Perla El Kettani +

Tadahisa Funaki
email: funaki@ms.u-tokyo.ac.jp

Danielle Hilhorst %
email: danielle.hilhorst@universite-paris-saclay.fr

Hyunjoon Park
email: hyunjoonps@gmail.com

AND Sunder Sethuraman
email: sethuram@math.arizona.edu

Singular limit of an Allen-Cahn equation with nonlinear diffusion

Keywords: MSC 2020 35K57, 35B40 Allen-Cahn equation, Mean curvature flow, Singular limit, Nonlinear diffusion, Interface, Surface tension

We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.

Introduction

The Allen-Cahn equation with linear diffusion

u t = ∆u - 1 ε 2 F (u)
was introduced to understand the phase separation phenomena which appears in the construction of polycrystalline materials [START_REF] Samuel | A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening[END_REF]. Here, u stands for the order parameter which describes the state of the material, F is a double-well potential with two distinct local minima α ± at two different phases, and the parameter ε > 0 corresponds to the interface width in the phase separation process. When ε is small, it is expected that u converges to either of the two states u = α + and u = α -. Thus, the limit ε ↓ 0 creates a steep interface dividing two phases; this is a phase separation phenomenon and the limiting interface is known to evolve according to mean curvature flow; see [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF][START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF].

In this paper, we prove generation and propagation of interface properties for an Allen-Cahn equation with nondegenerate nonlinear diffusion. More precisely, we study the problem

(P ε )          u t = ∆ϕ(u) + 1 ε 2 f (u) in D × R + ∂ϕ(u) ∂ν = 0 in ∂D × R + u(x, 0) = u 0 (x) for x ∈ D
where the unknown function u denotes a phase function, D is a smooth bounded domain in R N , N ≥ 2, ν is the outward unit normal vector to the boundary ∂D and ε > 0 is a small parameter. The nonlinear functions ϕ and f satisfy the following properties. We assume that f has exactly three zeros f (α -) = f (α + ) = f (0) = 0 where α -< 0 < α + , and

f ∈ C 2 (R), f (α -) < 0, f (α + ) < 0, f (0) > 0 (1)
so that f (s) > 0 for s < α -, f (s) < 0 for s > α + .

(

) 2
The conditions (1)-( 2) express the fact that f is a bistable function, so that the α ± are the stable steady states of the corresponding ordinary differential equation, whereas 0 is an unstable steady state. We suppose that

ϕ ∈ C 4 (R), ϕ ≥ C ϕ (3)
for some positive constant C ϕ , so that the partial differential equation (P ε ) is uniformly parabolic. We impose a relation between f and ϕ, namely

α + α - ϕ (s)f (s)ds = 0. ( 4
)
This condition implies the existence of the standing wave solution, which we will use extensively in the proof of Theorem 1.2. An experimental example is the case that ϕ(s) = e s , f (s) = e -s s(1 -s 2 ) [START_REF] Wagner | On the solution of diffusion problems involving concentration-dependent diffusion coefficients[END_REF]. Note that in the case of the standard Allen-Cahn equation with linear diffusion, the condition (4) becomes

α + α -f (s)ds = 0.
As for the initial condition u 0 (x) we assume that u 0 ∈ C 2 (D). Throughout the paper, we define C 0 and C 1 as follows:

C 0 := ||u 0 || C 0 (D) + ||∇u 0 || C 0 (D) + ||∆u 0 || C 0 (D) (5)
Furthermore, we define Γ 0 by Γ 0 := {x ∈ D : u 0 (x) = 0}.

In addition, we suppose Γ 0 is a C 4+ν , 0 < ν < 1 hypersurface without boundary, which is needed such that

Γ 0 D, ∇u 0 (x) • n(x) = 0 if x ∈ Γ 0 (7) u 0 > 0 in D + 0 , u 0 < 0 in D - 0 , (8)
where D - 0 denotes the region enclosed by Γ 0 , D + 0 is the region enclosed between ∂D and Γ 0 , and n is the outward normal vector to D - 0 . It is standard that the above formulation, referred to as Problem (P ε ), possesses a unique classical solution u ε . The regularity of the hypersurface Γ 0 is needed to insure the existence, up to a certain time, of a smooth interface Γ t which moves according to mean curvature [START_REF] Hilhorst | Dispersal towards food : the singular limit of an Allen-Cahn equation[END_REF]. The present paper is originally motivated by the study of the scaling limit of a Glauber+Zero-range particle system. In this microscopic system of interacting random walks, the Zero-range part governs the rates of jumps, while the Glauber part prescribes creation and annihilation rates of the particles. In a companion paper [START_REF] Kettani | Mean curvature interface limit from Glauber+Zero-range interacting particles[END_REF], we show that the system exhibits a phase separation and, under a certain spacetime scaling limit, an interface arises, in the limit macroscopic density field of particles, evolving in time according to the motion by mean curvature. The system is indeed well approximated from macroscopic viewpoint by the Allen-Cahn equation with nonlinear diffusion (P ε ), or more precisely by its discretized equation. Although, in this paper, we study (P ε ) under the Neumann boundary conditions, the formulation under periodic boundary conditions, used in the particle system setting in [START_REF] Kettani | Mean curvature interface limit from Glauber+Zero-range interacting particles[END_REF], can be treated similarly; see Remark 1 below.

In some other physical situations, it is expected that the mathematical modeling also involves nonlinear diffusion. In the experimental article [START_REF] Wagner | On the solution of diffusion problems involving concentration-dependent diffusion coefficients[END_REF], Wagner suggested that for metal alloys the diffusion depends on the concentration. In [START_REF] Alfaro | Generation of Interface for an Allen-Cahn Equation with Nonlinear Diffusion[END_REF][START_REF] Dal Passo | Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility[END_REF], the authors considered degenerate diffusion such as porous medium diffusion instead of linear diffusion. In [START_REF] Paul | Lacey : Motion by curvature in generalized Cahn-Allen models[END_REF], Fife and Lacey generalized the Allen-Cahn equation, which leads them to a parameter dependent diffusion Allen-Cahn equation. Recently, [START_REF] Folino | Plaza : Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity[END_REF] considered an Allen-Cahn equation with density dependent diffusion in 1 space dimension and showed a slow motion property. However, no rigorous proof on the motion of the interface in the nonlinear diffusion context has been given for larger space dimensions N ≥ 2.

In this context, the purpose of this article is to study the singular limit of u ε as ε ↓ 0. We first present a result on the generation of the interface. We use the following notation:

t ε = ε 2 | ln ε|/f (0), η 0 = min(|α -|, α + ). (9)
Theorem 1.1. Let u ε be the solution of the problem (P ε ), η be an arbitrary constant satisfying 0 < η < η 0 . Then, there exist positive constants ε 0 and M 0 such that, for all ε ∈ (0, ε 0 ), the following holds:

(i) for all x ∈ D α --η ≤ u ε (x, t ε ) ≤ α + + η; (10)
(ii) if u 0 (x) ≥ +M 0 ε, then u ε (x, t ε ) ≥ α + -η; (11)
(iii) if u 0 (x) ≤ -M 0 ε, then u ε (x, t ε ) ≤ α -+ η. ( 12
)
After the interface has been generated, the diffusion term has the same order as the reaction term. As a result the interface starts to propagate. Later, we will prove that the interface moves according to the following motion equation:

(IP ) V n = -(N -1)λ 0 κ on Γ t Γ t | t=0 = Γ 0 , (13)
where Γ t is the interface at time t > 0, V n is the normal velocity on the interface, κ denotes its mean curvature, and λ 0 is a positive constant which will be defined later (see ( 26) and (30)). A typical example is a case of spherical symmetry, where we assume that Γ 0 is the circle, namely Γ 0 = {x ∈ R N , |x| = 1}. Then the circle shrinks with the motion equation

   dR dt = - (N -1)λ 0 R , R(0) = 1,
where R(t) denotes the radius of the circle. It is well known that Problem (IP ) possesses locally in time a unique smooth solution. Fix T > 0 such that the solution of (IP ), in [START_REF] Hilhorst | Dispersal towards food : the singular limit of an Allen-Cahn equation[END_REF], exists in [0, T ] and denote the solution by Γ = ∪ 0≤t<T (Γ t × {t}). From Proposition 2.1 of [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF] such a T > 0 exists, and one can deduce that Γ ∈ C 4+ν, 4+ν 2 in [0, T ], given that Γ 0 ∈ C 4+ν . The second main theorem states a result on the generation and the propagation of the interface.

Theorem 1.2. Under the conditions given in Theorem 1.1, for any given 0 < η < η 0 there exist ε 0 > 0 and C > 0 such that

u ε ∈      [α --η, α + + η] for x ∈ D [α + -η, α + + η] if x ∈ D + t \ N Cε (Γ t ) [α --η, α -+ η] if x ∈ D - t \ N Cε (Γ t ) ( 14
)
for all ε ∈ (0, ε 0 ) and t ∈ [t ε , T ], where D - t denotes the region enclosed by Γ t , D + t is enclosed between ∂D and Γ t , and N r (Γ t ) := {x ∈ D, dist(x, Γ t ) < r}.

This theorem implies that, after generation, the interface propagates with a width of order O(ε). Note that Theorems 1.1 and 1.2 extend similar results known for linear diffusion Allen-Cahn equations [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF].

Figure 1: The interface of time t

We now state an approximation result inspired by a similar result proved in [START_REF] Alfaro | On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data[END_REF].

Theorem 1.3. (i) Let the assumptions of Theorem 1.2 hold and ρ > 1. Then, the solution u ε of (P ε ) satisfies

lim ε→0 sup ρt ε ≤t≤T, x∈D u ε (x, t) -U 0 d ε (x, t) ε = 0, ( 15
)
where U 0 is the standing wave solution defined in (20) and d ε denotes the signed distance function associated with Γ ε t := {x ∈ D : u ε (x, t) = 0}, defined by

d ε (x, t) = dist(x, Γ ε t ) if x ∈ D ε,+ t -dist(x, Γ ε t ) if x ∈ D ε,- t
where D ε,- t denotes the region enclosed by Γ ε t and D ε,+ t denotes the region between ∂D and Γ ε t .

(ii) For small enough ε > 0 and for any t ∈ [ρt ε , T ], Γ ε t can be expressed as a graph over Γ t .

Remark 1. Theorems 1.1, 1.2 and 1.3 hold not only for the Neumann boundary condition of Problem (P ε ) but also for periodic boundary conditions with D = T N , with similar proofs as given in Sections 3, 4 and 5.

The paper is organized as follows. In Section 2, the interface motion (IP ) is formally derived from the problem (P ε ) as ε ↓ 0. In particular, the constant λ 0 is obtained. Section 3 studies the generation of interface and gives the proof of Theorem 1.1. In a short time, the reaction term f governs the system and the solution of (P ε ) behaves close to that of an ordinary differential equation. Section 4 discusses the propagation of interface and Theorem 1.2 is proved. The sub-and super-solutions are constructed by means of two functions U 0 and U 1 formally introduced in asymptotic expansions in Section 2. Section 5 gives the proof of Theorem 1.3. Finally, in the Appendix, we define the mobility µ AC and the surface tension σ AC of the interface, especially in our nonlinear setting, and show the relation λ 0 = µ AC σ AC .

Formal derivation of the interface motion equation

In this section, we formally derive the interface motion equation corresponding to the Problem (P ε ) by applying the method of matched asymptotic expansions. To this purpose, we first define the interface Γ t and then derive its motion equation.

