M Al Haj

R Monneau

Existence of traveling waves for Lipschitz discrete dynamics. Monostable case as a limit of bistable cases

Keywords: Traveling waves, degenerate monostable non-linearity, KPP non-linearity, bistable non-linearity, Frenkel-Kontorova model, viscosity solutions, Perron's method

We study discrete monostable dynamics with general Lipschitz non-linearities. This includes also degenerate non-linearities. In the positive monostable case, we show the existence of a branch of traveling waves solutions for velocities c ≥ c + , with non existence of solutions for c < c + . We also give certain sufficient conditions to insure that c + ≥ 0 and we give an example when c + < 0. We as well prove a lower bound of c + , precisely we show that c + ≥ c * , where c * is associated to a linearized problem at infinity. On the other hand, under a KPP condition we show that c + ≤ c * . We also give an example where c + > c * .

This model of discrete dynamics can be seen as a generalized Frenkel-Kontorova model for which we can also add a driving force parameter σ. We show that σ can vary in an interval [σ -, σ + ]. For σ ∈ (σ -, σ + ) this corresponds to a bistable case, while for σ = σ + this is a positive monostable case, and for σ = σ -this is a negative monostable case. We study the velocity function c = c(σ) as σ varies in [σ -, σ + ]. In particular for σ = σ + (resp. σ = σ -), we find vertical branches of traveling waves solutions with c ≥ c + (resp. c ≤ c -).

Our method of proof is new and relies on viscosity solutions. Moreover, the monostable case with c = c + is seen advantageously as a limit situation of the bistable case. For c >> 1, the traveling waves are constructed as perturbations of solutions of an associated ODE. Finally to fill the gap between c = c + and large c, we use certain hull functions that are associated to correctors of a homogenization problem.

General motivation

Our initial motivation was to study the classical fully overdamped Frenkel-Kontorova model, which is a system of ordinary differential equations (1.1)

dX i dt = X i+1 -2X i + X i-1 + f (X i ) + σ,
where X i (t) ∈ R denotes the position of a particle i ∈ Z at time t, dX i dt is the velocity of this particle, f is the force created by a 1-periodic potential and σ represents the constant driving force. Such external force could be for example f (x) = 1cos(2πx) ≥ 0. This kind of system can be, for instance, used as a model of the motion of a dislocation defect in a crystal (see the book of Braun and Kivshar [START_REF] Braun | The Frenkel-Kontorova model, Concepts, Methods and Applications[END_REF]). This motion is described by particular solutions of the form (1.2)

X i (t) = φ(i + ct) with (1.3) φ ′ ≥ 0 and φ is bounded.

Such a solution, φ, is called a traveling wave solution and c denotes its velocity of propagation.

From (1.1) and (1.2), it is equivalent to look for solutions φ of (1.4) cφ ′ (z) = φ(z + 1) -2φ(z) + φ(z -1) + f (φ(z)) + σ with z = i + ct. For such a model, and under certain conditions on f, we show the existence of traveling waves for each value of σ in an interval [σ -, σ + ] (see Theorem 1.7). We also get the whole picture (see Figure 4 for qualitative properties of this picture) of the velocity function c = c(σ) with respect to the driving force σ, with vertical branches for σ = σ -or σ = σ + .

When f > 0 = f (0) = f (1) on (0, 1) and σ = 0, we can moreover normalize the limits of the profile φ as (1.5) φ(-∞) = 0 and φ(+∞) = 1.

This case is called a positive monostable case and is associated here to σ + = 0. Moreover, we can show the existence of a critical velocity c + such that the following holds. There exists a branch of traveling waves solutions for all velocity c ≥ c + and there are no solutions for c < c + . The goal of this paper is to present similar results in a framework more general than (1.4). To this end, given a real function F (whose properties will be specified in Subsections 1.2 and 1.3), we consider the following generalized equation with σ ∈ R (1.6) cφ ′ (z) = F (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) + σ,

where N ≥ 0 and r i ∈ R for i = 0, ..., N such that (1.7) r 0 = 0 and r i = r j if i = j, which does not restrict the generality. In (1.6), we are looking for both the profile φ and the velocity c.

Equation (1.1) can be seen as a discretization of the following reaction diffusion equation (1.8)

u t = ∆u + f (u).
In 1937, Fisher [START_REF] Fisher | The advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky and Piskunov [START_REF] Kolmogorov | Study of the diffusion equation with growth of quantity of matter and its application to biological problem[END_REF] studied the traveling waves for equation (1.8) which they proposed as a model describing the spreading of a gene throughout a population. Later, many works have been devoted for such equation that appears in biological models for developments of genes or populations dynamics and in combustion theory (see for instance, Aronson, Weinberger [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation[END_REF][START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] and Hadeler, Rothe [START_REF] Hadeler | Travelling fronts in nonlinear diffusion equations[END_REF]). For more developments and applications in biology of reaction diffusion equations, the reader may refer to [START_REF] Volpert | Reaction-diffusion waves in biology[END_REF] and to the references cited therein. There is also a considerable work on the existence, uniqueness and stability of traveling waves and their speed of propagation for the homogeneous KPP-Fisher non-linearity (see for example [START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF][START_REF] Hamel | Travelling fronts and entire solutions of the Fisher-KPP equation in R N[END_REF][START_REF] Hamel | Uniqueness and stability properties of monostable pulsating fronts[END_REF][START_REF] Hou | Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities[END_REF][START_REF] Zlatoš | Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations[END_REF]). Such results have been shown also for the inhomogeneous, heterogeneous and random KPP-Fisher non-linearity (see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | Propagation speed for reaction-diffusion equations in general domains[END_REF][START_REF] Nadin | Propagation phenomena for time heterogeneous KPP reaction-diffusion equations[END_REF]).

Traveling waves were studied also for discrete bistable reaction diffusion equations (see for instance [START_REF] Carpio | Wave solutions for a discrete reaction-diffusion equation[END_REF][START_REF] Chen | Traveling waves in discrete periodic media for bistable dynamics[END_REF]). See also [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF] and the references therein. In the monostable case, we distinguish [START_REF] Hudson | Existence of traveling waves for a generalized discrete Fisher's equation[END_REF] (for nonlocal non-linearities with integer shifts) and [START_REF] Coville | Non-local anisotropic dispersal with monostable nonlinearty[END_REF][START_REF] Li | Spreading speeds as slowest wave speeds for cooperative systems[END_REF][START_REF] Weinberger | Long-time behavior of a class of biological models[END_REF][START_REF] Yagisita | Existence and Nonexistence of traveling waves for a nonlocal monostable equation[END_REF] (for problems with linear nonlocal part and with integer shifts also). See also [START_REF] Guo | Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system[END_REF] for particular monostable non-linearities with irrational shifts. We also refer to [START_REF] Guo | Front propagation for discrete periodic monostable equations[END_REF][START_REF] Chen | Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices[END_REF][START_REF] Guo | Front propagation for a two dimensional periodic monostable lattice dynamical system[END_REF][START_REF] Chen | Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations[END_REF][START_REF] Chen | Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics[END_REF][START_REF] Hamel | Travelling fronts and entire solutions of the Fisher-KPP equation in R N[END_REF][START_REF] Zinner | Traveling wavefronts for the discrete Fisher's equation[END_REF] for different positive monostable non-linearities. In the monostable case, we have to underline the work of Hudson and Zinner [START_REF] Hudson | Existence of traveling waves for a generalized discrete Fisher's equation[END_REF] (see also [START_REF] Zinner | Traveling wavefronts for the discrete Fisher's equation[END_REF]), where they proved the existence of a branch of solutions c ≥ c * for general Lipschitz non-linearities (with possibly an infinite number of neighbors N = +∞, and possibly p types of different particles, while p = 1 in our study) but with integer shifts r i ∈ Z. However, they do not state the nonexistence of solutions for c < c * . Their method of proof relies on an approximation of the equation on a bounded domain (applying Brouwer's fixed point theorem) and an homotopy argument starting from a known solution. The full result is then obtained as the size of the domain goes to infinity. Here we underline that our results hold for the fully nonlinear case with real shifts r i ∈ R.

Several approaches were used to construct traveling waves for discrete monostable dynamics. We already described the homotopy method of Hudson and Zinner [START_REF] Hudson | Existence of traveling waves for a generalized discrete Fisher's equation[END_REF]. In a second approach, Chen and Guo [START_REF] Chen | Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations[END_REF] proved the existence of a solution starting from an approximated problem. They constructed a fixed point solution of an integral reformulation (approximated on a bounded domain) using the monotone iteration method (with sub and supersolutions). This approach was also used to get the existence of a solution in [START_REF] Fu | Traveling wave solutions for some discrete quasilinear parabolic equations[END_REF][START_REF] Chen | Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics[END_REF][START_REF] Guo | Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system[END_REF][START_REF] Guo | Front propagation for a two dimensional periodic monostable lattice dynamical system[END_REF]. A third approach based on recursive method for monotone discrete in time dynamical systems was used by Wienberger et al. [START_REF] Li | Spreading speeds as slowest wave speeds for cooperative systems[END_REF][START_REF] Weinberger | Long-time behavior of a class of biological models[END_REF]. See also [START_REF] Yagisita | Existence and Nonexistence of traveling waves for a nonlocal monostable equation[END_REF], where this method is used to solve problems with a linear nonlocal part. In a fourth approach [START_REF] Guo | Front propagation for discrete periodic monostable equations[END_REF], Guo and Hamel used global space-time sub and supersolutions to prove the existence of a solution for periodic monostable equations.

There is also a wide literature about the uniqueness and the asymptotics at infinity of a solution for a monostable non-linearities, see for instance [START_REF] Chen | Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices[END_REF][START_REF] Hou | Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities[END_REF] (for a degenerate case), [START_REF] Chen | Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations[END_REF][START_REF] Chen | Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics[END_REF] and the references therein. Let us also mention that certain delayed reaction diffusion equations with some KPP-Fisher non-linearities do not admit traveling waves (see for example [START_REF] Fu | Traveling wave solutions for some discrete quasilinear parabolic equations[END_REF][START_REF] Zinner | Traveling wavefronts for the discrete Fisher's equation[END_REF]).

Finally, we mention that our method opens new possibilities to be adapted to more general problems. For example, we can think to adapt our approach to a case with possibly p types of different particles similar to [START_REF] Forcadel | Homogenization of accelerated Frenkel-Kontorova models with n types particles[END_REF]. The case with an infinite number of neighbors N = +∞ could be also studied. We can also think to study fully nonlinear parabolic equations.

Main results in the monostable case

In this subsection, we consider equation (1.6) with σ = 0. We study the existence of traveling waves of equation (1.6) (with σ = 0) for positive degenerate monostable non-linearities and with conditions at infinity given by (1.5).

In order to present our results in this case, we have to introduce some assumptions on F : [0, 1] N +1 → R.

Assumption (A Lip ): i) Regularity: F ∈ Lip([0, 1] N +1 ).
ii) Monotonicity: F (X 0 , X 1 , ..., X N ) is non-decreasing w.r.t. each X i for i = 0.

Assumption (P Lip ):

Positive degenerate monostability: a Let f (v) = F (v, ..., v) such that f (0) = f (1) = 0, f > 0 in (0, 1). Our main result is: Theorem 1.1 (Monostable case: existence of a branch of traveling waves) Assume (A Lip ) and (P Lip ). Then there exists a real c + such that for all c ≥ c + there exists a traveling wave φ : R → R solution (in the viscosity sense (see Definition 2.1)) of (1.9)

1 f (v) v 0
    
cφ ′ (z) = F (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) on R φ is non-decreasing over R φ(-∞) = 0 and φ(+∞) = 1.

On the contrary for c < c + , there is no solution of (1.9).

Up to our knowledge, Theorem 1.1 is the first result for discrete dynamics with real shifts r i ∈ R in the fully nonlinear case. Even when r i ∈ Z, the only result that we know for fully nonlinear dynamics is the one of Hudson and Zinner [START_REF] Hudson | Existence of traveling waves for a generalized discrete Fisher's equation[END_REF]. However, the nonexistence of solutions for c < c + is not addressed in [START_REF] Hudson | Existence of traveling waves for a generalized discrete Fisher's equation[END_REF].

See Figure 2 for an explicit Lipschitz non-linearity example for which our result (Theorem 1.1) is still true, even if f ′ (0) is not defined. We also prove that the critical velocity c + is unstable in the following sense: Proposition 1.2 (Instability of the minimal velocity c + F ) There exists a function F satisfying (A Lip ) and (P Lip ) with a minimal velocity c + F such that there exists a sequence of functions F δ (satisfying (A Lip ) and (P Lip )) with associated critical velocity c +

F δ satisfying F δ → F in L ∞ ([0, 1] N +1 ) when δ → 0, but lim inf δ→0 c + F δ > c + F .
We believe that the critical velocity c + contains information about f ′ (0); similar to classical result in [START_REF] Kolmogorov | Study of the diffusion equation with growth of quantity of matter and its application to biological problem[END_REF] which asserts that the critical velocity of reaction diffusion equation (1.8) is c + = 2 f ′ (0). This shows that when F is only Lipschitz, it becomes very difficult to capture c + F and to show Theorem 1.1 (see its proof, Section 7).

Examples of functions F satisfying assumptions (A Lip ) and (P Lip ) are given for N = 2, r 0 = 0, r 1 = -1, r 2 = 1 by (1.10) F (X 0 , X 1 , X 2 ) = X 2 + X 1 -2X 0 + g(X 0 ), with for instance non-linearity g(x) = x(1x) or g(x) = x 2 (1x) 2 .

f (x) 1 0 1 2 θ λθ λ 2 λ 3 θ λ 2 θ λ 2 2 λ 3 2
x Figure 2: Lipschitz positive degenerate monostable non-linearity; the rest of the figure over [0, λ 3 2 ] is completed by dilation of center 0 and ratio λ.

In the next result, we prove that the critical velocity c + (given in Theorem 1.1) is non-negative for particular F, i.e. we need to assume some smoothness and strict monotonicity on F near {0} N +1 ; and this is given in assumption (P C 1 ) (which is stronger than (P Lip )):

Assumption (P C 1 ):
Positive degenerate monostability: a Let f (v) = F (v, ..., v) such that f (0) = 0 = f (1) and f > 0 in (0, 1).

Smoothness near {0} N +1 : a F is C 1 over a neighborhood of {0} N +1 in [0, 1] N +1 and f ′ (0) > 0.

Proposition 1.3 (Non-negative c + for particular F ) Consider a function F satisfying (A Lip ) and (P C 1 ). Let c + given by Theorem 1.1. Then we have c + ≥ 0, if one of the three following conditions i), ii) or iii) holds true:

i) Reflection symmetry of F Let X = (X i ) i∈{0,...,N } ∈ [0, 1] N +1 . Assume that for all i ∈ {0, ..., N } there exists i ∈ {0, ..., N } such that r i = -r i ; and F (X) = F (X) for all X ∈ [0, 1] N +1 , where X i = X i for i ∈ {0, ..., N }.

ii) All the r i 's "shifts" are non-negative Assume that r i ≥ 0 for all i ∈ {0, ..., N }.

iii) Strict monotonicity Let

(1.11) I = i ∈ {1, ..., N } such that there exists i ∈ {1, ..., N } with r i = -r i and assume that

(1.12) ∂F ∂X 0 (0) + i∈I min ∂F ∂X i (0), ∂F ∂X i (0) > 0.
Notice that because of the monotonicity of F in X j for j = 0, condition (1.12) is satisfied if

∂F ∂X 0 (0) > 0. Moreover, if (1.13) I = {1, ..., N } and ∂F ∂X i (0) = ∂F ∂X i (0) for all i ∈ I,
then condition (1.12) is equivalent to f ′ (0) > 0. In particular, under condition i) property (1.13) holds true. This shows that condition iii) is more general than condition i).

Remark that if we replace (P C 1 ) by (P Lip ) assuming i), ii) or iii), we do not know if c + ≥ 0.

Proposition 1.4 (Counter example with c + < 0) There exists a function F satisfying (A Lip ) and (P C 1 ) such that the associated critical velocity (given in Theorem 1.1) is negative, i.e. c + < 0.

In the following proposition, we give a lower bound of the critical velocity c + . To this end, assume that (1.14) ∃ i 0 ∈ {0, ..., N } such that r i 0 > 0 and ∂F ∂X i 0 (0, ..., 0) > 0,

or c + < 0, then c + ≥ c * ,
where

(1.15) c * := inf λ>0 P (λ) λ with P (λ) := N i=0 ∂F ∂X i (0, ..., 0)e λr i .
We can also get the result of Proposition 1.5 under conditions different from (1.14) (see Remark 9.1).

Here, it is natural to ask if we may have c + = c * in general or not. We give for instance in Lemma 9.3, an example of a non-linearity where we have c + > c * which answers the question. On the other hand, we can find a KPP type condition to insure the inequality c + ≤ c * , as show the following result:

Proposition 1.6 (KPP condition for c + ≤ c * )
Let F be a function satisfying (A Lip ) and (P Lip ). Let c + given by Theorem 1.1 and assume that F is differentiable at {0} N +1 in [0, 1] N +1 . If moreover F satisfies the KPP condition:

(1.16) F (X) ≤ N i=0 ∂F ∂X i (0, ..., 0)X i for every X ∈ [0, 1] N +1 , then c + ≤ c * with c * defined in (1.15).

Main result on the velocity function

In this subsection, we consider equation (1.6) with a constant parameter σ ∈ R and F : R N +1 → R. We are interested in the velocities c associated to σ (that we call roughly speaking the "velocity function").

For σ belonging to some interval [σ -, σ + ], we prove the existence of a traveling wave and we study the variation of its velocity c with respect to σ. Let E = (1, ..., 1), Θ = (θ, ..., θ) ∈ R N +1 and assume that the function F satisfies: Assumption ( ÃC 1 ):

Regularity: F is globally Lipschitz continuous over R N +1 and C 1 over a neighborhood in R N +1
of the two intervals ]0, Θ[ and ]Θ, E[.

Monotonicity: F (X 0 , ..., X N ) is non-decreasing w.r.t. each X i for i = 0.

Periodicity: F (X 0 + 1, ..., X N + 1) = F (X 0 , ..., X N ) for every X = (X 0 , ..., X N ) ∈ R N +1 .

Notice that, since F is periodic in E direction, then F is C 1 over a neighborhood of RE\(ZE ∪ ZΘ).

