(1+B 2 ) -1/2 min u -1 0 Γe -(-λ R 0 -ε)t R n(t, x, y)dy ≥ β, lim t→+∞ sup x≥(1+B 2 ) -1/2 max u -1 0 (γe -(-λ 0 +ε)t ) R n(t, x, y)dy = 0.
The upper bound of E n µ (t) in ( 19) is valid for all levels µ, and only requires the upper bound of v 0 in Assumption 2.4. However, the lower bound of E n µ (t) is valid for levels µ < β, and only requires the lower bound of v 0 in Assumption 2.4. Also note that the lower bound in [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] leads to 1 t min E n µ (t) → +∞ when t → +∞, thus we recover the acceleration.

We now give a sketch of the proof. The upper bound is much easier to prove since the nonlocal term is nonnegative. One constructs a supersolution of the form φ(t, X)Γ 0 (Y ) where φ satisfies ∂ t φ -∂ xx φ = (-λ 0 )φ with φ(0, •) displaying a heavy tail. The upper bound of Lemma 3.1 is still valid in this case, which leads to the result with an adequate control of the tails of n.

The proof of the lower bound is much more involved. Suppose first that

[-R, R] ⊂ [σ -, σ + ].
Then, after bounding the nonlocal term with a refinement of a Harnack inequality, we construct a subsolution of the form w(t, X, Y ) = u(t, X)Γ R 0 (Y ) where u satisfies the Fisher-KPP equation. Therefore applying Lemma 3.1 allows us to conclude. Note that this might not be a subsolution if R were too small, leading to λ R 0 being possibly nonnegative. In the general case we may have

[-R, R] ⊂ [σ -, σ + ]. In that event we construct a subsolution v(t, X, Y ) for t ∈ [0, 1], such that v(1, X, Y ) ≥ ρu 0 (X)Γ R 0 (Y ) on R × [-R, R] for some ρ > 0.
Then on [1, +∞) we consider a subsolution of the same form as w, which gives the result.

In particular, to prove acceleration under the hypothesis v 0 (X, Y ) ≥ u 0 (X)1 [σ-,σ+] (Y ), we have to use Γ R 0 instead of Γ 0 in order to construct the subsolution. Because of this, we obtain -λ R 0 in the lower bound of [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF]. Had we supposed the stronger hypothesis v 0 (X, Y ) ≥ u 0 (X)Γ 0 (Y ) instead, we could replace -λ R 0 with -λ 0 and take any ε ∈ (0, -λ 0 ). Let us also mention that β tends to zero as R → +∞, leading to a trade-off. Indeed, a large value of R provides a more precise location of the level sets, but also reduces the range of level sets being located.

We conclude this section by applying Theorem 2.7 in the cases where the functions u 0 and u 0 are of the forms ( 13)- [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF]. For simplicity, we only consider the lower bound.

Example 2.8. Suppose there exist X 0 , b > 0 and a ∈ (0, 1) such that u 0 (X) = Ce -bx a on [X 0 , +∞). Then if we select Γ = C, the lower bound in [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] becomes

min E n µ (t) ≥ 1 √ 1 + B 2 1 b (-λ R 0 -ε)t 1/a
, meaning the total population spreads with at least algebraic, superlinear speed.

Example 2.9. Suppose there exist X 0 , C, a > 0 such that u 0 (X) = CX -a on [X 0 , +∞). Then if we select Γ = C, the lower bound in [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] becomes

min E n µ (t) ≥ 1 √ 1 + B 2 exp 1 a (-λ R 0 -ε)t ,
thus the total population spreads with at least exponential speed.

Example 2.10. Suppose there exist X 0 > 1 and C, a > 0 such that u 0 (X) = C(ln x) -a on [X 0 , +∞). Then if we select Γ = C, the lower bound in [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] becomes

min E n µ (t) ≥ 1 √ 1 + B 2 exp exp 1 a (-λ R 0 -ε)t ,
that is the total population spreads with at least superexponential speed.

When the heavy tail is ill-directed

When the initial data admits a heavy tail in direction X → +∞, in the sense of Assumption 2.2, Theorem 2.6 proves the acceleration of the propagation. It is worth wondering if acceleration still occurs when considering heavy tail initial condition in a different direction than X → +∞. For the sake of clarity, we only consider the direction x → +∞, but the proof is easily adapted to any direction

X = x + B y √ 1 + B 2 → +∞, with B = B.
Theorem 2.11 (Ill-directed heavy tail prevents acceleration). Suppose λ 0 < 0. Suppose r, K satisfy Assumption 2.1. Suppose n 0 satisfies 0

≤ n 0 (x, y) ≤ u 0 (x)1 [σ-,σ+] (y), (20)
where u 0 ∈ L ∞ (R) and σ -< σ + . Let n be any global nonnegative solution of (1).

Then if we define c * := 2 -λ0 1+B 2 , there holds

lim sup t→+∞ R n(t, ct, y)dy = 0, ∀|c| > c * . ( 21
)
Notice that u 0 appearing in [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] is only assumed to be bounded. In particular, even if u 0 ≡ cst > 0, a much stronger assumption than a heavy tail, acceleration does not occur because of ill-orientation.

Before going further, let us mention that [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]Theorem 4.2] shows that, when r, K satisfy Assumption 2.1 and n 0 ≡ 0 is compactly supported, the spreading speed of the population is exactly c * , in the sense that

lim sup t→+∞ R n(t, ct, y)dy = 0, ∀|c| > c * , ( 22
)
lim inf t→+∞ R n(t, ct, y)dy ≥ β, ∀|c| < c * , (23)
for some β > 0 that may depend on c when |c| → c * . A consequence of Theorem 2.11 is that if n 0 ≡ 0 satisfies [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF], the population spreads exactly at speed c * , in the sense given by ( 22)- [START_REF] Prevost | Applications of partial differential equations and their numerical simulations of population dynamics[END_REF]. To prove that (23) holds, one cannot invoke the comparison principle because of the nonlocal term in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]. However, an essential element of the proof of Theorem 2.11 is the control of the tails (50). Using it, one can adapt the proof of [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]Theorem 4.2] to show that ( 23) is valid.

Outline of the paper. The rest of this article is organized as follows. In Section 3 we provide some materials necessary to the proof, that is an equivalent of Theorem 2.7 for the scalar Fisher-KPP equation, some properties of functions satisfying (Q), some principal eigenelements of elliptic operators, some estimates on the tails of n as well as a refinement of the parabolic Harnack inequality. Section 4 is devoted to the proof of Theorem 2.7, and presents a sketch of the proof of Theorem 2.6. Finally, Section 5 addresses the proof of Theorem 2.11.

Preliminaries

Acceleration in the scalar Fisher-KPP equation

We consider here the Fisher-KPP equation with a logistic reaction term :

∂ t u -∂ xx u = Λu(1 -u), t > 0, x ∈ R, u(0, x) = u 0 (x), x ∈ R, (24)
with Λ > 0. The function u 0 : R → [0, 1] is assumed to be uniformly continuous and asymptotically front-like, in the sense of [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed Ithe case of the whole space[END_REF], and to display a (one-dimensional) heavy tail in +∞, in the sense of [START_REF] Berestycki | Existence of self-accelerating fronts for a non-local reactiondiffusion equations[END_REF]. Under these assumptions, Hamel and Roques [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] proved that the level sets of u, defined for η ∈ (0, 1) by

E η (t) := {x ∈ R | u(t, x) = η} ,
propagate to the right by accelerating, that is min E η (t)/t → +∞ as t → +∞. Under assumption (Q), they also provide sharp estimates on the position of the level sets. This result, which will be an essential tool for our analysis, reads as follows.