Suppose that u ε converges to a step function u where

u(x, t) = α + in D + t α -in D - t . Let Γ t = D + t ∩ D - t , D + t ∪ D - t = D, t ∈ [0, T ].
Let also d(x, t) be the signed distance function to Γ t defined by

d(x, t) := -dist(x, Γ t ) for x ∈ D - t dist(x, Γ t ) for x ∈ D + t . (16)
Assume that u ε has the expansions

u ε (x, t) = α ± + εu ± 1 (x, t) + ε 2 u ± 2 (x, t) + • • •
away from the interface Γ and that

u ε (x, t) = U 0 (x, t, ξ) + εU 1 (x, t, ξ) + ε 2 U 2 (x, t, ξ) + • • • (17)
near Γ, where ξ = d ε . Here, the variable ξ is given to describe the rapid transition between the regions {u ε α + } and {u ε α -}. In addition, we normalize U 0 and U k so that

U 0 (x, t, 0) = 0 U k (x, t, 0) = 0. ( 18
)
To match the inner and outer expansions, we require that

U 0 (x, t, ±∞) = α ± , U k (x, t, ±∞) = u ± k (x, t) (19)
for all k ≥ 2.

After substituting the expansion ( 17) into (P ε ), we collect the ε -2 terms, to obtain

ϕ(U 0 ) zz + f (U 0 ) = 0.
Since this equation only depends on the variable z, we may assume that U 0 is only a function of the variable z, that is U 0 (x, t, z) = U 0 (z). In view of the conditions ( 18) and ( 19), we find that U 0 is the unique increasing solution of the following problem

(ϕ(U 0 )) zz + f (U 0 ) = 0 U 0 (-∞) = α -, U 0 (0) = 0, U 0 (+∞) = α + . (20)
In order to understand the nonlinear effect more clearly, we set

g(v) := f (ϕ -1 (v)),
where ϕ -1 is the inverse function of ϕ and define V 0 (z) := ϕ(U 0 (z)); note that such a transformation is possible by the condition (3). Substituting V 0 into equation (20) yields

V 0zz + g(V 0 ) = 0 V 0 (-∞) = ϕ(α -), V 0 (0) = ϕ(0), V 0 (+∞) = ϕ(α + ). (21)
Condition (4) then implies the existence of the unique increasing solution of (21).

Next we collect the ε -1 terms in the asymptotic expansion. In view of the definition of U 0 (z) and the condition [START_REF] Spohn | Interface motion in models with stochastic dynamics[END_REF], we obtain the following problem

(ϕ (U 0 )U 1 ) zz + f (U 0 )U 1 = d t U 0z -(ϕ(U 0 )) z ∆d U 1 (x, t, 0) = 0, ϕ (U 0 )U 1 ∈ L ∞ (R). ( 22
)
To prove the existence of solution to (22), we consider the function V 1 = ϕ (U 0 )U 1 , which satisfies the problem

   V 1zz + g (V 0 )V 1 = V 0z ϕ (ϕ -1 (V 0 )) d t -V 0z ∆d V 1 (x, t, 0) = 0, V 1 ∈ L ∞ (R). (23)
Now, Lemma 2.2 of [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF] implies the existence and uniqueness of

V 1 provided that R 1 ϕ (ϕ -1 (V 0 )) d t -∆d V 2 0z = 0. Substituting V 0 = ϕ(U 0 ) and V 0z = ϕ (U 0 )U 0z yields d t = R V 2 0z R V 2 0z ϕ (ϕ -1 (V 0 )) ∆d = R (ϕ (U 0 )U 0z ) 2 R ϕ (U 0 )U 2 0z ∆d. ( 24
)
It is known that d t = -V n , where V n is the normal velocity of the interface Γ t , and ∆d is equal to (N -1)κ, where κ is the mean curvature of Γ t . Thus, we obtain the motion equation Γ t ,

V n = -(N -1)λ 0 κ on Γ t , (25)
with

λ 0 = R (ϕ (U 0 )U 0z ) 2 R ϕ (U 0 )U 2 0z . (26)
In the appendix, the constant λ 0 is interpreted as the surface tension σ AC multiplied by the mobility µ AC of the interface. In particular, (IP ) coincides with the equation (1) in [START_REF] Samuel | A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening[END_REF]. Finally, we derive an explicit form of λ 0 . Indeed, we multiply the equation (20) by ϕ(U 0 ) z , yielding

ϕ(U 0 ) zz ϕ(U 0 ) z + f (U 0 )ϕ(U 0 ) z = 0 .
Integrating from -∞ to z, we obtain

1 2 ϕ(U 0 ) z 2 (z) + z -∞ f (U 0 )ϕ(U 0 ) z dz = 0 or alternatively 1 2 ϕ(U 0 ) z 2 (z) + U 0 (z) α - f (s)ϕ (s)ds = 0 . Hence, ϕ(U 0 ) z (z) = √ 2 W (U 0 (z)) , ( 27
)
where W is given by

W (u) = - u α - f (s)ϕ (s)ds = α + u f (s)ϕ (s)ds, (28)
where the last equality holds by [START_REF] Samuel | A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening[END_REF]. It follows that

R ϕ(U 0 ) z U 0z (z)dz = √ 2 R W (U 0 (z))U 0z (z)dz so that also R ϕ (U 0 )U 2 0z (z)dz = √ 2 α + α - W (u)du.
Similarly, since

R (ϕ (U 0 )U 0z ) 2 dz = √ 2 R (ϕ (U 0 ) W (U 0 (z))U 0z )dz, we get R (ϕ (U 0 )U 0z ) 2 dz = √ 2 α + α - ϕ (u) W (u)du, (29)
so that we finally obtain the formula

λ 0 = α + α -ϕ (u) W (u)du α + α - W (u)du . ( 30
)
Note that if ϕ(u) = u, the case of the linear diffusion Allen-Cahn equation, we recover the value λ 0 = 1 as expected.

Generation of the interface

In this section, we prove Theorem 1.1 on the generation of the interface. The main idea, based on the comparison principle Lemma 1, is to construct suitable sub-and super-solutions. The proof of Theorem 1.1 is given in Section 3.4.

Comparison principle

Lemma 1. Let v ∈ C 2,1 (D × R + ) satisfy (P )          v t ≥ ∆ϕ(v) + 1 ε 2 f (v) in D × R + ∂ϕ(v) ∂ν = 0 in ∂D × R + v(x, 0) ≥ u 0 (x) for x ∈ D.
Then, v is a super-solution of Problem (P ε ) and we have

v(x, t) ≥ u ε (x, t), (x, t) ∈ D × R + .
If v satisfies the opposite inequalities in Problem (P ), then v is a sub-solution of Problem (P ε ) and we have

v(x, t) ≤ u ε (x, t), (x, t) ∈ D × R + .
Proof. Consider the inequality satisfied for the difference of a super-solution v and a solution u ε . Apply the maximum principle to the function w := v -u ε to deduce that it is positive.

Solution of the corresponding ordinary differential equation

In the first stage of development, we expect that the solution behaves as that of the corresponding ordinary differential equation:

Y τ (τ, ζ) = f (Y (τ, ζ)) τ > 0 Y (0, ζ) = ζ ζ ∈ R. (31)
We deduce the following result from [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF].

Lemma 2. Let η ∈ (0, η 0 ) be arbitrary. Then, there exists a positive constant C Y = C Y (η) such that the following holds:

(i) There exists a positive constant µ such that for all τ > 0 and all ζ ∈ (-2C 0 , 2C 0 ),

e -µτ ≤ Y ζ (τ, ζ) ≤ C Y e f (0)τ . ( 32
) (ii) For all τ > 0 and all ζ ∈ (-2C 0 , 2C 0 ), Y ζζ (τ, ζ) Y ζ (τ, ζ) ≤ C Y (e f (0)τ -1

).

(iii) There exists a positive constants ε 0 such that, for all ε ∈ (0, ε 0 ), we have

(a) for all ζ ∈ (-2C 0 , 2C 0 ) α --η ≤ Y (f (0) -1 | ln ε|, ζ) ≤ α + + η; (33) (b) if ζ ≥ C Y ε, then Y (f (0) -1 | ln ε|, ζ) ≥ α + -η; (34) (c) if ζ ≤ -C Y ε, then Y (f (0) -1 | ln ε|, ζ) ≤ α -+ η.
Proof. These results can be found in Lemma 4.7 and Lemma 3.7 of [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF], except for (32). To prove (32), we follow similar computations as in Lemma 3.2 of [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF]. Differentiating (31) by ζ, we obtain

Y ζτ (τ, ζ) = f (Y (τ, ζ))Y ζ , τ > 0 Y ζ (0, ζ) = 1,
which yields the following equality,

Y ζ (τ, ζ) = exp τ 0 f (Y (s, ζ)) . (35)
Hence, for ζ = 0,

Y ζ (τ, 0) = exp τ 0 f (Y (s, 0)) = e f (0)τ ,
where the last equality follows since Y (τ, 0) = 0. Also, for ζ = α ± , by ( 1), we have

Y ζ (τ, α ± ) ≤ e f (0)τ . For ζ ∈ (α -+ η, α + -η) \ {0}, Lemma 3.4 of [1] guarantees the upper bound of Y ζ in (32). We only need to consider the case that ζ ∈ (-2C 0 , 2C 0 ) \ (α -+ η, α + -η).
It follows from (1) that we can choose a positive constant η and η such that

f (s) < 0 , s ∈ I, (36)
where

I := (α --η, α -+ η) ∪ (α + -η, α + + η). Moreover, (1) and (2) imply
Y (τ, ζ) ∈ J, (37)
for ζ ∈ J where J := (min{-2C 0 , α --η}, α -+ η) ∪ (α + -η, max{2C 0 , α + + η}). Thus, (35), ( 36) and (37) guarantee the upper bound of (32) for ζ ∈ I, which only leaves the case ζ ∈ (-2C 0 , 2C 0 ) \ I.

We consider now the case ζ ∈ (α + + η, 2C 0 ); the case of ζ ∈ (-2C 0 , α --η) can be analysed in a similar way. By (3.13) in [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF], we have

ln Y ζ (τ, ζ) = f (α + )τ + Y (τ,ζ) ζ f (s)ds, and f (s) = f (s) -f (α + ) f (s) . ( 38
) Note that f (s) → f (α + ) f (α + )
as s → α + , so that f may be extended as a continuous function. We define

F := f L ∞ (α + ,max{2C 0 ,α + +η}) . Since (2) yields Y (τ, ζ) > α + for ζ ∈ (α + + η, 2C 0 ), by (38) we can find a constant C Y large enough such that Y ζ (τ, ζ) ≤ C Y e f (α + )τ ≤ C Y e f (0)τ .
Thus, we obtain the upper bound of (32).

For the lower bound, we first define

µ := -min s∈I f (s), I = [-2C 0 , 2C 0 ] ∪ [α -, α + ].
Note that µ > 0 by [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF]. Thus, by (35), we obtain

Y ζ (τ, ζ) ≥ e -µτ .

Construction of sub-and super-solutions

We now construct sub-and super-solutions for the proof of Theorem 1.1. For simplicity, we first consider the case where

∂u 0 ∂ν = 0 on ∂D. (39)
In this case, we define sub-and super-solution as follows:

w ± ε (x, t) = Y t ε 2 , u 0 (x) ± ε 2 C 2 e µt/ε 2 -1 = Y t ε 2 , u 0 (x) ± P (t)
for some the constant C 2 . In the general case, where (39) does not necessarily hold, we need to modify w ± ε near the boundary ∂D. This will be discussed later in the proof of Theorem 1.1; see after equation (41).