Assumption ( BC 1 ): Define f (v) = F (v, ..., v) such that:

Bistability: f (0) = f (1)
and there exists θ ∈ (0, 1) such that

f ′ > 0 on (0, θ) f ′ < 0 on (θ, 1). 0 θ 1 Figure 3: Bistable non-linearity f
See Figure 3 for an example of f satisfying ( BC 1 ). Notice that assumptions ( ÃC 1 ) and ( BC 1 ) holds true in particular for the Frenkel-Kontorova model for β > 0 :

(1.17)

d dt X i = X i+1 + X i-1 -2X i -β sin 2π X i + 1 4 + σ.
Theorem 1.7 (General case: traveling waves and the velocity function) Under assumptions ( ÃC 1 ) and ( BC 1 ), define σ ± as

(1.18) σ + = -min f σ -= -max f. Associate for each σ ∈ [σ -, σ + ] the solutions m σ ∈ [θ -1, 0] and b σ ∈ [0, θ] of f (s) + σ = 0.
Then consider the following equation

(1.19)      cφ ′ (z) = F (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) + σ on R φ is non-decreasing over R φ(-∞) = m σ and φ(+∞) = m σ + 1,
1-Bistable case: traveling waves for σ ∈ (σ -, σ + ) We have (i) (Existence of a traveling wave) For any σ ∈ (σ -, σ + ), there exists a unique real c := c(σ), such that there exists a function φ σ : R → R solution of (1.19) in the viscosity sense.

(ii) (Continuity and monotonicity of the velocity function)

The map σ → c(σ) is continuous on (σ -, σ + ) and there exists a constant K > 0 such that the function c(σ) is non-decreasing and satisfies dc dσ ≥ K|c| on (σ -, σ + ) in the viscosity sense. In addition, there exists real numbers c -≤ c + such that

lim σ→σ - c(σ) = c -and lim σ→σ + c(σ) = c + .
Moreover, either c -= 0 = c + or c -< c + .

2-Monostable cases: vertical branches for σ = σ ± We have (i) (Existence of traveling waves for c ≥ c + when σ = σ + ) Let σ = σ + , then for every c ≥ c + there exists a traveling wave φ solution of

(1.20)      cφ ′ (z) = F (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) + σ + on R φ is non-decreasing over R φ(-∞) = 0 = m σ + and φ(+∞) = 1.
Moreover, for any c < c + , there is no solution φ of (1.20).

(ii) (Existence of traveling waves for c ≤ c -when σ = σ -) Let σ = σ -, then for every c ≤ c -, there exists a traveling wave φ solution of

(1.21)      cφ ′ (z) = F (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) + σ -on R φ is non-decreasing over R φ(-∞) = θ -1 = m σ -and φ(+∞) = θ.
Moreover, for any c > c -, there is no solution φ of (1.21).

Note that for the Frenkel Kontorova model (1.17), we have σ ± = ±1 and c + > 0 > c -(cf. Lemma 9.4), and Figure 4 illustrates the graph of the velocity c(σ) which has a plateau at the level c = 0 in particular if |σ| < β -1 (see Proposition 2.6).

In view of Theorem 1.7, we can ask the following: Open question 1. For a general F, what is the precise behavior of the function c(σ) close to the boundary of the plateau c = 0 and close to σ + and σ -? Open question 2. Can we construct a function F such that c + = 0 = c -? For indications in the direction of open question 1, see for instance [START_REF] Carpio | Wave solutions for a discrete reaction-diffusion equation[END_REF] (discussion on page 4 after Theorem 1.2).

Remark 1.8 (sign of c + and c -) If we can apply Proposition 1.3 for F + σ + , we deduce that c + ≥ 0. Similarly, by symmetry (see Lemma 3.7), it is possible to introduce similar assumptions to conclude that c -≤ 0. Notice that Theorem 1.1 is a generalization of Theorem 1.7-2 (i) for σ = σ + . Also, notice that Theorem 1.7-1 (i) is already proved in [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF] (see [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF]Proposition 2.3]).

As a notation, we set for a general function h:

F ((h(z + r i )) i=0,...,N ) = F (h(z + r 0 ), h(z + r 1 ), ..., h(z + r N ))
and we define

(1.23) r * = max i=0,...,N |r i |.
In the rest of the paper, we will use the notation introduced in Theorem 1.7.

Organization of the paper

Even if the main results of Subsections 1.2 and 1.3 are very different, the proofs are deeply related (because we use the results in the bistable case to deduce some results in the monostable case).

The paper is composed of three parts.

In a first part, we prove the existence of solutions for (1.9) for large velocities. This part is splitted into two sections (Sections 2 and 3). We recall, in Section 2, the notion of viscosity solutions and some useful results for monotone functions. Section 3 is devoted to the construction of a solution whenever we have positive supersolution with non-zero velocity. We also prove the existence of traveling waves solutions of (1.9) for c >> 1, which is applicable in particular for (1.20) and also for (1.21) when c << -1 (up to apply a suitable transformation).

We study in a second part the full range of velocities and it is decomposed into three sections (Sections 4, 5 and 6). Precisely, we revisit in Section 4 the results of [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF]. In a first subsection, we generalize and precise the result of existence of a traveling wave obtained in [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF]. We prove, in a second subsection, results about the passage to the limit in our equation and about the identification of the limits at infinity of the limit profile. In a third subsection, we apply the existence result of traveling waves obtained in Subsection 4.1 and we show the uniqueness of the velocity for solutions of (1.19) as a function of the driving force σ ∈ (σ -, σ + ). In Section 5, we prove the continuity and monotonicity of the velocity function over (σ -, σ + ) and we show that the velocity function attains finite limits c ± at σ ± . We also prove, in this section, the existence of solutions of (1.20) (resp. (1.21)) for c = c + (resp. c = c -). In Section 6, we fill the gap by proving the existence of solutions of (1.20) (resp. (1.21)) for every c ≥ c + (resp. c ≤ c -). Moreover, we show that for any c < c + (resp. c > c -) there is no solution of (1.20) (resp. (1.21)). We prove Theorem 1.7 at the end of Section 6.

The third part is also decomposed into three sections (Sections 7, 8 and 9) and it is dedicated to define and study the critical velocity. For instance, Theorem 1.1 is proved in Section 7, which we split in three subsections. In Subsection 7.1, we recall an extension result to R N +1 of a non-linearity defined on [0, 1] N +1 and then we prove Theorem 1.1 in the special case where the non-linearity is smooth. Under some additional assumptions, we prove the result of Theorem 1.1 using another approach in Subsection 7.2. In Subsection 7.3, we give the proof of Theorem 1.1 in full generality for Lipschitz non-linearities, where the construction of the critical velocity c + follows the lines of the proof of the regular case, but requires a lot of work to adapt it to this very delicate situation. In Section 8, we prove a strong maximum principle (Proposition 8.1), a lower bound (Proposition 8.3) and a Harnack type inequality (Proposition 8.4) for a profile that we use to prove that c + ≥ c * in Subsection 9.1. Section 9 is dedicated to properties of the critical velocity c + . Subsection 9.1 is specified for the proof of Proposition 1.5 where we show that c + ≥ c * . In this subsection, we also show that c + ≤ c * under a KPP type condition (precisely, we prove Proposition 1.6). We as well give an example (see Lemma 9.3) where c + > c * . In Subsection 9.2, we prove that c + is non-negative under certain assumptions, namely Proposition 1.3. While in Subsection 9.3, we construct a counter-example for which c + < 0, i.e. Proposition 1.4 and we prove the instability result of Proposition 1.2.

Finally in the Appendix (Section 10), we prove and state two kinds of results (which are used to prove that c + ≥ 0): first, extension by antisymmetry and antisymmetry-reflection (Propositions 10.1 and 10.4) and second, a comparison principle (Propositions 10.6 and 10.7).

Notations of our assumptions

In our paper, we introduce assumptions (A Lip ), (P Lip ) and (P C 1 ) in Section 1.2, assumptions ( ÃC 1 ) and ( BC Generically, assumptions of type A holds for F, assumptions of type P are positivity assumptions on f (v) = F (v, ..., v), and assumptions of type B are bistable assumptions for f.

Assumptions with tilde ( ˜) means that the functions F and f are considered on R N +1 and R respectively, and are assumed to be (1, ..., 1)-periodic and 1-periodic respectively. On the contrary, assumptions without tilde means assumptions for F and f on a finite box [0, 1] N +1 and [0, 1] respectively.

The subscript "Lip" means that we only require Lipschitz functions, while the subscript "C 1 " means that we require C 1 functions (at least on some part of their domain of definition).

Finally, assumptions with prime ( ′ ) are (locally in the paper) variant of the assumptions without prime.

Part I Vertical branches for large velocities 2 Preliminary results

We recall, in a first subsection, the definition of viscosity solutions, a stability result and Perron's method for constructing a solution. We state, in a second subsection, Helly's Lemma and the equivalence result between viscosity and almost everywhere solutions for non-decreasing functions. In a third subsection, we give an example with a discontinuous viscosity solution.

Viscosity solution

In the whole paper, we will use the notion of viscosity solutions that we introduce in this subsection. To this end, we recall that the upper and lower semi-continuous envelopes, u * and u * , of a locally bounded function u are defined as -The function u is a subsolution (resp. a supersolution) on

I ′ of (2.1) cu ′ (x) = F ((u(x + r i )) i=0,...,N ) + σ,
if u is upper semi-continuous (resp. lower semi-continuous) and if for all test function ψ ∈ C 1 (I) such that uψ attains a local maximum (resp. a local minimum) at x * ∈ I ′ , we have

cψ ′ (x * ) ≤ F ((u(x * + r i )) i=0,...,N ) + σ resp. cψ ′ (x * ) ≥ F ((u(x * + r i )) i=0,...,N ) + σ .
-A function u is a viscosity solution of (2.1) on I ′ if u * is a subsolution and u * is a supersolution on I ′ .

We also recall the stability result for viscosity solutions (see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]Theorem 4.1] and [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF]Proposition 2.4] for a similar proof).

Proposition 2.2 (Stability of viscosity solutions)

Consider a function F defined on R N +1 and satisfying ( ÃLip ) (introduced in Subsection 7.1). Assume that (u ε ) ε is a sequence of subsolutions (resp. supersolutions) of (2.1).

(i) Let u(x) = lim sup ε→0 * u ε (x) := lim sup (ε,y)→(0,x) u ε (y) and u(x) = lim inf ε→0 * u ε (x) := lim inf (ε,y)→(0,x) u ε (y),
be the relaxed upper and lower semi-limits. If u (resp. u) is finite, then u is a subsolution (resp. u is a supersolution) of (2.1).

(ii) Let T be a nonempty collection of subsolutions of (2.1) and set U (x) = sup u∈T u(x). If U * is finite then U * is a subsolution of (2.1). A similar result holds for supersolutions.

Next, we state Perron's method that we will use to construct a solution in Section 3. such that u and v are respectively a sub and a supersolution of (2.1) on I ′ . Let L be the set of all functions ṽ : I → R, such that u ≤ ṽ over I with ṽ supersolution of (2.1) on I ′ . For every z ∈ I, let w(z) = inf{ṽ(z) such that ṽ ∈ L}.

Then w is a solution of (2.1) over I ′ satisfying u ≤ w ≤ v over I.

Some results for monotone functions

In this subsection, we state Helly's Lemma for the convergence of a sequence of non-decreasing functions. We also recall the result about the equivalence between the viscosity and almost everywhere solutions. These results will be used later in Sections 4.

Example of discontinuous viscosity solution

We give in this section an example of a discontinuous viscosity solution.

Proposition 2.6 (Discontinuous viscosity solution) Consider β > 0, σ ∈ R and let (c, φ) be a solution of

(2.3)      cφ ′ (z) = φ(z + 1) -2φ(z) + φ(z -1) + β sin(2πφ(z)) + σ on R φ is non-decreasing φ(+∞) -φ(-∞) = 1.
Then σ ± = ±β. Moreover, if |σ| < β -1, then φ / ∈ C 0 and c = 0.

For the convenience of the reader we give the proof of this result (which is basically contained in Theorem 1.2 in Carpio et al. [START_REF] Carpio | Wave solutions for a discrete reaction-diffusion equation[END_REF]). Proof of Proposition 2.6

Clearly, we have σ ± = ±β (see Remark 1.10). Let |σ| < β -1 and let us show that φ / ∈ C 0 (R). Assume to the contrary that φ ∈ C 0 (R). Notice that because φ is non-decreasing and φ(+∞)φ(-∞) = 1, we deduce that

φ(z + 1) -2φ(z) + φ(z -1) ∈ [-1, 1]. Define now ψ(z) = φ(z + 1) -2φ(z) + φ(z -1) + β sin(2πφ(z)) + σ.
Because φ ∈ C 0 , then looking at the sup and inf of sin(2πφ), we deduce that

   sup R ψ ≥ β + σ -1 > 0 inf R ψ ≤ -β + σ + 1 < 0,
where the strict inequalities follow from |σ| < β -1. But cφ ′ = ψ which implies that cφ ′ changes sign. This is impossible because φ is non-decreasing. Therefore, φ / ∈ C 0 (R), which implies that c = 0.

Vertical branches for large velocities

We prove in this section that if (1.9) admits a positive supersolution φ (φ > 0), then there exists a solution of (1.9) (cf. Proposition 3.2). Conversely, we also show that if (c, φ) is a solution of (1.9), then (c, φ) is a supersolution of 1.9 for all c ≥ c (see Corollary 3.4). As a consequence of Proposition 3.2), we prove that system (1.9) admits a solution for all c >> 1 (cf. Proposition 3.5).

The result is applicable in particular for function F defined on R N +1 and satisfying ( ÃC 1 ) and ( BC 1 ) with σ = σ + , which can be always reduced to the case σ + = 0 by adding a constant to F, and hence we may get a solution for (1.20) for c >> 1. In this section, we show also the existence of solutions (1.21) for c << -1 which follows from the case σ = σ + using a transformation result (Lemma 3.7). Definition 3.1 (Supersolution of (1.9)) We say that (c, ψ) is a supersolution of (1.9

) if (c, ψ) satisfies      cψ ′ (z) ≥ F ((ψ(z + r i )) i=0,...,N ) on R ψ is non-decreasing over R ψ(-∞) = 0 and ψ(+∞) = 1.
Proposition 3.2 (Solution of (1.9) if it admits a positive supersolution) Consider a function F satisfying (A Lip ) and (P Lip ). Assume that there exists a continuous supersolution (c, ψ) of (1.9) with c = 0 and ψ > 0. Then there exists a traveling wave φ such that (c, φ) is a solution of (1.9).

Proof of Proposition 3.2

We have (c, ψ) is a supersolution of (1.9) with c = 0 and ψ > 0. Up to space translation, we may assume that ψ(0) = θ ∈ (0, 1). We will construct a solution using Perron's method.

Step 1: construction of a subsolution Consider the constant function ψ = ε with ε > 0 small enough fixed. Then

0 = cψ ′ (x) ≤ F ((ψ(x + r i )) i=0,...,N ) = f (ε).
Hence (c, ψ) is a subsolution of

(3.1) cw ′ (x) = F ((w(x + r i )) i=0,...,N ) on R.
Step 2: construction of local solution Since ψ(-∞) = 0, ψ(+∞) = 1, ψ > 0 and ψ is non-decreasing and continuous, then for ε small fixed and up to shift ψ, we can define k ε < 0 such that

(3.2) ψ(k ε ) = ε and ψ > ε on (k ε , +∞).
Then using Perron's method (Proposition 2.3), there exists a solution φ ε of (3.1) on (r

* + k ε , +∞) such that ε ≤ φ ε ≤ ψ on (k ε , +∞).
Step 3: φ ε is non-decreasing on (k ε , +∞) . Define for x ∈ (k ε , +∞) the function

φ(x) := inf p≥0 φ ε (x + p).
Clearly, since ε ≤ φ ε (x + p) for all p ≥ 0 and x ∈ (k ε , +∞) , we get ε ≤ φ(x) ≤ φ ε (x) ≤ ψ(x) for all x ∈ (k ε , +∞) . On the other hand, for all p ≥ 0, φ ε (x + p) is a solution of (3.1) over (r * + k ε , +∞) , then (φ) * is supersolution of (3.1) over (k ε + r * , +∞) (using Proposition 2.2 (ii)). Moreover, we have ε ≤ (φ) * ≤ ψ. But φ ε is defined as the infimum of supersolutions (recall Proposition 2.3 for Perron's method), thus φ ε ≤ (φ) * ≤ φ ≤ φ ε over (k ε , +∞) . Therefore, for every p ≥ 0,

φ ε (x) = φ(x) ≤ φ ε (x + p) over (k ε , +∞) ,
and hence φ ε is non-decreasing over (k ε , +∞) .

Step 4: passing to the limit ε → 0

Step 4.1: setting Since φ ε is a non-decreasing solution of (3.1) on (r * + k ε , +∞), then φ ε (+∞) has to solve f (x) = 0 (see (3.1)). But φ ε is a non-decreasing and 0 < ε ≤ φ ε ≤ ψ ≤ 1 over (k ε , +∞), we conclude that

φ ε (+∞) = 1.
Moreover, from equation (3.1) and c = 0, we deduce in particular that φ ε is Lipschitz on (r * + k ε , +∞) with |φ ′ ε | ≤ K 0 for a constant K 0 independent of ε. In addition, since φ ε (0) ≤ ψ(0) = θ and φ ε (+∞) = 1, then there exists x ε ≥ 0 such that φ ε (x ε ) = θ.

Notice also that for ε small enough, we have r * + k ε < 0 and we also have that k ε is increasing w.r.t. ε and

k ε → -∞ as ε → 0. Indeed, if k ε → k 0 ∈ R, then ψ(k 0 ) = 0 which is impossible since ψ > 0.
Step 4.2: global non-decreasing solution of (3.1) Let φε (x) := φ ε (x + x ε ) which is a solution of (3.1) on (-d ε , +∞), where

d ε = x ε -(r * + k ε ). We have φε (0) = θ and d ε → +∞ as ε → 0 because k ε → -∞ and x ε ≥ 0. We also have | φε | ≤ K 0 on (-d ε , +∞).
Thus passing to the limit ε → 0, φε converges (using Ascoli's Theorem) to some non-decreasing φ solution of

(3.3)      cφ ′ (x) = F ((φ(x + r i )) i=0,...,N ) 0 ≤ φ ′ ≤ K 0 on R 0 ≤ φ ≤ 1 and φ(0) = θ. Let a = φ(-∞) or φ(+∞). Then it is easy to see that 0 = f (a) which implies that φ(-∞) = 0 and φ(+∞) = 1. Therefore φ is a solution of (3.4)      cφ ′ (x) = F (φ(x + r i )) i=0,...,N ) on R φ is non-decreasing φ(-∞) = 0 and φ(+∞) = 1,
and this ends the proof.

Remark 3.3 (Relax of conditions of Proposition 3.2)

Up to adapt the proof of Proposition 3.2, it would be easy to relax the condition with c = 0 and/or ψ possibly discontinuous (but still monotone).

Corollary 3.4 (Half line of solutions)

Under the assumptions of Proposition 3.2, assume that (1.9) admits a solution (c, φ) with φ > 0. Then for all c ≥ c there exists a solution φ of (1.9).

