François Hamel

Luca Rossi

Séminaire Laurent Schwartz

Spreading sets and one-dimensional symmetry for reaction-diffusion equations

We consider reaction-diffusion equations ∂ t u = ∆u + f (u) in the whole space R N and we are interested in the large-time dynamics of solutions ranging in the interval [0, 1], with general unbounded initial support. Under the hypothesis of the existence of a traveling front connecting 0 and 1 with a positive speed, we discuss the existence of spreading speeds and spreading sets, which describe the large-time global shape of the level sets of the solutions. The spreading speed in any direction is expressed as a Freidlin-Gärtner type formula. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We then investigate the local properties of the level sets at large time. Some flattening properties of the level sets of the solutions, if initially supported on subgraphs, will be presented. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we state some asymptotic local one-dimensional and monotonicity symmetry properties for the elements of the Ω-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we present some logarithmic-in-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. Some related conjectures and open problems are also listed.

Framework and two main questions

We consider solutions of the reaction-diffusion equation

∂ t u = ∆u + f (u), t > 0, x ∈ R N , (1)
with N ≥ 2 and initial conditions u 0 having unbounded support. More precisely, the reaction term f : [0, 1] → R is of class C 1 ([0, 1]) with f (0) = f (1) = 0, and the initial conditions u 0 are assumed to be characteristic functions 1 U of sets U , i.e.

u 0 (x) = 1 if x ∈ U, 0 if x ∈ R N \U, (2)
where the initial support U is an unbounded measurable subset of R N (we use the term "initial support", with an abuse of notation, to refer to the set U in the definition of u 0 ). The Cauchy problem is well posed and, given u 0 , there is a unique bounded classical solution u of (1) such that u(t, •) → u 0 as t → 0 + in L 1 loc (R N ). More general initial conditions 0 ≤ u 0 ≤ 1 for which the upper level set {x ∈ R N : u 0 (x) ≥ θ} is at bounded Hausdorff distance from the support of u 0 , where θ ∈ (0, 1) is a suitable value depending on f , could be envisioned, at the expense of some further assumptions on the reaction term f . For the sake of simplicity of the presentation, we focus on initial conditions u 0 of the type [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF].

Due to diffusion, the solution u of ( 1)-( 2) is smooth at positive times and satisfies 0 < u < 1 in (0, +∞) × R N from the strong parabolic maximum principle, provided the Lebesgue measures of U and R N \ U are positive. However, from parabolic estimates, at each finite time, u stays close to 1 or 0 in subregions of U or R N \ U which are far away from ∂U . One first goal is to describe the shape at large time of the regions where u stays close to 1 or 0. How do these regions move and possibly spread in any direction? A fundamental issue is to understand whether and how the solution keeps a memory at large time of its initial support U . A basic question is the following:

Question 1. For a given vector e ∈ R N with Euclidean norm equal to 1, is there a spreading speed w(e) such that u(t, cte) → 1 as t → +∞ for every 0 ≤ c < w(e), u(t, cte) → 0 as t → +∞ for every c > w(e).

(

) 3
Can one find a formula for w(e) and, if any, how does w(e) depend on e and the initial support U ? Is there a uniformity with respect to e in (3) and are there spreading sets which describe the global shape of the level sets of u at large time?

The latter is expressed in terms of the notion of Ω-limit set, which is defined as follows: for a given bounded function u : R + × R N → R, the set Ω(u) := ψ ∈ L ∞ (R N ) : u(t n , x n + •) → ψ in L ∞ loc (R N ) as n → +∞, for some sequences (t n ) n∈N in R + diverging to +∞ and (x n ) n∈N in R N (4) is called the Ω-limit set of u. Roughly speaking, the Ω-limit set contains all possible asymptotic profiles of the function as t → +∞. For any bounded solution u of (1), the set Ω(u) is not empty and is included in C 2 (R N ), by standard parabolic estimates. Motivated by some known results in the literature, the following question naturally arises.

Question 2. Let u be a solution to [START_REF] Alfaro | Varying the direction of propagation in reaction-diffusion equations in periodic media[END_REF] emerging from an initial datum u 0 = 1 U .

Is it true that any function ψ ∈ Ω(u) is of the form

ψ(x) ≡ Ψ(x • e),
for some e ∈ S N -1 and Ψ : R → R? If the answer to the question is positive, we then say that u satisfies the asymptotic one-dimensional symmetry.

For the answer to Question 2 to possibly be affirmative, some conditions on f and U need to be imposed, as shown by some counter-examples presented in Section 5. We will also review in that section some known positive results which hold in the case where the initial support U is bounded, or when it is at finite Hausdorff distance from a halfspace, under some assumptions on f . We will see how such results can be extended for a nonlinearity f of the Fisher-KPP type, see condition [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF] below, giving a positive answer to Question 2 when U fulfills (in particular) a uniform interior ball condition and is convex, or, more generally, is at bounded Hausdorff distance from a convex set, see Theorem 16 below. These conditions on U are actually a very particular instance of the geometric hypotheses under which we derive our most general result about the asymptotic onedimensional symmetry, Theorem 17 below. Question 2 reclaims the De Giorgi conjecture about solutions of the Allen-Cahn equation (that is, stationary solutions of the reactiondiffusion equation ∆u + u(1 -u)(u -1/2) = 0 in R N , obtained after a change of unknown from the original Allen-Cahn equation), see [START_REF] Giorgi | Convergence problems for functionals and operators[END_REF].

The situation considered here can be viewed as a counterpart of many works devoted to the large-time dynamics of solutions of (1) with initial conditions u 0 that are compactly supported or converge to 0 at infinity. We refer to e.g. [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF][START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF][START_REF] Lewis | Allee dynamics and the spread of invading organisms[END_REF][START_REF] Muratov | Threshold phenomena for symmetric decreasing solutions of reactiondiffusion equations[END_REF][START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF][START_REF] Zlatoš | Sharp transition between extinction and propagation of reaction[END_REF] for extinction/invasion results in terms of the size and/or the amplitude of the initial condition u 0 for various functions f , and to [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF][START_REF] Du | Locally uniform convergence to an equilibrium for nonlinear parabolic equations on R N[END_REF][START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF][START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF][START_REF] Poláčik | Convergence and quasiconvergence properties of solutions of parabolic equations on the real line: an overview[END_REF] for general local convergence and quasiconvergence results. For the invading solutions u (that is, those converging to 1 locally uniformly in R N as t → +∞) with localized initial conditions, further estimates on the location and shape at large time of the level sets have been established in [START_REF] Ducrot | On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data[END_REF][START_REF] Gärtner | Location of wave fronts for the multi-dimensional KPP equation and Brownian first exit densities[END_REF][START_REF] Jones | Spherically symmetric solutions of a reaction-diffusion equation[END_REF][START_REF] Roquejoffre | Sharp large time behaviour in N -dimensional Fisher-KPP equations[END_REF][START_REF] Rossi | Symmetrization and anti-symmetrization in reaction-diffusion equations[END_REF][START_REF] Roussier | Stability of radially symmetric travelling waves in reaction-diffusion equations[END_REF][START_REF] Uchiyama | Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients[END_REF].

The case of general unbounded initial supports U has been much less investigated in the literature. One immediately sees that, for general unbounded sets U , Question 1 is much more intricate than in the case of bounded sets U , since the solutions u can spread from all regions of the initial support U , that is, not only from a single bounded region. The sets U themselves can be bounded in some directions and unbounded in others.

Two main hypotheses

In this section, we list some notations and hypotheses which are used in the various main results. The hypotheses are expressed in terms of the solutions of (1) with more general initial conditions than characteristic functions, or actually in terms of the reaction term f solely. We then discuss the logical link between these hypotheses. We let "| |" and " • " denote respectively the Euclidean norm and inner product in R N , B r (x) := {y ∈ R N : |y -x| < r} be the open Euclidean ball of center x ∈ R N and radius r > 0, B r := B r (0), and S N -1 := {e ∈ R N : |e| = 1} be the unit Euclidean sphere of R N . The distance of a point x ∈ R N from a set A ⊂ R N is given by dist(x, A) := inf |y -x| : y ∈ A , with the convention dist(x, ∅) = +∞. We also call (e 1 , • • • , e N ) the canonical basis of R N , that is,

e i := (0, • • • , 0, 1, 0, • • • , 0) for 1 ≤ i ≤ N , where 1 is the ith coordinate of e i .
Since both 0 and 1 are steady states, the question of the interplay between these two states and the diffusion is intricate. One way to differentiate the roles of 0 and 1 is to assume that the state 1 is more attractive than 0, in the sense that it attracts the solutions of (1) -not necessarily satisfying (2) -that are "large enough" in large balls at initial time.

Hypothesis 3. The invasion property occurs for any solution u of (1) with a "large enough" initial datum u 0 , that is, there exist θ ∈ (0, 1) and ρ > 0 such that if

θ 1 Bρ(x 0 ) ≤ u 0 ≤ 1 in R N , (5)
for some x 0 ∈ R N , then u(t, x) → 1 as t → +∞, locally uniformly with respect to x ∈ R N .