Lemma 3.1 (See [19, Theorem 1.1]). Let w satisfy (Q), see Definition 2.3. Let u(t, x) be the solution of [START_REF] Uchiyama | The behavior of solutions of some non-linear diffusion equations for large time[END_REF] with initial condition u 0 := w/||w|| ∞ . Then for any η ∈ (0, 1), ε ∈ (0, Λ), γ > 0 and Γ > 0, there exists

T = T η,ε,γ,Γ,Λ ≥ 0 so that E η (t) ⊂ w -1 γe -(Λ+ε)t , Γe -(Λ-ε)t , ∀t ≥ T.
In the sequel, in order to prove Theorem 2.7, we shall construct some sub-and super-solutions of the form u(t, X)Γ 0 (Y ) where u(t, X) solves [START_REF] Uchiyama | The behavior of solutions of some non-linear diffusion equations for large time[END_REF] or a linear version of [START_REF] Uchiyama | The behavior of solutions of some non-linear diffusion equations for large time[END_REF]. Then, for the estimates on E η (t) provided by Lemma 3.1 to transfer to estimates on E n µ (t) = x ∈ R | R n(t, x, y)dy = µ , we shall need a technical result which we now state. Proposition 3.2. Let w satisfy (Q). Then there exists ξ 1 > ξ 0 such that w(x) > w(ξ 1 ) for any x < ξ 1 .

In addition, for any 0 < a < b, Γ a > 0, Γ b > 0 and χ > 0, there exists t * ≥ 0 such that

min w -1 Γ a e -at + χ ≤ min w -1 Γ b e -bt , ∀t ≥ t * , ( 25
) max w -1 Γ a e -at + χ ≤ max w -1 Γ b e -bt , ∀t ≥ t * . ( 26
)
Proof. Set m := inf (-∞,ξ0] w. Since lim inf -∞ w > 0 and w > 0, it is easy to check that m > 0. Now, since lim +∞ w = 0, there exists x + > ξ 0 such that w(x + ) < m. Since w(x + ) > 0 and lim +∞ w = 0, we can find ξ 1 ≥ x + satisfying w (ξ 1 ) < 0. Finally, as w (ξ 1 ) < 0 and w is nonincreasing on [ξ 0 , +∞), we can readily check that, for any x < ξ 1 , there holds

w(ξ 1 ) ≤ w(x + ) < m ≤ w(x), if x ≤ ξ 0 , < w(x), if x ∈ (ξ 0 , ξ 1 ),
which proves the first assertion.

We now turn to the second assertion. We only give a proof of (25), seeing as the proof of ( 26) is identical. In the first place, set t ≥ 0 large enough such that for any t ≥ t

Γ a e -at , Γ b e -bt ∈ (0, m), ∀t ≥ t,
hence the sets w -1 (Γ a e -at ), w -1 Γ b e -bt are well-defined and compact. Next, suppose by contradiction that there exist 0 < a < b and positive constants Γ a , Γ b , χ such that

∀t * ≥ t, ∃t ≥ t * min w -1 Γ a e -at + χ > min w -1 Γ b e -bt .
As a result, we can construct an increasing sequence (t n ) n such that lim n→+∞ t n = +∞ and the above inequality holds for t = t n . In particular, there exists N ∈ N such that for any n ≥ N , there holds Γ a e -atn ≤ w(ξ 1 ), whence min w -1 Γ a e -atn ≥ ξ 1 > ξ 0 .

Meanwhile, since b > a, we can select N possibly even larger so that for any n ≥ N there holds

Γ b e -btn ≤ Γ a 2 e -atn < Γ a e -atn .
Both assertions imply, by monotony of w on [ξ 0 , +∞), that min w -1 Γ a e -atn < min w -1 Γ b e -btn . Now, from the mean value theorem, there is θ n ∈ min w -1 (Γ a e -atn ) , min w -1 Γ b e -btn such that

w (θ n ) = Γ a e -atn -Γ b e -btn min w -1 (Γ a e -atn ) -min w -1 (Γ b e -btn ) < 0, therefore |w (θ n )| ≥ Γ a e -atn -Γ b e -btn χ ≥ Γ a e -atn 2χ . (27)
However, since w satisfies (Q), there holds w (x) = o(w(x)) as x → +∞. As a consequence, there exists x χ ∈ R such that |w (x)| ≤ 1 4χ w(x) for any x ≥ x χ . As lim n→+∞ θ n = +∞, we obtain θ n > x χ for n large enough. For such n, we derive the following inequality :

|w (θ n )| ≤ 1 4χ w(θ n ) ≤ 1 4χ
Γ a e -atn , which contradicts (27). Thus (25) holds.

Some eigenvalue problems

We present here some useful eigenelements. This subsection is quoted from [1, Subsection 2.1], which was based on the results of [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF][START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed Ithe case of the whole space[END_REF]. The theory of generalized principal eigenvalue has been developed in [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF], and is well adapted to our problem when r, thus r, is bounded. Following [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF], we can then define, for r ∈ L ∞ (Ω) and Ω ⊂ R a (possibly unbounded) nonempty domain, the generalized principal eigenvalue

λ(r, Ω) := sup λ ∈ R | ∃φ ∈ H 2 loc (Ω), φ > 0, φ (Y ) + (r(Y ) + λ)φ(Y ) ≤ 0 . ( 28
)
As shown in [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF], if Ω is bounded, λ(r, Ω) coincides with the Dirichlet principal eigenvalue λ D , that is the unique real number such that there exists φ defined on Ω (unique up to multiplication by a scalar) satisfying

     -φ (Y ) -r(Y )φ(Y ) = λ D φ(Y ) a.e. in Ω, φ > 0 on Ω, φ = 0 on ∂Ω.
Note that λ(r, Ω) ≤ λ(r, Ω ) if Ω ⊃ Ω . The following proposition shows that λ(r, Ω) can be obtained as a limit of increasing domains. Proposition 3.3. Assume that r ∈ L ∞ (R). For any nonempty domain Ω ⊂ R and any sequence of nonempty domains

(Ω n ) n∈N such that Ω n ⊂ Ω n+1 , ∪ n∈N Ω n = Ω,
there holds λ(r, Ω n ) λ(r, Ω) as n → +∞. Furthermore, there exists a generalized principal eigenfunction, that is a positive function

Γ ∈ H 2 loc (R) such that -Γ (Y ) -r(Y )Γ(Y ) = λ(r, Ω)Γ(Y ),
a.e. in Ω.

Since our growth function r is only assumed to be bounded from above, we extend definition (28) to functions r in L ∞ loc (Ω) such that r ≤ r max on Ω, for some r max > 0. The set

Λ(r, Ω) := λ ∈ R | ∃φ ∈ H 2 loc (Ω), φ > 0, φ (Y ) + (r(Y ) + λ)φ(Y ) ≤ 0
is not empty since Λ(max(r, -r max ), Ω) ⊂ Λ(r, Ω), and is bounded from above, thanks to the monotony property of Ω → Λ(r, Ω). Finally, going back to the proof of [7, Proposition 4.2], we notice that Proposition 3.3 remains valid under the weaker assumption r ∈ L ∞ loc (Ω) is bounded from above. It follows from the above discussion that we are equipped with the generalized principal eigenvalue λ 0 ∈ R and a generalized principal eigenfunction

Γ 0 ∈ H 2 loc (R) such that      -Γ 0 (Y ) -r(Y )Γ 0 (Y ) = λ 0 Γ 0 (Y ) a.e. in Ω, Γ 0 > 0 on Ω, ||Γ 0 || L ∞ (R) = 1.
Let us also mention that, given that r satisfies Assumption 2.1, the function Γ 0 decays at least exponentially as |Y | → +∞. This result holds by using the comparison principle on {|Y | > Y 0 } with a supersolution of the form Ce -a|Y | , with Y 0 large enough (so that r + λ 0 ≤ -ε for some ε > 0), a small enough and C large enough.

Preliminary estimates

The following lemma gathers preliminary results from [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF], with n 0 satisfying Assumption 2.1 instead of being compactly supported. The proof of the following a priori estimates is easily adapted from [1, Lemmas 2.3 and 2.4] and is therefore omitted (see also the proof of Lemma 5.1).

Lemma 3.4 (Some a priori estimates). Let r, K, n 0 satisfy Assumption 2.1. Then, there exist N ∞ > 0, C > 0 and κ > 0 such that any global nonnegative solution of (1) satisfies

R n(t, x, y)dy ≤ N ∞ , (29)
n(t, x, y) ≤ Ce -κ|y-Bx| , (30)
for all t ≥ 0, x ∈ R, y ∈ R.