Lemma 3. Assume (39). Then, there exist positive constants ε 0 and

C 2 , C 2 independent of ε such that, for all ε ∈ (0, ε 0 ), w ± ε satisfies    L(w - ε ) < -C 2 e -µt ε 2 < C 2 e -µt ε 2 < L(w + ε ) in D × [0, t ε ] ∂w - ε ∂ν = ∂w + ε ∂ν = 0 on ∂D × [0, t ε ]. (40)
Proof. We only prove that w + ε is the desired super-solution; one can show that w - ε is a sub-solution in a similar way. The assumption (39) implies

∂w ± ε ∂ν = 0 on ∂D × R + .
Define the operator L by

Lu = u t -∆ϕ(u) - 1 ε 2 f (u). Then, direct computation with τ = t/ε 2 gives L(w + ε ) = 1 ε 2 Y τ + P (t)Y ζ -ϕ (w + ε )|∇u 0 | 2 (Y ζ ) 2 + ϕ (w + ε )∆u 0 Y ζ + ϕ (w + ε )|∇u 0 | 2 Y ζζ + 1 ε 2 f (Y ) = 1 ε 2 (Y τ -f (Y )) + Y ζ P (t) -ϕ (w + ε )|∇u 0 | 2 Y ζ + ϕ (w + ε )∆u 0 + ϕ (w + ε )|∇u 0 | 2 Y ζζ Y ζ .
By the definition of Y , the first term on the right-hand-side vanishes. By choosing ε 0 sufficiently small, for 0 ≤ t ≤ t ε , we have

P (t) ≤ P (t ε ) = ε 2 C 2 (e µt ε /ε 2 -1) = ε 2 C 2 (ε -1 -1) < C 0 .
Hence, |u 0 + P (t)| < 2C 0 . Applying Lemma 2, ( 5) and [START_REF] Brochet | Finite dimensional exponential attractor for the phase field model[END_REF] gives

Lw + ε ≥ Y ζ C 2 µe µt/ε 2 -(C 2 0 C 1 C Y e µt/ε 2 + C 0 C 1 + C 2 0 C 1 C Y (e µt/ε 2 -1)) = Y ζ (C 2 µ -C 2 0 C 1 C Y -C 2 0 C 1 C Y )e µt/ε 2 + C 2 0 C 1 C Y -C 0 C 1 .
By (32), for C 2 large enough, we can find a positive constant C 2 independent of ε such that

Lw + ε ≥ C 2 e -µt ε 2 .
Thus, w + ε is a super-solution for Problem (P ε ).

Proof of Theorem 1.1

We deduce from the comparison principle Lemma 1 and the construction of the sub-and super-solutions that

w - ε (x, t ε ) ≤ u ε (x, t ε ) ≤ w + ε (x, t ε ) (41)
under the condition (39). If (39) does not hold, one can modify the functions w ± as follows: from condition (8), there exist positive constants d 0 and ρ such that (i) the distance function d(x, ∂D) is smooth enough on {x ∈ D :

d(x, ∂D) < 2d 0 } and (ii) u 0 (x) ≥ ρ if d(x, ∂D) ≤ d 0 . Let ξ be a smooth cut-off function defined on [0, +∞) such that 0 ≤ ξ ≤ 1, ξ(0) = ξ (0) = 0 and ξ(z) = 1 for z ≥ d 0 . Define u + 0 := ξ(d(x, ∂D))u 0 (x) + [1 -ξ(d(x, ∂D))] max D u 0 u - 0 := ξ(d(x, ∂D))u 0 (x) + [1 -ξ(d(x, ∂D))] ρ.
Then, u - 0 ≤ u 0 ≤ u + 0 and u ± 0 satisfy the homogeneous Neumann boundary condition (39). Thus, by using a similar argument as in the proof of Lemma 3, we may find sub-and super-solutions as follows,

w ± ε (x, t) = Y t ε 2 , u ± 0 (x) ± ε 2 C 2 e f (0)t/ε 2 -1 .
We now show [START_REF] Krylov | Lectures on Elliptic and Parabolic Equations in Sobolev Spaces[END_REF], [START_REF] Paul | Lacey : Motion by curvature in generalized Cahn-Allen models[END_REF] and [START_REF] Folino | Plaza : Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity[END_REF]. By the definition of C 0 in (5), we have

-C 0 ≤ min x∈D u 0 (x) < ρ.
Thus, for ε 0 small enough, we have that

-2C 0 ≤ u ± 0 (x) ± (C 2 ε -C 2 ε 2 ) ≤ 2C 0 for x ∈ D
holds for any ε ∈ (0, ε 0 ). Thus, the assertion (10) is a direct consequence of (33) and (41).

For [START_REF] Paul | Lacey : Motion by curvature in generalized Cahn-Allen models[END_REF], first we choose M 0 large enough so that

M 0 ε -C 2 ε + C 2 ε 2 ≥ C Y ε. Then, for any x ∈ D such that u - 0 (x) ≥ M 0 ε, we have u - 0 (x) -ε 2 C 2 e f (0)t/ε 2 -1 ≥ u - 0 (x) -(C 2 ε -C 2 ε 2 ) ≥ M 0 ε -C 2 ε + C 2 ε 2 ≥ C Y ε.
Therefore, with (34) and (41), we see that [START_REF] Paul | Lacey : Motion by curvature in generalized Cahn-Allen models[END_REF]. Note that ( 12) can be shown in the same way. This completes the proof of Theorem 1.1.

u ε (x, t ε ) ≥ α + -η for any x ∈ D such that u - 0 (x) ≥ M 0 ε, which implies

Propagation of the interface

The main idea of the proof of Theorem 1.2 is that we proceed by imbrication: By the comparison principle Lemma 1, we show at the generation time that u + (x, 0) ≥ w + (x, t ε ) and u -(x, 0) ≤ w -(x, t ε ) so that we can pass continuously from the generation of interface sub-and super-solutions to the propagation of interface sub-and super-solutions.

To this end, we first introduce a modified signed distance function, and several estimates on the functions U 0 and U 1 useful in the sub-and super-solution construction, before showing Theorem 1.2 in Section 4.4.

A modified signed distance function

We introduce a useful cut off signed distance function d as follows. Recall the signed distance function d defined in [START_REF] Mielke | On evolutionary Γ-convergence for gradient systems[END_REF], and interface Γ t satisfying [START_REF] Hilhorst | Dispersal towards food : the singular limit of an Allen-Cahn equation[END_REF]. Choose d 0 > 0 small enough so that the signed distance function d is smooth in the set

{(x, t) ∈ D × [0, T ], |d(x, t)| < 3d 0 } and that dist(Γ t , ∂D) ≥ 3d 0 for all t ∈ [0, T ].
Let h(s) be a smooth non-decreasing function on R such that

h(s) =      s if |s| ≤ d 0 -2d 0 if s ≤ -2d 0 2d 0 if s ≥ 2d 0 .
We then define the cut-off signed distance function d by

d(x, t) = h(d(x, t)), (x, t) ∈ D × [0, T ].
Note, as d coincides with d in the region

{(x, t) ∈ D × [0, T ] : |d(x, t)| < d 0 },
that we have

d t = λ 0 ∆d on Γ t .
Moreover, d is constant near ∂D and the following properties hold.

Lemma 4. There exists a constant

C d > 0 such that (i) |d t | + |∇d| + |∆d| ≤ C d , (ii) |d t -λ 0 ∆d| ≤ C d |d| in D × [0, T ].
4.2 Estimates for the functions U 0 , U 1

Here, we give estimates for the functions which will be used to construct the sub-and super-solutions.

Recall that U 0 (cf. ( 20)) is a solution of the equation

(ϕ(U 0 )) zz + f (U 0 ) = 0.
We have the following results.

Lemma 5. There exists constants Ĉ0 , λ 1 > 0 such that for all z ∈ R,

(i) |U 0 |, |U 0z |, |U 0zz | ≤ Ĉ0 , (ii) |U 0z |, |U 0zz | ≤ Ĉ0 exp(-λ 1 |z|).
Proof. Recall that V 0 = ϕ(U 0 ) satisfies the equation ( 21) with ϕ ∈ C 4 (R). Lemma 2.1 of [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF] implies that there exist some positive constants C 0 and λ 1 such that, for all z ∈ R,

|V 0 |, |V 0z |, |V 0zz | ≤ C 0 ; |V 0z |, |V 0zz | ≤ C 0 exp(-λ 1 |z|),
and therefore similar bounds hold for U 0 .

In terms of the cut-off signed distance function d = d(x, t), for each (x, t) ∈ D × [0, T ], we define U 1 (x, t, •) : R → R as the solution of the following problem:

(ϕ (U 0 )U 1 ) zz + f (U 0 )U 1 = (λ 0 U 0z -(ϕ(U 0 )) z )∆d U 1 (x, t, 0) = 0, ϕ (U 0 )U 1 ∈ L ∞ (R). ( 42
)
Existence of the solution U 1 can be shown in the same way as that for U 1 in (22). Finally, we give the following estimates for U 1 = U 1 (x, t, z).

Lemma 6. There exist positive constants Ĉ1 and λ 1 such that for all z ∈ R

(i) |U 1 |, |U 1z |, |U 1zz |, |∇U 1z |, |∇U 1 |, |∆U 1 |, |U 1t | ≤ Ĉ1 , (ii) |U 1z |, |U 1zz |, |∇U 1z | ≤ Ĉ1 exp(-λ 1 |z|).
Here, the operators ∇ and ∆ act on the variable x.

Proof. Define V 1 (z) := ϕ (U 0 (z))U 1 (z).
As in (22), we obtain an equation for V 1 :

   V 1zz + g (V 0 )V 1 = λ 0 V 0z ϕ (ϕ -1 (V 0 )) -V 0z ∆d V 1 (x, t, 0) = 0, V 1 ∈ L ∞ (R). (43)
Applying Lemmas 2.2 and 2.3 of [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF] to (43) implies the boundedness of V 1 , V 1z , V 1zz . Moreover, since d is smooth in D × [0, T ], we can apply Lemma 2.2 of [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF] to obtain the boundedness of ∇V 1 , ∆V 1 . The desired estimates for the function U 1 now follows via the smoothness of ϕ as in the proof of Lemma 5.

Construction of sub-and super-solutions

We construct candidates sub-and super-solutions as follows: Given ε > 0, define

u ± (x, t) = U 0 d(x, t) ± εp(t) ε + εU 1 x, t, d(x, t) ± εp(t) ε ± q(t) (44)
where

p(t) = -e -βt/ε 2 + e Lt + K, q(t) = σ βe -βt/ε 2 + ε 2 Le Lt ,
in terms of positive constants ε, β, σ, L, K. Next, we give specific conditions for these constants which will be used to show that indeed u ± are sub-and super-solutions. We assume that the positive constant ε 0 obeys

ε 2 0 Le LT ≤ 1, ε 0 Ĉ1 ≤ 1 2 . ( 45
)
We first give a result on the boundedness of f (U 0 (z)) + (ϕ (U 0 (z)) zz .

Lemma 7. There exists b > 0 such that f (U 0 (z)) + (ϕ (U 0 )) zz < 0 on {z :

U 0 (z) ∈ [α -, α -+ b] ∪ [α + - b, α + ]}.
Proof. We can choose b 1 , F > 0 such that

f (U 0 (z)) < -F on {z : U 0 (z) ∈ [α -, α -+ b 1 ] ∪ [α + -b 1 , α + ]}. Note that (ϕ (U 0 )) zz = ϕ (U 0 )U 2 0z + ϕ (U 0 )U 0zz .
From Lemma 5, we can choose b 2 > 0 small enough so that

|(ϕ (U 0 )) z | < F, |(ϕ (U 0 )) zz | < F on {z : U 0 (z) ∈ [α -, α -+ b 2 ] ∪ [α + -b 2 , α + ]}. Define b := min{b 1 , b 2 }. Then, we have f (U 0 (z)) + (ϕ (U 0 )) zz < F -F = 0. Fix b > 0 which satisfies the result of Lemma 7. Denote J 1 := {z : U 0 (z) ∈ [α -, α -+ b] ∪ [α + - b, α + ]}, J 2 = {z : U 0 (z) ∈ [α -+ b, α + -b]}. Let β := -sup f (U 0 (z)) + (ϕ (U 0 (z))) zz 3 : z ∈ J 1 . (46)
The following result plays an important role in verifying sub-and super-solution properties.