If f satisfies the following conditions: f > 0 in (0, 1) and lim inf

s→0 + f (s) s 1+2/N > 0, (6)
then Hypothesis 3 is satisfied with any θ ∈ (0, 1) and ρ > 0, see [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]; this property is known as the hair trigger effect. If f > 0 in (0, 1) (without any further assumption on the behavior of f at 0 + ), then Hypothesis 3 is still satisfied with any θ ∈ (0, 1), and with ρ > 0 large enough. Hypothesis 3 holds as well if f is of the ignition type, that is, ∃ α ∈ (0, 1), f = 0 in [0, α] and f > 0 in (α, 1), [START_REF] Berestycki | The speed of propagation for KPP type problems. I. Periodic framework[END_REF] and θ in Hypothesis 3 can be any real number in the interval (α, 1), provided ρ > 0 is large enough. For a bistable function f satisfying

∃ α ∈ (0, 1), f < 0 in (0, α) and f > 0 in (α, 1), (8)
Hypothesis 3 is equivalent to [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF], and in that case θ in Hypothesis 3 can be any real number in (α, 1), provided ρ > 0 is large enough. However, without the lower bound in condition [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF], the solutions u may not converge to 1 at t → +∞ locally uniformly in R N , as easily seen for instance with functions f of the types [START_REF] Berestycki | The speed of propagation for KPP type problems. I. Periodic framework[END_REF] or [START_REF] Berestycki | Asymptotic spreading for general heterogeneous Fisher-KPP type equations[END_REF], when

1 0 f (s) ds > 0, see
u 0 L 1 (R N ) is small enough. For a tristable function f satisfying ∃ 0 < α < β < γ < 1, f < 0 in (0, α) ∪ (β, γ) and f > 0 in (α, β) ∪ (γ, 1), (9)
then it easily follows from [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that Hypothesis 3 is equivalent to the positivity of both integrals 1 β f and 1 0 f . More generally speaking, it actually turns out that Hypothesis 3 is equivalent to the following two simple simultaneous conditions on the function f , see [START_REF] Du | Locally uniform convergence to an equilibrium for nonlinear parabolic equations on R N[END_REF][START_REF] Poláčik | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R[END_REF]: [START_REF] Alfaro | Varying the direction of propagation in reaction-diffusion equations in periodic media[END_REF], [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF] and

∃ θ ∈ (0, 1), f > 0 in [θ,
∀ t ∈ [0, 1), 1 t f (s) ds > 0. (11)
Furthermore, θ can be chosen as the same real number in Hypothesis 3 and in [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF]. In particular, Hypothesis 3 is satisfied if f ≥ 0 in [0, 1] and if condition [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF] holds. Notice however that condition [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF] alone is not enough to guarantee Hypothesis 3, since functions f of the type ( 8) satisfy [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF] but do not satisfy Hypothesis 3 if [START_REF] Giorgi | Convergence problems for functionals and operators[END_REF] alone is not enough to guarantee Hypothesis 3, since there are C 1 ([0, 1]) functions f which vanish at 0 and 1 and satisfy [START_REF] Giorgi | Convergence problems for functionals and operators[END_REF] but not [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF]: consider for instance f defined by f (1) = 0 and f (s) = s(1 -s) 3 sin 2 (1/(1 -s)) for s ∈ [0, 1). Notice that, from the equivalence between Hypothesis 3 and ( 10)-( 11), Hypothesis 3 is then independent of the dimension N , whereas, for a function f which is positive in (0, 1), the hair trigger effect (that is, the arbitrariness of θ ∈ (0, 1) and ρ > 0 in Hypothesis 3) depends on N (for instance, for the function f (s) = s p (1 -s) with p ≥ 1, Hypothesis 3 holds in any dimension N ≥ 1, but the hair trigger effect holds if and only if p ≤ 1 + 2/N , see [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]).

1 0 f ≤ 0. Similarly, condition
In the large time dynamics of the solutions of the Cauchy problem (1), a crucial role is played by the traveling front solutions connecting the steady states 1 and 0, defined as

u(t, x) = ϕ(x • e -ct)
with c ∈ R, e ∈ S N -1 , and

1 = ϕ(-∞) > ϕ(z) > ϕ(+∞) = 0 for all z ∈ R. (12)
The level sets of these solutions are hyperplanes orthogonal to e, traveling with constant speed c in the direction e. If any, the profile ϕ is necessarily decreasing and unique up to shifts, for a given c. Most of the main results are derived under the following hypothesis:

Hypothesis 4. Equation (1) admits a traveling front connecting 1 to 0 with speed c 0 > 0.

Hypothesis 4 is fulfilled for instance if f > 0 in (0, 1), or if f is of the ignition type [START_REF] Berestycki | The speed of propagation for KPP type problems. I. Periodic framework[END_REF], or if f is of the bistable type (8) with 1 0 f (s) ds > 0 (in the last two cases, the speed c 0 is unique), see [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]. Hypothesis 4 is also satisfied for some functions f having multiple oscillations in the interval [0, 1], see Remark 6 below. It actually turns out that Hypothesis 4 is equivalent to the existence of a positive minimal speed c * of traveling fronts connecting 1 to 0, and that Hypothesis 4 also implies Hypothesis 3 and further spreading properties for the solutions of (1) fulfilling the conditions of Hypothesis 3: Proposition 5. [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] Assume Hypothesis 4. Then equation (1) admits a traveling front connecting 1 to 0 with minimal speed c * , and c * > 0. Furthermore, Hypothesis 3 is fulfilled and, for any solution u as in Hypothesis 3, it holds that

∀ c ∈ [0, c * ), min |x|≤ct u(t, x) → 1 as t → +∞. ( 13
)
Lastly, for any compactly supported initial datum 0 ≤ u 0 ≤ 1, the solution u of (1) satisfies

∀ c > c * , sup |x|≥ct u(t, x) → 0 as t → +∞.
The minimality of c * means that (1) in dimension N = 1 admits a solution of the form ϕ(x-c * t) satisfying [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF], and it does not admit a solution of the same type with c * replaced by a smaller quantity (notice that, necessarily, c * ≤ c 0 under the notation of Hypothesis 4). Proposition 5 answers affirmatively to Question 1 under Hypothesis 4, in the very special case of compactly supported initial data satisfying [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF], with w(e) = c * for all e ∈ S N -1 . This can be viewed as a natural extension of some results of the seminal paper [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF], which were originally obtained under more specific assumptions on f . Remark 6. Whereas Proposition 5 shows the implication "Hypothesis 4 =⇒ Hypothesis 3", the converse implication is false in general. For instance, consider equation [START_REF] Alfaro | Varying the direction of propagation in reaction-diffusion equations in periodic media[END_REF] with f satisfying (9) together with β 0 f > 0 and 1 β f > 0, and let c 1 and c 2 be the unique (positive) speeds of the traveling fronts ϕ 1 (x -c 1 t) and ϕ 2 (x -c 2 t) connecting β to 0 on the one hand, and 1 to β on the other hand. It follows from [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that, if c 1 ≥ c 2 , then Hypothesis 4 is not satisfied, whereas Hypothesis 3 is. In that case, it turns out that the compactly supported initial conditions u 0 giving rise to invading solutions u develop into a terrace of two expanding fronts with speeds c 1 and c 2 , in the sense that inf Bct u(t,

•) → 1 as t → +∞ if 0 < c < c 2 (resp. sup B c t \B c t |u(t, •) -β| → 0 as t → +∞ if c 2 < c < c < c 1 , resp. sup R N \Bct u(t, •) → 0 as t → +∞ if c > c 1 )
. We refer to [START_REF] Du | Radial terrace solutions and propagation profile of multistable reactiondiffusion equations over R N[END_REF][START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF][START_REF] Giletti | Pulsating solutions for multidimensional bistable and multistable equations[END_REF][START_REF] Poláčik | Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations[END_REF][START_REF] Poláčik | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R[END_REF] for further results on terraces in more general frameworks. On the other hand, still with (9) and the positivity of β 0 f and 1 β f , Hypothesis 4 is satisfied if (and, then, only if) c 1 < c 2 , see [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF].

Spreading speeds and spreading sets

In this section, under Hypothesis 4, we investigate the notions of asymptotic spreading speeds and spreading sets for the solutions u of ( 1)-( 2) with general unbounded sets U containing large enough balls. Such solutions u then converge to 1 as t → +∞ locally uniformly in R N , and even satisfy [START_REF] Du | Radial terrace solutions and propagation profile of multistable reactiondiffusion equations over R N[END_REF], with c * > 0 given by Proposition 5. We now want to provide a more precise description of the invasion of the state 0 by the state 1. The invasion cannot be uniform in all directions in general, since it shall strongly depend on the initial support U . For e ∈ S N -1 , we then look for a quantity w(e) ∈ (0, +∞] satisfying (3), referred to as the spreading speed and representing the asymptotic speed at which the level sets between 0 and 1 move along the direction e. If any, it satisfies w(e) ≥ c * by Proposition 5. However, contrary to the case of compactly supported initial data satisfying [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF], the spreading speed may not exist when the support of the initial condition is unbounded, see the comments at the end of this section.

Let us first introduce the notions of sets of directions "around which U is bounded" and "around which U is unbounded", for short the sets of bounded directions and of unbounded directions, defined by: any open cone C containing the ray R + {ξ}, the set U ∩ C is unbounded. We also define the notion of positive-distance-interior U ρ (with ρ > 0) of the set U as

B(U ) := ξ ∈ S N -1 : lim inf τ →+∞ dist(τ ξ, U ) τ > 0 and U(U ) := ξ ∈ S N -1 : lim τ →+∞ dist(τ ξ, U ) τ = 0 .
U ρ := x ∈ U : dist(x, ∂U ) ≥ ρ .
The first main result shows the existence of and a formula for the spreading speeds, providing a positive answer to the first part of Question 1. Theorem 7. [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] Assume that Hypothesis 4 holds, let c * > 0 and ρ > 0 be given by Proposition 5 and Hypothesis 3, and let u be the solution of (1)-( 2), with U ρ = ∅ and

B(U ) ∪ U(U ρ ) = S N -1 . ( 14
)
Then, for every e ∈ S N -1 , there exists w(e) ∈ [c * , +∞] such that (3) holds, and even

     lim t→+∞ min 0≤s≤c u(t, ste) = 1 for every 0 ≤ c < w(e), lim t→+∞ sup s≥c u(t, ste) = 0 for every c > w(e). (15)
Furthermore, w(e) is given explicitly by the variational formula

w(e) = sup ξ∈U (U ), ξ•e≥0 c * 1 -(ξ • e) 2 , (16)
with w(e) = c * if there is no ξ ∈ U(U ) such that ξ • e ≥ 0, and
w(e) = +∞ if e ∈ U(U ). Since U(U ) is closed in S N -1
, it follows from ( 16) and the above conventions that

    
w(e) = +∞ if and only if e ∈ U(U ), w(e) > c * if and only if there is ξ ∈ U(U ) such that ξ • e > 0, w(e) = c * if and only if there is no ξ ∈ U(U ) such that ξ • e > 0.