While Lemma 3.4 provides us with a uniform bound, we need more precise estimates on the nonlocal term R n(t, x, y)dy. To do so, we invoke a refinement of the parabolic Harnack inequality, as exposed in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]. For any (t, x) ∈ (0, +∞) × R N with N ≥ 1, we consider a solution u(t, x) of the following linear parabolic equation

∂ t u(t, x) - N i,j=1 a i,j (t, x)∂ xixj u(t, x) - N i=1 b i (t, x)∂ xi u(t, x) = f (t, x)u(t, x), t > 0, x ∈ R N (31)
where the coefficients are bounded, and (a i,j ) i,j=1,...,N is uniformly elliptic.

Theorem 3.5 (A refinement of the Harnack inequality [1, Theorem 2.7]).

Assume that all the coefficients (a i,j ) i,j=1,...,N , (b i ) i=1,...,N , f belong to L ∞ loc ((0, +∞) × R N ), and that (a i,j ) is uniformly positive definite on R N . Assume there exists K > 0 such that, for all 1 ≤ i, j ≤ N ,

a i,j (t, x) ≤ K, b i (t, x) ≤ K, f (t, x) ≤ K,
a.e. on (0, +∞) × R N .

Let R, δ, U, ε, ρ be positive constants.

There exists C > 0 such that for any t ≥ ε, any x ∈ R N and any nonnegative weak solution u ∈ H 1 ((0, +∞)× R N ) of (31) satisfying ||u|| L ∞ (R N ) ≤ U , there holds

max x∈B(x,R) u(t, x) ≤ C min x∈B(x,R) u(t, x) + δ.
Notice that, as seen from the proof of [1, Theorem 2.7], the constant C > 0 does not depend on t provided that t ≥ ε > 0, which validates the above setting.

Acceleration result

* µ,ε,Γ,R > 0 such that R n(t, x, y)dy < µ, ∀x ≥ (1 + B 2 ) -1/2 max u -1 0 γe -(-λ0+ε)t , ∀t ≥ T * µ,ε,γ , (32)
R n(t, x, y)dy > µ, ∀x ≤ (1 + B 2 ) -1/2 min u -1 0 Γe -(-λ R 0 -ε)t , ∀t ≥ T * µ,ε,Γ,R . (33)
Subsection 4.3 concludes the proof of our main result, namely Theorem 2.7, based on (32)-(33). Lastly, in subsection 4.4, we sketch the proof of Theorem 2.6.

In the rest of this section, in view of (3), we shall consider K ≡ 1 without loss of generality. Additionally, to alleviate notations, the function r(Y ) = r( √ 1 + B 2 Y ) will be denoted as r.

The upper bound (32)

This subsection is devoted to the proof of (32).

∂ t φ -∂ XX φ = Λφ, t > 0, X ∈ R, φ(0, X) = u 0 (X), X ∈ R, ( 34
)
where u 0 satisfies (Q). Set E φ η (t) = {X ∈ R | φ(t, X) = η} for any η > 0. Then for any η > 0, ε ∈ (0, Λ), γ > 0, there exists T = T η,ε,γ ≥ 0 such that for any t ≥ T the set E φ η (t) is nonempty, admits a maximum, and

max E φ η (t) ≤ max u -1 0 γe -(Λ+ε)t .
This means that the upper bound of Theorem 3.1 is still valid when the logistic reaction term Λu(1 -u) in ( 24) is replaced with Λu. Note that, as in Lemma 3.1, T η,ε,γ also depends on Λ. However, we ignore it here since we will fix Λ = -λ 0 . The proof of Lemma 4.1 is easily adapted from that of [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF]Theorem 1.1] and is consequently omitted.

Proof of (32). Set Λ = -λ 0 > 0, and

v(t, X, Y ) = φ(t, X)Γ 0 (Y ),
where φ is the solution of the Cauchy problem (34). We readily check that v is a supersolution of the operator

∂ t -∂ XX -∂ Y Y -r(Y ) : ∂ t v -∂ XX v -∂ Y Y v -r(Y )v = (∂ t φ -∂ XX φ + λ 0 φ) Γ 0 = 0.
Meanwhile, since v ≥ 0, it is clear that v is a subsolution of the same operator. Since [START_REF] Fisher | The wave of advance of advantageous genes[END_REF]

provides v 0 (X, Y ) ≤ v(0, X, Y ), we conclude with the maximum principle that v ≤ v on [0, +∞) × R 2 .
Next, let µ > 0, ε ∈ (0, Λ), and γ > 0. One can select η > 0 and δ > 0 such that

η 1 + B 2 R Γ 0 (Y )dY + δ < µ. ( 35
)
From our control of the tails (30), there exist C, κ > 0 such that

v(t, X, Y ) ≤ Ce -κ √ 1+B 2 |Y | , ∀t ≥ 0, ∀(X, Y ) ∈ R 2 .
From there, we can find ζ = ζ(δ) > 0 large enough, such that for any t ≥ 0 and x ∈ R, there holds

R n(t, x, y)dy = R v t, X(x, y), Y (x, y) dy = R v t, x + By √ 1 + B 2 , y -Bx √ 1 + B 2 dy = 1 + B 2 R v(t, 1 + B 2 x + Bs, s)ds ≤ 1 + B 2 -ζ -∞ Ce -κ √ 1+B 2 |s| ds + +∞ -ζ v(t, 1 + B 2 x + Bs, s)ds ≤ δ + 1 + B 2 +∞ -ζ φ(t, 1 + B 2 x + Bs)Γ 0 (s)ds. Now, set E φ η (t) = {x ∈ R | φ(t, X) = η}.
We will show that for any

x ≥ 1 √ 1 + B 2 max u -1 0 γe -(Λ+ε)t ,
and for t large enough there holds φ(t, √ 1 + B 2 x + Bs) ≤ η for any s ≥ -ζ. For now, let us only assume the condition on x. By applying Proposition 3.2 to u 0 with

     a = Λ + ε/2, b = Λ + ε, Γ a = γ, Γ b = γ, χ = Bζ, there exists t * (a, b, Γ a , Γ b , χ) = t * ε,γ,ζ ≥ 0 such that for all t ≥ t * ε,γ,ζ there holds max u -1 0 γe -(Λ+ε/2)t + Bζ ≤ max u -1 0 γe -(Λ+ε)t .
From there, Lemma 4.1 proves the existence of T η,ε/2,γ ≥ 0 such that for all t ≥ T * µ,ε,γ := max(t * ε,γ,ζ , T η,ε/2,γ ) the following holds

max E φ η (t) ≤ max u -1 0 γe -(Λ+ε/2)t ≤ max u -1 0 γe -(Λ+ε)t -Bζ ≤ 1 + B 2 x -Bζ.
It is then easily deduced that for any s > -ζ, there holds φ(t, √ 1 + B 2 x + Bs) < η. Indeed, assume by contradiction that there exists x 0 > √ 1 + B 2 x -Bζ such that φ(t, x 0 ) ≥ η. Since for any t ≥ 0 one has φ(t, X) → 0 as X → +∞, there would exist x 1 ∈ E φ η (t) ∩ [x 0 , +∞), which contradicts the above inequality. Finally, for any t ≥ T * µ,ε,γ and x ≥

1 √ 1+B 2 max u -1 0 γe -(Λ+ε)t , there holds : R n(t, x, y)dy ≤ δ + η 1 + B 2 +∞ -ζ Γ 0 (s)ds ≤ δ + η 1 + B 2 R Γ 0 (s)ds,
which, combined with (35), proves (32).

The lower bound (33)

This subsection is devoted to the proof of (33). Lemma 4.2. Let u 0 satisfy (Q), see Definition 2.3. Then there exists a function u 0 : R → R such that

• u 0 ≤ u 0 , • there exists ξ 2 ∈ R such that u 0 = u 0 on [ξ 2 , +∞), • u 0 satisfies (Q),
• u 0 is of class C 2 on R and there exists K ≥ 0 such that |u 0 | ≤ Ku 0 on R.

Proof of Lemma 4.2. From Proposition 3.2, there exists ξ 1 > ξ 0 so that u 0 (X) ≥ u 0 (ξ 1 ) for any X ≤ ξ 1 . Fix h ∈ (0, ξ 1 -ξ 0 ).