Proof of Corollary 3.4

Let (c, φ) be a solution (1.9) and c ≥ c, we have

Case 1: c = 0 cφ ′ (z) ≥ cφ ′ (z) = F ((φ(z + r i )) i=0,...,N ).
Hence (c, φ) is a supersolution of (1.9). Since φ > 0, then using Proposition 3.2, we deduce the existence of a solution of (1.9) for every c ≥ c, if c = 0.

Case 2: c = 0 If c = 0, then we consider a sequence of solutions (c n , φ n ) with c n = 0 and c n → 0 = c. Since φ n is monotone and bounded uniformly in n, then using Helly's Lemma (Lemma 2.4) and the diagonal extraction argument, φ n converges up to a subsequence to a non-decreasing function φ a.e. Moreover, we can assume (up to translation) that φ n (0) = 1 2 , and hence we get that φ(0) = 1 2 . In addition, We have

c n b 2 b 1 (φ n ) ′ (z)dz = b 2 b 1 F n ((φ n (z + r i )) i=0,...,N ) dz for every b 1 < b 2 . That is, c n (φ n (b 2 ) -φ n (b 1 )) = b 2 b 1 F n ((φ n (z + r i )) i=0,...,N ) dz. But |F ((φ n (z + r i )) i=0,...,N )| ≤ M 0 for some M 0 > 0 and F n ((φ n (z + r i )) i=0,...,N ) → F ((φ(z + r i )) i=0,...,N ) a.e.
Thus, using Lebesgue's dominated convergence theorem, we pass to the limit n → +∞ and get

b 2 b 1 F ((φ(z + r i )) i=0,...,N )dz which implies (since b 1 and b 2 are arbitrary) that (3.5) 0 = F ((φ(z + r i )) i=0,...,N )
almost everywhere. Then by Lemma 2.5, φ verifies (3.5) in the viscosity sense. and satisfies (4.6). Moreover, since φ(0) = 1 2 , we can deduce that φ is a solution of (1.9).

Proposition 3.5 (Existence of traveling waves for c >> 1)

Consider a function F satisfying (A Lip ) and (P Lip ). Then for c >> 1, there exists a traveling wave φ solution of (1.9).

Proof of Proposition 3.5

The strategy of the proof consists in constructing a positive supersolution for c >> 1 of a re-scaled form of the equation

(3.6) cφ ′ (y) = F (φ(y + r 0 ), φ(y + r 1 ), ..., φ(y + r N )) on R,
then we conclude by Proposition 3.2.

Step

1: re-scaling equation (3.6) If φ is a supersolution of (3.6) (with φ(-∞) = 0, φ(+∞) = 1), then for every z ∈ R, the function h defined as h(z) := φ(cz)
has to satisfy, knowing that c >> 1,

(3.7) h ′ (z) = F h z + r i c i=0,...,N on R.
Step 2: supersolution of (3.7)

In order to construct a supersolution of (3.7), we first mention some useful properties of the solution of the ODE

(3.8) h ′ 0 = F (h 0 , ..., h 0 ) = f (h 0 ) ≥ 0, with h 0 (0) = 1 2 .
Step 2.1: existence and monotonicity of h 0 Since f > 0 on (0, 1) and f is Lipschitz over [0, 1] (see assumptions (A Lip ) and (P Lip )), then there exists a C 1 solution h 0 of (3.8) defined on R, with values in [0, 1], satisfying

(3.9) h ′ 0 > 0 on R.
Since the constant functions 0 and 1 are respectively a sub and a supersolution of (3.8) (since

f (0) = f (1) = 0), then 0 ≤ h 0 (z) ≤ 1.
We also easily deduce that h 0 (-∞) = 0 and h 0 (+∞) = 1.

Step 2.2: supersolution of (3.7)

The proof is similar to Step 2.2. Let ε = 1 c and 0 < δ = M ε with M chosen large, and c chosen such that a = 1 + δ ≤ 2. Then consider the function

h(z) = h 0 (az)
that we want to show to be a supersolution of (3.7) on R, taking the advantage of the fact that (3.8) is a caricature of (3.7) for large c. We have

h(z + εr i ) = h 0 (az) + εar i L i with L i = 1 0 h ′ 0 (az + εar i t).
Because F ∈ Lip([0, 1] N +1 ) for some Lipschitz constant L, we get

F ((h(z + εr i )) i=0,...,N ) -f (h 0 (az)) = F ((h 0 (az) + aεr i L i )) i=0,...,N ) -F ((h 0 (az)) i=0,...,N ) ≤ εaL r 0 L 0 . . . r N L N , where r * = max i=0,...,N |r i | (recall (1.23)).
We now estimate the L i 's.

Case

1: f ∈ C 1 ([0, 1]) If f ∈ C 1 ([0, 1]), then for z ∈ R, we have h ′′ 0 (z) = f ′ (h 0 (z))h ′ 0 (z). As h ′ 0 > 0 on R and f ∈ C 1 ([0, 1]), we get for z ∈ R (ln(h ′ 0 (z))) ′ = f ′ (h 0 (z)),
where the absolute value of the right hand side is bounded by some constant K. Hence, using the continuity of h ′ 0 , for any b ∈ R and for all z ∈ R, we obtain

ln h ′ 0 (z + b) h ′ 0 (z) ≤ K|b|.
This implies that (3.10)

h ′ 0 (z + b) ≤ h ′ 0 (z)e K|b| for every z ∈ R. Case 2: f ∈ Lip([0, 1])
We want to show that (3.10) is still true if f ∈ Lip([0, 1]), and the point is to regularize by convolution the function f and then to pass to the limit. Using the extension result (cf. Lemma 7.1), there exists a function F defined over R N +1 and satisfying ( ÃLip ). Moreover, the function f (v) := F (v, ..., v) is nothing but the periodic extension of f with period 1.

Let

ρ ε (x) = 1 ε ρ( x ε )
, where ρ is a mollifier and define the function fε (x) := f ⋆ ρ ε (x). Then consider the ODE

(3.11)    h ′ ε = fε (h ε ) h ε (0) = 1 2 .
Since fε is C 1 , then there exists a unique regular solution h ε defined over R and satisfies

(3.12) h ′ ε (z + b) ≤ h ′ ε (z)e K|b| for every z ∈ R.
Moreover, since fε is periodic smooth, then there exists some C independent of ε such that

|h ′ ε | ≤ C on R.
Therefore, using Ascoli's theorem and the extraction diagonal argument, h ε converges locally uniformly to some h 1 that solves in the classical sense

(3.13)    h ′ 1 = f (h 1 ) h 1 (0) = 1 2 , and h ′ 1 (z + b) ≤ h ′ 1 (z)
e K|b| for every z ∈ R. But the constant functions 0 and 1 are respectively sub and supersolution of (3.13), then (3.8). Thus by uniqueness, we get that h 1 = h 0 , and hence h 0 satisfies (3.10).

0 ≤ h 1 ≤ 1, that is, h 1 is a solution of
Consequences in both Case 1 and Case 2 Now, we go back to estimate the L i 's. Using (3.10) for b = aεr i t and using the fact that a < 1, we get for every i ∈ {0, ..., N } that

0 ≤ L i = 1 0 h ′ 0 (az + aεr i t)dt ≤ h ′ 0 (az)e Kaε|r i | ≤ h ′ 0 (az)e Kεr * =: Kh ′ 0 (az).
This implies that

F ((h(z + εr i )) i=0,...,N ) -f (h 0 (az)) ≤ 2εL 1 r * Kh ′ 0 (az),
where we have used that a ≤ 2 and that L 1 := L 1 . . .

1

.

Therefore, we deduce that with

δ = M ε h ′ (z) -F ((h(z + εr i )) i=0,...,N ) = ah ′ 0 (az) -F ((h(z + εr i )) i=0,...,N ) = δh ′ 0 (az) -F ((h(z + εr i )) i=0,...,N ) -f (h 0 (az)) ≥ ε M -2L 1 r * K h ′ 0 (az) ≥ 0, if we choose M ≥ 2L 1 r * K. Therefore h is a supersolution of (3.7).
Step 3: solution of (3.7) for c >> 1 We have h(z) = h 0 (az) is a supersolution of (3.7). Moreover, since a > 0, h ′ 0 > 0 on R and

h 0 (-∞) = 0 and h 0 (+∞) = 1,
we deduce that 0 < h < 1.

Therefore, using Proposition 3.2, we get the existence of solution of (3.7) for c >> 1 and hence for (1.9).

Lemma 3.6 (Vertical branches for σ = σ ± ) Consider a function F satisfying ( ÃC 1 ) and ( BC 1 ). Assume that σ = σ + (resp. σ = σ -), then for c >> 1 (resp. c << -1), there exists a traveling wave solution of (1.20) (resp. (1.21)).

Proof of Lemma 3.6

Proving the existence of solution for c >> 1 when σ = σ + follows exactly from Proposition 3.5 where σ + = 0. However, the proof of the result for c << -1 when σ = σ -follows from the proof of the case σ = σ + and the transformation lemma below (Lemma 3.7).

Lemma 3.7 (Transformation of solutions)

Let φ be a solution of

(3.14) cφ ′ (z) = F ((φ(z + r i )) i=0,...,N ) + σ -over R, then φ(z) = θ -φ(-z)
is a solution of (3.14) with F, c, r i and σ -replaced respectively by

(3.15) F (X 0 , ..., X N ) = -F ((θ -X i ) i=0,...,N ) c = -c, r i = -r i and σ + = -σ - Moreover, if F satisfies ( ÃC 1 ) and ( BC 1 ) then F satisfies ( ÃC 1 ) and (B) (with f (v) = F (v, ..., v)).
Proof of Lemma 3.7

The proof of Lemma 3.7 is straightforward.

Part II

Study of the full range of velocities 4 Revisiting results of [1]

This section is divided into three subsections. In a first subsection, we generalize the result of existence of traveling waves obtained in [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF]. We present, in a second subsection, some techniques to pass to the limit in the equation. In a third subsection, we apply the existence result of first subsection for the bistable case when σ ∈ (σ -, σ + ).

Bistable case

We prove in this subsection the existence of traveling waves for the bistable non-linearity under weaker assumptions. This result is not proved in [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF] and it is more general. We will use this general result later in the proof of Theorem 1.1, Section 7.3, Step 1. This result will be used also to prove that c + ≥ 0 (proof of Proposition 1.3), Section 9.2.

In order to present our result, we assume that

Assumption (B Lip ): Let f (v) := F (v, ..

., v) and assume

Instability:

f (0) = 0 = f (1) and there exists b ∈ (0, 1) such that f (b) = 0, f | (0,b) < 0 and f | (b,1) > 0.
Strict monotonicity: There exists some η > 0 such that

F (X + (ω, ..., ω)) -F (X) ≥ η ω
for ω > 0 small enough and for all X close enough to (b, ..., b).

Proposition 4.1 (Existence of c for a Lipschitz bistable non-linearity) Consider a function F defined over [0, 1] N +1 and satisfying (A Lip ) and (B Lip ). Then there exist a real c and a function φ solution of (1.9) in the classical sense if c = 0 and almost everywhere if c = 0. Moreover, there is no a > r * and x ∈ R such that

(4.1) φ = b on [x -a, x + a].
This result is the analogue of the existence result of [1, Proposition 2.3], assuming that F is less regular near the instability b which is replaced by the strict monotonicity of F near b.

Proof of Proposition 4.1

As it is written above, the proof of Proposition 4.1 is a variant of the proof of [1, Proposition 2.3]. However, in this case, we obtain the contradiction using the strict monotonicity (Step 4.3) while the rest of the proof (Step 0 to Step 4.2 and Step 5) stays the same. We now prove the contradiction using the strict monotonicity, revisiting Step 4.3 of the proof of [1, Proposition 2.3].

Step 4.3: getting a contradiction We recall that we consider an approximation φ p of the profile φ, for some parameter p going to zero, which satisfies

c p φ ′ p (z) = F ((φ p (x + r i )) i=0,...,N ). We construct (see [1, Proposition 2.3]) a local minimum x * p of ψ p satisfying 0 < m p = ψ p (x * p ),
where

ψ p (x) = (φ p ) * (x + a) -(φ p ) * (x -a). Then it is possible to see as in [1, Proposition 2.3, Step 4.3], that 0 ≥ F ((a i ) i=0,...,N ) -F ((c i ) i=0,...,N ),
where

a i = k i if r i ≤ 0 k i + m p if r i > 0 and c i = k i -m p if r i ≤ 0 k i if r i > 0,
and

k i = (φ p ) * (x * p + a + r i ) if r i ≤ 0 (φ p ) * (x * p -a + r i ) if r i > 0.
Here, the notation c i is not ambiguous and has nothing to do with the velocity c p . Therefore, for p small enough, we have c i close to b and m p > 0 is small enough, thus using the strict monotonicity in (B Lip ), we deduce that

Since a i = c i + m p for every i ∈ {0, ..., N }, then 0 ≥ F ((c i + m p ) i=0,...,N ) -F ((c i ) i=0,...,N ).
0 ≥ F ((c i + m p ) i=0,...,N ) -F ((c i ) i=0,...,N ) ≥ ηm p > 0,
which is a contradiction.

Verification of (4.1) Assume that there exists a > r * and x 0 ∈ R such that (4.2)

φ σ = b σ on [x 0 -a, x 0 + a].
Then proceeding as in [

Results for passing to the limit

The main result of this subsection (Theorem 4.4) identifies the limits of a constructed profile. We also prove some results to pass to the limit in the equation, namely Lemma 4.2. We will use the results of this subsection to prove the continuity of the velocity function later in Proposition 5.4, Subsection 5.1. We start by introducing the following bistable notation:

Assumption ( Bm,b ): Let f (x) := F (x, ..., x) and m < b < m + 1, Bistability: f (m) = 0 = f (b) = f (m + 1), f < 0 on (m, b
) and f > 0 on (b, m + 1).

Lemma 4.2 (Passing to the limit) Consider a sequence of functions F n satisfying ( ÃLip ) and ( Bmn,bn ) (with m n ∈ [0, 1)) such that

(4.3) Lip(F n ) ≤ C independent on n.
Let (c n , φ n ) be a solution of

(4.4)      c n φ ′ n (z) = F n ((φ n (z + r i )) i=0,...,N ) over R φ n is non-decreasing on R φ n (-∞) = m n and φ n (+∞) = m n + 1.
Assume that

(4.5) |φ n | ≤ M for some M > 0 independent of n.
Assume moreover that there exists a real number c such that c n → c; and that F n → F locally uniformly and

(m n , b n ) → (m, b) as n → +∞.
Then, up to a subsequence, φ n converges almost everywhere to some φ that solves in the viscosity sense Proof of Lemma 4.2

(4.6)      cφ ′ (z) = F ((φ(z + r i )) i=0,...,N ) over R φ is non-decreasing on R m ≤ φ(-∞) and φ(+∞) ≤ m + 1.
Step 1: passing to the limit The proof of this result follows from [1, Proposition 2.3, Step 2]. For the convenience of the reader, we give the proof here.

Because of (4.3) and since φ n is bounded, we deduce that there exists a constant M 0 > 0 independent of n such that

(4.7) |F n ((φ n (z + r i )) i=0,...,N )| ≤ M 0 independent on n. Case 1: c = 0 Since |c n | ≥ |c| 2 for n large, then |φ ′ n | ≤ 2M 0 c for large n.
Thus φ n is uniformly Lipschitz. Using Ascoli's Theorem and the diagonal extraction argument, we get that φ n converges to φ (up to a subsequence) locally uniformly on R. Moreover φ is nondecreasing and satisfies (by stability of viscosity solutions)

(4.8) cφ ′ (x) = F ((φ(x + r i )) i=0,...,N ).
We easily deduce (4.6).

Case 2: c = 0 Since φ n is monotone and bounded (uniformly in n), then using Helly's Lemma (Lemma 2.4) and the diagonal extraction argument, φ n converges (up to a subsequence) to a non-decreasing φ a.e. Our goal is to show that (4.9) 0 = F ((φ(x + r i )) i=0,...,N ).

Subcase 2.1: c n = 0 for all n We first use the equivalence between viscosity solutions and almost everywhere solutions (Lemma 2.5) and then pass to the limit in (4.4) using Helly's lemma (Lemma 2.4). Hence, we get a solution φ of (4.9) almost everywhere. Again, we use Lemma 2.5 to conclude that φ is a viscosity solution of (4.9) and satisfies (4.6).

Subcase 2.2: c n = 0 for all n See Case 2 of the proof of Corollary 3.4 to deduce φ verifies (4.9) in the viscosity sense and satisfies (4.6).

Step

     (φ b ) * (0) ≤ m + 1 + b 2 (φ b ) * (0) ≥ m + 1 + b 2
Solutions φ a and φ b can be obtained as limits of We recall now the existence result of traveling waves whose a slightly different statement is given in [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF]Proposition 2.3]. In order to present the main result of this section, we need to introduce the following technical lemma.