For a set U satisfying U(U ) = ∅, formula ( 16) can be rephrased in a more geometric way:

w(e) = c * dist(e, R + U(U )) = c * sin ϑ , (17)
where ϑ ∈ [0, π/2] is the minimum between π/2 and the smallest angle between e and the directions in U(U ) (with the convention c * /0 = +∞). This formula immediately reveals that the map e → w(e) 14) is automatically satisfied and, if U ρ = ∅, then (15) holds with w(e) = c * for all e ∈ S N -1 , in agreement with Proposition 5. Formula ( 16) is called a Freidlin-Gärtner type formula, since Freidlin and Gärtner [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF] were the first to characterize the spreading speeds of solutions of reaction-diffusion equations in R N by a variational formula. They were actually concerned with spreading speeds for solutions of x-dependent reaction-diffusion equations of the Fisher-KPP type [START_REF] Fisher | The advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] (for which 0 < f (x, u)/u ≤ ∂f ∂u (x, 0) for all (x, u) ∈ R N × (0, 1)) with f (x, u) periodic with respect to x. More precisely, it follows from [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF], together with [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive media[END_REF][START_REF] Berestycki | The speed of propagation for KPP type problems. I. Periodic framework[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF], that (3) holds for these solutions, with

∈ [c * , +∞] is continuous in S N -1 . If U(U ) = ∅, then w(e) = c * for all e ∈ S N -1 . If U is bounded, condition (
w(e) = inf ξ∈S N -1 , ξ•e>0 c * (ξ) ξ • e (18)
for any e ∈ S N -1 , where c * (ξ) denotes the minimal speed of pulsating fronts connecting 1 to 0 in the direction ξ (a pulsating front connecting 1 to 0 with speed c in the direction ξ is a solution u : R × R N → (0, 1) such that u(t, x) = φ(x • ξ -ct, x), where φ(-∞, x) = 1, φ(+∞, x) = 0 uniformly in x ∈ R N , and φ has the same periodicity with respect to its second argument as the function f or other coefficients of the equation, see e.g. [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Shigesada | Traveling periodic waves in heterogeneous environments[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF][START_REF] Xin | Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity[END_REF][START_REF] Xin | Analysis and modeling of front propagation in heterogeneous media[END_REF]). Such formulas for the spreading speeds of solutions with compactly supported initial conditions have been recently extended to more general reaction terms in [START_REF] Rossi | The Freidlin-Gärtner formula for general reaction terms[END_REF].

For reaction-diffusion equations with spatially periodic coefficients, the spreading speed w(e) may depend on the direction e, even for solutions with compactly supported initial conditions u 0 . However, the continuity of the map e → w(e) still holds for monostable, ignition or bistable reactions f , as follows from [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Rossi | The Freidlin-Gärtner formula for general reaction terms[END_REF] and from the (semi-)continuity of the minimal or unique speeds of pulsating traveling fronts with respect to the direction, see [START_REF] Alfaro | Varying the direction of propagation in reaction-diffusion equations in periodic media[END_REF][START_REF] Gärtner | Location of wave fronts for the multi-dimensional KPP equation and Brownian first exit densities[END_REF][START_REF] Guo | Propagating speeds of bistable transition fronts in spatially periodic media[END_REF][START_REF] Rossi | The Freidlin-Gärtner formula for general reaction terms[END_REF] (but the continuity of the spreading speeds and even their existence do not hold in general when pulsating fronts connecting 1 to 0 do not exist anymore, see [START_REF] Giletti | Pulsating solutions for multidimensional bistable and multistable equations[END_REF]).

Remember that Hypothesis 4 holds if f > 0 in (0, 1), in the ignition case [START_REF] Berestycki | The speed of propagation for KPP type problems. I. Periodic framework[END_REF], and in the bistable case ( 8) with 1 0 f (s)ds > 0. In these cases, Theorem 7 yields the existence of the spreading speeds satisfying ( 3) and ( 15), given by ( 16) as soon as the initial datum u 0 = 1 U is associated with a set U ⊂ R N satisfying U ρ = ∅ and ( 14). Moreover, in the case of a positive nonlinearity satisfying ( 6), ρ > 0 can be arbitrarily small. On the other hand, the conclusions of Theorem 7 do not hold in general without Hypothesis 4: as in Remark 6, for a function f of the type ( 9) with c 1 > c 2 (where c 1 and c 2 are the positive speeds of the traveling fronts ϕ 1 (x -c 1 t) and ϕ 2 (x -c 2 t) connecting β to 0, and 1 to β, respectively), the solutions u of ( 1)-( 2) with U bounded (hence, ( 14) is satisfied) and U ρ = ∅ develop into a terrace of expanding fronts, ruling out the existence of w(e) satisfying (3).

The geometric assumption ( 14) is invariant under rigid transformations of U . It holds for instance if U is star-shaped and stays at a finite distance from its ρ-interior U ρ , by [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF].

The next result states the uniformity of (3) with respect to the directions e ∈ S N -1 , making more precise the answer to Question 1. Theorem 8. [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] Under the assumptions of Theorem 7, for any compact set

C ⊂ R N ,      lim t→+∞ min x∈C u(t, tx) = 1 if C ⊂ W, lim t→+∞ max x∈C u(t, tx) = 0 if C ⊂ R N \ W, ( 19
)
where W is the envelop set of the speeds w(e)'s, that is,

W := re : e ∈ S N -1 , 0 ≤ r < w(e) . (20)
Formula ( 17) reveals that W has the following simple geometric expression:

W = R + U(U ) + B c * (21)
(with the convention that

R + ∅ + B c * = B c * if U(U ) = ∅).
Indeed, on the one hand, if U(U ) = ∅, then w(e) ≡ c * and W = B c * . On the other hand, if U(U ) = ∅, for any e and r ≥ 0, one has dist(re, R + U(U )) = r dist(e, R + U(U )) = rc * /w(e) by ( 17) (using the convention rc * /(+∞) = 0), and therefore the equivalence between ( 20) and ( 21) follows. Formula ( 21) means that W is the c * -neighborhood of the positive cone generated by the directions U(U ). It implies that W is an open set which is either unbounded (when

U(U ) = ∅)
, or it coincides with B c * . For periodic Fisher-KPP equations, formula [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] for the spreading speeds of solutions with compactly supported initial conditions means that the closure of the set W defined by [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF] coincides with the Wulff shape of the envelop set of the minimal speeds c * (ξ) of pulsating fronts and, since the map ξ → c * (ξ) ∈ (0, +∞) is continuous in S N -1 , the set W would therefore be a convex compact set. For our problem (1)-( 2), the set W defined in ( 20)-( 21) is not bounded as soon as U(U ) = ∅. Furthermore, it is not convex in general. For instance, if U = ∅ is a non-convex closed cone, say with vertex 0, then R + U(U ) ∪ {0} = U and, from (21), W is not convex either. Nevertheless, if U is a general convex set, then R + U(U ) ∪ {0} and W are convex, from [START_REF] Gärtner | Location of wave fronts for the multi-dimensional KPP equation and Brownian first exit densities[END_REF] again. More generally speaking, if there is a convex set U which lies at a finite Hausdorff distance from U , then U(U ) = U(U ) and therefore W is convex, even if U itself is not.

Having in mind [START_REF] Fisher | The advance of advantageous genes[END_REF], W is called a spreading set for (1)-( 2). We point out that ( 19) is stronger than (3), owing to the continuity of the map e → w(e) in S N -1 . It also yields the first line of [START_REF] Ducrot | On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data[END_REF]. Compared to the first lines of ( 3) and ( 15), the first line of ( 19) provides an additional uniformity with respect to the directions e. It also follows from [START_REF] Fisher | The advance of advantageous genes[END_REF] and the continuity of the map e → w(e) ∈ [c * , +∞] that, for any σ ∈ (0, 1) and A > 0, min

x ∈ σW∩B A u(t, tx) → 1 as t → +∞.
Formulas similar to [START_REF] Fisher | The advance of advantageous genes[END_REF] have been established for the solutions of more general heterogeneous equations or systems with compactly supported initial conditions and Fisher-KPP reactions [START_REF] Barles | Wavefront propagation for reaction-diffusion systems of partial diffusion equations[END_REF][START_REF] Berestycki | Asymptotic spreading for general heterogeneous Fisher-KPP type equations[END_REF][START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF][START_REF] Liang | Maximizing the spreading speed of KPP fronts in two-dimensional stratified media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF], bistable reactions [START_REF] Xin | Analysis and modeling of front propagation in heterogeneous media[END_REF], or even more general terms [START_REF] Rossi | The Freidlin-Gärtner formula for general reaction terms[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF]. The main difference is that, in these references, the spreading speeds and sets are bounded, unlike the spreading set W defined in ( 20)-( 21), which is unbounded as soon as U(U ) = ∅.

Theorems 7-8 are concerned with the convergence towards 1 and 0 as t → +∞ along some rays or some dilated sets. The next two results provide a description of the asymptotic shape of the upper level sets of a solution u, defined for λ ∈ (0, 1) and t > 0 by

E λ (t) := x ∈ R N : u(t, x) > λ . ( 22
)
That description involves the Hausdorff distance between some sets depending on E λ (t) and tW. The Hausdorff distance is defined, for any pair of subsets A, B ⊂ R N , by

d H (A, B) := max sup x∈A dist(x, B), sup y∈B dist(y, A) , with the conventions that d H (A, ∅) = d H (∅, A) = +∞ if A = ∅ and d H (∅, ∅) = 0.
Theorem 9. [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] Under the assumptions of Theorems 7-8, it holds that

∀ R > 0, ∀ λ ∈ (0, 1), d H B R ∩ 1 t E λ (t) , B R ∩ W -→ t→+∞ 0. ( 23
)
Theorem 9 gives the approximation of t -1 E λ (t) by W as t → +∞, locally with respect to the Hausdorff distance. But the convergence is not global in general, and d H (t -1 E λ (t), W) → 0 as t → +∞ in general, see the comments at the end of this section. However, it is global if U is bounded and U ρ = ∅, by Proposition 5, with W = B c * in this case.