Since u 0 satisfies (Q), there is ξ 2 > ξ 1 such that u 0 (ξ 2 ) < u 0 (ξ 1 ) and |u 0 (X)| ≤ u 0 (X) for all X ≥ ξ 2 . Next, one can construct a nondecreasing, concave function φ : R

+ → R + of class C 2 satisfying φ(x) = x, ∀x ≤ u 0 (ξ 2 ), m, ∀x ≥ u 0 (ξ 1 ),
for some m < u 0 (ξ 1 ). Finally, set u 0 = φ • u 0 . Let us prove that u 0 satisfies all the desired properties. Given that φ is concave, we have φ(x) ≤ x on R + , thus u 0 ≤ u 0 on R. Meanwhile, for any X ≥ ξ 2 , there holds u 0 (X) ≤ u 0 (ξ 2 ), which implies u 0 (X) = u 0 (X). Furthermore, the function

u 0 is of class C 2 on R. Indeed, by composition u 0 is C 2 on (ξ 1 -h, +∞), whereas on (-∞, ξ 1 ], u 0 is constant from our choice of ξ 1 .
Let us check that the function u 0 , which is clearly bounded, satisfies condition (Q), see Definition 2.3. Given that φ and u 0 are uniformly continuous, so is u 0 . Then, since u 0 = u 0 on [ξ 2 , +∞) and u 0 satisfies (Q), we collect for free the properties corresponding to the third and fourth lines of condition (Q), as well as lim +∞ u 0 = 0. Meanwhile, φ > 0 leads to u 0 > 0. Eventually, given our choice of ξ 1 , one deduces lim inf -∞ u 0 ≥ u 0 (ξ 1 ), whence lim inf -∞ u 0 = m > 0. As a result, u 0 does satisfy (Q).

It remains to prove the existence of a real K ≥ 0 such that |u 0 | ≤ Ku 0 on R. From our choice of ξ 2 , there holds

|u 0 (X)| = 0 < u 0 (X) ∀X ≤ ξ 1 , = |u 0 (X)| ≤ u 0 (X) = u 0 (X) ∀X ≥ ξ 2 .
Meanwhile, u 0 is C 2 and positive on [ξ 1 , ξ 2 ], thus there exists K ≥ 0 such that for any

X in [ξ 1 , ξ 2 ] |u 0 (X)| ≤ K min [ξ1,ξ2] u 0 ≤ K u 0 (X).
Thus it suffices to choose K = max(1, K ).

We now turn to the derivation of estimate (33).

Proof of (33). The proof involves three steps. First we construct a subsolution v on

[0, 1] × R × [σ --α, σ + + α],
which, for α > 0 large enough, provides a lower bound of the form v(1, X, Y ) ≥ ρ u 0 (X)Γ R 0 (Y ) for some ρ > 0, where u 0 is constructed from u 0 as in Lemma 4.2.

Next, equipped with this lower bound provided by v, we construct a second subsolution w on [1, +∞) × R 2 , which spreads and accelerates in the direction X → +∞. Let us recall that due to the nonlocal term, equation (1) does not satisfy the comparison principle. Thus, to prove that v(t, •, •) ≥ w(t, •, •) for t ≥ 1, we invoke a refinement of the parabolic Harnack inequality, that is Theorem 3.5.

The last step consists in transferring the estimates on the level sets of w into estimates on E n µ (t), similarly to the proof of (32) in subsection 4.1.

First subsolution. Let α > 0 large enough so that [-R, R] ⊂ (σ --α, σ + + α). Let p(t, Y ) be the solution of the initial boundary value problem

     ∂ t p -∂ Y Y p = 0, t ∈ (0, 1], Y ∈ (σ --α, σ + + α), p(t, Y ) = 0, t ∈ (0, 1], Y = σ ± ± α, p(0, Y ) = p 0 (Y ), Y ∈ (σ -, σ + ),
with p 0 the quadratic polynomial that satisfies p 0 (σ ± ) = 0 and p 0 σ++σ-2 = 1. In other words, p(t, Y ) solves the one-dimensional heat equation on [0, 1] × [σ --α, σ + + α], with zero Dirichlet conditions imposed on the boundary. Let us recall that N ∞ , C and κ are positive real numbers such that (29) and (30) hold. Moreover, since r ∈ L ∞ loc (R), there exists r min ≤ 0 such that r(Y ) ≥ r min for any Y ∈ [σ --α, σ + + α]. We define the following subsolution v(t, X, Y ) := e -kt u 0 (X)p(t, Y ),

k := K -r min + N ∞ , (36)
Ω := {(t, X, Y ) | 0 < t < 1, X ∈ R, Y ∈ (σ --α, σ + + α)} , (37)
where u 0 is constructed from u 0 as in Lemma 4.2, with the associated constant K ≥ 0, so that k ≥ N ∞ > 0.

Let us prove that v ≥ v on Ω. We first check that v ≤ v on the parabolic boundary of Ω, that is

∂ p Ω = {0} × R × [σ -, σ + ] D + D -,
where

D ± := (t, X, σ ± ± α) | t ∈ (0, 1], X ∈ R . On the one hand, it follows from (17) that v 0 (X, Y ) ≥ v(0, X, Y ) for X ∈ R and Y ∈ [σ -, σ + ]. On the other hand, on D + ∪ D -, one has v = 0 ≤ v. Thus v ≥ v on ∂ p Ω. It remains to show that v -v is
a supersolution of a parabolic problem on Ω. First, for t ∈ (0, 1), there holds

∂ t v -∂ XX v -∂ Y Y v = r(Y )v -v R v(t, χ, ψ)dy ≥ (r min -N ∞ )v. Meanwhile, since |u 0 | ≤ Ku 0 , one obtains ∂ t v -∂ XX v -∂ Y Y v = e -kt -ku 0 p -u 0 p = e -kt (r min -N ∞ )u 0 p + (-Ku 0 -u 0 )p ≤ (r min -N ∞ )v.
We conclude by the maximum principle that v ≥ v on Ω. We have thus completed the first step of the proof.

Second subsolution. We now turn to the construction of a second subsolution. The maximum principle shows

that p(1, Y ) is positive on (σ --α, σ + + α). Since [-R, R] ⊂ (σ --α, σ + + α), there exists p min > 0 so that p(1, Y ) ≥ p min for any Y ∈ [-R, R]. Thus, for any X ∈ R and Y ∈ [-R, R], there holds v(1, X, Y ) ≥ v(1, X, Y ) ≥ e -k p min u 0 (X) ≥ e -k p min u 0 (X)Γ R 0 (Y ),
where Γ R 0 solves [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF]. Fix now any real number ρ > 0 small enough so that ρ < ||u 0 || ∞ e -k p min . We allow ourselves to take ρ even smaller if needed.

Set Λ R ε = -λ R 0 -ε/2 > 0. For any t ≥ 1 and X, Y ∈ R, define w(t, X, Y ) := ρu(t, X)Γ R 0 (Y ),
where u(t, X) solves the Fisher-KPP equation

u t -u XX = Λ R ε u(1 -u), t > 1, X ∈ R, u(1, X) = u 0 (X)/||u 0 || ∞ , X ∈ R.
In particular, u(t, X) ∈ [0, 1] thanks to the maximum principle.

We shall now prove that v(t, X, Y

) > w(t, X, Y ) on [1, +∞) × R 2 . Given our choice of ρ, we indeed have v(1, X, Y ) > w(1, X, Y ) on R 2 .
Assume by contradiction that the closed set

E = t > 1 | ∃(X, Y ) ∈ R 2 , v(t, X, Y ) = w(t, X, Y ) ,
is nonempty. Set t 0 := min E > 1 and (X 0 , Y 0 ) ∈ R 2 the point where v(t 0 , X 0 , Y 0 ) = w(t 0 , X 0 , Y 0 ). Note that this implies Y 0 ∈ (-R, R), since the maximum principle yields v(t, •, •) > 0 for any t > 0. Before going further, we first use Theorem 3.5 to estimate the nonlocal term in [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF]. Let (x 0 , y 0 ) be the corresponding coordinates of (X 0 , Y 0 ) obtained through the change of variable [START_REF] Berestycki | Traveling fronts guided by the environment for reaction-diffusion equations[END_REF]. Fix M > 0, large enough so that 3C κ e -κM ≤ ε/8 and |y 0 -

Bx 0 | = √ 1 + B 2 |Y 0 | ≤ M .
Thanks to the control of the tails (30), there holds R v(t 0 , χ(X 0 , Y 0 , y), ψ(X 0 , Y 0 , y)

)dy = R v t 0 , X0-BY0 √ 1+B 2 + By √ 1 + B 2 , -B X0-BY0 √ 1+B 2 + y √ 1 + B 2 dy = R n(t 0 , x 0 , y)dy = R n(t 0 , x 0 , Bx 0 + y)dy ≤ 2M max y∈[-M,M ] n(t 0 , x 0 , Bx 0 + y) + [-M,M ] c
Ce -κ|y| dy.