φ a n (x) = φ n (x + a n ) and φ b n (x) = φ n (x + b n ) for suitable shifts a n , b n such that

Lemma 4.3 (Controlling the finite difference)

Consider F satisfying ( ÃC 1 ), σ 0 ∈ (σ -, σ + ) fixed and β > 0. Let a > r * (r * is given by (1.23)) and M 0 > 0, then for all σ ∈ [σ 0β, σ 0 + β] ⊂ (σ -, σ + ) and for every ε > 0, there exists δ > 0 such that for all function φ (viscosity) solution of

               cφ ′ (x) = F ((φ(x + r i )) i=0,...,N ) + σ on R φ ′ ≥ 0 φ(x + 1) ≤ φ(x) + 1 |c| ≤ M 0 |cφ ′ | ≤ M 0 ,
and for all

x 0 ∈ R satisfying φ * (x 0 + a) -φ * (x 0 -a) ≤ δ, we have dist α, {m σ , b σ } + Z < ε for all α ∈ [φ * (x 0 ), φ * (x 0 )].
Proof of Lemma 4.3

The proof of this lemma follows from a straightforward generalization of [1, Proposition 3.2] for the function F replaced by

F + σ and (0, b) replaced by (m σ , b σ ) for σ ∈ [σ 0 -β, σ 0 + β] ⊂ (σ -, σ + )
and for some β > 0. We similarly show that for every ε > 0 there exists δ σ (ε) > 0 such that the result holds true. However, we can show that δ σ (ε) = δ(ε) can be chosen independent of σ and the proof of this generalization follows exactly the same lines. Indeed, we proceed by contradiction assuming that the statement is false for a sequence σ n ∈ [σ 0β, σ 0 + β], and consider a sequence of solutions φ n . The presence of σ n does not create any additional difficulty in the passage to the limit in the equation. ) and ( BC 1 ). We assume moreover that the solution (c n , φ n ) of (4.4) is given by Proposition 4.5 for σ n ∈ (σ -, σ + ). Let (c ∞ , φ ∞ ) be the solution of (4.6) constructed in Lemma 4.2. If σ ∞ ∈ (σ -, σ + ), then we have moreover

φ ∞ (-∞) = m σ∞ and φ ∞ (+∞) = m σ∞ + 1.
Proof of Theorem 4.4 Let (c n , φ n ) be a solution of (4.4) given by Proposition 4.5 and (c ∞ , φ ∞ ) be a solution of (4.6) for σ ∞ ∈ (σ -, σ + ), obtained by passing to the limit n → ∞. Our aim is to show that

φ ∞ (-∞) = m σ∞ and φ ∞ (+∞) = m σ∞ + 1. For ε > 0 small enough (ε < 1 2 min(b σn -m σn , m σn + 1 -b σn )), let z n , y n ∈ R such that (4.12) (φ n ) * (z n ) ≥ b σn + ε (φ n ) * (z n ) ≤ b σn + ε and (4.13) (φ n ) * (y n ) ≥ b σn -ε (φ n ) * (y n ) ≤ b σn -ε.
Assume moreover that up to translate φ n , we have

(φ n ) * (0) ≤ b σn (φ n ) * (0) ≥ b σn .
For every x ∈ R, set with a > r *

ψ n (x) := (φ n ) * (x + a) -(φ n ) * (x -a) ≥ 0
and denote by m n = min

[yn,zn] ψ n (x) = ψ n (x n ) ≥ 0, for some x n ∈ [y n , z n ] since ψ n is lower semi-continuous. We claim that m n > 0. Indeed, if m n = 0, then since ψ n (y n ), ψ n (z n ) ≥ δ(ε)
> 0 (because of (4.12), (4.13) and using Lemma 4.3), we get

x n ∈ (y n , z n ).
Moreover, we have that

0 = ψ n (x n ) = (φ n ) * (x n + a) -(φ n ) * (x n -a)
and φ n is non-decreasing, hence

φ n = const over (x n -a, x n + a),
and φ n solves f + σ n = 0. Now, since b σn -ε ≤ (φ n ) * (y n ) ≤ φ n (x n ) ≤ (φ n ) * (z n ) ≤ b σn + ε,
we get that φ n = b σn over (x na, x n + a).

Therefore, for r * < a < a, we have

φ n = b σn over [x n -a, x n + a],
which is in contradiction with Proposition 4.5. Therefore, m n > 0 and the proof of the identification of limits of the profile proceeds similarly as in [1, Proposition 2.3], where now Step 5 is no longer necessary. In particular we avoid the case φ(±∞) = b σ∞ .

4.3 Application to the existence of traveling waves for σ ∈ (σ -, σ + )

In this section we prove, for every σ ∈ (σ -, σ + ), the existence of a unique velocity c = c(σ) and the existence of a traveling wave φ = φ σ solution of (1.19).

The main result of this section is: Assume that F satisfies ( ÃC 1 ), ( BC 1 ) and let σ ∈ (σ -, σ + ). Then there exists a unique real c(σ) (simply denoted by c σ ) such that there exists a function φ σ : R → R solution of (1.19) for c = c σ (in the viscosity sense). Moreover, this solution satisfies the following property: there is no a > r * (r * is given in (1.23)) and x ∈ R such that

(4.14) φ σ = b σ on [x -a, x + a],
where b σ , m σ are defined in Theorem 1.7.

In order to prove Proposition 4.5, we introduce the following lemma: Under the assumptions ( ÃC 1 ) and ( BC 1 ), the two maps

[σ -, σ + ] → [θ -1, 0] σ → m σ and [σ -, σ + ] → [0, θ] σ → b σ ,
are continuous. Moreover, the map m σ is increasing in σ, while b σ is decreasing.

The proof of Lemma 4.6 is straightforward from the definition of σ ± and from assumption ( BC 1 ).

Proof of Proposition 4.5

Let σ ∈ (σ -, σ + ). Let m σ ∈ (θ -1, 0) and b σ ∈ (0, θ) (since σ = σ ± ) be the solutions of f (s)+σ = 0. Because of assumption ( BC 1 ), the function (f + σ) | [mσ ,mσ +1] is of bistable type, that is f + σ satisfies ( B′ C 1 ) f (v) + σ = 0 for v = m σ , b σ and m σ + 1 (f + σ) | (mσ ,bσ ) < 0, (f + σ) | (bσ ,mσ +1) > 0 and f ′ (b σ ) > 0.
Step 1: existence of a traveling wave Since F satisfies ( ÃC 1 ) and b σ ∈ (0, θ)

(because σ = σ ± ), then F is C 1 near {b σ } N +1
. Therefore, for ω > 0 small enough, X close enough to {b σ } N +1 and for all ε > 0, we have

F (X + (ω, ..., ω)) -F (X) = 1 0 dt N i=0 ∂F ∂X i (X + t(ω, ..., ω))ω ≥ (N + 1)(f ′ (b σ ) -ε)ω ≥ (N + 1) f ′ (b σ ) 2 ω for ε ≤ f ′ (bσ) 2 = ηω with η = (N + 1) f ′ (bσ) 2 .
Again, since F satisfies ( ÃC 1 ), which implies in particular that F satisfies (A Lip ), then using Proposition 4.1, there exists a traveling wave φ σ and a velocity c σ solution of (1.19).

Step 2: uniqueness of the velocity c σ under (M ) Assume that F is decreasing close to {m σ } N +1 and {m σ + 1} N +1 in the direction E = (1, ..., 1). That is, there exists ε > 0 small such that F satisfies:

(M ) F (X + (a, ..., a)) < F (X) for all a > 0 such that X, X + (a, ..., a) ∈ [m σ , m σ + ε] N +1
F (X + (a, ..., a)) < F (X) for all a > 0 such that X, X + (a, ..., a)

∈ [m σ + 1 -ε, m σ + 1] N +1 .
Then under assumptions ( ÃC 1 ) and (M ), the velocity c σ is unique, (as a consequence of [1, Theorem 1.5 (a)])).

Step 3: checking that F satisfies (M ) Since F is C 1 over a neighborhood of RE\(ZE ∪ ZΘ), then for every δ > 0 there exists

ε = ε(δ) > 0 such that if X, X + (a, ..., a) ∈ [m σ , m σ + ε] N +1 , then
F (X + (a, ..., a)) -F (X) -f ′ (m σ )a = 1 0 dt N i=0 ∂F ∂X i (X + t(a, ..., a)) - ∂F ∂X i (m σ , ..., m σ ) a ≤ (N + 1)aδ. Now, since f ′ (m σ ) < 0, we deduce that F (X + (a, ..., a)) -F (X) ≤ (f ′ (m σ ) + (N + 1)δ)a < 0
for δ > 0 small enough. Similarly, we show that F is decreasing close to {m σ + 1} N +1 . Note that, the proof of (4.14) follows exactly as the proof of (4.1).

Properties of the velocity

We split this section into two subsections. We dedicate a first subsection to the proof of monotonicity and continuity of the velocity function c(σ) over (σ -, σ + ). In a second subsection, we prove that the velocity function attains finite limits c ± as σ goes to σ ± respectively. We also prove the existence of traveling waves solutions of (1.20) (resp. (1.21)) for c = c + (resp. c = c -).

Monotonicity and continuity of the velocity

This subsection consists in two results. The monotonicity (Corollary 5.2 and Lemma 5.5) and the continuity (Proposition 5.4) of the velocity function on (σ -, σ + ). We start with the following result.

     φ 1 (-∞) < φ 2 (-∞) φ 1 (+∞) < φ 2 (+∞) φ 1 (+∞) > φ 2 (-∞).
Then

c 1 ≤ c 2 .
Proof of Proposition 5.1 Assume to the contrary that c 2 < c 1 . Let a ∈ R and define φ a 2 (x) = φ 2 (x + a). Hence, for a ≥ 0 large enough fixed, we get

φ a 2 ≥ φ 1 over R. Next, set u 1 (t, x) = φ 1 (x + c 1 t) u 2 (t, x) = φ a 2 (x + c 2 t
), then the u j are respectively a sub and a supersolution for j = 1, 2 of the following equation

(5.2) ∂ t u j (t, x) = F ((u j (t, x + r i )) i=0,...,N ) + σ j .
Moreover, at time t = 0, we have

u 2 (0, x) = φ a 2 (x) ≥ φ 1 (x) = u 1 (0, x) over R.
Thus applying the comparison principle for equation (5.2) (see [16, Propositions 2.5 and 2.6]), we get u 2 (t, x) ≥ u 1 (t, x) for all (t, x) ∈ [0, +∞) × R.

Taking x = yc 2 t, we get φ a 2 (y) ≥ φ 1 (y + (c 1c 2 )t) for all t ≥ 0 and y ∈ R.

Using that c 1 > c 2 and passing to the limit t → ∞, we get φ a 2 (y) ≥ φ 1 (+∞) for all y ∈ R.

But φ a 2 (-∞) < φ 1 (+∞) (see (5.1)), hence a contradiction. Therefore c 1 ≤ c 2 .

Corollary 5.2 (Monotonicity of the velocity over [σ -, σ + ]) Assume ( ÃC 1 ), ( BC 1 ) and let

σ 1 , σ 2 ∈ [σ -, σ + ] such that σ 1 < σ 2 . Let i = 1, 2 and associate for each σ = σ i a solution (c i , φ i ) of (1.19).
Then

c 1 ≤ c 2 . Proof of Corollary 5.2 Let σ 1 , σ 2 ∈ [σ -, σ + ] such that σ 1 < σ 2 . Since (c 1 , φ 1
) and (c 2 , φ 2 ) are two solutions of (1.19), then φ 1 and φ 2 are respectively a sub and a supersolution of

cφ ′ (x) = F ((φ(x + r i )) i=0,...,N ) + σ 2 .
Moreover, for m σ i denoted by m i , we have (see Lemma 4.6)

φ 1 (-∞) = m 1 < m 2 = φ 2 (-∞) φ 1 (+∞) = m 1 + 1 < m 2 + 1 = φ 2 (+∞) φ 1 (+∞) = m 1 + 1 > m 2 = φ 2 (-∞).
Therefore, the result of Corollary 5.2 follows from Proposition 5.1.

Then we have the straightforward consequence of Proposition 5.1.

Corollary 5.3 (Monotonicity and limits of c(σ))

Assume ( ÃC 1 ), ( BC 1 ). For σ ∈ (σ -, σ + ), let (c(σ), φ σ ) be a solution of (1.19) given in Proposition 4.5. Then the velocity function is non-decreasing on (σ -, σ + ). Moreover, the limits

lim σ→σ - c(σ) = c -and lim σ→σ + c(σ) = c + exist and satisfy -∞ ≤ c -≤ c + ≤ +∞.

Proposition 5.4 (Continuity of the velocity function)

Suppose that F satisfies ( ÃC 1 ), ( BC 1 ) and let σ ∈ (σ -, σ + ). Let (c(σ), φ σ ) be a solution of (1.19) given in Proposition 4.5. Then the map σ → c(σ) is continuous on (σ -, σ + ).

Proof of Proposition 5.4

Let σ 0 ∈ (σ -, σ + ) and c 0 := c(σ 0 ) be the associated velocity given in Proposition 4.5. Let σ n ∈ (σ -, σ + ) be a sequence such that σ n → σ 0 and let c n = c(σ n ). We want to show that c n → c 0 .

Assume that φ 0 and φ n (for each n) are solutions of (1.19) associated respectively to σ 0 and σ n (for each n).

Step 1: passing to the limit n → +∞ As a consequence of the monotonicity of c(σ) (Proposition 5.1) and the fact that σ 0 , σ n ∈ (σ -, σ + ) for all n, we get that c n is bounded. Thus, up to a subsequence, we set c = lim n→+∞ c n .

Recall that (c n , φ n ) solves

c n φ ′ n (z) = F ((φ n (z + r i )) i=0,...,N ) + σ n and θ -1 < m σn ≤ φ n ≤ m σn + 1 < 1.
Therefore, passing to the limit n → +∞ (see Lemma 4.2), φ n converges to a function φ almost everywhere, and φ solves (in the viscosity sense)

(5.3) cφ ′ (x) = F ((φ(x + r i )) i=0,...,N ) + σ 0 .
Moreover, Theorem 4.4 implies that (c, φ) solves (1.19) for σ = σ 0 .

Step 2: conclusion From the uniqueness of the velocity on (σ -, σ + ) (Proposition 4.5) and the fact that c 0 and c are associated to σ 0 ∈ (σ -, σ + ), we deduce that c = c 0 . From the uniqueness of the limit c (whatever is the subsequence σ n → σ 0 ), we deduce the continuity of the velocity function c. Suppose that c 1 > 0. Since F ∈ Lip(R N +1 ) and φ 1 is bounded, then there exists some

C > 0 such that |F ((φ 1 (x + r i )) i=0,...,N )| ≤ C. Therefore 0 ≤ φ ′ 1 ≤ c -1 1 (|σ 1 | + C). Hence for δ = c 1 (|σ 1 | + C) -1 , we have (using (5.5)) (c 1 + δ(σ 2 -σ 1 ))φ ′ 1 ≤ σ 2 + F ((φ 1 (x + r i )) i=0,...,N ).
But, this means that (c, φ 1 ), with c = c 1 + δ(σ 2σ 1 ), is a subsolution of (5.5) with σ = σ 2 .

Comparing φ 1 (x + ct) to φ 2 (x + c 2 t) as in Proposition 5.1, we deduce that c ≤ c 2 , that is,

(5.6) c 2 -c 1 σ 2 -σ 1 ≥ c 1 (|σ 1 | + C) -1 =: Kc 1 (σ 1 ∈ (σ -, σ + ) bounded).
Now letting σ 1 → σ 2 , and using the continuity of c(σ), inequality (5.4) follows (in the sense of viscosity) in case c > 0. Similarly, we prove that c(σ) verifies (5.4) for c < 0.

Consider σ ∈ (σ -, σ + ) and let (c σ , φ σ ) be a solution of (1.19), namely

(5.8)

     c σ φ ′ σ (z) = F (φ σ (z + r 0 ), φ σ (z + r 1 ), ..., φ σ (z + r N )) + σ on R. φ σ is non-decreasing over R φ σ (-∞) = m σ and φ σ (+∞) = m σ + 1.
As in the proof of Proposition 5.4, there exists some constant M > 0 independent on σ such that

|F (φ σ (z + r 0 ), φ σ (z + r 1 ), ..., φ σ (z + r N )) + σ + | ≤ M for all σ ∈ (σ -, σ + ).
Moreover, up to translate φ σ , we can assume that (because m σ → 0 as σ → σ + ) (5.9)

(φ σ ) * (0) ≤ 1 2 ≤ φ * σ (0).
Step 1: passing to the limit σ → σ + Applying Lemma 4.2, we deduce that there exists some function φ = φ + which satisfies, in viscosity sense

(5.10)

     c + (φ) ′ (z) = F (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) + σ + on R. φ is non-decreasing over R 0 = m σ + ≤ φ ≤ m σ + + 1 = 1.
Step 2: limits of the profile φ Passing to the limit in (5.9), we get In order to prove Proposition 6.1, we will need the following preliminary result that is proved in [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF]. Let F be a given function satisfying assumption ( ÃC 1 ), p > 0 and σ ∈ R. There exists a unique λ(σ, p) = λ p (σ) such that there exists a locally bounded function h p : R → R satisfying (in the viscosity sense):

0 ≤ φ(-∞) ≤ φ * (0) ≤ 1 2 ≤ (φ) * (0) ≤ φ(+∞) = 1.
(6.1)            λ p h ′ p (z) = F ((h p (z + pr i )) i=0,...,N ) + σ on R h p (z + 1) = h p (z) + 1 h ′ p (y) ≥ 0 |h p (z + z ′ ) -h p (z) -z ′ | ≤ 1 for any z, z ′ ∈ R.
Moreover, there exists a constant K > 0, independent on p and σ, such that

(6.2) |λ p -σ| ≤ K(1 + p)
and the function

λ p : R → R σ → λ p (σ)
is continuous with λ p (±∞) = ±∞.

For the proof of Lemma 6.2, we refer the reader to [16, Theorems 1.5 and 1.6]. However, proving that λ p (±∞) = ±∞ follows from (6.2).

Corollary 6.3 (Existence of φ p )

Let F be a given function satisfying assumption ( ÃC 1 ), p > 0 and c ∈ (c + , +∞) fixed. Then there exists σ = σ(c, p) ∈ R such that there exists a function φ p : R → R that satisfies in the viscosity sense:

(6.3)          cφ ′ p (z) = F ((φ p (z + r i )) i=0,...,N ) + σ(c, p) on R φ ′ p non-decreasing φ p z + 1 p = φ p (z) + 1.
Proof of Corollary 6.3 Let σ = σ(c, p) such that (6.4) λ p (σ) = cp and define the function φ p as:

(6.5) φ p (x) = h p (px),
where h p is given by Lemma 6.2. This gives the result. Now, we give the proof of Proposition 6.1.

Proof of Proposition 6.1 Choose c > c + and let δ 0 > 0 such that c > c + + δ 0 .

Step 1: preliminary Choose η > 0 small and let σ +η ≤ σ η < σ + . From Proposition 4.5, we know that for σ η , there exits a solution (c ση , φ ση ) of (1. [START_REF] Guo | Front propagation for discrete periodic monostable equations[END_REF]) such that

c ση ≤ c + .
Moreover, as c ση = lim p→0 c(σ η , p) with c(σ η , p) = λ(σ η , p) p (see the proof of existence of [1, Proposition 2.3]), then there exists p η such that for all 0 < p ≤ p η , we have

(6.6) |c(σ η , p) -c ση | ≤ δ 0 .
Thus, for 0 < p ≤ p η , we get

(6.7) c(σ η , p) ≤ c ση + δ 0 ≤ c + + δ 0 < c.
Moreover, since the map σ → λ(σ, p) = c(σ, p)p is continuous with λ(±∞, p) = ±∞ (see Lemma 6.2), then for such 0 < p ≤ p η , there exists σ p ∈ R and a function φ p : R → R (see Corollary 6.3) such that c(σ p , p) = c and (c, φ p ) solves (6.3). Hence from (6.7), we get c(σ η , p) < c(σ p , p).

In addition, since λ(σ, p) is non-decreasing with respect to σ, then (6.8)

σ p > σ η ≥ σ + -η for 0 < p ≤ p η .
Step 2: passing to the limit p → 0 Since lim p→0 λ(σ p , p) = lim p→0 cp = 0, we deduce from (6.2) that there exists some L 0 > 0 independent of p such that (6.9)

|σ p | ≤ L 0 for 0 < p ≤ p η .
Thus σ p → σ 0 as p → 0 (up to a subsequence).