Lemma 8. There exists a constant σ 0 small enough such that for every 0 < σ < σ 0 , we have

U 0z -σ(f (U 0 ) + (ϕ (U 0 )) zz ) ≥ 3σβ.
Proof. To show the assertion, it is sufficient to show that there exists σ 0 such that, for all 0 < σ < σ 0 ,

U 0z σ -(f (U 0 ) + (ϕ (U 0 )) zz ) ≥ 3β. ( 47
)
We prove the result on each of the sets J 1 , J 2 .

On the set J 1 , note that U 0z > 0 on R. If z ∈ J 1 , for any σ > 0 we have

U 0z σ -(f (U 0 ) + (ϕ (U 0 )) zz ) > -sup z∈J 1 (f (U 0 ) + (ϕ (U 0 )) zz ) = 3β.
On the set J 2 , which is compact in R, there exist positive constants c 1 , c 2 such that

U 0z ≥ c 1 , |f (U 0 ) + (ϕ (U 0 )) zz | ≤ c 2 .
Therefore, we have

U 0z σ -(f (U 0 ) + (ϕ (U 0 )) zz ) ≥ c 1 σ -c 2 → ∞ as σ ↓ 0,
which implies (47) on J 2 for σ small enough.

Before we present the rigorous proof that u ± are sub-and super-solutions, we first give detailed computations needed in the sequel. Recall (44). First, note, with U 0 and U 1 corresponding to u + , that

ϕ(u + ) = ϕ(U 0 ) + (εU 1 + q)ϕ (U 0 ) + (εU 1 + q) 2 1 0 (1 -s)ϕ (U 0 + (εU 1 + q)s)ds f (u + ) = f (U 0 ) + (εU 1 + q)f (U 0 ) + (εU 1 + q) 2 2 f (θ(x, t)), (48)
where θ is a function satisfying θ(x, t) ∈ (U 0 , U 0 + εU 1 + q(t)). Straightforward computations yield

(u + ) t = U 0z d t + εp t ε + εU 1t + U 1z (d t + εp t ) + q t ∆ϕ(u + ) = ∇ • (ϕ(U 0 )) z ∇d ε + U 1z ϕ (U 0 )∇d + ε∇U 1 ϕ (U 0 ) + (εU 1 + q)(ϕ (U 0 )) z ∇d ε + ∇R = (ϕ(U 0 )) zz |∇d| 2 ε 2 + (ϕ(U 0 )) z ∆d ε + (U 1z ϕ (U 0 )) z |∇d| 2 ε + U 1z ϕ (U 0 )∆d + 2∇U 1z ϕ (U 0 ) • ∇d + ∇U 1 (ϕ (U 0 )) z • ∇d + ε∆U 1 ϕ (U 0 ) + (U 1 ϕ (U 0 ) z ) z |∇d| 2 ε + q(ϕ (U 0 )) zz |∇d| 2 ε 2 + ∇U 1 (ϕ (U 0 )) z • ∇d + (εU 1 + q)(ϕ (U 0 )) z ∆d ε + ∆R (49)
where R(x, t) = (εU

1 + q) 2 1 0 (1 -s)ϕ (U 0 + (εU 1 + q)s)ds. Define r(x, t) = 1 0 (1 -s)ϕ (U 0 + (εU 1 + q)s)ds. Then, we have ∆R(x, t) = ∇ • ∇ (εU 1 ) 2 + 2εqU 1 + q 2 r = ∇ • 2εU 1 (U 1z ∇d + ε∇U 1 ) + 2q (U 1z ∇d + ε∇U 1 ) r(x, t) + (εU 1 ) 2 + 2εqU 1 + q 2 ∇r(x, t) = 2 (U 1z ∇d + ε∇U 1 ) 2 + 2εU 1 U 1zz |∇d| 2 ε + U 1z ∆d + 2∇U 1z • ∇d + ε∆U 1 r(x, t) + 2q U 1zz |∇d| 2 ε + U 1z ∆d + 2∇U 1z • ∇d + ε∆U 1 r(x, t) + 2 2εU 1 (U 1z ∇d + ε∇U 1 ) + 2q (U 1z ∇d + ε∇U 1 ) ∇r(x, t) + (εU 1 ) 2 + 2εqU 1 + q 2 ∆r(x, t) (50)
where

∇r(x, t) = 1 0 (1 -s)ϕ (U 0 + (εU 1 + q)s) (U 0 + εU 1 s) z ∇d ε + ε∇U 1 s ds ∆r(x, t) = 1 0 (1 -s)ϕ (U 0 + (εU 1 + q)s) (U 0 + εU 1 s) z ∆d ε +(U 0 + εU 1 s) zz |∇d| 2 ε 2 + (2∇U 1z • ∇d + ε∆U 1 )s ds + 1 0 (1 -s)ϕ (4) (U 0 + (εU 1 + q)s) (U 0 + εU 1 s) z ∇d ε + ε∇U 1 s 2 ds.
Define l(x, t), r i (x, t) for i = 1, 2, 3 as follows:

l(x, t) = U 1zz |∇d| 2 ε + U 1z ∆d + 2∇U 1z • ∇d + ε∆U 1 r 1 (x, t) = 2 (U 1z ∇d + ε∇U 1 )
2 + 2εU 1 l(x, t) r(x, t) + 4εU 1 (U 1z ∇d + ε∇U 1 ) ∇r(x, t) + (εU 1 ) 2 ∆r(x, t) r 2 (x, t) = 2ql(x, t)r(x, t) + 4q (U 1z ∇d + ε∇U 1 ) ∇r(x, t) + 2εqU 1 ∆r(x, t) r 3 (x, t) = q 2 ∆r(x, t).

Thus,

∆R = r 1 + r 2 + r 3 . (51)
We have the following properties for r i .

Lemma 9. There exists C r > 0 independent of ε such that

|r 1 | ≤ C r , |r 2 | ≤ q ε C r , |r 3 | ≤ q 2 ε 2 C r . (52)
Proof. Note that, by Lemmas 5, 6 and (45) the term U a := U 0 + (εU 1 + q)s is uniformly bounded. Hence, the terms ϕ (U a ), ϕ (U a ), ϕ (4) (U a ) are uniformly bounded, and in particular r is bounded. By a similar reasoning for ∇r and ∆r, it follows that there exists some positive constants c ∇ , c ∆ such that

|∇r| ≤ c ∇ ε , |∆r| ≤ c ∆ ε 2 .
Moreover, by Lemmas 5, 6 there exists a positive constant c l such that

|l(x, t)| ≤ c l ε .
Combining these estimates yields (52).

Let σ a fixed constant satisfying

0 < σ ≤ min{σ 0 , σ 1 , σ 2 }, (53)
where σ 0 is the constant defined in Lemma 8, and σ 1 and σ 2 are given by

σ 1 = 1 2(β + 1) , σ 2 = β (F + C r )(β + 1) , F = ||f || L ∞ (α --1,α + +1) . (54)
Note that, since σ < σ 1 and (45), we have

α --1 ≤ |u ± | ≤ α + + 1.
Lemma 10. Let β be given by (46) and let σ satisfy (53). Then, there exists ε 0 > 0 and a positive constant C p , which does not depend on ε, such that

   L(u -) < -C p < C p < L(u + ) in D × [0, T ] ∂u - ∂ν = ∂u + ∂ν = 0 on ∂D × [0, T ] (55)
for every ε ∈ (0, ε 0 ).

Proof. In the following, we only show that u + is a super solution; one can show that u -is a sub-solution in a similar way. Combining the computations above in (48), ( 49), ( 50) and (51), we obtain

Lu + = (u + ) t -∆(ϕ(u + )) - 1 ε 2 f (u + ) = E 1 + E 2 + E 3 + E 4 + E 5 + E 6 ,
where

E 1 = - 1 ε 2 (ϕ(U 0 )) zz |∇d| 2 + f (U 0 ) - |∇d| 2 -1 ε 2 q(ϕ (U 0 )) zz - |∇d| 2 -1 ε (U 1 ϕ (U 0 )) zz E 2 = 1 ε U 0z d t - 1 ε ((ϕ(U 0 )) z ∆d + (U 1z ϕ (U 0 )) z + (U 1 ϕ (U 0 ) z ) z + U 1 f (U 0 )) E 3 = [U 0z p t + q t ] - 1 ε 2 qf (U 0 ) + q(ϕ (U 0 )) zz + q 2 2 f (θ) -r 3 (x, t) E 4 = εU 1z p t - q ε (ϕ (U 0 )) z ∆d + U 1 f (θ) -r 2 (x, t) E 5 = εU 1t -ε∆U 1 ϕ (U 0 ) E 6 = U 1z d t -2∇U 1z ϕ (U 0 ) • ∇d -2∇U 1 (ϕ (U 0 )) z • ∇d -(U 1 ϕ (U 0 )) z ∆d -r 1 (x, t) - (U 1 ) 2 2 f (θ).
Estimate of the term E 1 . Using (20) we write E 1 in the form

E 1 = - |∇d| 2 -1 ε 2 (ϕ(U 0 )) zz + q(ϕ (U 0 )) zz - |∇d| 2 -1 ε (U 1 ϕ (U 0 )) zz .
We only consider the term e 1 := |∇d| 2 -1 ε (U 1 ϕ (U 0 )) zz ; the other terms can be bounded similarly. In the region where |d| ≤ d 0 , we have |∇d| = 1 so that e 1 = 0. If, however |∇d| = 1, we have

|(U 1 ϕ (U 0 )) zz | ε ≤ Ĉ1 ε e -λ 1| d ε +p(t)| ≤ Ĉ1 ε e -λ 1[ d 0 ε -p(t)] ≤ Ĉ1 ε e -λ 1[ d 0 ε -(1+e LT +K)] .
Choosing ε 0 small enough such that

d 0 2ε 0 -1 + e LT + K ≥ 0, we deduce |(U 1 ϕ (U 0 )) zz | ε ≤ Ĉ1 ε e -λ 1 d 0 2ε → 0 as ε ↓ 0. Thus, 1 ε |(U 1 ϕ (U 0
)) zz | is uniformly bounded, so that there exists Ĉ2 independent of ε, L such that

|e 1 | ≤ Ĉ2 .
Finally, as a consequence, we deduce that there exists C1 independent of ε, L such that

|E 1 | ≤ C1 . ( 56
)
Estimate of the term E 2 . Using (42), we write E 2 in the form

E 2 = 1 ε U 0z d t - 1 ε λ 0 U 0z ∆d = U 0z ε (d t -λ 0 ∆d).
Applying Lemma 4, 5 and 6 gives

|E 2 | ≤ C d Ĉ0 |d| ε e -λ 1| d ε +p| ≤ C d Ĉ0 max ξ∈R |ξ|e -λ 1 |ξ+p| .
Note that max ξ∈R |ξ|e -λ 1 |ξ+p| ≤ |p| + 1 λ 1 (cf. [START_REF] Hilhorst | Dispersal towards food : the singular limit of an Allen-Cahn equation[END_REF]). Thus, there exists C2 such that

|E 2 | ≤ C2 (1 + e LT ). ( 57
)
Estimate of the term E 3 . Substituting p t = q ε 2 σ and then replacing q by its explicit form (cf. ( 44)) gives

E 3 = q ε 2 σ U 0z -σ(f (U 0 ) + (ϕ (U 0 )) zz ) -σq 1 2 f (θ) + ε 2 q 2 r 3 + q t = 1 ε 2 βe -βt ε 2 + ε 2 Le Lt U 0z -σ(f (U 0 ) + (ϕ (U 0 )) zz ) -σ 2 (βe -βt ε 2 + Lε 2 e Lt ) 1 2 f (θ) + ε 2 q 2 r 3 - 1 ε 2 σβ 2 e -βt ε 2 + ε 2 σL 2 e Lt = 1 ε 2 βe -βt ε 2 (I -σβ) + Le Lt [I + ε 2 σL]
where

I := U 0z -σ(f (U 0 ) + (ϕ (U 0 )) zz ) -σ 2 (βe -βt ε 2 + Lε 2 e Lt ) 1 2 f (θ) + ε 2 q 2 r 3 .
Applying Lemma 8, using ( 45) and ( 53), yields