The following result provides an asymptotic global approximation of t -1 E λ (t) by the family of sets t -1 U + B c * , under a different assumption on U . Theorem 10. [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] Assume that Hypothesis 4 holds, let c * > 0 and ρ > 0 be given by Proposition 5 and Hypothesis 3, and let u be the solution of (1)-( 2), with U ρ = ∅ and

d H (U, U ρ ) < +∞. ( 24
)
Then, d H (E λ (t), U + B c * t ) = o(t) as t → +∞ for every λ ∈ (0, 1), that is,

∀ λ ∈ (0, 1), d H 1 t E λ (t) , 1 t U + B c * -→ t→+∞ 0. ( 25
)
Property [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF] means that E λ (t) behaves at large time t as the set U thickened by c * t. A sufficient condition for [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF] to hold is that the set U fulfills the uniform interior sphere condition of radius ρ: in such case d H (U, U ρ ) ≤ 2ρ. In particular, if f satisfies condition [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive media[END_REF], then Theorem 10 applies to any non-empty set U which is uniformly C 1,1 .

We point out that a single formula like [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF] valid for all λ ∈ (0, 1) does not hold in general without Hypothesis 4. For instance, as in Remark 6, consider a tristable function f of the type ( 9) with c 1 > c 2 (where c 1 and c 2 are the positive speeds of the traveling fronts ϕ 1 (x -c 1 t) and ϕ 2 (x -c 2 t) connecting β to 0, and 1 to β, respectively). Then, as follows from [START_REF] Du | Radial terrace solutions and propagation profile of multistable reactiondiffusion equations over R N[END_REF][START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF][START_REF] Poláčik | Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations[END_REF], the solutions u of ( 1)-( 2) with U bounded and

U ρ = ∅ (hence, (24) is satisfied) are such that d H (E λ (t), U + B c 2 t ) = o(t) as t → +∞ if β < λ < 1, respectively d H (E λ (t), U + B c 1 t ) = o(t) as t → +∞ if 0 < λ < β.
In order to enlighten our results stated above, we consider the important class of unbounded sets U given by subgraphs of some functions (see also the next section for further results). Namely, we consider

U := x = (x , x N ) ∈ R N -1 × R : x N ≤ γ(x ) , (26)
with γ : R N -1 → R belonging to L ∞ loc (R N -1 ). Assume for instance that γ is of the form

γ(x ) = α |x | + o(|x |) as |x | → +∞,
for some α ∈ R. We see that U ρ = ∅ for any ρ > 0 and that B(U ) = e ∈ S N -1 : e N > α|e | and U(U ) = U(U ρ ) = e ∈ S N -1 : e N ≤ α|e | . Thus ( 14) is fulfilled, hence (3), ( 15), ( 19) and ( 23) hold under Hypothesis 4 on f , by Theorems 7-9. However, the shape of the envelop set W given by ( 20)-( 21) strongly depends on the sign of α. If α > 0, then

W = {x ∈ R N : x N < α |x | + c * √ 1 + α 2 }:
it is a translation of the interior of the cone R + U(U ), hence it is non-convex and not C 1 . If α < 0 then W is still given by the c * -neighborhood of the same cone R + U(U ), which now becomes "rounded" in its upper part; indeed in such a case w(e) = c * if e N ≥ |e |/|α|, and W is convex and

C 1 (but not C 2 ). If α = 0 (which includes the case γ bounded) then W = {x ∈ R N : x N < c * } is a half-space, with w(e) = +∞ if e N ≤ 0, and w(e) = c * /e N if e N > 0. If γ in (26) satisfies γ(x ) |x | → -∞ as |x | → +∞, then B(U ) = S N -1 \{-e N } and U(U ) = U(U ρ ) = {-e N }
, with e N = (0, • • • , 0, 1). Hence ( 14) is fulfilled and therefore, under Hypothesis 4, properties (3), ( 15), ( 19) and ( 23) hold by Theorems 7-9, with

W = -R + e N + B c * = x ∈ R N : |x | < c * , x N ≤ 0 ∪ B c *
. This is a cylinder with a "rounded" top, which is convex and C 1 , but not C 2 . Lastly, if γ has uniformly bounded local oscillations, that is, if there is M > 0 such that |γ(x )-γ(y )| ≤ M for all x ∈ R N -1 , y ∈ R N -1 with |x -y | ≤ 1 (this is the case if γ is globally Lipschitzcontinous), then condition ( 24) is fulfilled and (25) holds, by Theorem 10.

To complete this section, we present a list of situations where one or both hypotheses ( 14) and ( 24) of Theorems 7-10 do not hold and the conclusions (3), ( 15), ( 19), ( 23) and ( 25) fail. The examples also show that the conditions ( 14) and ( 24) on U cannot be compared in general. We also discuss the validity of the following convergences:

lim t→+∞ 1 t E λ (t) = W = lim t→+∞ 1 t U + B c * , (27)
that one may expect to hold but that actually fail in general. The convergences [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] would be understood with respect to the Hausdorff distance (which does not guarantee the uniqueness of the limit). Notice first that, if ( 14) is fulfilled together with U ρ = ∅ and Hypothesis 4, then (23) holds and the limit of t -1 E λ (t), if any, must be the set W (that is, the Hausdorff distance between the limit set and W must be 0). All of the following instances refer to the equation ( 1) with logistic term f (u) = u(1 -u), for which Hypothesis 4 holds, as well as the hair trigger effect, i.e., θ ∈ (0, 1) and ρ > 0 can be arbitrary in Hypothesis 3 and Proposition 5, in which c * = 2. The sets U listed below have non-empty interiors, and ( 14) and ( 24) are understood here with ρ > 0 arbitrarily small.

• The set U := n∈N B 2 n +1 \ B 2 n -1 fulfills (24) (hence (25) holds), but it violates [START_REF] Du | Locally uniform convergence to an equilibrium for nonlinear parabolic equations on R N[END_REF], and ( 3), ( 15), ( 19) and ( 23) all fail, for any function w : S N -1 → [0, +∞] and any star-shaped, open set W ⊂ R N , and moreover both limits in [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] do not exist.

• The set

U := U 1 ∪ U 2 with U 1 := x ∈ R N : x 1 ≥ 0 and x 2 2 + • • • + x 2 N ≤ 1 and U 2 := x ∈ R N : x 1 ≥ 0 and (x 2 - √ x 1 ) 2 + x 2 3 + • • • + x 2 N ≤ e -x 2 1 , fulfills (14)
(hence (3), ( 15), ( 19) and ( 23) hold), but it violates [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF], and (25) fails. Moreover, the first limit in [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] exists whereas the second one does not.

• The set U := x ∈ R N : |x N | ≤ e -|x | 2 violates both ( 14) and [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF], and (3), ( 15), ( 19), ( 23) and (25) all fail, with w(e) and W given by ( 16) and [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF]. Moreover, the two limits in [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] exist but do not coincide.

• The set U := x ∈ R N : x N ≤ |x | fulfills ( 14) and (24) (hence (3), ( 15), ( 19), ( 23) and (25) all hold), but both limits in [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF] do not exist and d H (t -1 E λ (t), W) = +∞ for all λ ∈ (0, 1) and t > 0.

The details about the above counter-examples can be found in [START_REF] Hamel | Spreading speeds and spreading sets of reaction-diffusion equations[END_REF]Section 6].

Flattening properties in the case of subgraphs

In this section, we focus on the important class of initial conditions which are characteristic functions of subgraphs in R N . Up to rotation, let us consider graphs in the direction x N , hence initial conditions u 0 given by

u 0 (x , x N ) = 1 if x N ≤ γ(x ), 0 otherwise, (28)
that is, u 0 = 1 U with U given by ( 26) and γ ∈ L ∞ loc (R N -1 ). From parabolic estimates, u(t, x , x N ) → 0 as x N → +∞ and u(t, x , x N ) → 1 as x N → -∞, locally uniformly in

(t, x ) ∈ [0, +∞) × R N -1 . Furthermore, ∂ x N u < 0 in (0, +∞) × R N ,
by the strong parabolic maximum principle. As a consequence, for every t > 0, x ∈ R N -1 and λ ∈ (0, 1), there exists a unique X λ (t, x ) ∈ R such that

u(t, x , X λ (t, x )) = λ, (29)
and the function (λ, t, x ) → X λ (t, x ) is actually continuous in (0, 1) × (0, +∞) × R N -1 .

In other words, the sets E λ (t) given in [START_REF] Giletti | Pulsating solutions for multidimensional bistable and multistable equations[END_REF] are the open subgraphs of x → X λ (t, x ). Theorems 7-10 applied to this case give some information on the shape of the graphs of X λ (t, •) at large time and large space in terms of γ, provided the assumptions of these theorems are fulfilled (see the previous section). We are now interested in the local-inspace behavior of the graphs of X λ (t, •) at large time. Let us first point out that, because of the asymmetry of the roles of the steady states 0 and 1 (assuming Hypothesis 4), the behavior of the graphs of X λ (t, •) will be radically different depending on the profile of the function γ at infinity. Consider the particular case γ(x ) = α |x |. Whatever α may be, the graphs of the functions X λ (t, •) look like the sets {x ∈ R N : dist(x, U ) = c * t} at large time t, by Theorem 10. For each t > 0, the set {x ∈ R N : dist(x, U ) = c * t} is a shift of ∂U in the direction x N and therefore has a vertex if α > 0, whereas it is C 1 if α ≤ 0. Of course, for each t > 0, in both cases α > 0 and α ≤ 0, each level set of u (that is, each graph of X λ (t, •)) is of class C 2 from the implicit function theorem and the negativity of ∂ x N u. Nevertheless, the previous observations imply that there should be a difference between the flattening properties of the level sets of u according to the coercivity of γ at infinity. The following result deals with the non-coercive case, i.e., lim sup |x |→+∞ γ(x )/|x | ≤ 0.

Theorem 11. [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF] Assume that Hypothesis 4 holds. Let u be the solution of (1) with an initial datum u 0 given by [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF].