Recall that φ p is non-decreasing and that

φ p x + 1 2p -φ p x + -1 2p = 1.
We can also assume that

     (φ p ) * (0) ≥ 1 2 (φ p ) * (0) ≤ 1 2 .
Therefore, since F ∈ Lip(R N +1 ) and due to (6.9), we deduce (as in the proof of [1, Lemma 2.8]) that there exists some M > 0 independent on n such that

|F ((φ p (x + r i ) i=0,...,N )) + σ n | ≤ M.
Applying arguments similar to the ones of the proof of Lemma 4.2, we see that φ p converges to some φ almost everywhere and φ is a viscosity solution of (6.10)

     cφ ′ (x) = F ((φ(x + r i )) i=0,...,N ) + σ 0 φ non-decreasing and bounded φ(+∞) -φ(-∞) ≤ 1,
and φ satisfies

     φ * (0) ≥ 1 2 φ * (0) ≤ 1 2 .
In addition, we have σ 0 ≥ σ +η (because of (6.8)).

But η > 0 is arbitrary, hence

σ 0 ≥ σ + .
Moreover, since σ 0 ≤ σ + (otherwise, (6.10) admits no solution, see Remark 1.10), thus

σ 0 = σ + .
Finally, since φ(±∞) solves f + σ + = 0, then we conclude that φ(-∞) = m σ + = 0 and φ(+∞) = 1, which ends the proof. Proof of Lemma 6.4 Let σ = σ + and (c, φ) be a solution of (1.20). We want to prove that c ≥ c + (similarly, we show that there is no solution of (1.21) for c > c -when σ = σ -).

It is known from Theorem 1.7-1, that for every σ ∈ (σ -, σ + ), there exists (c(σ), φ σ ) solution of (1.19). Let σ n ∈ (σ -, σ + ) be a sequence such that σ n → σ + , c(σ n ) → c + and (c(σ n ), φ σn ) is a solution of (1.19). Since σ n < σ + , Proposition 5.1 implies that c(σ n ) ≤ c. Therefore, passing to the limit σ n → σ + , we get that c + ≤ c, which ends the proof.

Lemma 6.5 (Strict inequality between threshold velocities)

Consider a function F satisfying ( ÃC 1 ), ( BC 1 ) and let c -, c + given by Corollary 5.7. If c -= 0 or c + = 0, then c -< c + .

Proof of Lemma 6.5 This is a straightforward consequence of (5.4).

Proof of Theorem 1.7 Theorem 1.7 is proved in several propositions and lemmata. In Propositions 4.5, 6.1, 5.4 and Lemma 5.5, we prove, for σ ∈ (σ -, σ + ), the existence of traveling waves and the monotonicity and the continuity of the velocity of propagation respectively. Existence of vertical branches of solutions (when σ = σ ± ) is proved in Lemma 3.6, where we show the existence of traveling waves for c >> 1 and for c << -1; and in Corollary 5.7, Lemma 5.8, Proposition 6.1, Lemma 6.4 and Lemma 6.5, where we respectively show the existence of finite critical limits c ± of the velocity function when σ goes to σ ± , the existence of solutions for the critical limits of velocity, fill the gap and prove the non-existence of solution when c < c + and σ = σ + or when c > c -and σ = σ -, and finally prove the inequality between c + and c -.

Part III

Definition and study of the critical velocity 7 Definition of the critical velocity: proof of Theorem 1.1

We devote this section to the proof of Theorem 1.1 and we split it into two subsections. We recall in a first subsection an extension result over R N +1 . For pedagogical reasons, we also prove in this subsection the result of Theorem 1.1 in a simple case where the non-linearity F is assumed to be smooth (cf. Proposition 7.2). we prove, in a second subsection, the existence of branch of solution using an approach different from that we use to prove Theorem 1.1 but under some addition assumptions (cf. Proposition 7.3). This result is less general then Theorem 1.1. In a third subsection, we give the proof of Theorem 1.1 in full generality for Lipschitz non-linearities F.

To prove the result (in any case), we first show the existence of traveling waves for c >> 1 by applying Proposition 3.5. The next step is to define the critical velocity c + and then we prove, for all c ≥ c + , the existence of traveling wave solutions of system (1.9). Finally, We show the non-existence of solutions of (1.9) for any c < c + .

Preliminary results

We start this subsection by recalling an extension result of the function F defined on [0, 1] N +1 into a function F over R N +1 . We also prove the result of Theorem 1.1 in a simple case.

Lemma 7.1 (Extension of F )

Consider a function F defined over [0, 1] N +1 and satisfying (A Lip ) such that F (0, ..., 0) = F (1, ..., 1) = 0. There exists an extension F defined over R N +1 such that Monotonicity: F (X 0 , ..., X N ) is non-decreasing w.r.t. each X i for i = 0.

F| [0,1] N +1 = F
Periodicity: F (X 0 + 1, ..., X N + 1) = F (X 0 , ..., X N ) for every X = (X 0 , ..., X N ) ∈ R N +1 . Lemma 7.1 corresponds to Lemma 2.1 in [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF] whose proof is given in the appendix A of [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF]. Notice that the function f (v) := F (v, ..., v) is nothing but a periodic extension of f on R with period 1, that is f|

[0,1] = f, hence f (0) = f (1) = 0. Notice also that φ is a solution of      cφ ′ (z) = F ((φ(z + r i )) i=0,...,N ) on R φ is non-decreasing over R φ(-∞) = 0 and φ(+∞) = 1
if and only if φ is a solution of (7.1)

     cφ ′ (z) = F ((φ(z + r i )) i=0,...,N ) on R φ is non-decreasing over R φ(-∞) = 0 and φ(+∞) = 1,
since F| [0,1] N +1 = F. In particular F satisfies (P Lip ) if F satisfies (P Lip ).

In order to prove Theorem 1.1 in a special case when F is smooth (see Proposition 7.2), we need to introduce precise assumptions.

Assumption (A C 1 ): Regularity: F ∈ C 1 ([0, 1] N +1 ).
Monotonicity: F (X 0 , ..., X N ) is non-decreasing w.r.t. each X i for i = 0.

Assumption (P ′ C 1 ): Positive degenerate monostability: a Let f (v) = F (v, ..., v) such that f (0) = 0 = f (1)
and f > 0 in (0, 1).

Smoothness near {0} N +1 and {1} N +1 : a There exists δ > 0 such that f ′ > 0 on (0, δ)

f ′ < 0 on (1 -δ, 1)
Proposition 7.2 (Vertical branch, simple case) Consider a function F satisfying (A C 1 ) and (P ′ C 1 ). Then the result of Theorem 1.1 holds true.

Proof of Proposition 7.2 Note that σ + = 0 in this case. Using Proposition 3.5, we deduce that for c >> 1, there exists a solution of (1.9). Next, from the extension lemma (Lemma 7.1), we see that if F satisfies (A C 1 ) (which implies (A Lip )), then the extended function F satisfies ( ÃC 1 ). Because of assumption (P ′ C 1 ) and f is 1-periodic with f = f on [0, 1], there exists ε 0 > 0 small enough, such that for -ε 0 < σ < 0, f + σ has a bistable shape over (m σ , m σ + 1) where m σ is defined exactly as in Theorem 1.7. Precisely, by bistable shape we mean that there exists m σ and b σ solutions of

f + σ = 0 satisfying -1 < m σ < 0 < b σ < m σ + 1 < 1 and      f + σ < 0 on (m σ , b σ ) f + σ > 0 on (b σ , m σ + 1) f ′ (b σ ) > 0 and f ′ (m σ ) = f ′ (m σ + 1) < 0.
For σ ∈ (-ε 0 , 0), using Proposition 4.5 (which stays true with ( BC 1 ) replaced by (P ′ C 1 ) and σ ∈ (-ε 0 , 0) instead of σ ∈ (σ -, σ + )), we show the existence of a unique velocity c σ such that there exists a profile φ σ solution of system (1.19) with F replaced by F . From Propositions 5.1 and 5.4 (which stay true similarly for ( BC 1 ) replaced by (P ′ C 1 ) and σ ∈ (-ε 0 , 0)), we get that the map σ → c σ is monotone continuous on (-ε 0 , 0) and we define as in Corollary 5.3 the critical velocity c + as lim

σ→0 - c σ = c + .
Again, up to replace ( BC 1 ) by (P ′ C 1 ) and σ ∈ (σ -, σ + ) by σ ∈ (-ε 0 , 0), we can use Lemma 5.6, Corollary 5.7 and Lemma 5.8, and show that c + < +∞ and that (1.9) admits a solution for c = c + . We use Proposition 6.1 (again with ( BC 1 ) replaced by (P ′ C 1 )) to fill the gap and get the existence of solution (c, φ) for each c ≥ c + . Finally, the non-existence of solutions for c < c + follows from Lemma 6.4 (with ( BC 1 ) replaced by (P ′ C 1 )).

Another approach of the proof of Theorem 1.1 under additional assumptions

We introduce in this subsection another proof for the existence of branch of solutions of (1.9) under some additional assumptions. This result is less general then the result of Theorem 1.1, but more general then Proposition 7.2.

Proposition 7.3 (Existence of branch of solutions under additional assumptions)

We work under the assumptions of Theorem 1.1. Let

(7.2) c + = inf E with E := {c ∈ R such that ∃ (c, φ) solution of (1.9)}.
Then c + > -∞ and c + ∈ E. Moreover, if F is increasing in X i + with r i + > 0 and c + = 0, then for every c ≥ c + there exists a solution of (1.9).

Remark 7.4 (The set E is nonempty) Proposition 3.5 implies directly that E = ∅. Moreover, from the definition of c + , we see that for all c < c + there is no solution of (1.9).

Proof of Proposition 7.3 Let c + be defined in (7.2). We first want to shoe that c + ∈ E.

Step 1: c + ∈ E Assume by contradiction that c + / ∈ E. From the definition of c + (see (7.2)), there exists a sequence c n ∈ E such that c n → c + and (c n , φ n ) is a solution of (1.9).

Case 1: c + = -∞ Set φ n (x) = φ n (|c n |x), then we have (7.3) -φ ′ n (y) = F φ n y + r i |c n | i=0,...,N .
Since φ n is invariant with respect to space translation and F is Lipschitz, we may assume that

φ n (0) = 1 2 .
Moreover, since F Lipschitz over [0, 1] N +1 , then we can show that there exists a constant M > 0 independent of c n such that |φ ′ n | ≤ M. Using Ascoli's Theorem, we pass to the limit c n → -∞ in (7.3) and we get that φ n converges (up to a subsequence) locally uniformly to φ which solves

-φ ′ = F (φ(y), ..., φ(y)) = f (φ(y))
and satisfies

φ(0) = 1 2 . But φ ′ ≥ 0 (since φ ′ c + ≥ 0), hence 0 ≥ -φ ′ (0) = f (φ(0)) = f 1 2 > 0.
Contradiction. Thus c + > -∞.

Case 2: c + > -∞
Case 2.1: c + = 0 If c + = 0, then passing to the limit using Ascoli's theorem as in Case 1, we can deduce that there exists a solution (c + , φ + ) of (1.9) and hence c + ∈ E.

Case 2.2: c + = 0 See Case 2 of the proof of Corollary 3.4.

Step 2: filling the gap Let c > c + , we want to prove the existence of a solution of (1.9) for c.

Step 2.1: c + = 0 Let φ + be a solution of 1.9 associated for c + . We first show that φ + > 0. We distinguish the following two cases:

Case 1: c + < 0 Assume that φ + (x 0 ) = 0 for some x 0 ∈ R. Since c + < 0, then using the strong maximum principle ([1, Lemma 6.1]) we get that φ + = 0 on [x 0 , +∞), which is a contradiction since φ + (+∞) = 1.

Case 2: c + > 0 Assume that φ + (x 0 ) = 0 for some x 0 ∈ R. Using again the strong maximum principle for c + > 0 ([1, Lemma 6.2]), which is based on the fact that F is increasing in X i + with r i + > 0, we get a similar contradiction.

Step 2.2: conclusion:

E = [c + , +∞) Since (c + , φ +
) is a solution of (1.9) with φ + > 0, then we deduce from Corollary 3.4 that there exists a solution of (1.9) for every c > c + . This implies that E = [c + , +∞).

Proof of Theorem 1.1

This subsection is devoted for the proof of Theorem 1.1.

Proof of Theorem 1.1

Let us consider a general function F : [0, 1] N +1 → R and f (v) = F (v, ..., v) satisfying (A Lip ) and (P Lip ). We have to adapt the proof of Proposition 7.2 with a much lower regularity of F (here F is only Lipschitz). To this end, we will introduce an approximation F δ of F.

Step 0: a δ-approximation Define for X = (X 0 , ..., X N ) ∈ [0, 1] N +1 and δ > 0 small

F δ (X) = F (X) -f (X 0 ) + f δ (X 0 ),
where

f δ (v) =      max f (δ) + L 0 (v -δ), 0 on [0, δ] max f (1 -δ) -L 0 (v -(1 -δ)), 0 on [1 -δ, 1] f on [δ, 1 -δ],
with a constant

L 0 > 0 satisfying L 0 > 2Lip(F ) =: 2L ∞ F . Clearly, we have F δ (v, ..., v) = f δ (v). Set (7.4)        b δ = δ - f (δ) L 0 > 0 m δ = 1 -δ + f (1 -δ) L 0 < 1 which satisfies (7.5) 0 < b δ < δ < 1 -δ < m δ < 1,
and

f δ (b δ ) = 0 = f δ (m δ ) and f δ > 0 on (b δ , m δ ).
Let F and Fδ defined on R N +1 be the extension functions of F and F δ (which are defined on [0, 1] N +1 ) respectively constructed by Lemma 7.1. Define fδ (v) = Fδ (v, ..., v) and f (v) = F (v, ..., v), then fδ and f are 1-periodic with ( fδ )

| [0,1] = f δ and ( f ) | [0,1] = f. Moreover, since fδ ≤ f , we get that (7.6) Fδ ≤ F over R N +1 . Now, for σ < 0 small fixed (0 < -σ < min [δ,1-δ] f ), define 0 < b δ,σ < m δ,σ < 1 such that (7.7)      fδ + σ (b δ,σ ) = 0 = fδ + σ (m δ,σ ) = fδ + σ (m δ,σ -1) fδ + σ < 0 on (m δ,σ -1, b δ,σ ) fδ + σ > 0 on (b δ,σ , m δ,σ ). Notice that m δ,σ → m δ b δ,σ → b δ as σ → 0 -.
For simplicity, we will denote F , Fδ , f and fδ by F, F δ , f and f δ respectively.

Step 1: existence of a solution of the approximated non-linearity F δ From the definition of f δ , we see that (for 0 < -σ < min

[δ,1-δ] f ) (7.8) b δ < b δ,σ < δ.
Now, because of (7.8) and using the definition of F δ with the fact that F is L ∞ F -Lipschitz, then for X close to {b δ,σ } N +1 and ω > 0 small enough, we get that (7.9) ), we deduce that there exists a solution φ δ,σ that solves in viscosity sense (7.10)

F δ (X + (ω, ..., ω)) -F δ (X) = F (X + (ω, ..., ω))-F (X)-f (X 0 + ω)+f (X 0 )+f δ (X 0 + ω)-f δ (X 0 ) ≥ -2ωL ∞ F + ωL 0 = ω(-2L ∞ F + L 0 ) = ωη >
     c δ,σ φ ′ δ,σ (x) = F δ ((φ δ,σ (x + r i )) i=0,...,N ) + σ on R φ δ,σ is non-decreasing over R φ δ,σ (-∞) = m δ,σ -1 and φ δ,σ (+∞) = m δ,σ .
More precisely, we have used the fact that F δ (• + {m δ,σ -1} N +1 ) + σ satisfies (A Lip ) and (P Lip ) on [0, 1] N +1 with b defined by b δ,σ = b + m δ,σ -1, and Proposition 4.1 provides a profile φ : R → [0, 1] such that φ + m δ,σ -1 =: φ δ,σ .

Step 2: c δ,σ is non-decreasing in σ for δ fixed Here, this is a variant of the proof of Proposition 5.1. Let δ > 0 fixed,min

[δ,1-δ] f < σ 1 < σ 2 < 0
and set (c δ,σ 1 , φ δ,σ 1 ), (c δ,σ 2 , φ δ,σ 2 ) be the associated solutions of (7.10) for σ 1 and σ 2 respectively.

We have

m δ,σ 1 -1 < m δ,σ 2 -1 < m δ,σ 1 < m δ,σ 2 ;
that is φ δ,σ 1 (±∞) < φ δ,σ 2 (±∞), and φ δ,σ 1 (+∞) > φ δ,σ 2 (-∞). Thus using the proof of Proposition 5.1, we deduce that c δ,σ 1 ≤ c δ,σ 2 .

Step 3: c δ,σ is non-increasing in δ for σ fixed For δ 2 > δ 1 > 0, fix σ such thatmin

[δ 1 ,1-δ 1 ]
f < σ < 0 and associate respectively the two solutions (c δ 2 ,σ , φ δ 2 ,σ ) and (c δ 1 ,σ , φ δ 1 ,σ ) of (7.10). From the definition of F δ , m δ,σ and b δ,σ (see Step 0), we see that

F δ 2 ≤ F δ 1 , hence (c δ 2 ,σ , φ δ 2 ,σ
) is a subsolution of (7.10) for F δ replaced by F δ 1 . Moreover, we also have that

m δ 2 ,σ -1 < m δ 1 ,σ -1 < m δ 2 ,σ < m δ 1 ,σ , hence φ δ 2 ,σ (±∞) < φ δ 1 ,σ (±∞) and φ δ 2 ,σ (+∞) > φ δ 1 ,σ (-∞).
Using the proof of Proposition 5.1 (which is still true for sub and supersolutions), we deduce that c δ 2 ,σ ≤ c δ 1 ,σ .

Step 4: passing to the limit σ → 0 -= σ + For δ > 0 fixed, let (c δ,σ , φ δ,σ ) be a solution of (7.10). Since F δ ≤ F (see Step 0), we deduce that (c δ,σ , φ δ,σ ) is a subsolution for (7.10), with F δ replaced by F.

On the other hand, let us consider any solution φ c 0 of (7.11)

     c 0 φ ′ c 0 (x) = F ((φ c 0 (x + r i )) i=0,...,N ) on R φ c 0 is non-decreasing over R φ c 0 (-∞) = 0 and φ c 0 (+∞) = 1.
From Proposition 3.5, we know that such a solution does exist at least for c 0 >> 1.