I ≥ 3σβ -σσ 2 β + Lε 2 e Lt |f (θ)| + ε 2 q 2 |r 3 | ≥ 3σβ -σσ 2 (β + 1) |f (θ)| + ε 2 q 2 |r 3 | ≥ 2σβ,
where the last inequality follows from (54). This implies that

E 3 ≥ σβ 2 ε 2 e -βt ε 2 + 2σβLe Lt . ( 58
)
Estimate of the term E 4 . Substituting again p t = q ε 2 σ , with q in its explicit form (44) gives

E 4 = q εσ U 1z -σ((ϕ (U 0 )) z ∆d + U 1 f (θ)) -σ ε q r 2 = 1 ε βe -βt ε 2 + ε 2 Le Lt U 1z -σ((ϕ (U 0 )) z ∆d + U 1 f (θ)) -σ ε q r 2 .
Applying Lemma 4, 5, 6 and 9 gives the uniform boundedness of the last factor in parenthesis. Thus, there exists a constant C4 such that

|E 4 | ≤ C4 1 ε βe -βt ε 2 + ε 2 Le Lt . ( 59
)
Estimate of the terms E 5 and E 6 . Applying Lemma 4, 5 and 6, it follows that there exists C5 such that

|E 5 | + |E 6 | ≤ C5 . ( 60
)
Combination of the above estimates. Collecting the estimates (56),( 57),( 58),( 59),(60), we obtain

L(u + ) ≥ σβ 2 ε 2 -C4 β ε e -βt ε 2 + 2σβL -ε C4 L -C2 e Lt -C1 -C2 -C5 ≥ σβ 2 ε 2 -C4 β ε e -βt ε 2 + 2σβL 3 -ε C4 L e Lt + 2σβL 3 -C2 e Lt + 2σβL 3 -C6
where C6 = C1 + C2 + C5 . Choose ε 0 small enough and L large enough so that

σβ > 3 C4 ε 0 , σβL > 3 max{ C2 , C6 }.
Then, we deduce that there exists a positive constant C p , independent of ε, such that L(u + ) ≥ C p .

Proof of Theorem 1.2

The proof of Theorem 1.2 is divided in two steps: (i) For large enough K > 0, we prove that u

-(x, t) ≤ u ε (x, t + t ε ) ≤ u + (x, t) for x ∈ D, t ∈ [0, T -t ε
] and (ii) we employ (i) to show the desired result.

Step 1. Let η ∈ (0, η 0 ) be arbitrary. Fix σ, β that satisfy (46), (53) and σβ < η 2 .

Theorem 1.1 implies the existence of constants ε 0 and M 0 such that ( 10)-( 12) are satisfied with the constant η replaced by σβ/4. Conditions ( 7) and ( 8) imply that there exists a positive constant M 1 such that

if d(x, 0) ≤ -M 1 ε, then u 0 (x) ≤ -M 0 ε, if d(x, 0) ≥ M 1 ε, then u 0 (x) ≥ +M 0 ε.
Hence, we deduce, by applying [START_REF] Krylov | Lectures on Elliptic and Parabolic Equations in Sobolev Spaces[END_REF], [START_REF] Folino | Plaza : Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity[END_REF], that

u ε (x, t ε ) ≤ H + (x) :=      α + + σβ 4 if d(x, 0) ≥ -M 1 ε α -+ σβ 4 if d(x, 0) < -M 1 ε.
Also, by applying ( 10), [START_REF] Paul | Lacey : Motion by curvature in generalized Cahn-Allen models[END_REF],

u ε (x, t ε ) ≥ H -(x) :=      α + - σβ 4 if d(x, 0) > M 1 ε α -- σβ 4 if d(x, 0) ≤ M 1 ε.
Next, we fix a sufficient large constant K such that

U 0 (M 1 -K) ≤ α -+ σβ 4 and U 0 (-M 1 + K) ≥ α + - σβ 4 .
For such a constant K, Lemma 10 implies the existence of coefficients ε 0 and L such that the inequalities in (55) holds. We claim that

u -(x, 0) ≤ H -(x) ≤ H + (x) ≤ u + (x, 0). ( 61
)
We only prove the last inequality since the first inequality can be proved similarly. By Lemma 6, we have

|U 1 | ≤ Ĉ1
. Thus, we can choose ε 0 small enough so that, for ε ∈ (0, ε 0 ), we have ε Ĉ1 ≤ σβ 4 . Then, noting (44),

u + (x, 0) ≥ U 0 d(x, 0) + εp(0) ε -ε Ĉ1 + σβ + ε 2 σL > U 0 d(x, 0) ε + K + 3 4 σβ.
In the set {x ∈ D : d(x, 0) ≥ -M 1 ε}, the inequalities above, and the fact that U 0 is an increasing function imply

u + (x, 0) > U 0 (-M 1 + K) + 3 4 σβ ≥ α + + σβ 2 > H + (x).
Moreover, since U 0 ≥ α -in the set {x ∈ D : d(x, 0) < -M 1 ε}, we have

u + (x, 0) > α -+ 3 4 σβ > H + (x).
Thus, we proved the first inequality in (61) above.

The inequalities (61) and Lemma 10 now permit to apply the comparison principle Lemma 1, so that we have

u -(x, t) ≤ u ε (x, t + t ε ) ≤ u + (x, t) for x ∈ D, t ∈ [0, T -t ε ]. (62)
Step 2. Choose C > 0 so that

U 0 (C -e LT -K) ≥ α + - η 2 and U 0 (-C + e LT + K) ≤ α -+ η 2 .
Then, we deduce from (62), noting (44), that

if d(x, t) ≥ εC, then u ε (x, t + t ε ) ≥ α + -η if d(x, t) ≤ -εC, then u ε (x, t + t ε ) ≤ α -+ η
and since α ± ± η are respectively sub-and super-solutions of (P ε ), we conclude that

u ε (x, t + t ε ) ∈ [α --η, α + + η] for all (x, t) ∈ D × [0, T -t ε ], ε ∈ (0, ε 0 ).
Remark 2. These sub and super solutions guarantee that u ε α + (respectively, u ε α -) for d(x, t) ≥ c(respectively, d(x, t) ≤ -c) with t > ρt ε , ρ > 1 and ε > 0 small enough. In fact, by the definition of q(t), we expect

εU 1 ± q(t) = O(ε)
for t > (ρ -1)t ε . Also, by Lemma 5, we expect

0 < U 0 (z) -α -< cε for z > c ε , 0 < α + -U 0 (z) < cε for z < - c ε .
These estimates yield that there exists a positive constant c such that

|u ε (x, t) -α + | ≤ c ε for d(x, t) > c, |u ε (x, t) -α -| ≤ c ε for d(x, t) < -c for t > ρt ε .
5 Proof of Theorem 1.3

We now introduce the concept of an eternal solution. A solution of an evolution equation is called eternal if it is defined for all positive and negative times. In our problem, we study the nonlinear diffusion problem

w τ = ∆ϕ(w) + f (w), ((z , z (N ) ), τ ) ∈ R N × R, (63)
where z ∈ R N -1 and z (N ) ∈ R. In order to prove Theorem 1.3, we first present two lemmas.

Lemma 11. Let S be a domain of R N × R and let u be a bounded function on S satisfying

u t = ∆ϕ(u) + f (u), (x, t) ∈ S, (64)
where ϕ, f satisfy conditions (1), (3). Then, for any smooth bounded subset S ⊂ S separated from ∂S by a positive constant d we have

u C 2+θ,1+θ/2 (S ) ≤ C , (65)
for any positive constants 0 < θ < 1 and C which depends on u L ∞ (S) , ϕ, f , d, θ and the size of S , where

u C k+θ,k +θ (S ) = u C k,k (S ) + N i,j=1 sup (x,t),(y,t)∈S ,x =y |D k x u(x, t) -D k x u(y, t)| |x -y| θ + sup (x,t),(x,t )∈S ,t =t |D k t u(x, t) -D k t u(x, t )| |t -t | θ
where k, k are non-negative integers and 0 < θ, θ < 1.

Proof. Since S is separated from ∂S by a positive distance, we can find subsets S 1 , S 2 such that S ⊂ S 2 ⊂ S 1 ⊂ S and such that ∂S, ∂S , ∂S i are separated by a positive distance less than d. By condition (3) the regularity of u(x, t) is the same as the regularity of v(x, t) = ϕ(u(x, t)). Note that by (64) v satisfies

v t = ϕ (ϕ -1 (v))[∆v + g(v)] , g(s) = f (ϕ -1 (s))
on S. By Theorem 3.1 p. 437-438 of [START_REF] Ladyženskaja | ceva : Linear and quasilinear equations of parabolic type[END_REF], there exists a positive constant c 1 such that

|∇v| ≤ c 1 in S 1
where c 1 depends only on N, ϕ, ||u|| L ∞ (S) and the distance between S and S 1 . This, together with Theorem 5, p 122 of [START_REF] Krylov | Lectures on Elliptic and Parabolic Equations in Sobolev Spaces[END_REF], imply that

v W 2,1 p (S 2 ) ≤ c 2 ( v L p (S 1 ) + ϕ (ϕ -1 (v))g(v) L p (S 1
) ) for any p > N + 2 where c 2 is a constant that depends on c 1 , p, N, ϕ. With this, by fixing p large enough, the Sobolev embedding theorem in chapter 2, section 3 of [START_REF] Ladyženskaja | ceva : Linear and quasilinear equations of parabolic type[END_REF] yields

v C 1+θ,(1+θ)/2 (S 2 ) ≤ c 3 v W 2,1 p (S 2 )
where 0 < θ < 1 -N +2 p and c 3 depends on c 2 and p. This implies that ϕ (ϕ -1 (v)), g(v) are bounded uniformly in C 1+θ,(1+θ)/2 (S 2 ). Therefore, by Theorem 10.1 p 351-352 of [START_REF] Ladyženskaja | ceva : Linear and quasilinear equations of parabolic type[END_REF] we obtain

v C 2+θ,1+θ/2 (S ) ≤ c 4 v C 1+θ,(1+θ)/2 (S 2 )
where c 4 depends on c 2 , f and ϕ.

Remark 3. Lemma 11 implies the uniform C 2,1 boundedness of the entire solution w in the whole space. This can be derived as follows: Let

S (a,b) = {(x, t) ∈ R N × R, |x -a| 2 + (t -b) 2 ≤ 2}, S (a,b) = {(x, t) ∈ R N × R, |x -a| 2 + (t -b) 2 ≤ 1}
where (a, b) ∈ R N × R. Then, Lemma 11 implies the uniform C 2,1 boundedness of w within S (a,b) where the upper bound is fixed by (65). Since this upper bound is independent to the choice of (a, b), we have uniform C 2,1 bound of w in the whole space.

Next, we present a result inspired by a similar one in [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF].

Lemma 12. Let w((z , z (N ) ), τ ) be a bounded eternal solution of (63) satisfying lim inf

z (N ) →-∞ inf z ∈R N -1 ,τ ∈R w((z , z (N ) ), τ ) = α -, lim sup z (N ) →∞ sup z ∈R N -1 ,τ ∈R w((z , z (N ) ), τ ) = α + , (66)
where z = (z (1) , z (2) ,

• • • z (N -1)
). Then, there exists a constant z * ∈ R such that

w((z , z (N ) ), τ ) = U 0 (z (N ) -z * ).
Proof. We prove the lemma in two steps. First we show w is an increasing function with respect to the z (N ) variable. Then, we prove that w only depends on z (N ) , which means that there exists a function

ψ : R → (α -, α + ) such that w((z , z (N ) ), τ ) = ψ(z (N ) ), ((z , z (N ) ), τ ) ∈ R N × R.
From the increasing property with respect to z (N ) , this allows us to identify ψ as the unique standing wave solution U 0 of the problem (20) up to a translation factor z * . We deduce from (66) that there exist A > 0 and η ∈ (0, η 0 ) such that

α + -η ≤ w((z , z (N ) ), τ ) ≤ α + + η, z (N ) ≥ A α --η ≤ w((z , z (N ) ), τ ) ≤ α -+ η, z (N ) ≤ -A (67)
where η 0 is defined in [START_REF] Kettani | Mean curvature interface limit from Glauber+Zero-range interacting particles[END_REF]. Let τ ∈ R, ρ ∈ R N -1 be arbitrary. Define

w s ((z , z (N ) ), τ ) := w((z + ρ, z (N ) + s), τ + τ )
where s ∈ R. Fix χ ≥ 2A and define

b * := inf b > 0 : ϕ(w χ ) + b ≥ ϕ(w) in R N × R . ( 68
)
We will prove that b * = 0, which will imply that w χ ≥ w in R N × R since ϕ is a strictly increasing function.