Next, in order to estimate the first term of (39), let us recall that the solutions are uniformly bounded, as implied by (30). This allows us to use the refinement of the Harnack inequality, namely Theorem 3.5, with δ = C

2M κ e -κM > 0. Thus there exists a constant C M > 0 such that

max (x,y)∈[-M,M ] 2 n(t 0 , x 0 + x, Bx 0 + y) ≤ C M min (x,y)∈[-M,M ] 2 n(t 0 , x 0 + x, Bx 0 + y) + δ ≤ C M n(t 0 , x 0 , y 0 ) + δ,
which we plug into (39) to obtain

R v t 0 , X0-BY0 √ 1+B 2 + By √ 1 + B 2 , -B X0-BY0 √ 1+B 2 + y √ 1 + B 2 dy ≤ 2M C M v(t 0 , X 0 , Y 0 ) + 3C κ e -κM .
Going back to our proof by contradiction, since (w -v) is negative on [1, t 0 ) × R 2 , it reaches its maximum on [1, t 0 ] × R 2 at the point (t 0 , X 0 , Y 0 ). Thus

∂ t (w -v) -∂ XX (w -v) -∂ Y Y (w -v) -r(Y 0 )(w -v) (t 0 , X 0 , Y 0 ) ≥ 0, (40)
On the one hand, there holds

-[∂ t v -∂ XX v -∂ Y Y v -r(Y )v] (t 0 , X 0 , Y 0 ) = v(t 0 , X 0 , Y 0 ) R v(t 0 , χ, ψ)dy ≤ 2M C M w(t 0 , X 0 , Y 0 ) 2 + 3C κ e -κM w(t 0 , X 0 , Y 0 ).
On the other hand, one has

∂ t w -∂ XX w -∂ Y Y w -r(Y )w = ρ ∂ t u -∂ XX u + λ R 0 u Γ R 0 ≤ ρ Λ R ε u + λ R 0 u Γ R 0 ≤ - ε 2 w.
Thus, (40) leads to

0 ≤ - ε 2 w(t 0 , X 0 , Y 0 ) + 2M C M w(t 0 , X 0 , Y 0 ) 2 + 3C κ e -κM w(t 0 , X 0 , Y 0 ) ≤ - ε 2 + 2M C M ρ + 3C κ e -κM w(t 0 , X 0 , Y 0 ) ≤ - ε 2 + ε 8 + ε 8 w(t 0 , X 0 , Y 0 ),
provided we select ρ small enough so that 2M C M ρ ≤ ε/8. Note that reducing ρ may change the values of t 0 , X 0 , Y 0 but all above estimates remain true since there always holds Y 0 ∈ (-R, R) and C M does not depend on t 0 ≥ 1. In the end, one has -εw(t 0 , X 0 , Y 0 )/4 ≥ 0. This implies w(t 0 , X 0 , Y 0 ) ≤ 0, which is absurd. As a result, w(t, X, Y ) < v(t, X, Y ) for any t ≥ 1 and X, Y ∈ R.

Conclusion.

Before going further, let us mention that, for any t > 1 and X ∈ R, there holds u(t, X) < 1, thus R w(t, X, Y )dy < ρ

R Γ R 0 (Y )dy = ρ 1 + B 2 R Γ R 0 (y)dy =: β.
Therefore, the lower bound v(t, X, Y ) ≥ w(t, X, Y ) on [1, +∞) × R 2 does not provide any information on the location of E n µ (t) for levels µ ≥ β. As a result the location of larger levels are seemingly out of reach, which is typical of equations without comparison principle, as already mentioned in the introduction and subsection 2.5. Now, given any µ ∈ (0, β), one can select η = η(µ) ∈ (0, 1) such that η > µ/β. Since v ≥ w on [1, +∞) × R 2 , there holds for any t

≥ 1 R n(t, x, y)dy ≥ R w(t, X, Y )dy ≥ R ρu(t, X)Γ R 0 (Y )dy ≥ ρ R u t, x + By √ 1 + B 2 Γ R 0 y -Bx √ 1 + B 2 dy ≥ ρ 1 + B 2 R u(t, 1 + B 2 x + Bs)Γ R 0 (s)ds ≥ ρ 1 + B 2 R -R u(t, 1 + B 2 x + Bs)Γ R 0 (s)ds.
We will show that for any

x ≤ 1 √ 1 + B 2 min u -1 0 Γe -(Λ R ε -ε/2)t ,
and for t large enough there holds u(t, √ 1 + B 2 x + Bs) ≥ η for |s| ≤ R. For now, let us only assume the condition on x. By applying Proposition 3.2 to u 0 with

     a = Λ R ε -ε/2, b = Λ R ε -ε/4, Γ a = Γ, Γ b = Γ, χ = BR,
we deduce that there exists t * a,b,Γa,Γ b ,χ = t * Γ,R,ε ≥ 0 such that for any s ∈ [-R, R] and t ≥ t * Γ,R,ε , there holds

1 + B 2 x + Bs ≤ min u -1 0 Γe -(Λ R ε -ε/2)t + Bs ≤ min u -1 0 Γe -(Λ R ε -ε/2)t + BR ≤ min u -1 0 Γe -(Λ R ε -ε/4)t .
Since η ∈ (0, 1), Lemma 3.1 gives the existence of a real

T η,ε/4,Γ,Λ R ε ≥ 0 such that min E η (t) ≥ min u -1 0 Γe -(Λ R ε -ε/4)t , ∀t ≥ T η,ε/4,Γ,Λ R ε ,
where E η (t) = {x ∈ R | u(t, x) = η}. However, Lemma 4.2 provides some ξ 2 ∈ R such that u 0 (X) = u 0 (X) for X ≥ ξ 2 , and since u 0 satisfies (Q) (see Definition 2.3), one can take T η,ε/4,Γ,Λ R ε possibly even larger so that min u -1

0 Γe -(Λ R ε -ε/4)t ≥ ξ 2 . Also, since u 0 ≤ u 0 , there holds min u -1 0 Γe -(Λ R ε -ε/4)t ≥ ξ 2 , so that min u -1 0 Γe -(Λ R ε -ε/4)t = min u -1 0 Γe -(Λ R ε -ε/4)t , ∀t ≥ T η,ε/4,Γ,Λ R ε . Additionally, seeing that inf X≤0 u(t, X) → 1 as t → +∞, there is t η,R,ε ≥ 0 such that lim inf X→-∞ u(t, X) > η, ∀t ≥ t η,R,ε .
Finally, set

T * η,ε,Γ,R := max(t * Γ,R,ε , T η,ε/4,Γ,Λ R ε , t η,R,ε ), then for any x ≤ 1 √ 1+B 2 min u -1 0 Γe -(Λ R ε -ε/2)t and t ≥ T * η,ε,Γ,R , there holds min E η (t) ≥ min u -1 0 Γe -(Λ R ε -ε/4)t ≥ 1 + B 2 x + BR. (41)
Thus for any s ∈ (-R, R), one has u(t, √ 1 + B 2 x + Bs) > η. Indeed, assume by contradiction that there exists s

0 ∈ (-R, R) such that u(t, √ 1 + B 2 x + Bs 0 ) ≤ η. Given that t ≥ t η,R,ε , there would exist x 1 ∈ E η (t) ∩ (-∞, √ 1 + B 2 x + Bs 0 ), which is absurd considering (41). Consequently, whenever t ≥ T * η,ε,Γ,R and x ≤ 1 √ 1+B 2 min u -1 0 Γe -(Λ R ε -ε/2)t , there holds R n(t, x, y)dy ≥ ρ 1 + B 2 R -R ηΓ R 0 (s)ds ≥ ηβ > µ. Since Λ R ε -ε/2 = -λ R 0 -ε,
this concludes the proof of (33).