Since φ δ,σ satisfies

φ δ,σ (-∞) = m δ,σ -1 and φ δ,σ (+∞) = m δ,σ ,
then φ δ,σ (±∞) < φ c 0 (±∞) and φ δ,σ (+∞) > φ c 0 (-∞). Thus using the proof of Proposition 5.1 (which is still true for sub and supersolutions), we deduce that

c δ,σ ≤ c 0 for all σ ∈ (-min [δ,1-δ] f, 0).
Since the map σ → c δ,σ is non-decreasing, then

c δ,σ → c + δ as σ → 0 -.
Therefore, passing to the limit σ → 0 -, using Lemma 4.2, φ δ,σ converges almost everywhere to some φ δ that solves in the viscosity sense (7.12)

     c + δ φ ′ δ (x) = F δ ((φ δ (x + r i )) i=0,...,N ) on R φ δ is non-decreasing over R m δ -1 ≤ φ δ (-∞) and φ δ (+∞) ≤ m δ .
We can insure that φ δ is non constant, assuming that

     (φ δ,σ ) * (0) ≥ b δ + m δ 2 (φ δ,σ ) * (0) ≤ b δ + m δ 2 ,
and this implies in addition that φ δ (-∞) ≤ b δ and φ δ (+∞) = m δ .

Step 5: passing to the limit δ → 0 + Since c δ,σ ≤ c 0 for any δ > 0 and σ ∈ (-min

[δ,1-δ] f, 0), we get (7.13) c + δ ≤ c 0 for all δ ∈ 0, 1 2 .
Moreover, since c δ,σ is non-increasing in δ, then c + δ is non-increasing in δ. Hence from (7.13), we get (7.14) lim

δ→0 + c + δ = c + ≤ c 0 .
We can also assume, up to translation, that the solution φ δ of (7.12) satisfies

     (φ δ ) * (0) ≥ 1 2 (φ δ ) * (0) ≤ 1 2 .
Thus passing to the limit δ → 0 + , using again Lemma 4.2, then φ δ converges, up to a subsequence, almost everywhere to some φ which solves in viscosity sense (7.15)

     c + φ ′ (x) = F ((φ(x + r i )) i=0,...,N ) on R φ is non-decreasing over R 0 ≤ φ(-∞) and φ(+∞) ≤ 1
and satisfies (7.16)

     (φ) * (0) ≥ 1 2 (φ) * (0) ≤ 1 2 .
But φ(±∞) is a solution of f = 0, then we get φ(-∞) = 0 and φ(+∞) = 1.

This implies that if (c 0 , φ c 0 ) is a solution of (7.11), then c 0 ≥ c + and moreover there exists such a solution (c 0 , φ c 0 ) = (c + , φ). We also recall that we have solutions of (7.11) for c 0 >> 1. Our goal now is to fill the gap and to show that we have solutions for all c ≥ c + .

Step 6: filling the gap This step is analogous to the proof of Proposition 6.1. Fix c > c + and let β 0 > 0 such that (7.17) c > c + + β 0 .

Step 6.1 construction of a solution (c, φ) associated to some σ Substep 6.1.1:

c + = lim δ→0 - c + δ
We know from Steps 4, 5 that there exists a non trivial solution (c + δ , φ δ ) of (7.12) and that c + = lim δ→0 - c + δ . Thus there exists some δ 0 > 0 such that (7.18)

|c + δ -c + | ≤ β 0 3 for all 0 < δ ≤ δ 0 . Substep 6.1.2: c + δ = lim σ→0 - c δ,σ
Similarly, we know from Steps 1, 4 that, for every 0 < δ ≤ δ 0 , there exists a solution (c δ,σ , φ δ,σ ) of (7.10) and that c + δ = lim σ→0 - c δ,σ . Thus there exists some σ δ > 0 such that

(7.19) |c δ,σ -c + δ | ≤ β 0 3 for all 0 < -σ ≤ σ δ . Substep 6.1.3: c δ,σ = lim p→0 + c δ,σ,p
Based on the proof of [1, Proposition 2.3], there exists (for every 0 < δ ≤ δ 0 and 0 < -σ ≤ σ δ such that (7.19) holds true) a velocity c δ,σ,p , a profile φ δ,σ,p and some p δ,σ > 0 such that c δ,σ,p converges up to a subsequence to c δ,σ as p → 0 and

(7.20) |c δ,σ,p -c δ,σ | ≤ β 0 3
for all p of the subsequence such that 0 < p ≤ p δ,σ , where (c δ,σ,p , φ δ,σ,p ) is a solution of (7.21)

         c δ,σ,p (φ δ,σ,p ) ′ (x) = F δ ((φ δ,σ,p (x + r i )) i=0,...,N ) + σ on R (φ δ,σ,p ) ′ ≥ 0 φ δ,σ,p x + 1 p = 1 + φ δ,σ,p (x).
           cφ ′ (x) = F δ ((φ(x + r i )) i=0,...,N ) + σ on R φ ′ ≥ 0 φ x + 1 p = 1 + φ(x).
Substep 6.1.5: consequence of Substeps 6.1.1-6.1.4 For every 0 < δ ≤ δ 0 , 0 < -σ ≤ σ δ and 0 < p ≤ p δ,σ , (7.17), (7.18), (7.19) and (7.20) hold true, thus we get c δ,σ,p ≤ c + + β 0 < c = c δ,σ,p .

But the map σ → c δ,σ,p is non-decreasing (see Lemma 6.2 and (7.22)), hence we obtain that (7.24) σ < σ = σ δ,p .

Step 6.2: getting a profile for the original problem with velocity c Substep 6.2.0: a priori estimate on σ The couple (c, φ δ,σ,p ) is a solution of (7.23), thus for p < 1, we get

φ δ,σ,p (x + 1) -φ δ,σ,p (x) ≤ 1;
and hence we can show that there exists a constant M 0 independent of p and δ such that

|F δ | ≤ M 0 .
Thus integrating (7.23) over [0, 1], implies that there exists a constant K > 0 such that |σ| ≤ K for all δ < 1 2 and p ≤ 1.

Substep 6.2.1: passing to the limit p → 0 Since |f δf | ≤ o δ (1), then we can assume, up to translation, that

(7.25) (φ δ,σ,p ) * (0) ≥ γ δ,σ (φ δ,σ,p ) * (0) ≤ γ δ,σ with |f δ (γ δ,σ ) + σ| ≥ 1 4 osc(f ), with for instance γ δ,σ ∈ [m δ -1, m δ ].
Hence using the proof of Lemma 4.2 and the last equality of (7.23), we pass to the limit p → 0 and φ δ,σ,p converges up to subsequence to a non trivial (because of (7.25)) solution φ δ,σ δ,0 of (7.26)

     cφ ′ δ,σ δ,0 (z) = F δ ((φ δ,σ δ,0 (z + r i )) i=0,...,N ) + σ δ,0 on R φ δ,σ δ,0 is non-decreasing on R φ δ,σ δ,0 (+∞) -φ δ,σ δ,0 (-∞) ≤ 1, where σ δ,p → σ δ,0
and |σ δ,0 | ≤ K.

Substep 6.2.2: establishing σ δ,0 = 0 Since σ < σ δ,p (see (7.24)), then we get σ ≤ σ δ,0 . Thus passing to the limit σ → 0, we get σ δ,0 ≥ 0, without any change in equation (7.26). Moreover, since we have

0 = f δ (φ δ,σ δ,0 (±∞)) + σ δ,0
and f δ ≥ 0, then we get that σ δ,0 = 0.

Therefore, because of (7.25), φ δ := φ δ,σ δ,0 =0 satisfies (7.12) with c + δ replaced by c.

Substep 6.2.3: passing to the limit δ → 0 Up to translation, we assume that

     (φ δ ) * (0) ≥ b δ + m δ 2 (φ δ ) * (0) ≤ b δ + m δ 2 ,
Therefore, passing to the limit using once more Lemma 4.2, φ δ converges up to a subsequence to a solution φ of (7.15) and (7.16), with c + replaced by c. This φ is non trivial because of (7.16). Moreover, since φ(±∞) solves f = 0, we deduce that φ is a solution of (1.9) associated for the velocity c.

Step 7: no solution for c < c + This step is analogous to Lemma 6.4. Let (c, φ) be a solution of (1.9). Then as a solution of (7.11), we can choose (c 0 , φ 0 ) = (c, φ). Therefore, the choice c 0 = c in (7.14), implies that

c + ≤ c,
and then there is no a solution of (1.9) for c < c + .

For every ε > 0, define the function

ψ ε (x, t) := φ(t) - 1 ε (x -x 0 ) 2 . Then ψ ε (x 0 , t 0 ) = φ(t 0 ) = v * (t 0 ) = u * (x 0 , t 0 ).
Using the definition of ψ ε and (8.6), we deduce that for any r ε > 0 small enough such that [t 0r ε , t 0 + r ε ] ⊂ (0, T ), we have

     ψ ε (x 0 ± r ε , t) = φ(t) - r 2 ε ε ≤ v * (t) - r 2 ε ε = u * (x 0 , t) - r 2 ε ε < u * (x 0 , t) ψ ε (x, t 0 ± r ε ) = φ(t 0 ± r ε ) - 1 ε (x -x 0 ) 2 < v * (t 0 ± r ε ) = u * (x 0 , t 0 ± r ε )
Therefore, since u * is lower semi-continuous, then for every ε > 0 there exists c ε ≥ 0 such that

ψ ε -c ε ≤ u * on (x 0 -r ε , x 0 + r ε ) × (t 0 -r ε , t 0 + r ε ) = at P ε = (x ε , t ε ) ∈ (x 0 -r ε , x 0 + r ε ) × (t 0 -r ε , t 0 + r ε ),
with P ε = (x ε , t ε ) → (x 0 , t 0 ) when ε → 0 and r ε → 0. Now, since u satisfies (8.4) in the viscosity sense and ψ εc ε is a test function, then we deduce that (8.7)

φ t (t ε ) = (ψ ε ) t (P ε ) ≥ -Lu * (P ε ). This implies that φ t (t 0 ) ≥ -L lim inf ε→0 u * (P ε ) = -Lu * (x 0 , t 0 ) = -Lv * (t 0 ).
Thus v satisfies (8.5) in the viscosity sense and hence u(x 0 , •) satisfies (8.4) on (0, T ) in the viscosity sense.

Step 1.3: conclusion Let 0 ≤ s 0 < t 0 and set w(t) = e -L(t-s 0 ) v * (s 0 ) which is a solution of w t = -Lw. Because v * (s 0 ) ≥ w * (s 0 ), we deduce from the comparison principle that

(8.8) v(t) ≥ w(t) on [s 0 , T ).
In particular, evaluating (8.8) at t = t 0 , we get

0 = v(t 0 ) ≥ e -L(t 0 -s 0 ) v * (s 0 ), which implies that 0 ≥ v * (s 0 ) = v(s 0 ) = u(x 0 , s 0 ),
and this is true for any s 0 ∈ [0, t 0 ]. Because u ≥ 0, we deduce that u(x 0 , s) = 0 for all 0 ≤ s ≤ t 0 .

Step 2: u(x 0 + r i 0 , t 0 ) = 0 Note that for the test function φ ≡ 0, we have

u(x, t) ≥ φ(x, t) for all (x, t) ∈ R × (0, T ) u(x 0 , t 0 ) = φ(x 0 , t 0 ) for (x 0 , t 0 ) ∈ R × (0, T ).
Therefore, the supersolution viscosity inequality implies that

0 = φ t (x 0 , t 0 ) ≥ N i=0 ∂F ∂X i (0, ..., 0)u(x 0 + r i , t 0 ) ≥ ∂F ∂X 0 (0, ..., 0)u(x 0 , t 0 ) + ∂F ∂X i 0 (0, ..., 0)u(x 0 + r i 0 , t 0 ),
where we have used (8.3) and the fact that u ≥ 0. Because u(x 0 , t 0 ) = 0, we conclude that 0 ≥ ∂F ∂X i 0 (0, ..., 0)u(x 0 + r i 0 , t 0 ).

By assumption (8.1), we recall that ∂F ∂X i 0 (0, ..., 0) > 0. Therefore, since u ≥ 0, we deduce that u(x 0 + r i 0 , t 0 ) = 0.

Step 3: u(x 0 + kr i 0 , s) = 0 for k ∈ N and 0 ≤ s ≤ t 0 Since u(x 0 + r i 0 , t 0 ) = 0, then by Step 2, we deduce that u(x 0 + kr i 0 , t 0 ) = 0 for k ∈ N. Using Step 1, we get that u(x 0 + kr i 0 , s) = 0 for all 0 ≤ s ≤ t 0 and k ∈ N.

Now, we give a lower bound for a solution of the nonlinear problem.

Lemma 8.2 (Existence of a solution for the nonlinear problem) Consider a function F satisfying ( ÃLip ), (P Lip ) and let ε ∈ (0, 1]. Then there exists ψ : R × (0, +∞) → R a viscosity solution of (8.9)

ψ t (x, t) = F ((ψ(x + r i , t)) i=0,...,N ) on R × (0, +∞)
with initial condition satisfying Proof of Lemma 8.2

The proof is done in steps.

Step 1: construction of ψ δ solution of (8.9) Let δ > 0 and define

H δ =        0 if x ≤ -δ x δ + 1 if x ∈ [-δ, 0] 1 if x ≥ 0
Then for every x ∈ R, we have H δ (x) is non-increasing as δ decreases to zero and we also have

H δ (x) ≥ H(x).
Since for any given δ > 0, the function H δ is bounded uniformly continuous, then using [16, Corollary 2.9], we deduce that for every δ > 0, there exists a unique continuous solution ψ δ of (8.9) satisfying (8.11)

ψ δ (x, 0) = εH δ (x).
Step 2: properties of ψ δ Since H δ (x) is non-increasing when δ decreases to zero and H δ (x) ≥ 0, then using the comparison principle (see [16, Proposition 2.5]), we deduce that ψ δ is non-increasing as δ decreases to zero and ψ δ (x, t) ≥ 0 for all (x, t) ∈ R × (0, +∞). Moreover, since H δ (x + h) ≥ H δ (x) for every h ≥ 0 and δ > 0 fixed, then by comparison principle ([16, Proposition 2.5]), we deduce that

ψ δ (x + h, t) ≥ ψ δ (x, t),
i.e. ψ δ is non-decreasing w.r.t. x. Also, since 0 and 1 are two solutions of (8.9) and 0 ≤ εH δ ≤ 1, then from the comparison principle we get that

0 ≤ ψ δ ≤ 1. Now, let C 0 = sup [0,1] N +1 |F | and for h ≥ 0, we set ψ ± δ (x, t) := ψ δ (x, h) ± C 0 t for t ≥ 0. Then ψ + δ
is a supersolution and ψ - δ is a subsolution of (8.9) with

ψ - δ (x, 0) ≤ ψ δ (x, h) ≤ ψ + δ (x, 0).
Hence, using the comparison principle, we get for all t ≥ 0 (8.12)

ψ - δ (x, t) ≤ ψ δ (x, h + t) ≤ ψ + δ (x, t), i.e. ψ δ (x, h) -C 0 t ≤ ψ δ (x, h + t) ≤ ψ δ (x, h) + C 0 t.
Because this true for any t, h ≥ 0, we deduce that (8.13)

|ψ δ (x, t) -ψ δ (x, s)| ≤ C 0 |t -s| for all x ∈ R, t, s ∈ [0, +∞).
Step 3: the limit δ → 0 Since ψ δ is non-increasing as δ decreases to zero and ψ δ (x, t) ≥ 0 for all (x, t) ∈ R × (0, +∞). Then ψ + δ converges pointwisely to some function ψ ≥ 0, as δ → 0. Using the stability of viscosity solutions (Proposition 2.2 (ii), applied for sup -ψ δ ), we deduce that ψ * is a supersolution of (8.9). Moreover, since ψ δ is non-decreasing w.r.t. x and satisfies (8.13), then

ψ is non-decreasing w.r.t. x |ψ(x, t) -ψ(x, s)| ≤ C 0 |t -s| for all x ∈ R, t, s ∈ [0, +∞).
This implies that ψ * = lim sup δ→0 * ψ δ .

Hence, using Proposition 2.2 (i), we deduce that ψ * is a subsolution of (8.9). Therefore, ψ solves (8.9) in the viscosity sense.

In addition, since H η (x) ≥ H(x) ≥ H δ (xδ), for every η, δ > 0, then ψ η (x, t) ≥ ψ δ (xδ, t) for every η, δ > 0.

Passing to the limit η → 0, we obtain ψ(x, t) ≥ ψ δ (xδ, t) for every δ > 0, this implies that for every δ > 0, we have

(8.14) ψ δ (x, t) ≥ ψ(x, t) ≥ ψ δ (x -δ, t) for every (x, t) ∈ R × [0, +∞).
Moreover, we have ψ δ ∈ C 0 and

ψ δ (x, 0) = 0 = ψ δ (x -δ, 0) for x ≤ -δ ψ δ (x, 0) = ε = ψ δ (x -δ, 0) for x ≥ δ.
Hence, for every δ > 0, we get

ψ * (x, 0) = ψ * (x, 0) = 0 for x ≤ -δ ε for x ≥ δ.
Therefore, we obtain that

ψ * (x, 0) = ψ * (x, 0) = 0 for x < 0 ε for x > 0.
Using again (8.14), we get for (x, t) = (0, 0) that

ε ≥ ψ * (0, 0) ≥ ψ * (0, 0) ≥ 0.
Finally, since ψ * is upper semi-continuous and ψ * is lower semi-continuous, we deduce that ψ * (x, 0) = εH * (x) and ψ * (x, 0) = εH * (x).

Proposition 8.3 (Lower bound on a solution of the evolution nonlinear problem) Consider a function F satisfying ( ÃLip ) and (P Lip ). Assume moreover that F is C 1 over a neighborhood of {0} N +1 in [0, 1] N +1 and (8.15) ∃ i 0 ∈ {0, ..., N } such that r i 0 > 0 and ∂F ∂X i 0 (0, ..., 0) > 0.

Then there exists ε 0 ∈ (0, 1] and T 0 > 0 such that for all δ ∈ (0, T 0 ) and R > 0, there exists κ = κ(δ, R) > 0 such that for every 0 < ε ≤ ε 0 , the function ψ = ψ ε given by Lemma 8.2 with initial conditions (8.10) satisfies

(8.16) ψ ε (x, t) ≥ κε for all (x, t) ∈ [-R, R] × [δ, T 0 ].

Proof of Proposition 8.3

We first give an upper bound on the solution ψ of (8.9) and then we prove Proposition 8.3 by contradiction.

Step 0: upper bound on ψ on (0, 2T 0 ) Let M (t) := sup x∈R ψ(x, t).