To see this, we assume that this does not hold, that is b * > 0. Note, by (66) and (67) we have

w χ ≥ α + -η > α -+ η ≥ w if z (N ) = -A, lim z (N ) →±∞ ϕ(w χ ) -ϕ(w) → 0. ( 69
) Let E = {(x, t) ∈ R N × R, ϕ(w) -ϕ(w χ ) > 0}.
Define a function Z on E as follows Z((z , z (N ) ), τ ) := e -C Z τ [ϕ(w) -ϕ(w χ )]((z , z (N ) ), τ ),

C Z := max sup x∈E [ϕ (w) -ϕ (w χ )]∆ϕ(w) + [ϕ (w)f (w) -ϕ (w χ )f (w χ )] ϕ(w) -ϕ(w χ ) , 0 ≥ 0.
Note that C Z is bounded, since w χ is bounded uniformly in C 2,1 (E) by Remark 3 and in view of ( 1) and

(3) we have

lim x→y ϕ (x) -ϕ (y) ϕ(x) -ϕ(y) = ϕ (y) ϕ (y) < ∞, lim x→y ϕ (x)f (x) -ϕ (y)f (y) ϕ(x) -ϕ(y) = (ϕ f ) (y) ϕ (y) < ∞. (70)
Direct computations give

Z τ -ϕ (w χ )∆Z = e -C Z τ ϕ (w)[∆ϕ(w) + f (w)] -e -C Z τ ϕ (w χ )[∆ϕ(w χ ) + f (w χ )] -C Z Z -e -C Z τ ϕ (w χ )[∆ϕ(w) -∆ϕ(w χ )] = [ϕ (w) -ϕ (w χ )]∆ϕ(w) + [ϕ (w)f (w) -ϕ (w χ )f (w χ )] e -Czτ -C Z Z ≤ C Z Z -C Z Z = 0 in E.
Then, the maximum principle [START_REF] Murray | Maximum Principles in Differential Equations[END_REF] Theorem 5 p.173 yields that the maximum of Z is located at the boundary of E. By the definition of E, Z = 0 on the boundary of E which implies Z ≤ 0 in E. This contradicts the definition of E. Thus, we conclude that b * = 0. Next, we prove that w ≤ w χ for any χ > 0 (see (73) below). For this purpose, define

χ * := inf χ ∈ R, w χ ≥ w for all χ ≥ χ . (71)
Then, our goal can be obtained by proving that χ * ≤ 0. By the previous argument, we already know that χ * ≤ 2A. Since w((z , -∞), τ ) = α -, it follows from (66) that χ * > -∞, since otherwise we would have

α -= w -∞ ((z , z (N ) ), τ ) ≥ w,
leading to a contradiction since w((z , +∞), τ ) = α + > α -. Thus, we conclude that -∞ < χ * ≤ 2A. Assume that χ * > 0, and define

E := {((z , z (N ) ), τ ) ∈ R N × R; |z (N ) | ≤ A}.
If inf E (w χ * -w) > 0, then there exists δ 0 ∈ (0, χ * ) such that w ≤ w χ * -δ in E for all δ ∈ (0, δ 0 ). Since w ≤ w χ * -δ on ∂E , we deduce from a similar argument as above that w ≤ w χ * -δ in {((z , z (N ) ), τ ) ∈ R N × R; |z (N ) | ≥ A}. This contradicts the definition of χ * in (71) so that inf E (w χ * -w) = 0. Thus, we must have a sequence

((z n , z n ), t n ) and z∞ ∈ [-A, A] such that w((z n , z n ), t n ) -w χ * ((z n , z n ), t n ) → 0, z n → z ∞ as n → ∞. Define w n ((z , z (N ) ), τ ) := w((z + z n , z (N ) ), τ + t n ). Since w n is bounded uniformly in C 2+θ,1+θ/2 (R N × R) by Lemma 11, w n converges in C 2,1
loc to a solution w ∞ of (63). Define Z by Z((z , z (N ) ), τ

) := [ϕ(w χ * ∞ ) -ϕ(w ∞ )]((z , z (N ) ), τ ).
Since ϕ is strictly increasing, by (71) we have

Z((z , z (N ) ), τ ) ≥ 0 in R N × R Z((0, z ∞ ), 0) = lim n→∞ [ϕ(w χ * n ) -ϕ(w n )]((0, z n ), 0) = lim n→∞ [ϕ(w χ * ) -ϕ(w)]((z n , z n ), t n ) = 0. (72)
Then, direct computation gives

Zτ -ϕ (w χ * ∞ )∆ Z = ϕ (w χ * ∞ )[∆ϕ(w χ * ∞ ) + f (w χ * ∞ )] -ϕ (w ∞ )[∆ϕ(w ∞ ) + f (w ∞ )] -ϕ (w χ * ∞ )[∆ϕ(w χ * ∞ ) -∆ϕ(w ∞ )] = [ϕ (w χ * ∞ ) -ϕ (w ∞ )]∆ϕ(w ∞ ) + [ϕ (w χ * ∞ )f (w χ * ∞ ) -ϕ (w ∞ )f (w ∞ )],
If Z = 0 we obtain Zτ -ϕ (w χ * ∞ )∆ Z = 0. If Z > 0, we obtain

Zτ -ϕ (w χ * ∞ )∆ Z = [ϕ (w χ * ∞ ) -ϕ (w ∞ )]∆ϕ(w ∞ ) + [ϕ (w χ * ∞ )f (w χ * ∞ ) -ϕ (w ∞ )f (w ∞ )] ϕ(w χ * ∞ ) -ϕ(w ∞ ) Z ≥ -C Z,
for some positive constant C, where the last inequality follows from (70) and the fact that ∆ϕ(w ∞ ) is uniformly bounded in the whole space. Since by (72) Z attains a non-positive minimum at ((0, z ∞ ), 0), we deduce from the maximum principle applied on the domain R N × (-∞, 0] that Z = 0 for all (z , z

(N ) ) ∈ R N , τ ≤ 0. Hence, Z ≡ 0 in R N × R. This implies that w ∞ ((0, 0), 0) = w ∞ ((ρ, χ * ), τ ) = w ∞ ((2ρ, 2χ * ), 2τ ) = • • • = w ∞ ((kρ, kχ * ), kτ )
for all k ∈ Z, contradicting the fact that w ∞ ((kρ, kχ * ), kτ ) → α + as k → ∞ and w ∞ ((kρ, kχ * ), kτ ) → α - as k → -∞. Thus, we have χ * ≤ 0, and therefore

w((z , z (N ) ), τ ) ≤ w 0 ((z , z (N ) ), τ ) = w((z + ρ, z (N ) ), τ + τ ) (73)
hold for any ρ ∈ R N -1 , τ ∈ R.

We now show that w only depends on z (N ) . Suppose w depends on z and τ . Then, there exist

z 1 , z 2 ∈ R N -1 , z (N ) ∈ R and t 1 , t 2 ∈ R such that w((z 1 , z (N ) ), t 1 ) < w((z 2 , z (N ) ), t 2 ). (74)
Then, by letting z = z 2 , ρ = z 1 -z 2 and τ = t 2 , τ = t 1 -t 2 in the inequality (73), we deduce w((z 2 , z (N ) ), t 2 ) ≤ w((z 1 , z (N ) ), t 1 ), contradicting (74). This implies that w only depends on z (N ) , namely w((z , z (N ) ), τ ) = ψ(z (N ) ). Finally, from the definition of χ * , we have that ψ is increasing.

Proof of Theorem 1.3. We first prove (i). Recall that d(x, t) is the cut-off signed distance function to the interface Γ t moving according to equation [START_REF] Hilhorst | Dispersal towards food : the singular limit of an Allen-Cahn equation[END_REF], and d ε (x, t) is the signed distance function corresponding to the interface

Γ ε t := {x ∈ D, u ε (x, t) = 0}.
Let T 1 be an arbitrary constant such that T 2 < T 1 < T . Assume by contradiction that (15) does not hold. Then, there exist η > 0 and sequences

ε k ↓ 0, t k ∈ [ρt ε k , T ], x k ∈ D such that α + -η > 0 > α -+ η and u ε k (x k , t k ) -U 0 d ε k (x k , t k ) ε k ≥ η. (75)
For the inequality (75) to hold, by Theorem 1.2 and U 0 (±∞) = α ± , we need

d ε k (x k , t k ) = O(ε k ).
With these observations, and also by Theorem 1.2, there exists a positive constant C such that

|d(x k , t k )| ≤ Cε k (76)
for ε k small enough.

If x k ∈ Γ ε k t k , then the left-hand side of (75) vanishes, which contradicts this inequality. Since the sign can either be positive or negative, by extracting a subsequence if necessary we may assume that

u ε k (x k , t k ) > 0 for all k ∈ N, (77)
which is equivalent to

d ε k (x k , t k ) > 0 for all k ∈ N.
By (76), each x k has a unique orthogonal projection p k := p(x k , t k ) ∈ Γ t k . Let y k be a point on Γ ε k t k that has the smallest distance from x k , and therefore u ε k (y k , t k ) = 0. Moreover, we have

u ε k (x, t k ) > 0 if x -x k < y k -x k . (78)
We now rescale u ε k around (p k , t k ). Define

w k (z, τ ) := u ε k (p k + ε k R k z, t k + ε 2 k τ ), (79)
where R k is a orthogonal matrix in SO(N, R) that rotates the z (N ) axis, namely the vector (0,

• • • , 0, 1) ∈ R N onto the unit normal vector to Γ t k at p k ∈ Γ t k , say x k -p k x k -p k .
To prove our result, we use Theorem 1.2 which gives information about u ε k for t k + ε 2 k τ ≥ t ε k . Then, since Γ t is separated from ∂D by some positive distance, w k is well-defined at least on the box