If lim sup |x |→+∞ γ(x ) |x | ≤ 0, (30)
then, for every λ ∈ [θ, 1) and every basis (e 1 , • • • , e N -1 ) of R N -1 , there holds

lim inf t→+∞ min |x |≤R, 1≤i≤N -1 |∇ x X λ (t, x ) • e i | -→ 0 as R → +∞, (31)
where θ ∈ (0, 1) is given by Proposition 5 and Hypothesis 3, and X λ by [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF]. In particular, if

N = 2, lim inf t→+∞ min [-R,R] |∂ x 1 X λ (t, •)| → 0 as R → +∞, for every λ ∈ [θ, 1).
Roughly speaking, Theorem 11 says that the level set of any value λ ∈ [θ, 1) becomes almost flat in some directions along sequences of points of R N and along sequences of times diverging to +∞. We point out that the estimates on ∇ x X λ (t, x ) immediately imply analogous ones on ∇ x u(t, x , X λ (t, x )), because

∇ x u(t, x , X λ (t, x )) = -∂ x N u(t, x , X λ (t, x ))∇ x X λ (t, x ) and ∂ x N u is bounded in [1, +∞) × R N by parabolic estimates. Hence, (31) implies that lim inf t→+∞ min |x |≤R, 1≤i≤N -1 |∇ x u(t, x , X λ (t, x )) • e i | -→ 0 as R → +∞.
for every λ ∈ [θ, 1) and every basis (e 1 , • • • , e N -1 ) of R N -1 . The proof of ( 31) is done in [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF] by way of contradiction and uses the fact that λ is larger than or equal to the quantity θ ∈ (0, 1) given by Proposition 5 and Hypothesis 3. If f > 0 in (0, 1), Hypothesis 3 is satisfied for any θ ∈ (0, 1) and thus the conclusion (31) of Theorem 11 holds for any λ ∈ (0, 1). Furthermore, if f satisfies (6), then (31) can be strengthened as lim inf t→+∞ min |x |≤R |∇ x X λ (t, x )| → 0 as R → +∞ for every λ ∈ (0, 1), see [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF].

Without [START_REF] Ishige | Parabolic power concavity and parabolic boundary value problems[END_REF], property [START_REF] Ishige | To logconcavity and beyond[END_REF] does not hold in general (immediate couterexamples are solutions whose level sets are parallel hyperplanes which are not orthogonal to e N ). Moreover, if one assumes lim inf |x |→+∞ γ(x )/|x | ≥ 0 instead of (30), the conclusion (31) does not hold either in general (counter-examples are given by rotated bistable V -shaped fronts, from [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF][START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF][START_REF] Roquejoffre | Nontrivial large-time behaviour in bistable reactiondiffusion equations[END_REF]). However, with [START_REF] Ishige | Parabolic power concavity and parabolic boundary value problems[END_REF], we expect that the liminf of the min can be replaced by a limit in [START_REF] Ishige | To logconcavity and beyond[END_REF], without reference to the size R, namely, we propose the following.

Conjecture 12. Under the assumptions of Theorem 11, the conclusion (31) can be strengthened by the limit, for every λ ∈ [θ, 1),

∇ x X λ (t, x ) -→ 0 as t → +∞, locally uniformly in x ∈ R N -1 . ( 32
)
We emphasize that, even under the assumption [START_REF] Ishige | Parabolic power concavity and parabolic boundary value problems[END_REF], property [START_REF] Jones | Spherically symmetric solutions of a reaction-diffusion equation[END_REF] does not hold in general uniformly with respect to x ∈ R N -1 (for instance, in dimension N = 2, (32

) fails if γ = 1 in (-∞, -1] and γ = -1 in [1, +∞)).
On the other hand, a strong support to the validity of Conjecture 12 is provided by the conclusion of Theorem 10. Indeed, it asserts that, for any λ ∈ (0, 1), E λ (t) ∼ U + B c * t for t large, in the sense of [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF], and one can check that condition [START_REF] Ishige | Parabolic power concavity and parabolic boundary value problems[END_REF] entails that the exterior unit normals to the set U + B c * t at the points (x , x N ) ∈ ∂(U + B c * t ) (whenever they exist) approach the vertical direction e N = (0, • • • , 0, 1) as t → +∞, locally uniformly with respect to x ∈ R N -1 . Hence the same is expected to hold for the sets E λ (t), which is what [START_REF] Jones | Spherically symmetric solutions of a reaction-diffusion equation[END_REF] asserts. This kind of arguments can be made rigorous, leading to a result which is a weaker statement than Conjecture 12, that is, min |x |≤βt |∇ x X λ (t, x )| → 0 as t → +∞ for every β > 0, see [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF]. A weaker statement than Conjecture 12 is also derived in [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF] in the case where f satisfies Fisher-KPP condition (39) below, namely, there holds that lim inf t→+∞ max |x |≤A |∇ x X λ (t, x )| = 0 for every λ ∈ (0, 1) and A > 0 (see also Theorem 21 below).

As for the full Conjecture 12, the following result shows that, under assumption (30), the conclusion (32) holds for initial data u 0 having an asymptotically conical support, or being asymptotically x -spherically-symmetric and nonincreasing. Notice that the following result uses the weaker Hypothesis 3 instead of Hypothesis 4.

Theorem 13. [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF] Assume that Hypothesis 3 holds. Let u be the solution of (1) with an initial datum u 0 given by [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF], where γ satisfies one of the following assumptions:

(i) either γ is of class C 1 outside a compact set and there is ≥ 0 such that    γ (x 1 ) → ∓ as x 1 → ±∞ if N = 2, ∇γ(x ) = - x |x | + O(|x | -1-η ) as |x | → +∞, for some η > 0, if N ≥ 3; (33)
(ii) or γ is continuous outside a compact set and γ(x

)/|x | → -∞ as |x | → +∞;
(iii) or γ(x ) = Γ(|x -x 0 |) outside a compact set, for some x 0 ∈ R N -1 and some continuous nonincreasing function Γ : R + → R;

(iv) or γ(x ) = Γ(|x -x 0 |) outside a compact set, for some x 0 ∈ R N -1 and some C 1 function Γ : R + → R such that Γ (r) → 0 as r → +∞.

Then, for every λ 0 ∈ (0, 1), there holds that ∇ x X λ (t, x ) -→ 0 as t → +∞, locally in x ∈ R N -1 and uniformly in λ ∈ (0, λ 0 ] (34)

and moreover ∇ x u(t, x , x N ) -→ 0 as t → +∞, locally in x ∈ R N -1 and uniformly in x N ∈ R.

In dimension N = 3, by writing γ(x ) = γ(r, ϑ) in the standard polar coordinates, condition [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]

means that ∂ r γ(r, ϑ) = -+ O(r -1-η ) and ∂ ϑ γ(r, ϑ) = O(r -η ) as r → +∞.
It is easy to see that, even under Hypothesis 4 (which is stronger than Hypothesis 3), if [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] holds with > 0, then the convergence in ( 34) cannot be uniform with respect to x ∈ R N -1 . In other words, if the initial interface between the states 0 and 1 has a non-zero slope at infinity, then the level sets cannot become uniformly flat at large time. This observation naturally leads to the following conjecture.

Conjecture 14. Assume that Hypothesis 4 holds. Let u be the solution of (1) with an initial datum u 0 given by [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF].

If lim |x |→+∞ ∇γ(x ) = 0, (35)
then, for every λ 0 ∈ (0, 1),

∇ x X λ (t, x ) -→ 0 as t → +∞, uniformly in x ∈ R N -1 and in λ ∈ (0, λ 0 ] ( 36
)
and moreover

∇ x u(t, x) -→ 0 as t → +∞, uniformly in x ∈ R N . ( 37
)
Properties ( 36)- [START_REF] Matano | Large time behavior of disturbed planar fronts in the Allen-Cahn equation[END_REF] obviously hold if γ is constant. Furthermore, if condition ( 35) is replaced by the boundedness of γ, then, at least for some classes of functions f , properties (36) (with λ ∈ (0, λ 0 ] replaced by λ ∈ [a, b], for some fixed 0 < a ≤ b < 1) and [START_REF] Matano | Large time behavior of disturbed planar fronts in the Allen-Cahn equation[END_REF] hold: more precisely, if the function f is of the bistable type [START_REF] Berestycki | Asymptotic spreading for general heterogeneous Fisher-KPP type equations[END_REF], these properties follow from some results in [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF], and the same conclusions hold for more general functions f of the multistable type [START_REF] Poláčik | Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations[END_REF] or for KPP type functions f satisfying (39) below or slightly weaker conditions, see [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF][START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov[END_REF][START_REF] Uchiyama | The behavior of solutions of some semilinear diffusion equation for large time[END_REF]. Further estimates on the exact position of the level sets X λ in the bistable or KPP cases have been established in [START_REF] Matano | Large time behavior of disturbed planar fronts in the Allen-Cahn equation[END_REF][START_REF] Matano | Stability of planar waves in the Allen-Cahn equation[END_REF][START_REF] Roquejoffre | Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations[END_REF]. However, by considering some functions γ with large local oscillations at infinity, it turns out that both conclusions of Conjecture 14 cannot hold if [START_REF] Lewis | Allee dynamics and the spread of invading organisms[END_REF] is replaced by the weaker condition lim |x |→+∞ γ(x )/|x | = 0, see [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF].

To complete this section, let us point out that, under the assumptions of Theorems 7 and 11, the solution u of ( 1) with [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] propagates with speed c * in the direction e N = (0, • • • , 0, 1), owing to Theorem 7, that is, w(e N ) = c * in ( 15)- [START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF]. We conjecture that the solution u then locally converges along its level sets to the front profile ϕ with minimal speed c * . Conjecture 15. Under the assumptions of Theorems 7 and 11, it holds, for every λ ∈ (0, 1), for every sequence (t n ) n∈N diverging to +∞, and for every bounded sequence

(x n ) n∈N in R N -1 , u(t n + t, x n + x , X λ (t n , x n ) + x N ) -→ ϕ(x N -c * t + ϕ -1 (λ)) as n → +∞, ( 38
)
in C 1;2 loc (R t × R N -1 x
) and uniformly with respect to x N ∈ R. If one further assumes (35), then the above limit holds for every sequence (x n ) n∈N in R N -1 , bounded or not.