Then M (0) = ε (since ψ * (x, 0) = εH * (x)). Since ψ is a solution of (8.9) then, using the viscosity techniques, we can show that M * is a subsolution, i.e. satisfies in the viscosity sense

v t (t) ≤ F (M * (t), ..., M * (t)) = f (M * (t)).
Using the comparison principle for the ODE x ′ = f (x), we deduce that (8.17)

M * (t) ≤ M 0 (t) over [0, ∞),
where M 0 is a solution of

M ′ 0 (t) = f (M 0 (t)) ≥ 0 for (0, +∞) M 0 (0) = ε. Now, because M 0 is non-decreasing, if M 0 (t) ≤ 2ε then M ′ 0 (t) ≤ sup [0,2ε] f ≤ 2L 1 ε,
where L 1 is the Lipschitz constant of f (because f (0) = 0). Thus we get

M 0 (t) ≤ ε + 2tL 1 ε < 2ε if t < 1 2L 1 .
Therefore for (8.18)

T 0 = 1 4L 1 , we get M * (t) ≤ M 0 (t) ≤ 2ε on [0, 2T 0 ], which implies that ψ ε = ψ satisfies (8.19) ψ ε (x, t) ≤ 2ε for (x, t) ∈ R × [0, 2T 0 ].
Step 1: establishing (8.16) Assume to that contrary that (8.16) is false. Then there exist δ ∈ (0, T 0 ) (with T 0 given in (8.18)), R > 0 and two sequences ε n → 0, κ n → 0 as n → +∞ and points (8.20)

P n = (x n , t n ) ∈ [-R, R] × [δ, T 0 ] such that ψ εn (P n ) ≤ κ n ε n . Define ψ n (x, t) := 1 ε n ψ εn (x, t) for all (x, t) ∈ R × (0, 2T 0 ).
Then we have (using

(8.19)),      0 ≤ ψ n ≤ 2 over R × [0, 2T 0 ) ψ n (P n ) ≤ κ n → 0 (ψ n ) * (x, t = 0) = H * (x) and (8.21) (ψ n ) t (x, t) = 1 ε n F (ε n (ψ n (x + r i , t)) i=0,...,N ).
Step 1.1: uniform lower bound of ψ n Denote by Z = (ψ n (x + r i , t)) i=0,...,N . Since F is C 1 over a neighborhood of {0} N +1 , then for ε n small enough, we can show that

(ψ n ) t (x, t) = 1 ε n F (ε n (ψ n (x + r i , t)) i=0,...,N ) = 1 0 ∂F ∂X 0 (sε n Z)ψ n (x, t)ds + N i=1 1 0 ∂F ∂X i (sε n Z)ψ n (x + r i , t)ds ≥ -Lψ n (x, t) + 1 2 ∂F ∂X i 0 (0, ..., 0)ψ n (x + r i 0 , t),
where we have used the fact that ψ n ≥ 0 and ∂F ∂X i ≥ 0 for all i = 0. Hence ψ n is a supersolution of the equation (8.22) w t (x, t) = -Lw(x, t) + 1 2 ∂F ∂X i 0 (0, ..., 0)w(x + r i 0 , t).

Now, let

H η (x) =          0 if x < 0 1 η x if 0 ≤ x ≤ η 1 if x ≥ 1
for η > 0 small. Since ∂F ∂X i 0 (0, ..., 0) ≥ 0, then by a simple calculation, we can show that the function

φ(x, t) = e -Lt H η (x)
(with L the Lipschitz constant of F ) is a subsolution of (8.22). Moreover, we have

(ψ n ) * (x, t = 0) = H * (x) ≥ H η (x) = φ(x, t = 0).
Therefore, using a comparison principle for (8.22), we deduce that (8.23) e -Lt H η (x) ≤ ψ n (x, t) for all (x, t) ∈ R × [0, 2T 0 ).

Step 1.2: passing to the limit and getting a contradiction Since ψ n (x, t) is uniformly bounded on R × [0, 2T 0 ) and

(ψ n ) t (x, t) ≥ N i=0 1 0 ∂F ∂X i (sε n Z)ψ n (x + r i , t)ds,
then using the fact that F is C 1 over a neighborhood of {0} N +1 and ε n → 0, we deduce that

ψ ∞ = lim inf n→+∞ * ψ n satisfies in the viscosity sense on R × [0, 2T 0 )        (ψ ∞ ) t (x, t) ≥ N i=0 ∂F ∂X i (0, ..., 0)ψ ∞ (x + r i , t) 0 ≤ ψ ∞ ≤ 2 and (8.24) e -Lt H η (x) ≤ ψ ∞ (x, t) for all (x, t) ∈ R × [0, 2T 0 ).
In addition, we also have

P n → P ∞ = (x ∞ , t ∞ ) in [-R, R] × [δ, T 0 ] , hence using the fact that ψ n (P n ) → 0, we get ψ ∞ (P ∞ ) = 0.
Using the strong maximum principle (Proposition 8.1) that holds for supersolutions, we deduce for k ∈ N that ψ ∞ (x ∞ + kr i 0 , t) = 0 for all 0 ≤ t ≤ t ∞ .

But r i 0 > 0, hence for t = 0, k >> 1 and using (8.24), we get

1 = H η (x ∞ + kr i 0 ) ≤ ψ ∞ (x ∞ + kr i 0 , 0) = 0,
which is a contradiction.

In the following proposition, we give a Harnack type inequality.

In particular, we have u(x, 0) ≥ v * (x, 0) for all x ∈ R.

Using the comparison principle (see [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF]Proposition 2.5]), we deduce that (8.33) u ≥ v for all (x, t) ∈ R × (0, +∞).

From (8.32), we deduce that

(8.34) u ≥ κ 1 u(x 0 ) on [x 0 -R, x 0 + R] × [δ, T 0 ],
with κ 1 = ε 0 κ (using ε ∈ (0, 1], u(x 0 ) ∈ (0, 1] and the definition of ε). Because u(x, t) = u(x + ct), we conclude that inf

(x,t)∈[x 0 -R,x 0 +R]×[δ,T 0 ] u(x + ct) ≥ κ 1 u(x 0 ).
Now, for any r > 0, we can find R r > 0 large enough such that B r (x 0 ) ⊂ B Rr (x 0 ) + c[δ, T 0 ]. Therefore, since u is continuous and non-decreasing, then (8.35) u(x 0r) = inf

x∈Br(x 0 ) u(x) ≥ inf (x,t)∈[x 0 -R,x 0 +R]×[δ,T 0 ] u(x + ct) ≥ κ 1 u(x 0 )
with κ 1 = κ 1 (r). Let ρ = r 2 and choose y 0 such that B ρ (y 0 ) = (x 0r, x 0 ), i.e. y 0ρ = x 0r and y 0 + ρ = x 0 . Thus, using again the fact that u is non-decreasing, we get sup

Bρ(y 0 ) u = u(y 0 + ρ) = u(x 0 ) and u(x 0 -r) = u(y 0 -ρ) = inf Bρ(y 0 )
u.

Therefore, we deduce from (8.35) that (8.36) sup

Bρ(y 0 ) u ≤ κ 1 inf Bρ(y 0 ) u with κ 1 = 1 κ 1 .
Using (8.36) for 2ρ ≥ r * and κ 0 = κ 1 (r * ) = ε 0 κ(δ, R r * ) -1 , setting z 0 = y 0ρ and using the monotonicity of u, we get

(8.37) u(z 0 + r * ) ≤ u(z 0 + 2ρ) = u(y 0 + ρ) = sup Bρ(y 0 ) u ≤ κ 0 inf Bρ(y 0 ) u = κ 0 u(y 0 -ρ) = κ 0 u(z 0 ).
Finally, since x 0 is chosen arbitrary at the beginning of the reasoning, we deduce that (8.36) and (8.37) do hold for any y 0 , z 0 . This shows (8.27) and (8.28), and ends the proof.

Properties of the critical velocity

In a first subsection, we prove that c + ≥ c * , precisely Proposition 1.5. We also show, if F satisfies the KPP condition (1.16), that c * ≥ c + (see Proposition 1.6). In this subsection, we also give an example where c + > c * (Lemma 9.3). We prove in a second subsection Proposition 1.3 which asserts that the critical velocity satisfies c + ≥ 0 under additional assumptions. In a third subsection, we give an example (Proposition 1.4) that shows that we can have c + < 0 when the additional assumptions are not satisfied. We also prove the instability of the critical velocity, namely Proposition 1.2.

Lower bound for c +

In this subsection, we prove a lower bound for the critical velocity c + given in Theorem 1.1. Precisely, we show in Proposition 1.5 that c + ≥ c * . In Lemma 9.3, we give an example where c + > c * . In this subsection, we also prove that and c * ≥ c + under a KPP condition (see Proposition 1.6).

We start with the proof of Proposition 1.5

Proof of Proposition 1.5 Under assumptions (A Lip ) and (P Lip ), let c + given by Theorem 1.1. We want to show that c + ≥ c * with c * given in (1.15).

Part I: proving that c + ≥ c * under the assumption (1.14) Let c ≥ c + such that c = 0 and let us prove that c ≥ c * . Associate for c a profile φ such that (c, φ) is a solution of (1.9) (this is always possible since c ≥ c + , see Theorem 1.1).

Step I.1: φ ′ (x) φ(x) is globally bounded From Harnack inequality (8.27), we deduce that if φ(x 0 ) = 0 at some point x 0 ∈ R, then φ ≡ 0 which is impossible for a solution of (1.9). Therefore φ > 0.

We have

c φ ′ (x) φ(x) = 1 φ(x)
F ((φ(x + r i )) i=0,...,N ).

We also know, using the monotonicity of F w.r.t. X i for i = 0 and F (0, ..., 0) = 0, that

F (φ(x), φ(x + r 1 ), ..., φ(x + r N )) = F (φ(x), φ(x + r 1 ), ..., φ(x + r N )) -F (0, ..., 0) ≤ F (φ(x), φ(x + r * ), ..., φ(x + r * )) -F (0, ..., 0
≤ φ ′ (x) φ(x) ≤ 1 |c| L 1 φ(x + r * ) φ(x) .
From Proposition 8.4, we know that there exists a constant κ 0 > 1 such that (9.1) φ(x + r * ) ≤ κ 0 φ(x), therefore, we deduce that

(9.2) 0 ≤ φ ′ (x) φ(x) ≤ M := κ 0 L 1 |c| .
Step I.2: proving that c ≥ c * Since φ satisfies (9.2), then lim sup

x→-∞ φ ′ (x) φ(x) = λ exists and λ = lim n→+∞ φ ′ (x n ) φ(x n ) for some x n → -∞ as n → +∞. Let φ n (x) := φ(x + x n ) φ(x n ) ≥ 0, then φ n (0) = 1 and φ n satisfies (9.3) cφ ′ n (x) = 1 φ(x n )
F ((φ(x + x n + r i )) i=0,...,N ) on R. Now, since for all i, φ(x + x n + r i ) → 0 as n → +∞, F (0, ..., 0) = 0 and F is C 1 over a neighborhood of {0} N +1 in [0, 1] N +1 , then we see that we can write for n large enough

(9.4) cφ ′ n (x) = N i=0 1 0 ∂F ∂X i (sφ(x + x n + r i ))φ n (x + r i )ds on R.
From (8.28), we deduce that for k ∈ N\{0}, we have

φ(x n + kr * ) ≤ (κ 0 ) k φ(x n ) and φ(x + r * ) ≤ κ 0 φ(x),
with κ 0 > 1. Hence for x ∈ [(k -1)r * , kr * ], we get

(9.5) 0 ≤ φ n (x) = φ(x + x n ) φ(x n ) ≤ (κ 0 ) k ≤ (κ 0 ) x r * +1 ≤ κ 0 e µx with µ = ln κ 0 r * .
This implies that 0 ≤ φ n (x) ≤ κ(x) := κ 0 e µx + .

From (9.2), we have

0 ≤ φ ′ n φ n ≤ M, which implies that (9.6) 0 ≤ φ ′ n (x) ≤ Mκ(x).
Therefore, using Ascoli's Theorem and the extraction diagonal argument, we deduce that φ n converges locally uniformly to some φ ∞ which satisfies (in the viscosity sense) (9.7)

                 cφ ′ ∞ (x) = N i=0 ∂F ∂X i (0, ..., 0)φ ∞ (x + r i ) on R φ ′ ∞ ≥ 0 φ ∞ (0) = 1 φ ∞ (x + r * ) ≤ κ 0 φ ∞ (x).
Therefore, using Lemma 9.2 below (with a 0 = r * > 0), we deduce that (9.8) c ≥ c * .

Step I.3: conclusion (c + ≥ c * ) Since (9.8) holds true for any c ≥ c * with c = 0, we deduce that c + ≥ c * .

Part II: proving c + ≥ c * if c + < 0 Since c + < 0, we deduce from Proposition 1.3 ii) that there exists some r i 1 < 0. Let ε > 0 and define the function F ε (X 0 , ..., X N ) := F (X 0 , ..., X N ) + ε(X i 1 -X 0 ).

Using Theorem 1.1, there exists a critical velocity c + ε such that for every c ≥ c + ε there exists a solution of (1.9) with F replaced by F ε .

Let (c + , φ + ) be a solution of (1.9) given by Theorem 1.1. Since φ + is non-decreasing and r i 1 < 0 (φ + (x + r i 1 )φ + (x) ≤ 0), then

c + (φ + ) ′ (x) = F ((φ + (x + r i )) i=0,...,N ) ≥ F ε ((φ + (x + r i )) i=0,...,N ).
In addition, if φ + (x 0 ) = 0 for some x 0 ∈ R, then using the strong maximum principle ([1, Lemma 6.1]), we deduce that φ + = 0 on [x 0 , +∞) because c + < 0, which is a contradiction since φ + (+∞) = 1. Therefore, (c + , φ + ) is a supersolution of (1.9) for F replaced by F ε with φ + > 0.

Using now Proposition 3.2, we deduce that there exists a solution (c + , φ) of (1.9) with F replaced by F ε . But c + ε is the minimal velocity associated to F ε , thus we deduce that

c + ≥ c + ε .
This implies in particular that c + ε < 0, and since

∂F ε ∂X i 1 (0, ..., 0) = ∂F ∂X i 1 (0, ..., 0) + ε > 0,
then by Remark 9.1 below, we get that (9.9)

c + ≥ c + ε ≥ c * ε .
However from (1.15), we have

c * ε = inf λ>0 P ε (λ) λ with P ε (λ) = N i=0 ∂F ε ∂X i (0, ..., 0)e λr i = N i=0
∂F ∂X i (0, ..., 0)e λr i + ε(e λr i 1 -1)

= P (λ) + ε(e λr i 1 -1).

Hence

c * ε = inf λ>0 P (λ) λ + ε λ (e λr i 1 -1) .
Thus passing to the limit ε → 0, we get that

lim ε→0 c * ε = inf λ>0 P (λ) λ = c * .
Therefore, we deduce from (9.9) that c + ≥ c * , which ends the proof. In order to see it, we can prove a lower bound (analogue to Proposition 8.3) with

ψ ε ≥ κε on [δ, R] × [δ, T 0 ]
for δ > 0 (this lower bound is obtained with a variant of the strong maximum principle, Proposition 8.1).

From this, we can deduce a Harnack inequality for solution of (8.26) with c < 0 (analogue to Proposition 8.4). Again using this Harnack inequality, we can conclude that c + ≥ c * as in the proof of Proposition 1.5. Lemma 9.2 (Lower bound for c + for linear problem) Let F be a function satisfying (A Lip ) and differentiable at {0} N +1 in [0, 1] N +1 . Assume moreover that F satisfies (1.14) and

(9.10) f ′ (0) = N i=0 ∂F ∂X i (0, ..., 0) > 0,
where we recall that f (v) = F (v, ..., v). Let c = 0 and assume that there exists a 0 > 0 and C 0 > 0 such that φ is a solution of (9.11)

                     cφ ′ (x) = N i=0 ∂F ∂X i (0, ..., 0)φ(x + r i ) on R φ ′ ≥ 0 φ > 0 1 ≤ φ(x + a 0 ) φ(x) ≤ C 0 for all x ∈ R. Then c ≥ c * ,
where c * is given in (1.15).

Proof of Lemma 9.2

Step 0: preliminary Let a ∈ (0, a 0 ) and let

K * = inf E with E = {k ≥ 1 such that kφ(x) ≥ φ(x + a) for all x ∈ R}.
We deduce from (9.11) that E = ∅ because C 0 ∈ E. By definition of K * , we have (9.12) K * φ(x) ≥ φ(x + a) for every x ∈ R.

We have K * ≥ 1. If K * = 1, then φ is constant and the first equation of (9.11) gives

0 = N i=0 ∂F ∂X i (0, ..., 0) = f ′ (0)
which is a contradiction with (9.10). Therefore K * > 1, and there exists λ > 0 such that (9.13)

K * = e λa .
Again by definition of K * , for every ε > 0, there exists x ε ∈ R such that

(9.14) (K * -ε)φ(x ε ) < φ(x ε + a). Let φ ε (x) := φ(x + x ε ) φ(x ε ) . Then φ ε (0) = 1, (9.15) K * φ ε (x) ≥ φ ε (x + a)
and (9.14) can be rewritten as

(9.16) (K * -ε)φ ε (0) < φ ε (a).
Step 1: passing to limit ε → 0 Since c = 0, we can bound both φ ε and φ ′ ε on any bounded interval uniformly w.r.t. ε (as in Step 2 of the proof of Proposition 1.5). Therefore, using Ascoli Theorem and the extraction diagonal argument, we deduce that φ ε converges to some φ 0 locally uniformly and φ 0 satisfies (in the viscosity sense) (9.17)

                     cφ ′ 0 (x) = N i=0 ∂F ∂X i (0, ..., 0)φ 0 (x + r i ) on R φ ′ 0 ≥ 0 φ 0 (0) = 1
K * φ 0 (0) ≤ φ 0 (a) (using (9.16)) K * φ 0 (x) ≥ φ 0 (x + a) (using (9.15)). Now, let w(x) = K * φ 0 (x)φ 0 (x + a). Then from (9.17), we deduce that w satisfies (9.18)

           cw ′ (x) = N i=0 ∂F ∂X i (0, ..., 0)w(x + r i ) on R w ≥ 0 on R w(0) = 0.
Then using the half strong maximum principle [1, Lemma 6.1], we get that w(x) = 0 for all cx ≤ 0, i.e.

(9.19) k * φ 0 (x) = φ 0 (x + a) for all cx ≤ 0.

Step 2: establishing c ≥ c * Let φ 0,n (x) := φ 0 (xcn) φ 0 (-cn) .

Then φ 0,n (0) = 1. Moreover, using (9.19), we have

K * φ 0 (x -cn) φ(-cn) = φ 0 (x -cn + a) φ 0 (-cn) for all c(x -cn) ≤ 0.
Hence (9.20) K * φ 0,n (x) = φ 0,n (x + a) for all cx ≤ c 2 n.

Step 2.1: passing to the limit n → +∞ As before, we can pass to the limit and show that φ 0,n → φ 0,∞ with (9.21)

           cφ ′ 0,∞ (x) = N i=0 ∂F ∂X i (0, ..., 0)φ 0,∞ (x + r i ) on R φ ′ 0,∞ ≥ 0 φ 0,∞ (0) = 1.
Moreover, passing to the limit in (9.20), we deduce that (9.22)

K * φ 0,∞ (x) = φ 0,∞ (x + a) for all x ∈ R.
Step 2.2: conclusion Let

z(x) = φ 0,∞ (x) e λx .
Recall that φ 0,∞ ∈ C 1 (because c = 0). Then z ∈ C 1 and satisfies

(9.23) cz ′ (x) + cλz(x) = N i=0 ∂F ∂X i (0, ..., 0)e λr i z(x + r i ) on R.
We also have

z(x + a) = φ 0,∞ (x + a) e λ(x+a) = K * φ 0,∞ (x)
e λa e λx = z(x), where we have used (9.22) and (9.13).