No acceleration for ill-directed heavy tails

In contradistinction with Section 4, we will keep the notation r(Y ) = r(y -Bx) = r( √ 1 + B 2 Y ), since the two coordinate systems (x, y) and (X, Y ) will be used in conjunction during the proof.

Proof of Theorem 2.11. We only give the proof for c > c * , as the case c < -c * is similar. We first set the following positive constants s

* := c * 1 + B 2 = 2 -λ 0 , γ := -λ 0 .
Fix any s > s * , and define ψ(t, x, y) := Ce -γ(X-st) Γ 0 (Y ), ϕ(t, x, y) := e -αt u(t, x)p(t, y),

where X, Y are given by ( 6), α ≥ γ 2 1 + 1 B 2 may be chosen arbitrarily large, C > 0 is a positive constant to be determined later, and the functions u(t, x), p(t, y) respectively solve

∂ t u -∂ xx u = u(||u 0 || ∞ -u) t > 0, x ∈ R, u(0, x) = u 0 (x) x ∈ R, (42)
∂ t p -∂ yy p = 0 t > 0, y ∈ R, p(0, y) = 1 [σ-,σ+] (y) y ∈ R. ( 43
)
Since p solves the one-dimensional heat equation, it is expressed as the convolution

p(t, y) = 1 √ 4πt σ+ σ- e -(y-z) 2 /(4t) dz. (44)
We shall prove that ψ + ϕ ≥ n using the maximum principle. Notice that one clearly has ψ(0, •, •) ≥ 0, and ϕ(0, x, y) ≥ n 0 (x, y) with [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]. Also, since nonnegative, n is a subsolution of the linear local operator :

L := ∂ t -∂ xx -∂ yy -r(y -Bx), = ∂ t -∂ XX -∂ Y Y -r(Y ), thus it suffices to prove that L(ψ + ϕ) ≥ 0 on (0, +∞) × R 2 to conclude. Given that -∂ Y Y Γ 0 -r(Y )Γ 0 (Y ) = λ 0 Γ 0 (Y ), we have Lψ = γs -γ 2 + λ 0 ψ > 0,
since γs * -γ 2 + λ 0 = 0. In particular, ψ is a supersolution.

Let us now turn our attention to the function ϕ. Since r satisfies Assumption 2.1, there exists

Y 0 > 0 such that r(Y ) ≤ -α whenever |Y | ≥ Y 0 . Set Ω 0 := (x, y) ∈ R 2 |Y | = 1 √ 1 + B 2 |y -Bx| < Y 0 .
On Ω c 0 , there holds -r(Y ) = -r(y -Bx) ≥ α. Therefore this implies

Lϕ = -αϕ + (||u 0 || ∞ -u)ϕ -r(y -Bx)ϕ ≥ (||u 0 || ∞ -u)ϕ ≥ 0.
Thus L(ψ + ϕ) ≥ 0 on (0, +∞) × Ω c 0 . Next, let us consider the domain Ω 0 . On this domain, ϕ may no longer be a supersolution because r may be greater than -α. However we shall prove that ψ + ϕ is a supersolution. Indeed, there holds

Lψ ≥ Q 1 e -γ(X-st) , (45a)
Lϕ ≥ -Q 2 e -αt p(t, y),

where

Q 1 := γs -γ 2 + λ 0 C min |Y |≤Y0 Γ 0 (Y ) > 0, (46a)
Q 2 := (α + r max )||u 0 || ∞ . (46b)
Now, let us divide Ω 0 into two parts :

Ω -:= Ω 0 ∩ {(x, y) | BX -Y 0 ≤ θ}, Ω + := Ω 0 ∩ {(x, y) | BX -Y 0 > θ},
where θ := max 1, (σ + + 1) 1 + B 2 > 0.

On the domain Ω -, one has

Lψ ≥ Q 1 e -γX ≥ Q 1 e -γ(Y0+θ)/B Lϕ ≥ -Q 2 e -αt ||p(0, •)|| ∞ ≥ -Q 2 .
In view of (46) it suffices to take C large enough to reach L(ψ + ϕ) ≥ 0 on (0, +∞) × Ω -. It remains to prove that L(ψ + ϕ) ≥ 0 on (0, +∞) × Ω + . For any (x, y) ∈ Ω + , we have

y = BX + Y √ 1 + B 2 ≥ BX -Y 0 √ 1 + B 2 ≥ θ √ 1 + B 2 > σ + .
Consequently, from (44) we obtain

p(t, y) ≤ 1 √ 4πt (σ + -σ -)e -(y-σ+) 2 /(4t) = 1 √ 4πt (σ + -σ -) exp - 1 4t BX + Y √ 1 + B 2 -σ + 2 ≤ 1 √ 4πt (σ + -σ -)e -Z 2 /4t ,
where

Z := BX -Y 0 √ 1 + B 2 -σ + ≥ θ √ 1 + B 2 -σ + ≥ 1.
As a result, from (45), there holds

L(ψ + ϕ) ≥ Q 1 e -γX e γst -Q 2 e -αt p(t, y) ≥ Q 1 e -γZ √ 1+1/B 2 e γst -Q 2 t -1/2 e -αt e -Z 2 /4t ,
where

Q 1 := Q 1 e -γ(Y0+ √ 1+B 2 σ+)/B , Q 2 := Q 2 (σ + -σ -) 1 √ 4π . Thus L(ψ + ϕ) ≥ 0 on (0, +∞) × Ω + if Q 1 e -γZ √ 1+B 2 /B ≥ Q 2 t -1/2 e -Z 2 /4t e (-α-γs)t , (47)
for every Z ≥ 1 and t > 0. For any such Z, the function g : t → t -1/2 e -Z 2 /4t e (-α-γs)t , defined on (0, +∞), attains its maximum at

t max = 1 + 4(γs + α)Z 2 -1 4(γs + α) > 0, which, since √ 1 + a 2 -1 ≤ a ≤ √ 1 + a 2 for any a ≥ 0, leads to g(t) ≤ Q 3 exp - (γs + α)Z 2 1 + 4(γs + α)Z 2 -1 exp - 1 + 4(γs + α)Z 2 4 ≤ Q 3 e - √ γs+αZ ,
with Q 3 := t -1/2 max e 1/4 > 0. Therefore, (47) amounts to

Q 1 e -γZ √ 1+1/B 2 ≥ Q 2 Q 3 e - √ γs+αZ .
Now, let us recall that since α ≥ γ 2 1 + 1 B 2 , we have

γ 1 + 1 B 2 ≤ √ γs + α.
Finally, by increasing C > 0, if necessary, one has Q 1 ≥ Q 2 Q 3 , thus L(ψ + ϕ) ≥ 0 on (0, +∞) × Ω + . Putting all together, we have thus proved L(ψ + ϕ) ≥ 0 on (0, +∞) × R 2 and, from the comparison principle, n(t, x, y) ≤ ψ(t, x, y) + ϕ(t, x, y) on (0, +∞) × R 2 . In other words, for any s > s * and α ≥ γ 2 1 + 1 B 2 , there exists C > 0 such that for any t > 0 and (x, y) ∈ R 2 , there holds n(t, x, y) ≤ Ce -γ(X-st) Γ 0 (Y ) + e -αt u(t, x)p(t, y).

(48)

We are now in the position to complete the proof of ( 21). This requires an additional control of the tails of n which is postponed to Lemma 5.1. In the sequel, we select C > 0 and κ > 0 such that ( 50 Additionally, there exists T α,µ ≥ 0 such that for any t ≥ T α,µ and for all x ∈ R, there holds R e -αt u(t, x)p(t, y)dy ≤ e -αt ||u 0 || ∞ (σ + -σ -) ≤ µ 2 .