By [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF], the second conclusion does not hold in general if assumption [START_REF] Lewis | Allee dynamics and the spread of invading organisms[END_REF] is replaced by lim |x |→+∞ γ(x )/|x | = 0. On the other hand, Conjecture 15, and especially its second part, holds if γ is bounded, for some classes of functions f , see [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Matano | Large time behavior of disturbed planar fronts in the Allen-Cahn equation[END_REF][START_REF] Matano | Stability of planar waves in the Allen-Cahn equation[END_REF][START_REF] Poláčik | Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations[END_REF][START_REF] Roquejoffre | Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations[END_REF].

Asymptotic one-dimensional symmetry

Let us now present some results about the asymptotic one-dimensional symmetry, related to the Question 2 presented in Section 1. They concern Fisher-KPP [START_REF] Fisher | The advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] functions f , that is, satisfying

f (0) = f (1) = 0, f (s) > 0 for all s ∈ (0, 1), and s → f (s) s is nonincreasing in (0, 1]. ( 39
)
In this case the hair trigger effect holds, i.e., Hypothesis 3 is fulfilled for any θ, ρ > 0, moreover Hypothesis 4 also holds and the minimal speed with the properties stated in Proposition 5 is explicit: c * = 2 f (0), see [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF].

Theorem 16. [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] Assume that f is of the Fisher-KPP type [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF]. Let u be the solution of (1) with an initial datum

u 0 = 1 U such that U ⊂ R N satisfies ∃ δ > 0, d H (U, U δ ) < +∞. ( 40
)
Assume moreover that U is convex, or more generally, that there is a convex set U such that d H (U, U ) < +∞. Then, any function ψ ∈ Ω(u), with Ω(u) defined by (4), is of the form ψ(x) = Ψ(x • e), for some e ∈ S N -1 and a function Ψ : R → R which is either constant or strictly monotone.

Theorem 16 extends the asymptotic one-dimensional symmetry property known to hold when U is bounded, as a consequence of some results of Jones [START_REF] Jones | Spherically symmetric solutions of a reaction-diffusion equation[END_REF] valid for even more general functions f provided U contains a large ball, as well as when U is the subgraph of a bounded function, by [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF][START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov[END_REF][START_REF] Uchiyama | The behavior of solutions of some semilinear diffusion equation for large time[END_REF]. Conversely, the asymptotic one-dimensional symmetry fails when U is "V -shaped", i.e., the union of two non-parallel half-spaces, which fulfills [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF] but is not at a finite Hausdorff distance from a convex set nor it satisfies [START_REF] Muratov | Threshold phenomena for symmetric decreasing solutions of reactiondiffusion equations[END_REF]. For such an initial datum, the Ω-limit set of the solution contains elements which are not one-dimensional.

Condition [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF] means that there exists R > 0 such that, for any x ∈ U , there is a ball B δ (x 0 ) ⊂ U with |x -x 0 | < R. It is fulfilled in particular if U satisfies a uniform interior ball condition. One can show that, in dimension N = 2, for a convex set U , property [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF] is equivalent to require that U has nonempty interior. The role of ( 40) is cutting off regions of U playing a negligible role in the large-time behavior of the solutions. This assumption is necessary, otherwise one could consider a V -shaped set U and then take

U := U ∪ k∈Z N B e -|k| 2 (k)
, which is at finite Hausdorff distance from the convex set R N but does not satisfy [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF], and the associated solution violates the asymptotic one-dimensional symmetry (because it essentially behaves at large time as the solution associated with U ) see [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF]. The aforementioned two examples show that the answer to Question 2 cannot be positive without any assumption on U .

The idea of the proof of Theorem 16 in [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] consists in reducing to a case where it is possible to apply the reflection argument of Jones [START_REF] Jones | Spherically symmetric solutions of a reaction-diffusion equation[END_REF], which is valid for more general functions f but fails when U is unbounded. This is achieved by an approximation of the solution through a suitable truncation of its initial support. In order to control the error, new types of supersolutions initially supported in exterior domains are used, which are also employed in the proofs of the results of Section 3.

As a matter of fact, the convex-proximity assumption on U in Theorem 16 is a very special case of a geometric hypothesis under which the one-dimensional symmetry holds. Namely, for a given nonempty set U ⊂ R N and a given point x ∈ R N , we let π x denote the set of orthogonal projections of x onto U , i.e.,

π x := ξ ∈ U : |x -ξ| = dist(x, U ) ,
and, for x / ∈ U , we define the opening function as follows:

O(x) := sup ξ∈πx, y∈U \{ξ} x -ξ |x -ξ| • y -ξ |y -ξ| , with the convention that O(x) = -∞ if U = ∅ or U is a singleton (otherwise -1 ≤ O(x) ≤ 1). When O(x) = -∞, one has O(x) = cos α,
where α is the infimum among all ξ ∈ π x of half the opening of the largest exterior cone to U at ξ having axis x-ξ.

Here is our most general asymptotic symmetry result.

Theorem 17. [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] Assume that f is of the Fisher-KPP type [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF]. Let u be a solution of (1) with an initial datum u 0 = 1 U such that U ⊂ R N satisfies (40) and moreover

lim R→+∞ sup x∈R N , dist(x,U )=R O(x) ≤ 0. (41)
Then any function in Ω(u) is one-dimensional and, in addition, it is either constant or strictly monotone, in the sense of Theorem 16.

It is understood that the left-hand side in condition ( 41) is equal to -∞ (hence the condition is fulfilled) if sup x∈R N dist(x, U ) < +∞ (and indeed in such a case one has by [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF] that sup x∈R N dist(x, U δ ) < +∞ for some δ > 0, which implies that u(t, x) → 1 uniformly in x ∈ R N as t → +∞ due to Proposition 5, hence in particular the asymptotic one-dimensional symmetry holds). We also point out that the limit in (41) always exists, because the involved quantity is nonincreasing with respect to R, see [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF]. Hypothesis [START_REF] Muratov | Threshold phenomena for symmetric decreasing solutions of reactiondiffusion equations[END_REF] means that the angle α in the definition of O(x) tends to a value larger than or equal to π/2 (i.e., the exterior cone contains a half-space) as dist(x, U ) → +∞. Theorem 17 yields Theorem 16 because, firstly, convex sets satisfy O(x) ≤ 0 for every x / ∈ U (actually, they are characterized by such condition in the class of closed sets) and, secondly, if (41) holds for a given set, then it holds true for any set at finite Hausdorff distance from it, by [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF].

However, the class of sets satisfying (41) is wider. It contains for instance the subgraphs of functions with vanishing global mean, i.e., U of the type [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]

with γ ∈ L ∞ loc (R N -1 ) such that γ(x ) -γ(y ) |x -y | -→ 0 as |x -y | → +∞ (42)
(see Corollary 20 below for the precise result in this case). Actually, with ( 26) and ( 42), any ψ ∈ Ω(u) is of the form ψ(x) = Ψ(x N ), Theorems 16 and 17 are concerned with locally uniform convergence properties along sequences of times (t n ) n∈N diverging to +∞ and sequences of points (x n ) n∈N . As a matter of fact, a uniform asymptotic property can be derived, from the conclusions of Theorems 16 or 17. It is expressed in terms of the eigenvalues of the Hessian matrices D 2 u(t, x) (with respect to the x variables). For a symmetric real-valued matrix A of size N × N , let λ 1 (A) ≤ • • • ≤ λ N (A) denote its eigenvalues, and let

σ k (A) := 1≤j 1 <•••<j k ≤N λ j 1 (A) × • • • × λ j k (A), 1 ≤ k ≤ N,
be the elementary symmetric polynomials of eigenvalues of A (σ k (D 2 u(t, x)) is also called k-Hessian).

Theorem 18. [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] Let f and u be as in Theorem 17. Then, for every

2 ≤ k ≤ N , σ k (D 2 u(t, x)) → 0 as t → +∞ uniformly in x ∈ R N .
Theorem 18 is proved using the asymptotic local one-dimensional symmetry given by Theorem 17, together with standard parabolic estimates. We point out that, if ψ : R N → R is of class C 2 and one-dimensional, then σ k (D 2 ψ(x)) = 0 for all 2 ≤ k ≤ N and x ∈ R N , since the quantities σ k (D 2 ψ(x)) involve sums of products of at least two eigenvalues of D 2 ψ(x) (but σ 1 (D 2 ψ(x)) = 0 in general). However, the converse property is immediately not true (for instance, the function ψ : (x 1 , x 2 ) → x 2 1 + x 2 satisfies σ 2 (D 2 ψ(x 1 , x 2 )) = 0 for all (x 1 , x 2 ) ∈ R 2 , but it is not one-dimensional).

Once the asymptotic one-dimensional symmetry and monotonicity properties are established, it is natural to ask what are the directions in which the solution actually becomes locally one-dimensional. Namely, we investigate the set

E := e ∈ S N -1 : ∃ ψ ∈ Ω(u) such that ψ(x) ≡ Ψ(x • e)
for some strictly decreasing function Ψ ∈ C 2 (R) .

Under the assumptions of Theorems 16 or 17, the set E is then the set of the directions of decreasing monotonicity of all non-constant elements of Ω(u) (by the direction of decreasing monotonicity of a -necessarily one-dimensional by Theorems 16 or 17-non-constant function ψ ∈ Ω(u), we mean the unique e ∈ S N -1 such that ψ(x) = Ψ(x • e) for all x ∈ R N , with Ψ decreasing). The constant functions ψ are excluded in the above definition, this is necessary because they are one-dimensional in every direction. Thus, a direction e belongs to E only if, along some diverging sequences of times, the solution flattens in the directions orthogonal to e but not in the direction e, along some sequence of points. We characterize the set E in terms of the initial support U .

Theorem 19. [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] Let f and u be as in Theorem 17. Then E defined in [START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF] is given by

E = e ∈ S N -1 : x n -ξ n |x n -ξ n | → e as n → +∞, for some sequences (x n ) n∈N , (ξ n ) n∈N in R N
such that dist(x n , U ) → +∞ as n → +∞ and ξ n ∈ π xn for all n ∈ N .