B k := (z, τ ) ∈ R N × R : |z| ≤ c ε k , -(ρ -1) | ln ε k | f (0) ≤ τ ≤ T -T 1 ε 2 k , for some c > 0. We remark that B k ⊂ B k+1 , k ∈ N and lim k→∞ B k = R N × R. Writing R k = (r ij ) 1≤i,j≤N , we remark that R -1 k = R T k , which implies that N i=1 r 2 i = 1, N i=1,j =m r ji r i = 0. ( 80
) Since ∂ 2 z i ϕ(w k ) = ε 2 k N j=1 N =1 r ji r i ∂ x x j ϕ(u ε k ), we have ∆ϕ(w k ) = ε 2 k N i=1 ∂ 2 z i ϕ(w k ) = ε 2 k N i=1 N =1 r 2 i ∂ 2 x ϕ(u ε k ) + ε 2 k N i=1 N j, =1,j = r ji r i ∂ x x j ϕ(u ε k ) = ε 2 k ∆ϕ(u ε k ).
Thus, we obtain

w k τ = ∆ϕ(w k ) + f (w k ) in B k .
From the propagation result in Theorem 1.2 and the fact that the rotation matrix R k maps the z (N ) axis to the unit normal vector of Γ t at p k , there exists a constant C > 0 such that

z (N ) ≥ C ⇒ w k (z, τ ) ≥ α + -η > 0, z (N ) ≤ -C ⇒ w k (z, τ ) ≤ α -+ η < 0 (81)
as long as (z, τ ) ∈ B k . It follows from the first line of ( 14) that α --η 0 ≤ w k ≤ α + + η 0 for k large enough. Then, by Lemma 11 we can find a subsequence of (w k ) converging to some w ∈ C 2,1 (R N × R) which satisfies

w τ = ∆ϕ(w) + f (w) on R N × R.
From Remark 2 we can deduce (66). Then, by Lemma 12, there exists z * ∈ R such that

w(z, τ ) = U 0 (z (N ) -z * ). ( 82
)
Define sequences of points {z k }, {z k } by

z k := 1 ε k R -1 k (x k -p k ), zk := 1 ε k R -1 k (y k -p k ). (83)
From ( 76) and Theorem 1.2, we have 2), which implies that the sequences z k and zk are bounded. Thus, there exist subsequences of {z k }, {z k } and z ∞ , z∞ ∈ R N such that Since the normal vector to Γ t k at p k is equal to x k -p k , and the mapping R -1 k sends the unit normal vector to Γ t k at p k to the vector (0, • • • 0, 1) ∈ R N , we conclude z ∞ must lie on the z (N ) axis so that we can write

|d(x k , t k )| = x k -p k = O(ε k ), y k -p k ≤ y k -x k + x k -p k = |d ε k (x k , t k )| + |d(x k , t k )| = O(ε k ) (see Figure
z kn → z ∞ , zkn → z∞ , as k → ∞.
z ∞ = (0, • • • , 0, z (N ) ∞ ).
Since, by (77), w(z ∞ , 0) = lim kn→∞ w kn (z kn , 0) = lim kn→∞ u ε kn (x kn , t kn ) ≥ 0, we deduce from (82) and the fact that U 0 is an increasing function that

w(z ∞ , 0) = U 0 (z (N ) ∞ -z * ) ≥ 0 ⇒ z (N ) ∞ ≥ z * .
where α ±,δ are the two stable solutions of f (u) + δ = 0. Then, we define the mobility by

µ AC := - c (0) α + -α - ,
with a normalization factor α + -α -as in [START_REF] Spohn | Interface motion in models with stochastic dynamics[END_REF]; compare (4.6) and (4.7) in [START_REF] Spohn | Interface motion in models with stochastic dynamics[END_REF] noting that the boundary conditions at ±∞ are switched so that we have a negative sign for µ AC .

To derive the formula for µ AC , we multiply ϕ(U δ ) z to (88) and integrate it over R to obtain Thus, the mobility is given by the formula

µ AC = ϕ * ± α + α - 2W (u)du = ϕ * ± R ϕ (U 0 )U 2 0z (z)dz , (90)
where

ϕ * ± = ϕ(α + ) -ϕ(α -) α + -α - .
On the other hand, surface tension is defined as a gap of the energy of the microscopic transition surface from α -to α + in the normal direction and that of the constant profile α -or α + . More precisely, define the energy of a profile u = {u(z)} z∈R by

E(u) = R 1 2 ϕ(u) z 2 + W (u) dz.
Recall that the potential W is defined by (28), and W ≥ 0 and W (α ± ) = 0 hold. In particular, W is normalized as min u∈R W (u) = 0 so that min u=u(•) E(u) = 0. Then, the surface tension is defined as

σ AC := 1 ϕ * ± min u:u(±∞)=α ± E(u),
by normalizing the energy by ϕ * ± . We observe that E is defined through ϕ. Note that the nonlinear Allen-Cahn equation, that is (87) with δ = 0, is a distorted gradient flow associated with E(u):

u t = - δE(u) δϕ(u) , z ∈ R,
where the right hand side is defined as the functional derivative of E(u) in ϕ(u), which is given by δE(u) δϕ(u) = -ϕ(u) zz -f (u(z)).

Indeed, to see the second term -f (u(z)), setting v = ϕ(u), one can rewrite W (u) = W (ϕ -1 (v)) as a function of v so that

W (ϕ -1 (v)) = W (ϕ -1 (v)) ϕ -1 (v) = -f (ϕ -1 (v))ϕ (ϕ -1 (v)) 1 ϕ (ϕ -1 (v)) = -f (ϕ -1 (v)) = -f (u).
We call the flow "distorted", since the functional derivative is taken in ϕ(u) and not in u. One can rephrase this in terms of the change of variables v(z) = ϕ(u(z)). Indeed, we have E(u) = E(v) under this change, where

E(v) = R 1 2 v 2 z + W (ϕ -1 (v)) dz,
and δ E δv = -v zz -f (ϕ -1 (v)) = -v zz -g(v).

Therefore, in the variable v(z), the nonlinear Allen-Cahn equation can be rewritten as

v t = ϕ (u)u t = -ϕ (ϕ -1 (v)) • δ E δv = ϕ (ϕ -1 (v)) v zz + g(v)}.
This type of distorted equation for v is sometimes called Onsager equation; see [START_REF] Mielke | On evolutionary Γ-convergence for gradient systems[END_REF]. Now we come back to the computation of the surface tension σ AC . In fact, it is given by

σ AC = 1 ϕ * ± R V 2 0z dz = 1 ϕ * ± α + α - ϕ (u) 2W (u)du, (91)
where V 0 = ϕ(U 0 ) and satisfies (21). Indeed, the second equality follows from (29). To see the first equality, by definition,

σ AC = 1 ϕ * ± min u:u(±∞)=α ± E(u) = 1 ϕ * ± min v:v(±∞)=ϕ(α ± ) E(v)
and the minimizers of E under the condition v(±∞) = ϕ(α ± ) are given by V 0 and its spatial shifts. Thus,

σ AC = 1 ϕ * ± E(V 0 ) = 1 ϕ * ± R 1 2 V 2 0z + W (ϕ -1 (V 0 )) dz.
However, since V 0z = 2W (U 0 (z)) by ( 27), we have R W (ϕ -1 (V 0 ))dz = R 1 2 V 2 0z dz. In particular, this implies the first equality of (91).

By (30) combined with (90) and (91), we see that λ 0 = µ AC σ AC .

C 1 :

 1 = max |s|≤I |ϕ(s)| + max |s|≤I ϕ (s) + max |s|≤I |ϕ (s)|, I = C 0 + max(|α -|, α + ).

  (a) Points x k , y k , p k and interfaces Γ t k , Γ ε k t k inside the box B k . (b) Points z ∞ and z∞ and hyperplanes z (N ) = z * , z (N ) = z (N ) ∞ .

Figure 2 :

 2 Figure 2: In (a) the distance between Γ t k and Γ ε k t k is of O(ε k ). In (b), since we rescale space by ε -1 , the distance between two hyperplanes is of O(1).

U 2 z

 2 δz ϕ(U δ ) z dz + δ(ϕ(α + ) -ϕ(α -)) = O(δ 2 ), (89) by noting that R ϕ(U δ ) zz ϕ(U δ ) z dz = 1 2 R ϕ(U δ ) z dz = 0, R ϕ(U δ ) z dz = ϕ(α +,δ ) -ϕ(α -,δ ) = ϕ(α + ) -ϕ(α -) + O(δ), R f (U δ )ϕ(U δ ) z dz = R f (U δ )ϕ (U δ )U δz dz = -α +,δ α -,δ W (u)du = O(δ 2 ).The last line follows by the change of variable u = U δ (z), W (u) = -f (u)ϕ (u) (recall (28)),α + α -W (u)du = 0 and W (α ± ) = 0, W ∈ C 1 . However, since one can at least formally expect U δ = U 0 + O(δ) (recall (20) for U 0 ), by (27), R U δz ϕ(U δ ) z dz = R U 0z ϕ(U 0 ) z dz + O(δ) = R U 0z 2W (U 0 (z))dz + O(δ) = α + α - 2W (u)du + O(δ),by the change of variable u = U 0 (z) again. This combined with (89) leads to c (0) = -ϕ(α + ) -ϕ(α -)α + α - 2W (u)du .

Acknowledgement

The authors are grateful to the professors Henri Berestycki and Francois Hamel for a useful discussion. P. El Kettani, D. Hilhorst and H.J. Park thank IRN ReaDiNet as well as the French-Korean project STAR. T. Funaki was supported in part by JSPS KAKENHI, Grant-in-Aid for Scientific Researches (A) 18H03672 and (S) 16H06338, and also thanks IRN ReaDiNet. S. Sethuraman was supported by grant ARO W911NF-181-0311, a Simons Foundation Sabbatical grant, and by a JSPS Fellowship, and thanks Waseda U. for the kind hospitality during a sabbatical visit.

From the definition of y kn and (79), we have w(z ∞ , 0) = lim k→∞ w kn (z kn , 0) = lim k→∞ u ε kn (y kn , t kn ) = 0.

(84)

Next, we show that

Choose z ∈ R N satisfying z -z ∞ ≤ z∞ -z ∞ and a sequence a kn ∈ R + such that a kn → z -z ∞ and ε kn a kn ≤ x kn -y kn as k → ∞. Then, we define sequences n kn and b kn by

Note that b kn → z as k → ∞. Then, by (83), we obtain

where the last inequality holds by (78). Note that (82

Thus, we have either z ∞ = z∞ or, in view of (82) , ( 84) and (85), that the ball of radius ||z ∞ -z ∞ || centered at z ∞ is tangent to the hyperplane z (N ) = z * at z∞ . Hence, z∞ is a point on z (N ) axis. With this observation and (82), we have

This last property implies

We have therefore reached a contradiction since, by (76), ( 82) and (86),

For the proof of (ii), we use the same method as in [START_REF] Alfaro | On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data[END_REF].

Appendix: Mobility and surface tension

Mobility is defined as a linear response of the speed of traveling wave to the external force. More precisely, motivated by (4.1) and (4.2) in [START_REF] Spohn | Interface motion in models with stochastic dynamics[END_REF], let us consider the nonlinear Allen-Cahn equation with external force δ on R for small enough |δ|:

and the corresponding traveling wave solution U = U δ (z) with speed c(δ):

Remark 4. The linear case ϕ(u) = Ku is discussed by Spohn [START_REF] Spohn | Interface motion in models with stochastic dynamics[END_REF], in which K is denoted by κ. In this case, since ϕ = K and ϕ * ± = K, by (90) and (91), we have µ AC = R U 2 0z dz -1 and σ AC = K R U 2 0z dz. These formulas coincide with (4.7) and (4.8) in [START_REF] Spohn | Interface motion in models with stochastic dynamics[END_REF] by noting that U 0 is the same as w in [START_REF] Spohn | Interface motion in models with stochastic dynamics[END_REF] in the linear case except that the direction is switched due to the choice of the boundary conditions.


E Labarthe

A J Bougrine

H Delalu

J Berthet

J J Counioux

EQUILIBRIUM STUDY BETWEEN CONDENSED PHASES BY ISOPLETHIC THERMAL ANALYSIS WHEN A MISCIBILITY GAP IS OBSERVED

Keywords: ternary diagram, isoplethic thermal analysis, miscibility gap, critical point, piperidine, water, sodium sulfate

Isoplethic thermal analysis was used to determine the solid-liquid-liquid equilibria in the ternary system water-sodium sulfatepiperidine. The changes in state observed on the thermogram recorded during the displacement in a quasi-binary section permit the identification of the different phases and the delimitation of the corresponding equilibrium domains. Two isotherms were established at 25°C and 40°C because these temperatures frame the peritectic decomposition of the sodium sulfate decahydrate. Miscibility gaps were completely delimited and each critical point was calculated. This study permitted to determine the optimal conditions for separating the organic phase by liquid-liquid extraction.