Then, set ξ := -ζ + Bct. For any t ≥ T α,µ , combining (48) and (50) one obtains R Γ 0 (z)dz > 0. From there we deduce lim sup t→+∞ R n(t, ct, y)dy ≤ µ. This proves (21) since µ may be taken arbitrarily small. To conclude the above proof, we require the control (50). Lemma 5.1. Suppose r, K and n 0 satisfy the assumptions of Theorem 2.11. Let n be any global nonnegative solution of (1). Then there exists N ∞ > 0 such that R n(t, x, y)dy ≤ N ∞ .

(49)

Additionally, for every α > 0 there exist C , κ > 0 such that n(t, x, y) ≤ C e -κ|y-Bx| + e -αt u(t, x)p(t, y),

where u and p respectively solve (42)-(43).

Proof of Lemma 5.1. The proof, very similar to that of [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]Lemma 2.4], is included here for the sake of completeness. The first assertion is straightforward. If we define the mass N (t, x) := R n(t, x, y)dy, an integration of (1) along the y variable provides the inequality

∂ t N -∂ xx N ≤ N (r max -k -N ).
Since N 0 (x) ≤ ||u 0 || ∞ (σ + -σ -), it follows from the maximum principle that the mass is uniformly bounded :

N (t, x) ≤ N ∞ := max ||u 0 || ∞ (σ + -σ -), r max k - ,
which proves (49).

Let us now turn to the second assertion. Fix R > 0 large enough such that r(Y ) ≤ -α whenever |Y | ≥ R. Set

Ω R := (x, y) ∈ R 2 | |Y | = |y -Bx| √ 1 + B 2 < R .
Let us prove that n is uniformly bounded on Ω R . In view of Assumption 2.1, there exists M > 0 such that for all (x, y) ∈ Ω R+1 , there holds r(y -Bx) -R K(t, x, y, y )n(t, x, y )dy ≤ r L ∞ (Ω R ) + k + N ∞ =: M.

As a consequence, n is the solution of a linear parabolic problem with bounded coefficients on Ω R+1 (the nonlocal term being treated as a function of (t, x, y)), which allows us to apply the parabolic Harnack inequality (see [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF] for instance). Fix any τ > 0. There exists C H = C H (τ, R) > 0 such that for all t > 0 and x ∈ R : Seeing that C H does not depend on x, the population n(t, x, y) is uniformly bounded by C H N∞ 2R on R + × Ω R . To conclude, define, for any α > 0, ϕ(t, x, y) := Ce -κ(|y-Bx|-R √ 1+B 2 ) + e -αt u(t, x)p(t, y), where C, κ are positive constants, and u, p solve (42) and (43) respectively. Let us check that n(t, x, y) ≤ ϕ(t, x, y) on R + ×R 2 . The inequality holds for t = 0 by [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] which is nonnegative if we fix any κ ≤ α 1+B 2 . Since n ≥ 0, it is a subsolution of the same operator. The maximum principle allows us to conclude that n(t, x, y) ≤ ϕ(t, x, y) on (0, +∞) × Ω c R . We deduce (50) by setting C := Ce κR √ 1+B 2 .

max (x,

Figure 1 :

 1 Figure1: In grey, the region where v 0 (X, Y ) is assumed to be greater than u 0 (X), that is a one-dimensional heavy tail in the X direction. The dotted lines delimit the area where r > 0.

Subsections 4 .

 4 1 and 4.2 are devoted to prove the following : under the hypotheses of Theorem 2.7, there exist T * µ,ε,γ > 0 and T

Lemma 4 . 1 .

 41 Let Λ > 0 and φ the solution of the Cauchy problem

  ) holds. Choose any c > c * = s * / √ 1 + B 2 and s ∈ (s * , c √ 1 + B 2 ). Now, fix µ > 0 and select ζ > 0 large enough so that C

1+B 2

 2 κ|y-Bct| + e -αt u(t, ct)p(t, y) dy+ +∞ ξ [ϕ(t, ct, y) + ψ(t, ct, y)] dy ≤ κ|y| dy + R e -αt u(t, ct)p(t, y)dy + ct+Bz-st) Γ 0 (z)dz ≤ µ + Ce -γ( √ 1+B 2 c-s)t where C = C √ 1 + B 2 e γBζ/ √ 1+B 2

  If we choose any C ≥ C H N∞ 2R , there holds n(t, x, y) ≤ ϕ(t, x, y) on R + × Ω R . Next, on the remaining region (0, +∞) × Ω c R , we have r(y -Bx) ≤ -α, thus ϕ satisfies∂ t ϕ -∂ xx ϕ -∂ yy ϕ -r(y -Bx)ϕ = -κ 2 (1 + B 2 ) -r(y -Bx) Ce -κ(|y-Bx|-R √ 1+B 2 ) + (-α + ||u 0 || ∞ -u(t, x) -r(y -Bx))e -αt u(t, x)p(t, y), ≥ α -κ 2 (1 + B 2 ) Ce κR √ 1+B 2 e -κ|Y | ,

  R+1 denotes the closed ball of radius R of center (x, Bx). This yields

	max (x,y)∈B R (x)	n(t, x, y) ≤	C H 2R R	n(t + τ, x, y)dy =	C H 2R	N (t + τ, x) ≤	C H N ∞ 2R	.

y)∈B R (x) n(t, x, y) ≤ C H min (x,y)∈B R (x) n(t + τ, x, y),

where B R (x) ⊂ Ω

Acknowledgements. This research was supported by the ANR I-SITE MUSE, project MICHEL 170544IA (n°ANR-IDEX-0006). The author would like to thank J. Coville for raising the issue of ill-directed heavy tails, and his advisor M. Alfaro, for his regular support and useful remarks.

Conclusion

Proof of Theorem 2.7. Let us recall that (32)-(33) have been established in subsections 4.1 and 4.2 respectively. We can now complete the proof of Theorem 2.7. Set N t (x) := R n(t, x, y)dy, for any t ≥ 0 and x ∈ R. Set T * µ,ε,γ,Γ,R := max(T * µ,ε,γ , T * µ,ε,Γ,R ). One can reformulate (32)-(33) as follows : there exists β > 0 so that for any µ ∈ (0, β), ε ∈ (0, -λ R 0 ), Γ > 0 and γ > 0, there exists T * µ,ε,γ,Γ,R ≥ 0 such that for all t ≥ T * µ,ε,γ,Γ,R , there holds

Since N t is continuous for any t > 0, and

for any t ≥ T * µ,ε,γ,Γ,R . This concludes the proof of Theorem 2.7.

A heavy tail induces acceleration

In this short subsection, we give a sketch of the proof of Theorem 2.6, as it follows the same lines as the proof of (33) done in subsection 4.2.

By assumption, there holds λ 0 < 0. Select R > 0 large enough so that λ R 0 < 0 (see subsection 2.2). Choose any c > 2 -λ R 0 and set α c ∈ (0, -λ R 0 ) the only real satisfying c = α c + -λ R 0 αc . Since there exists m > 0 such that u 0 (X) ≥ min(m, e -αcX ), we can construct, as in the proof of Lemma 4.2, a C 2 function u 0 : R → [0, 1] that satisfies

• u 0 is asymptotically front-like, i.e. satisfies [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed Ithe case of the whole space[END_REF],

• u 0 (X) = e -αcX for X large enough,

• there exists K ≥ 0 such that |u 0 | ≤ Ku 0 on R.

Then, select α > 0 large enough so that [-R, R] ⊂ (σ --α, σ + + α). Set v as in (36)-(38). From ( 16), the maximum principle allows us to conclude that

Next, as in subsection 4.2, we can construct a second subsolution on [1, +∞) × R 2 of the form w(t, X, Y ) = ρu(t, X)Γ R 0 (Y ) for some ρ > 0, where Γ R 0 solves [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] and u(t, X) solves

However, since u 0 (X) decays as e -αcX , the function u(t, X) converges to a shift of the front ϕ c (X -ct) solution of the Fisher-KPP equation u t -u XX = -λ R 0 u(1 -u), see [START_REF] Uchiyama | The behavior of solutions of some non-linear diffusion equations for large time[END_REF]. As a consequence, with the same calculation as in subsection 4.2, we deduce the existence of a level β > 0 such that for any µ ∈ (0, β), there holds

Since c > 2 -λ R 0 may be chosen arbitrarily large, there holds min E n µ (t)/t → +∞ as t → +∞, for all µ ∈ (0, β), leading to the result of Theorem 2.6.