In particular, E = ∅ if and only if U is relatively dense in R N or U = ∅. Moreover, for any e ∈ E, any sequence (x n ) n∈N in R N , and any sequence (t n ) n∈N diverging to +∞ such that u(t n , x n + x) → Ψ(x • e) as n → +∞ locally uniformly in x ∈ R N , with Ψ strictly decreasing, one necessarily has dist(x n , U ) ∼ c * t n as n → +∞.

We remark that, without the assumption [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF], the last statement of Theorem 19 may immediately fail. Indeed, if U = {0} then u(t, x) ≡ 0 for all t > 0,

x ∈ R N , hence E = ∅, but U = ∅ is not relatively dense in R N .
When U is bounded with non-empty interior, it follows from Theorem 19 that E = S N -1 . On the one hand, this conclusion gives an additional property -namely the strict monotonicity-with respect to the asymptotic symmetry result contained in [START_REF] Jones | Spherically symmetric solutions of a reaction-diffusion equation[END_REF]. On the other hand, still when U is bounded, the same conclusion is also a consequence of [START_REF] Ducrot | On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data[END_REF][START_REF] Roquejoffre | Sharp large time behaviour in N -dimensional Fisher-KPP equations[END_REF], where it is proved by a completely different argument than in [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF]. The characterization of the directions of asymptotic strict monotonicity in the case of unbounded sets U is more involved. The proof of Theorem 19 is based on an argument by contradiction and on the acceleration of the solutions when they become less and less steep.

Theorem 19 implies that if U is of class C 1 then E is contained in the closure of the set of the outward unit normal vectors to U . If U is convex then E coincides with the closure of the set of outward unit normal vectors to all half-spaces containing U . When U is the subgraph of a function γ with vanishing global mean, i.e. satisfying [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF], then it turns out that E = {e N }, with e N = (0, • • • , 0, 1), as the following result shows. Corollary 20. [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] Assume that f is of the Fisher-KPP type [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF]. Let u be the solution of (1) with an initial datum u 0 = 1 U , where U is given by [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF] with γ ∈ L ∞ loc (R N -1 ) satisfying [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF]. Then any function ψ ∈ Ω(u) is of the form ψ(x , x N ) ≡ Ψ(x N ) for all (x , x N ) ∈ R N -1 × R, with Ψ either constant or strictly decreasing. Moreover, E = {e N }.

Since, by parabolic estimates, the convergence in the definition (4) of the Ω-limit set holds true in C 2 loc (R N ) up to subsequences, Corollary 20 implies that

∇ x u(t, x , x N ) → 0 as t → +∞, uniformly with respect to (x , x N ) ∈ R N -1 × R.
A way to interpret this result is that the oscillations of the initial datum are "damped" as time goes on through some kind of averaging process. We point out that Corollary 20 does not imply the existence of a function Ψ : R + × R → R such that u(t, x , x N ) -Ψ(t, x N ) → 0 as t → +∞ uniformly in (x , x N ) ∈ R N -1 × R, and indeed such a function Ψ does not exist in general (as shown in [START_REF] Roquejoffre | Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations[END_REF] when N = 2 and the limits lim x →±∞ γ(x ) exist but do not coincide). Condition [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF] is satisfied in particular when γ is bounded, and in such a case the conclusion of Corollary 20 can also be deduced from [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF][START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov[END_REF][START_REF] Uchiyama | The behavior of solutions of some semilinear diffusion equation for large time[END_REF]. It is possible to relax the uniform mean condition [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF] of γ in Corollary 20, at the price of restricting the Ω-limit set. Similarly, the arguments of the proof of Theorem 17 can somehow be localized. Loosely speaking, if one focuses on the asymptotic one-dimensional property around a given direction, the global geometric assumption (41) can be restricted to the points x around that direction, and hypothesis (40) can be relaxed too. This leads us to introduce the notion of Ω-limit set in a direction e ∈ S N -1 of a solution u, defined as

Ω e (u) := ψ ∈ L ∞ (R N ) : u(t n , x n + •) → ψ in L ∞ loc (R N ) for some sequences (t n ) n∈N in R + diverging to +∞, and (x n ) n∈N in R N \ {0} such that x n /|x n | → e as n → +∞ ⊂ Ω(u).
Theorem 21. [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF] Assume that f is of the Fisher-KPP type [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF]. Let u be the solution of (1) with an initial condition u 0 = 1 U , where U has nonempty interior and is such that

U ⊂ (x , x N ) ∈ R N -1 × R : x N ≤ γ(x ) , (44)
for a function γ ∈ L ∞ loc (R N -1 ) satisfying [START_REF] Ishige | Parabolic power concavity and parabolic boundary value problems[END_REF]. Then, any function ψ ∈ Ω e N (u) satisfies ψ(x , x N ) ≡ Ψ(x N ) in R N , with Ψ either constant or strictly decreasing. In particular, ∇ x u(t, x , x N ) → 0 as t → +∞ locally in x ∈ R N -1 and uniformly in x N ∈ [R, +∞), for any R ∈ R. Moreover if the inclusion is replaced by an equality in (44), then ∇ x u(t, x , x N ) → 0 as t → +∞ locally in x ∈ R N -1 and uniformly in x N ∈ R.

To complete this section, we propose a list of open questions and conjectures related to the previous results. First of all, under the Fisher-KPP assumption [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF], let ϕ be the traveling front profile with minimal speed, that is, for each e ∈ S N -1 , ϕ(x • e -c * t) satisfies (1) with 0 = ϕ(+∞) < ϕ < ϕ(-∞) = 1 and c * = 2 f (0). Based on Theorems 17 and 19, and together with the definition ( 43) of E, we propose the following.

Conjecture 22. Let f and u be as in Theorem 17. Then,

Ω(u) = 0, 1, x → ϕ(x • e + a) : e ∈ E, a ∈ R .
The above conjecture is known to be true in the case where U is bounded with nonempty interior, by [START_REF] Ducrot | On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data[END_REF][START_REF] Roquejoffre | Sharp large time behaviour in N -dimensional Fisher-KPP equations[END_REF], as well as when U is the subgraph of a bounded function, or more generally when there are two half-spaces H and H -necessarily with parallel boundaries-such that H ⊂ U ⊂ H , by [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF][START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov[END_REF][START_REF] Uchiyama | The behavior of solutions of some semilinear diffusion equation for large time[END_REF].

We remark that the assumption (41) of Theorem 17 is stable by bounded perturbations of the sets U . We could then wonder whether the asymptotic one-dimensional symmetry is also stable with respect to bounded perturbations of the initial support. Namely, if the solution to (1) with an initial datum 1 U satisfying ( 40) is asymptotically locally planar, and if U ⊂ R N satisfies [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF] and d H (U , U ) < +∞, then is the solution to (1) with initial datum 1 U asymptotically locally planar as well?

One can also wonder whether the reciprocal of Theorem 17 is true, in the following sense: if the asymptotic one-dimensional symmetry holds for a solution u of (1) with initial datum 1 U and U satisfying [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF], does necessarily U fulfill ( 41)? The answer is immediately seen to be negative in general: take for instance U given by U = n∈N [2 n , 2 n + 1] × R N -1 , which fulfills [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities[END_REF] but not [START_REF] Muratov | Threshold phenomena for symmetric decreasing solutions of reactiondiffusion equations[END_REF], while u -hence any element of Ω(u) -is one-dimensional, depending on the variable x 1 only. However, the question is open if U is connected.

The results of this section are concerned with the Fisher-KPP equation, i.e. when f satisfies [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF]. However, the same question about the asymptotic one-dimensional symmetry can be asked for more general reaction terms f , still with f (0) = f (1) = 0 and satisfying Hypothesis 4, or simply the invasion property stated in Hypothesis 3. First of all, the condition (40) should be strengthened, by requiring δ to be larger than the quantity ρ in Hypothesis 3 (indeed, if f is for instance of the bistable type (8) with 1 0 f (s)ds > 0, then by [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF][START_REF] Zlatoš | Sharp transition between extinction and propagation of reaction[END_REF] there is δ 0 > 0 such that the solution to (1) with initial condition u 0 = 1 B δ 0 converges uniformly as t → +∞ to a ground state, that is, a positive radial solution converging to 0 as |x| → +∞, hence u is not asymptotically locally planar). For general functions f for which Hypothesis 3 holds, if U is bounded and U ρ = ∅, then the solutions to (1) with initial condition 1 U are known to be asymptotically locally planar, by [START_REF] Jones | Spherically symmetric solutions of a reaction-diffusion equation[END_REF]. The same conclusion holds for bistable functions f of the type (8) if there are two half-spaces H and H -necessarily with parallel boundaries-such that H ⊂ U ⊂ H , by [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF][START_REF] Matano | Large time behavior of disturbed planar fronts in the Allen-Cahn equation[END_REF][START_REF] Matano | Stability of planar waves in the Allen-Cahn equation[END_REF] (see also [START_REF] Poláčik | Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations[END_REF] for the case of more general functions f ). On the other hand, still for bistable functions f of the type (8) for which 1 0 f (s)ds > 0, the solutions u to (1) with initial condition 1 U are not asymptotically locally planar if U is V-shaped, that is, if it is the union of two half-spaces with non-parallel boundaries, by [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF][START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF][START_REF] Roquejoffre | Nontrivial large-time behaviour in bistable reactiondiffusion equations[END_REF]. These known results lead us to formulate the following De Giorgi type conjecture for the solutions of the reaction-diffusion equation ( 1) beyond the Fisher-KPP case.

Conjecture 23. Assume that the invasion property, namely Hypothesis 3, holds for some ρ > 0. Let u be the solution to (1) with an initial datum u 0 = 1 U such that U ⊂ R N satisfies d H (U, U ρ ) < +∞ and (41). Then any function in Ω(u) is one-dimensional and, in addition, it is either constant or strictly monotone.

We point out that, when u 0 is given by ( 28) with γ satisfying [START_REF] Lewis | Allee dynamics and the spread of invading organisms[END_REF], Conjecture 14 would imply the validity of the asymptotic one-dimensional symmetry.