Introduction

Studying a high order system requires long and painstaking experimentations when using traditional analysis techniques. Therefore, the laboratory has developed automated isoplethic methods that consist in progressively modifying the global composition of the system, monitoring the evolution of an intensive variable and using the information obtained. The first works [1] , which were based on successive and then continuous conductimetric measurements, demonstrated how it was possible to determine all the characteristics of an isotherm: solubility curves, invariant points, nature and composition of solid phases.

Direct or differential thermal analysis rests on the exploitation of the thermal effects that occur when a sample is subjected to a regular variation of its heat energy. The progressive addition of a given solution in a mixture modifies its global composition. The molar and mass compositions of each constituent vary linearly through time and thus define an isoplethic section of the system formed by all the independent constituents. Isoplethic Thermal Analysis [2][3] , or ITA, consists in using the thermal phenomena observed under almost constant pressure and temperature conditions.

Experimental

In an invariant isotherm-isobar system, the number of phases in equilibrium is equal to the number of independent constituents c. If an infinitesimal quantity da (molar or mass unit) of a mixture A, with composition vector A , is added to an invariant system composed of c phases j of composition vector j F in quantity f j , only the quantities df j of these phases will be modified. According to the properties of the isobarycentric transformations [4] , these thermodynamic variables satisfy the relation (1):

j c j j j F df A da . . 1 ∑ = = = (1)
which can also be written as: 0 . The resolution of this linear equation system permits the calculation of each df j for a given value of da. If H A is the enthalpy of the mixture added and H j is the enthalpy of the phase j, the enthalpy difference (molar or in mass) of the system is given by:

A j c j 1 j j H . da H . df dH - = ∑ = = (4)
In the case of an invariant system, dH depends only on the added quantity da. In particular, this variation of enthalpy is not a function of the mass and composition of the initial system but it is directly proportional to da.

If the mixture A is introduced at a constant flow rate, q =da / dt, the heat quantity involved per time unit is constant:

dH / dt = k 1 .q (5)
In addition, if the measurement cell is of isoperibolic type, the heat flow Φ, to which the system is subjected, is proportional to the variation of temperature, according to Newton's law:

T k dt Q ∆ = = Φ . 2 δ ( 6
)
Where ∆T is the temperature difference between the mixture temperature T M and the thermostatic chamber temperature T C .

C M T T T - = ∆ (7)
Hence the relation:

∆T = (k 1 /k 2 ).q = K.q (8)
Finally, for an invariant isotherm-isobar system, a given mixture and injection speed, the variation of temperature between the mixture and the thermostatic chamber is constant and a temperature plateau should be observed. The displacement of an indifferent equilibrium is therefore expressed by ITA as a plateau of invariance, as in direct thermal analysis.

In practice, the variation of the temperature ∆T is recorded as a function of the volume v of solution added. Each equilibrium domain has a corresponding specific variation law, allowing the determination of phase transformations.

An experimental set-up was built [5] according to the principle defined previously and adjusted in particular to a temperature measurement sensitivity in the region of mK. This set-up is completely automated and is composed of a platinum probe Pt100, a thermostat Hetofrig type CB11, a reactor, an automatic injection system and a magnetic agitator.

The injection of the solvent in the stirred mixture is performed by a computer controlled automatic injection system. The injection range used is within 1 and 4 mL/hour. The temperature values are stored at regular time intervals. The phase transformations are determined by computer processing of the numerical data by linear extrapolation of the variation laws. This device was applied to the study of equilibria between the solid-liquid phases and liquid-liquid phases of the ternary system watersodium sulfate-piperidine, used as the basis for extracting organic compounds in a complex synthetic mixture.

The ternary system studied is delimited by three binary systems. The Na 2 SO 4 -C 5 H 10 NH system is unknown, while in the temperature field studied the piperidine is totally miscible in water. On the other hand, two hydrates are present in the binary mixture H 2 O-Na 2 SO 4 [6-7] : a metastable heptahydrate and a stable decahydrate (Glauber salt) which undergoes a peritectic decomposition at 32.4°C.

In order to analyse the impact of the decahydrate on the equilibria, we established two isotherms at 25°C and 40°C, that frame its decomposition temperature. Figure 2 gives an example of a thermogram obtained from a ternary mixture whose composition point A is located in Figure 3. It shows the evolution of the temperature mixture when a volume of pure water is added. Segment AB corresponds to the progressive hydration of anhydrous sodium sulfate into decahydrated salt. Reaching steady state was terminated with a short plateau of invariance. From B to C, crossing the two phases equilibrium domain liquid+Na 2 SO 4 .10H 2 O is monotonous. After reaching steady state C at C', crossing the triphasic domain L1+L2+Na 2 SO 4 .10H 2 O is shown by a plateau of invariance up to point D. Point D, beyond which no solid phase can be observed, corresponds to the limit conode of the miscibility gap. By using this technique, the different equilibrium domains are clearly defined and permit the full determination of the isotherm. The 25°C isotherm is characterized by the crystallization of two solid phases, the anhydrous and the sodium sulfate decahydrate, and the existence of a miscibility gap in the crystallization area of the decahydrate. The liquid-liquid equilibrium is delimited by the conode L1-L2 and the critical point κ. The solubility of the anhydrous sulfate in concentrated aqueous solutions of piperidine is very weak and the liquidus curve is almost confounded with the limit binary system H 2 O-C 5 H 10 NH. At 40°C, the decahydrate is no more stable and the liquidus curve of the anhydrous salt is interrupted by the demixing curve. One of the parts of the liquidus is almost confounded with the point L1.

Results and discussion

The liquid-liquid area is stimgly dependent on temperature. The critical line, that separates the two half surfaces of the binodal, is a fundamental characteristic of the miscibility gap. However, even at a given temperature, its composition is difficult to define experimentally. Ideally, when the two branches of the binodal curve are almost symmetrical, determining the coordinates of the critical point by using the diameter method is possible when a series of conodes is known. Its application to the two isotherms studied shows, unfortunately, that the curvatures are too important to allow reliable extrapolation and we have adapted this method to non-symmetrical curves by barycentric weighting of two liquid phases. For each conode (i), the point P i obtained satisfies the relation:

i i i i i L w L w P 2 2 1 1 . . + = (9)
where w 1i and w 2i represent respectively the mass fractions of the liquid phases of compositions The value of w corresponding to the best correlation coefficient is chosen to then define the coordinates of the critical point by intersection with the binodal.

Conclusions

After a theoretical analysis of the bases of the Isoplethic Thermal Analysis intended to improve the use of thermograms, this technique was applied to the study of the water-sodium sulfatepiperidine system. The two isotherms 25°C and 40°C, framing the peritectic decomposition of the sodium sulfate decahydrate, were both plotted in full and, in particular, the miscibility gap which appears was completely delimited: the critical point of the binodal and the limit conode, corresponding to the appearance of anhydrous and decahydrated sodium sulfate were identified. Lastly, the experimental results obtained constitute fundamental data for treating more or less complex solutions containing the three constituents taken into account. In particular, this study highlighted that increasing temperature favours extraction potential by liquid-liquid separation and determines the associated conditions.

Fig. 1 :

 1 Fig.1:Example of thermogram registred during addition of pure water in the ternary mixture A.

Figures 2 and 3

 3 Figures 2 and 3 represent isotherms 25°C and 40°C respectively of the ternary system H 2 O-Na 2 SO 4 -C 5 H 10 NH while tables 1 and 2 give the associated experimental values, expressed in mass fractions.

Fig. 2 :

 2 Fig.2: Ternary system H 2 O-Na 2 SO 4 -C 5 H 10 NH : isotherm 25°C (mass fractions) Solid phase I: Na 2 SO 4 Solid phase II: Na 2 SO 4 .10H 2 O

Fig. 3 :

 3 Fig.3: Ternary system H 2 O-Na 2 SO 4 -C 5 H 10 NH: isotherm 40°C (mass fractions) Solid phase I: Na 2 SO 4

i L 1 and i L 2 .

 2 The correlation coefficients R 2 of the regression lines are calculated as a function of the parameter w = w 2i , then the curve R 2 = f(w) is plotted as shown in the example in Figure4.

Fig. 4 :

 4 Fig.4: Correlation coefficient R 2 as a function of the mass fraction of liquid phase (L2 i )

Table 1 :

 1 Isotherm 25°Cdetermined by ATI, chemical analysis(*)or generalized diameter method (**)

	2 O

Table 2 :

 2 Isotherm 40°C determined by ATI, chemical analysis (*) or generalized diameter method (**)

	w(C 5 H 10 NH)	w(Na 2 SO 4 )	Phenomenon
	0.0009	0.314	Invariant point L1*
	0.008	0.272	binodal
	0.008	0.259	″
	0.022	0.182	″
	0.039	0.142	″
	0.078	0.118	″
	0.12 0.137 0.291	0.10 0.092 0.041	Critical point κ** binodal ″
	0.375 0.465 0.563 0.04 0.272 0.474 0.61 0.628	0.027 0.011 0.006 0.30 0.181 0.067 0.014 0.002	″ ″ Limit conode L1-L2 ″ ″ ″ Invariant point L2*


Nelia Luviano

Marie Lopez

Fleur Gawehns

Cristian Chaparro

Paola B Arimondo

Koen Verhoeven

Benoit Pujol

Benjamin Gourbal

David Duval

Wannaporn Ittiprasert

Jean-Francois Allienne

Damien Pouzol

Geneviève Tavernier

Céline Cosseau

Christoph Grunau

New insights into the role of DNA methylation in Biomphalaria glabrata infection susceptibility

Introduction

As for many transmissible diseases non-pharmacological interventions are one of the most effective ways to control schistosoma infections. One of theses, the control of the intermediate host, is also recognized by WHO as a key element to eradication of the disease in humans. Biomphalaria snails can show remarkable resistance to schistosomes and prevalence is low in snail populations even in regions where the parasite is highly present in the vertebrate host. Resistance is a heritable trait, but it shows also high plasticity e.g., snails can be become resistant to the same parasite strain after a primary infection. One of the components of heritable plasticity is epigenetic information that refers to the heritable but reversible changes in gene function, and one of the bearers of epigenetic information is DNA methylation. We reasoned that modifying DNA methylation in the snail would have potentially positive effects on resistance traits.

Methods

We developed chemical DNA methylation modulators to which the snails were exposed to modify their pan-genomic methylation and we established a method for targeted DNA methylation modifications based on dCas9/Crispr. For heritability of resistance measure we used epigenetic recombinant inbred lines (EpiRILs) and experimental infections.

Results

We applied a dCas9-SunTag-DNMT3A complex and synthetic sgRNA to transfect B. glabrata embryos and observed an increase of CpG methylation at the target site in 50% of the hatching snails. DNA methylations were also introduced in an inbred B.glabrata line by chemical treatment in F0. This led to phenotypic variation in F3 and methylation differences in F3. Both increase and decrease (up to 100% and down to 20% prevalence from 86% control) of infection success indicating variation in complex resistance/compatibility trait.

Conclusions

We show that DNA methylation can be modified in B.glabrata both chemically and dCasbased, and pan-genomic or targeted. These modifications produce heritable phenotypic variants and have an influence on infection susceptibility confirming the epigenetic component of this trait. A rational of DNA methylation-based vector control will be presented.

Acknowledgment for financial support :

This work was supported by Wellcome Trust strategic award (107475/Z/15/Z) and by a PhD grant to NL from the Region Occitanie (EPIPARA project) and the University of Perpignan Via Domitia graduate school ED305. With the support of LabEx CeMEB, an ANR "Investissements d'avenir" program (ANR-10-LABX-04-01) through the Environmental Epigenomics Platform and the "projets de recherche exploratoires du CeMEB 2018" project "Epigenetics of inbreeding depression (EPID)". This study is set within the framework of the "Laboratoires d'Excellences (LABEX)" TULIP (ANR-10-LABX-41). BQR UPVD provided funding for the Nanoject III Programmable Nanoliter Injector.

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The 20-min presentation is part of the Roundtable 10, theme "New approaches on the intermediate host studies" and it is scheduled for November 23rd from 09:00 to 10:40 hours.