Hugo Brument

Rebecca Fribourg

Gerard Gallagher

Thomas Howard

Flavien Lécuyer

Tiffany Luong

Victor Mercado

Etienne Peillard

Xavier De Tinguy

Maud Marchal

Pyramid Escape: Design of Novel Passive Haptics Interactions for an Immersive and Modular Scenario

In this paper, we present the design of ten different 3D user interactions using passive haptics and embedded in an escape game scenario in which users have to escape from a pyramid in a limited time. Our solution is innovative by its modularity, allowing interactions with virtual objects using tangible props manipulated either directly using the hands and feet or indirectly through a single prop held in the hand, in order to perform several interactions with the virtual environment (VE). We also propose a navigation technique based on the "impossible spaces" design, allowing users to naturally walk through several overlapping rooms of the VE. All together, our different interaction techniques allow the users to solve several enigmas built into a challenging scenario inside a pyramid.

GENERAL DESCRIPTION

Passive haptic scenarios significantly enhance VEs [START_REF] Insko | Passive Haptics Significantly Enhances Virtual Environments[END_REF]. However, allowing the succession of many passive haptics interactions in one scenario remains challenging due to the distinctive nature of each interaction. In this paper, we present the design of ten different 3D user interactions using passive haptics. These interactions, further described in Section 3, are the following: pushing, sliding, rotating, pressing with the foot and with the hand, throwing, plugging, rotating, joining, squeezing, identifying the attributes of an object and human feedback. They all include passive haptic feedback, be it through direct manipulation performed by the hands or the feet, or indirectly through the reuse of a single handheld prop allowing for the execution of several interactions in the VE. The main characteristic of our design is the re-usability of the same prop for several interactions as well as the use of our handheld prop to mediate the interaction with other tangible props, each constraining the user's movement and providing adequate physical feedback to the hand. In addition to the manipulation techniques, we used the "impossible spaces" design [START_REF] Suma | Impossible spaces: Maximizing natural walking in virtual environments with self-overlapping architecture[END_REF] to maximize the size of the VE, allowing users to navigate the VE by naturally walking through several overlapping rooms. Our different interaction techniques are embedded into an exciting escape game scenario in which users have to solve enigmas to go from one room of a pyramid to another until they finally find the exit.

APPARATUS

Users are immersed in the VE using a head mounted display. Their feet and hands are tracked using two HTC Vive trackers and a Leap Motion. The location of each static tangible object is set and known at all times thanks to an initial calibration phase. They are fixed on the ground or a table and their position and orientation are registered using a Vive tracker fixed to a calibration prop which can be plugged onto the other props. The navigation space is 2 by 3 meters. * e-mail: firstname.lastname@inria.fr

3D USER INTERACTIONS

As a scenario, we propose an escape room game inside a pyramid. The user has to solve several enigmas in four different rooms inside the pyramid. We designed ten different user interactions involving passive haptics and described in this section in a chronological order.

Pushing

Users start the adventure lying down on the ground without seeing anything in the VE. They are told that they are in a sarcophagus and thus, their first interaction is to push open the sarcophagus lid in order to get out. In the real environment, they actually push a lid made of cardboard attached to a light structure surrounding them (see Figure 1, left pictures). Users are then attracted by a light emitted by a scepter that they are supposed to grab and that they will further use to interact with the VE. In the real word, the scepter is a 3D-printed object which will be the main prop used to perform physical interactions with other tangible props (see Figure 1, right pictures).

Sliding

With the scepter in hand, users need to find a stone where they can plug the scepter into a hole, allowing them to slide it, triggering the opening of a door to another room. The prop is made of a fixed runner and a moving part constrained along a single axis. The user is able to slide the moving part with the bottom of the scepter and feels a resisting force generated by rubber bands (see Figure 2).

Pressing with the Foot

In the second room, users need to find a code on the wall that they then must enter by pressing four stone buttons located on the ground using their foot. Physical buttons are made of several layers of foam (see Figure 3). The code opens a door and presents the user with an additional element that they can join to their scepter. This element is required for further interactions in the next room.

Throwing

In the third room, users encounter three containers, including one containing fire. They must collect a fire projectile from this container using the top of their scepter and throw it into the other containers in order to kindle them. The prop for the throwing task is made of a ping pong ball held in a container and attached to a counterweight by a wire that returns the ball to the container after it is released. Users are able to pick up the ball with the spoon-like part at the top of the handheld prop, thanks to magnets that attract metal elements inside the ball. The ball is detached whenever the wire is fully stretched or the prop velocity is high enough (see Figure 4).

Plugging

Once the throwing task is completed, users see some hieroglyphs on a table that they can grab with the scepter thanks to the additional joined element obtained in the previous room. The physical prop associated with the hieroglyph is joined to the scepter and then fixed to the support using magnets and variable gaps. The users are thus able to plug each hieroglyph into its corresponding location, unlocking a rotation mechanism by physically removing magnetic pins (see Figure 5).

Rotating

Users then have to plug their scepter into a hole on a circular plate and rotate it, moving a ray of light that in turn illuminates a gem once the correct number of rotations has been performed. The user can plug the lower part of the handheld prop onto a 3D-printed part fixed to a wooden bar that constrains the rotation around a screw (see Figure 6). The rotation unlocks the door of the fourth room. Before leaving the third room, users are invited to put down the scepter, freeing their hands.

Squeezing

In the fourth room, users face a table on which there are three receptacles, each has a button in front of it. Users have to press the buttons in a certain order indicated on a surface covered by milling beetles. To see the order, users must squash the beetles with their hand, making them disappear. In the real world, their hand squeezes bubble wrap, giving them a sensation of pressure (see Figure 7, left pictures).

Material Recognizing

The code indicated on the surface informs about an order between materials contained in the three receptacles. Users cannot see into the receptacles, so they have to put their hand into them to recognize each material. In the real world, users put their hand into physical receptacles that contain water, sand or water bubbles (see Figure 7, right pictures).

Pressing with the Hand

After the identification of the materials, users can press the buttons corresponding to each receptacle in the correct order. The physical buttons corresponding to the virtual ones were 3D printed and contain foam to generate a passive pressure sensation (see Figure 8). The resolution of this enigma opens the final door that frees the users.

Shaking-hand Bonus

Finally out of the pyramid, the user faces a mysterious pharaoh walking towards him. When close enough to the user, the pharaoh extends his arm in the direction of the user, who can then experience a handshake with physical feedback provided by the experimenter's hand (see Figure 9).

Figure 1 :

 1 Figure 1: (Left) pushing the sarcophagus lid (virtual and real), (Right) the main prop (virtual scepter and real prop).

Figure 2 :

 2 Figure 2: Sliding an object through rails: (Left) virtual environment with the sliding task, (Right) user motion with the real prop.

Figure 3 :

 3 Figure 3: Pressing the buttons with the feet: (Left) virtual representation of the feet and the buttons, (Right) real interaction with the foot.

Figure 4 :

 4 Figure 4: Throwing fire projectiles: (Left) virtual environment with the containers, (Right) details of the real prop for throwing the fire projectile.

Figure 5 :

 5 Figure 5: Plugging a hieroglyph into its receptacle: (Left) virtual representation, (Right) details of the hieroglyph manipulation.

Figure 6 :

 6 Figure 6: Rotating circulate plate: (Left) virtual representation of the circular plate, (Right) details of the rotation mechanism.

Figure 7 :

 7 Figure 7: (Left) squeezing insects task, (Right) identification of three different materials.

Figure 8 :

 8 Figure 8: Pressing buttons with the hand: (Left) virtual scene with the three buttons, (Right) details of the real buttons.

Figure 9 :

 9 Figure 9: Shaking hands with the pharaoh: (Left) VE with the pharaoh, (Right) experimenter's hand for human-touch feedback.

ACKNOWLEDGMENTS

Funded by the European Union's Horizon 2020 research and innovation programme grant No 801413 ("H-Reality"). The authors would also like to thank Alexandre Audinot for his help.