Let us also mention another natural question related to the preservation of the convexity of the upper level sets of u when u 0 = 1 U and U is convex. It is known from [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF][START_REF] Ishige | To logconcavity and beyond[END_REF] that, if U is convex, then the solution of the heat equation ∂ t u = ∆u is quasi-concave at each t > 0, that is, for each t > 0 and λ ∈ R, the upper level set {x ∈ R N : u(t, x) > λ} is convex. The same conclusion holds for (1) set in bounded convex domains instead of R N , and under some additional assumptions on f , by [START_REF] Ishige | Parabolic power concavity and parabolic boundary value problems[END_REF]. A natural question is to wonder for which class of functions f this property still holds for (1) in R N . Notice finally that, for any solution u to (1), for any sequence (t n ) n∈N diverging to +∞, and for any sequence (x n ) n∈N in R N , the functions u(t n +•, x n +•) converge locally uniformly in R×R N , up to extraction of a subsequence, to an entire solution to (1) (that is, a solution for all t ∈ R). Remembering Theorem 16 on the asymptotic one-dimensional symmetry for the solutions to (1) with u 0 = 1 U and U convex, and having in mind the question of the previous paragraph on the convexity of the upper level sets, it is then natural to ask the following: if an entire solution v : R × R N → [0, 1] to (1) is quasi-concave for every t ∈ R, is v(t, •) necessarily one-dimensional for every t ∈ R? [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive media[END_REF] The logarithmic lag in the KPP case Assume in this section that f satisfies the Fisher-KPP condition [START_REF] Matano | Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences[END_REF]. We recall that Hypotheses 3 and 4 are fulfilled, and the minimal speed c * of traveling fronts connecting 1 to 0 is given by c * = 2 f (0). It is known from [START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF][START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov[END_REF][START_REF] Nolen | Convergence to a single wave in the Fisher-KPP equation[END_REF][START_REF] Uchiyama | The behavior of solutions of some semilinear diffusion equation for large time[END_REF] that, in the one-dimensional case, the solution u of (1) with initial condition u 0 = 1 R -is such that sup x∈R u(t, x) -ϕ x -c * t + 3 c * ln t + x 0 → 0 as t → +∞, for some x 0 ∈ R. Hence, there is a lag by (3/c * ) ln t of the position of the level sets of u behind the position c * t given by the spreading speed. In dimension N = 2, for initial conditions trapped between two shifts of 1 R×R -, there is a bounded function a such that sup (x 1 ,x 2 )∈R 2 u(t, x 1 , x 2 ) -ϕ x 2 -c * t + 3 c * ln t + a(t, x 1 ) → 0 as t → +∞, see [START_REF] Roquejoffre | Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations[END_REF]. In any dimension N ≥ 2, if the nonnegative initial condition 0 ≡ u 0 ≤ 1 is compactly supported, there is a Lipschitz continuous function a : S N -1 → R such that sup see [START_REF] Ducrot | On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data[END_REF][START_REF] Gärtner | Location of wave fronts for the multi-dimensional KPP equation and Brownian first exit densities[END_REF][START_REF] Roquejoffre | Sharp large time behaviour in N -dimensional Fisher-KPP equations[END_REF]. Notice that N + 2 = 3 + (N -1) corresponds to an additional lag by ((N -1)/c * ) ln t, compared with the 1-dimensional case, which is due to the curvature of the level sets inherited from the fact that the initial condition is compactly supported.

Let us now consider the case of a solution to (1) with an initial condition given by ( 28) and investigate the lag between the position of the level sets of u behind c * t in the direction x N . Assuming that γ is bounded from above, one infers by comparison that, up to an additive constant, the lag is between (3/c * ) ln t (the lag in the 1-dimensional case) and ((N + 2)/c * ) ln t (the lag in the case of compactly supported initial conditions): namely, for every λ ∈ (0, 1) and x ∈ R N -1 , under the notations (29), the lag c * t -X λ (t, x ) satisfies

3 c * ln t + O(1) ≤ c * t -X λ (t, x ) ≤ N + 2 c * ln t + O(1) as t → +∞. ( 45
)
But it is not clear in principle whether or not this lag is equal to one of these bounds or whether it takes intermediate values. The main result of this section states that the actual lag coincides with the upper bound in [START_REF] Poláčik | Convergence and quasiconvergence properties of solutions of parabolic equations on the real line: an overview[END_REF] provided that γ tends to -∞ at infinity faster than a logarithm with a suitable negative coefficient. Thus, in such a case, the position of the level sets of u in the direction x N is the same as when the initial condition is compactly supported. Here is the precise result.

Theorem 24. [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF] Assume that f is of the Fisher-KPP type (39) and let u be the solution of (1) with an initial condition u 0 satisfying [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF]

locally uniformly with respect to λ ∈ (0, 1) and x ∈ R N -1 , and the inequality "≤" holds true in the above formula locally uniformly in λ ∈ (0, 1) and uniformly in x ∈ R N -1 .

If the upper bound for γ in ( 46) is relaxed, we expect the lag of the solution with respect to the critical front to differ from the one associated with compactly supported initial data, that is ((N + 2)/c * ) ln t. We derive the following lower bound for the lag. Proposition 25. [START_REF] Hamel | Flattening and logarithmic lag of level sets in reaction-diffusion equations[END_REF] Assume that f is of the Fisher-KPP type (39) and let u be the solution of (1) with an initial condition u 0 satisfying [START_REF] Hamel | Asymptotic one-dimensional symmetry for the Fisher-KPP equation[END_REF]. If there is σ ≥ -(N -1) such that

lim sup |x |→+∞ γ(x ) ln |x | ≤ 2σ c * , (48)

  The sets B(U ) and U(U ) are respectively open and closed relatively to S N -1 . The condition ξ ∈ B(U ) is equivalent to the existence of an open cone C containing the ray R + {ξ} = {τ ξ : τ > 0} such that U ∩ C is bounded. Conversely, for any ξ ∈ U(U ) and

  x∈R N \{0} u(t, x) -ϕ |x| -c * t + N + 2 c * ln t + a x |x| → 0 as t → +∞,

  . If

		lim sup |x |→+∞	γ(x ) ln(|x |)	< -	2(N -1) c *	,	(46)
	then	X λ (t, x ) = c * t -	N + 2 c

* ln t + O(1) as t → +∞,

* This work has received funding from Excellence Initiative of Aix-Marseille Université -A*MIDEX, a French "Investissements d'Avenir" programme, and from the French ANR RESISTE (ANR-18-CE45-0019) project. The first author acknowledges support of the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université), LabEx CARMIN (ANR-10-LABX-59-01), and Università degli Studi di Roma La Sapienza, where he was Sapienza Visiting Professor and where part of this work was done.

then, for any λ ∈ (0, 1), X λ (t, x ) ≤ c * t -3 -σ c * ln t + o(ln t) as t → +∞, [START_REF] Roquejoffre | Nontrivial large-time behaviour in bistable reactiondiffusion equations[END_REF] locally uniformly with respect to x ∈ R N -1 .

Property [START_REF] Roquejoffre | Nontrivial large-time behaviour in bistable reactiondiffusion equations[END_REF] means that the lag c * t -X λ (t, x ) is at least ((3 -σ)/c * ) ln t + o(ln t) as t → +∞. Notice that this holds even for positive σ. We conjecture that, if the limsup is replaced by a limit in [START_REF] Roquejoffre | Sharp large time behaviour in N -dimensional Fisher-KPP equations[END_REF] and the inequality by an equality, then the estimate (49) should be sharp, namely, the lag should be

for every λ ∈ (0, 1) and x ∈ R N -1 . We emphasize that when σ = 0, this formula would be coherent with the 1-dimensional lag. This formula would also mean that the constant -2(N -1)/c * in (46) would be optimal for the lag to be equivalent to that of solutions with compactly supported initial conditions. Lastly, it would provide a continuum of lags with logarithmic factors ranging in the whole half-line (-∞, (N +2)/c * ]. In particular, solutions with initial conditions of the type (28) with γ(x ) ∼ (6/c * ) ln |x | as |x | → +∞ would have no logarithmic lag, i.e., the same position c * t along the x N -axis as the planar front moving in the direction e N , up to a o(ln t) term as t → +∞. Furthermore, if γ(x ) ∼ κ ln |x | as |x | → +∞ for some κ > (6/c * ), then the logarithmic lag would be negative, i.e., the position of the solution would be ahead of that of the front by a logarithmic-in-time term (observe that the term is linear in time when γ(x ) ∼ α|x | as |x | → +∞ with α > 0, by Theorem 7 and ( 15)-( 16)). Theorem 24 allows us to prove part of Conjecture 12 about the flattening of the level sets under the hypotheses of that theorem.

Corollary 26. [29]

Assume that f is of the Fisher-KPP type (39) and let u be the solution of (1) with an initial condition u 0 satisfying (28) and [START_REF] Poláčik | Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations[END_REF]. Then the following hold:

(i) the conclusion (32) of Conjecture 12 holds, and even locally in λ ∈ (0, 1), that is, ∇ x X λ (t, x ) → 0 as t → +∞, locally uniformly in x ∈ R N -1 and λ ∈ (0, 1);

(ii) for any λ ∈ (0, 1) and x 0 ∈ R N -1 , the function

which exists (up to subsequences) locally uniformly in (t, x , x N ) ∈ R×R N , is independent of x and satisfies lim x N →-∞ u(t, x N + c * t) = 1 and lim x N →+∞ u(t, x N + c * t) = 0, uniformly with respect to t ∈ R.

Corollary 26 shows that, as t → +∞, the solution approaches a one-dimensional entire solution whose level sets move in the direction e N with average velocity equal to c * . It is then natural to expect that u(t, x N ) = ϕ(x N -c * t + ϕ -1 (λ)) for all (t, x N ) ∈ R 2 , where ϕ is the front connecting 1 and 0 with minimal speed c * . That would correspond to property [START_REF] Matano | Stability of planar waves in the Allen-Cahn equation[END_REF] in Conjecture 15. By comparison and some arguments based on the number of intersections of solutions to (1) in dimension 1, it can be shown that u(t, x N ) ≥ ϕ(x N -c * t+ζ) in R 2 , for some ζ ∈ R. But the proof of (38) would still require additional arguments.