Because z is a-periodic (and continuous), there exists x 0 ∈ R such that z attain it's minimum at x 0 . We claim that z(x 0 ) = 0. Indeed, if z(x 0 ) = 0, then we deduce from (9.23) that N i=1 ∂F ∂X i (0, ..., 0)e λr i z(x 0 + r i ) = 0.

Since ∂F ∂X i (0, ..., 0) ≥ 0 for all i = 1, ..., N and F satisfies (1.14), we deduce that z(x 0 + r i 0 ) = 0.

Repeating the same process, we get that z = 0 on x 0 + r i 0 N. Since z is a-periodic, then z = 0 on x 0 + r i 0 N + aZ ≡ x 0 + a(

r i 0 a N + Z). Since a ∈ (0, a 0 ) is arbitrary, then we can choose a ∈ (0, a 0 ) such that r i 0 a ∈ R\Q. Therefore, x 0 + a( r i 0 a N + Z) is dense in R. This implies, since z is continuous, that z = 0 on R, which is a contradiction with z(0) = 1.
Therefore, z(x 0 ) = 0. Again, since z(x 0 ) = min z ≥ 0, then using (9.23), we get that cλz(x 0 ) = ∂F ∂X 0 (0, ..., 0)e λr 0 z(x 0 ) + N i=1 ∂F ∂X i (0, ..., 0)e λr i z(x 0 + r i )

≥ ∂F ∂X 0 (0, ..., 0)e λr 0 z(x 0 ) + N i=1 ∂F ∂X i (0, ..., 0)e λr i z(x 0 ) = z(x 0 ) N i=0 ∂F ∂X i (0, ..., 0)e λr i .
Using the fact that z(x 0 ) = 0, we deduce that

cλ ≥ N i=0 ∂F ∂X i (0, ..., 0)e λr i .
Recall that λ > 0. Therefore, we get

c ≥ P (λ) λ ≥ inf λ>0 P (λ) λ = c * ,
where P (λ) = N i=0 ∂F ∂X i (0, ..., 0)e λr i . This ends the proof. Now, we give the proof of Proposition 1.6, where we show that c + ≤ c * under a KPP type condition.

Proof of Proposition 1.6

The goal is to prove that for any real c > c * (c * < +∞), we have c + ≤ c.

For such c, we have c > c * = inf λ>0 P (λ)

λ , hence there exists some λ 0 > 0 such that c > P (λ 0 ) λ 0 .

This implies that φ(x) = e λ 0 x satisfies (9.24)

cφ ′ (x) > G((φ(x + r i )) i=0,...,N ),
where G(X) = N i=0 ∂F ∂X i (0, ..., 0)X i . Let F be the extension over R N +1 of F (given by Lemma 7.1).

The goal is now to construct a supersolution of (9.25) cw ′ (x) = F ((w(x + r i )) i=0,...,N ) on R.

Step 1: φ(x) := min(1, φ(x)) is a supersolution of (9.25) We recall that φ(0) = 1. Let x < 0, we have

φ(x + r i ) = φ(x + r i ) for r i ≤ 0 φ(x + r i ) ≤ φ(x + r i ) for r i > 0.
Since F is non-decreasing w.r.t. X i for i = 0, then G satisfies the same property, hence

G((φ(x + r i )) i=0,...,N ) ≥ G((φ(x + r i )) i=0,...,N ) ≥ F ((φ(x + r i )) i=0,...,N ),
where we have used (1.16) and the fact that 0 ≤ φ(x) ≤ 1. But φ(x) = φ(x) is a test function for x < 0 and φ satisfies (9.24), thus we get for x < 0 :

cφ ′ (x) = cφ ′ (x) > G((φ(x + r i )) i=0,...,N ) ≥ F ((φ(x + r i )) i=0,...,N ).
Similarly for x > 0, we have φ(x + r i ) ≤ 1 for r i < 0 φ(x + r i ) = 1 for r i ≥ 0.

Moreover, since φ(x) = 1 is a test function for x > 0, we get

cφ ′ (x) = 0 = F (1, ..., 1) ≥ F ((φ(x + r i )) i=0,...,N ).
Now for x = 0, we have φ(0) = 1 = φ(0) is a supersolution of (9.25) because there is no test function touching φ from below at x = 0 (see Definition 2.1). Finally, since 0 ≤ φ(x) ≤ 1, then F ((φ(x + r i )) i=0,...,N ) = F ((φ(x + r i )) i=0,...,N ) and hence φ is a supersolution of (9.25).

Step 2: subsolution of (9.25) Let (c + , φ + ) be a solution of (9.25) given by Theorem 1.1. We know, from the proof of Theorem 1.1 (see (7.10)), that

c + = lim δ→0 ( lim σ→0 - c δ,σ ) and φ + = lim δ→0 ( lim σ→0 - φ δ,σ )
where δ > 0, σ < 0 are small enough and (c δ,σ , φ δ,σ ) is a solution of (with Fδ = F δ )

c δ,σ φ ′ δ,σ (x) = F δ ((φ δ,σ (x + r i )) i=0,...,N ) + σ and φ δ,σ (-∞) = m δ,σ -1, φ δ,σ (+∞) = m δ,σ with m δ,σ -1 < 0 < m δ,σ < 1.
Since F δ = Fδ ≤ F (see (7.6)) and σ < 0, then we deduce that (c δ,σ , φ δ,σ ) is a subsolution of (9.25) with (c, w) is replaced by (c δ,σ , φ δ,σ ).

Step 3: establishing c + ≤ c * Using the proof of Proposition 5.1, we deduce that c δ,σ ≤ c.

Passing to the limit σ → 0 -and then δ → 0 (as in the proof of Theorem 1.1), we deduce that

0 : [0, 1] 3 → R defined as F 0 (X 0 , X -1 , X 1 ) := g(X 1 ) + g(X -1 ) -2g(X 0 ) + f (X 0 ), with r 0 = 0, r ±1 = ±1 and f, g : [0, 1] → R are C 1 over a neighborhood of 0, Lipschitz on [0, 1] and satisfying      f (0) = f (1) = 0 f > 0 on (0, 1) f ′ (0) > 0 and      g ′ (0) = 0 g(1) = 1 + g(0) g ′ ≥ 0.
Let c + given by Theorem 1.1 (with F replaced by F 0 ), then

c + > c * ,
where c * is defined in (1.15).

An example of such g is g(x) = x -1 2π sin(2πx). Proof of Lemma 9.3 Since g ′ (0) = 0 and f ′ (0) > 0, then P (λ) = f ′ (0) > 0. Thus we get that c * = inf λ>0 P (λ) λ = 0. By Proposition 1.3 i), we have that c + ≥ 0 = c * . We want to show that c + = 0.

Assume to the contrary that c + = 0 and let φ be a solution of (1.9) with F replaced by F 0 . Using the equivalence between the viscosity solution and almost everywhere solutions (see Lemma 2.5), we deduce that φ is an almost everywhere solution of (9.27) 0 = F ((φ(z + r i )) i=0,...,N ).

That is there exists a set N of measure zero such that for every z / ∈ N , equation (9.27) holds true. Let N 0 = ∪ k∈Z (N + k) and choose z 0 ∈ R\N 0 (set N 0 has also a zero measure), then equation (9.27) holds true for every z 0 + k with k ∈ Z. Hence (9.28) g(φ(z

0 + k + 1)) + g(φ(z 0 + k -1)) -2g(φ(z 0 + k)) = -f (φ(z 0 + k)) ≤ 0 for every k ∈ Z.
Let h be the piecewise affine function which is affine on each interval [k, k + 1] and satisfying h(z 0 + k) = g(φ(z 0 + k)) with k ∈ Z. Thus, it is easy to conclude using (9.28) that h is concave. Moreover, h is bounded because g is bounded on [0, 1] and 0 ≤ φ ≤ 1. Therefore, h is constant. This implies that g(φ(z 0 )) = g(φ(z 0 + k)) = const for all k ∈ Z.

Moreover, since g ′ ≥ 0, φ(-∞) = 0 and φ(+∞) = 1, we conclude that g = const on [0, 1], which is a contradiction with g(1) = 1 + g(0). Hence, we get c + > 0 = c * .

Critical velocity c + is non-negative

This subsection is devoted for the proof of Proposition 1.3. Independently, we also show that c -< 0 < c + for the Frenkel-Kontorova model (1.17).

Proof of Proposition 1.3 Let (c, φ) be a solution of (1.9) given in Theorem 1.1 with c fixed. Our goal is to show that c ≥ 0; and hence c + ≥ 0. We perform the proof in several steps.

Step 0: preliminary Define for X = (X 0 , ..., X N ) ∈ [0, 1] N +1 and δ > 0 small the function

(9.29) F δ (X) = F (X) -f (X 0 ) + f δ (X 0 ),
where

f δ (v) = f on [0, 1 -δ] max (f (1 -δ) -L 0 (v -(1 -δ)), 0) on [1 -δ, 1],
velocity satisfies c + < 0. To this end, we will construct a function f ∈ Lip([0, 1]), which is linear in a neighborhood of zero with f ′ (0) > 0, such that there exists a couple (c, φ) with c < 0 solution of (9.38)

     cφ ′ (x) = φ(x -1) -φ(x) + f (φ(x)) on R φ ′ ≥ 0 φ(-∞) = 0 and φ(+∞) = 1.
Let c = -µ with 0 < µ < 1 and

φ(x) =      1 2 e γx on (-∞, 0] 1 - 1 2 e -γx on [0, +∞)
with γ > 0. We claim that φ ∈ C 1 (R) and (-µ, φ) solves (9.39)

     0 < φ(x) -φ(x -1) -µφ ′ (x) on R φ ′ > 0 φ(-∞) = 0 and φ(+∞) = 1,
which is possible to check for 0 < γ << 1. Therefore, it is sufficient to define the function f as

(9.40) f (φ(x)) := φ(x) -φ(x -1) -µφ ′ (x) > 0 for all x ∈ R.
Notice that, when x → +∞, φ(+∞) = 1 and φ ′ (x) → 0, thus f (1) = 0. Similarly, we have f (0) = 0. Moreover, since φ ∈ C 1,1 (R), we have that f ∈ Lip((0, 1)). In fact, by a direct tedious calculation, one can deduce that

f (x) =                  (1 -e -γ -µγ)x for x ∈ 0, 1 2 1 + (1 + µγ)(x -1) + e -γ 4(x -1) for x ∈ 1 2 , 1 - 1 2 e -γ (1 -e γ + µγ)(x -1) for x ∈ 1 - 1 2 e -γ , 1 ,
and this implies that f ∈ Lip([0, 1]) and 1 > f ′ (0) > 0. We can even check that f is concave and C 1 except at the point x = 1 2 , where it is neither concave nor C 1 .

Remark that to get more regular non-linearities, one can consider (9.41)

f ε (x) := φ(•) -φ(• -1) -µφ ′ (•) ⋆ ρ ε (x),
where ρ ε satisfies ρ ε ≥ 0, ρ ε (x) = 1 ε ρ( x ε ) (ρ is a mollifier) and supp ρ ε ⊂ B ε (0). However, in this case, ρ ε ⋆ φ is a solution of (9.38), with f replaced by f ε , and then f ε ∈ C ∞ ([0, 1]) with f ′ ε (0) > 0. Now, we give the proof of the instability result, namely Proposition 1.2.

Proof of Proposition 1.2

We have seen, in Proposition 1.4, that there exists a function F satisfying (A Lip ) and (P C 1 ) such that the associated critical velocity c + F := c + satisfies (9.42) c + F < 0.

Our goal is to build a sequence of functions F δ with a critical velocity c + F δ such that

F δ → F in L ∞ ([0, 1] N +1 )
as δ → 0, and prove that (9.43) lim inf δ→0 c + F δ > c + F .

Step 1: construction of F δ Define for X = (X 0 , ..., X N ) ∈ [0, 1] and δ > 0 small the function (9.44) F δ (X) = F (X)f (X 0 )f δ (X 0 ), where (9.45)

f δ (v) = max f (δ) + L 0 (v -δ), 0 on [0, δ] f on [δ, 1],
with a constant L 0 > 0 satisfying L 0 > 2Lip(F ) =: 2L ∞ F . By construction of f δ , we clearly have

F δ -F L ∞ = f -f δ L ∞ → 0 as δ → 0.
Step 2: existence of c + Step 3: establishing (9.43) Our aim is to show that c + F δ ≥ 0. Since F δ is non-decreasing w.r.t. X i for all i = 0, then for X = (X 0 , X ′ ) ∈ [0 δ , 1] N +1 , we have F δ (X 0 , X ′ ) ≥ F δ (X 0 , 0 δ , ..., 0 δ ) := A(X 0 ). Moreover, for X 0 X 0 + h ∈ [0 δ , δ] with h > 0, we have A(X 0 + h) -A(X 0 ) = F (X 0 + h, 0 δ + h, ..., 0 δ + h) -F (X 0 , 0 δ , ..., 0 δ )f (X 0 + h)

+ f (X 0 ) + f δ (X 0 + h) -f δ (X 0 ) ≥ -2hL ∞ F + hL 0 = h(L 0 -2L ∞ F ) > 0,
where we have used that F is L ∞ F -Lipschitz (in the second line) and that L 0 > 2L ∞ F in the last inequality. This implies that A is increasing over [0 δ , δ], but A(0 δ ) = F δ (0 δ , 0 δ , ..., 0 δ ) = 0. Hence, we get A ≥ 0 over [0 δ , δ].

Therefore, we deduce that F δ ≥ 0 over [0 δ , δ] × [0 δ , 1] N . Now since φ(-∞) = 0 δ , then for z << -1 very negative, we get that φ(z + r 0 ) = φ(z) ∈ [0 δ , δ]. Hence, for all φ(z) ∈ [0 δ , δ], we obtain from (9.46) that c + F δ φ ′ (z) = F δ (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) ≥ 0, but φ ′ ≥ 0, thus we deduce that c + F δ ≥ 0. This implies that (because of (9.42))

lim inf δ→0 c + F δ ≥ 0 > c + F .
Step 4: conclusion Let Therefore, up to rename F δ as F δ , this ends the proof of Proposition 1.2.

10 Appendix: Useful results used for the proof of c + ≥ 0

This subsection is dedicated for the useful tools that we use to prove that the critical velocity is non-negative, i.e c + ≥ 0.

Proposition 10.1 (Extension by antisymmetry)

Let F be a function defined over Q = [0, 1] N +1 satisfying (A Lip ) and such that F (0, ..., 0) = 0. Then there exists an antisymmetric extension G defined over [-1, 1] N +1 such that

G | Q = F G(-X) = -G(X)
and G satisfies (A Lip ) over [-1, 1] N +1 . Moreover, if F is C 1 over a neigborhood of {0} N +1 in [0, 1] N +1 and f ′ (0) > 0 (f (v) := F (v, ..., v)), then there exists η > 0 such that for every a > 0 small and X = (X 0 , ..., X N ) ∈ [-1, 1] N +1 such that X, X + (a, ..., a) are close enough to {0} N +1 , we have (10.1) G(X + (a, ..., a)) -G(X) ≥ ηa.

Remark 10.2 (Reflection)

Note that if F is invariant by reflection symmetry, then it is possible to show that G also; precisely, we mean that if F (X) = F (X) for X i = X i with r i = -r i , then G(X) = G(X).

We recall before proving Proposition 10.1 the following properties of the orthogonal projection which can be easily shown: ii) "Antisymmetry" where we recall that Q ′ = [-1, 0] N +1 . For X ∈ Q, we clearly have G(X) = F (X).

Let Q ′ = [-1, 0] N +1 = -Q, then P roj | Q ′ (-X) = -P roj | Q (X).
Step 1: G(-X) = -G(X)

We have

G(-X) = F (P roj | Q (-X)) -F (-P roj | Q ′ (-X)) = F (-P roj | Q ′ (X)) -F (P roj | Q (X)) = -G(X),
where we have used in the second line the antisymmetry in Lemma 10.3.

Step 2: G satisfies (A Lip ) Since F is globally Lipschitz and the orthogonal projection is 1-Lipschitz, then G is globally Lipschitz on [-1, 1] N +1 . We now prove that G is non-decreasing w.r.t. X i for all i = 0. Let X = (X i ) i=0,...,N , Y = (Y i ) i=0,...,N ∈ [-1, 1] N +1 such that X i ≥ Y i for all i ∈ {1, ..., N } X 0 = Y 0 , Now, since F is C 1 over a neighborhood of X (X close to {0} N +1 ), then we get G(X + (a, ..., a)) -G(X) = a f ′ (0) + o(X + ) + o(X -) + o(1) ≥ a f ′ (0) 2 > 0 for X close enough to {0} N +1 .

Case 2: X ∈ Q Σ and X + aE ∈ Q Σ There exists an integer p ≥ 1 such that

G(X + aE) -G(X) = p k=0 G(X + t k E) -G(X + t k-1 E) ,
where 0 = t 0 < t 1 < ... < t p = a such that for k = 1, ..., p, we have X + [t k-1 , t k ]E ∈ Q Σ k , with Σ = Σ 0 and Σ = Σ p . Therefore, using Case 1 for each segment, we deduce that G(X + aE) -G(X) ≥ ηa, with η = f ′ (0) 2 > 0.

We now introduce an extension by antisymmetry-reflection of F : Proposition 10.4 (Extension by antisymmetry-reflection) Let F be a function defined on Q = [0, 1] N +1 satisfying (A Lip ) and such that F (0, ..., 0) = 0. Let X = (X i ) i=0,...,N ∈ [0, 1] N +1 and assume that (10.3) for all i ∈ {1, ..., N } there exists i ∈ {1, ..., N } such that r i = -r i .

Then there exists a function G defined on [-1, 1] N +1 which satisfies (A Lip ) on [-1, 1] N +1 such that

G | Q = F G(-X) = -G(X) (antisymmetric-reflection),
where we recall that X i = X i with r i = -r i . Moreover, if F is C 1 over a neighborhood of {0} N +1 and (10.4) ∂F ∂X 0 (0) + N i=1 min ∂F ∂X i (0), ∂F ∂X i (0) > 0, then there exists η > 0 such that for every a > 0 small and X = (X 0 , ..., X N ) ∈ [-1, 1] N +1 such that X, X + (a, ..., a) are close enough to {0} N +1 , we have Therefore, for each i ∈ {1, ..., N + M } there exists i ∈ {1, ..., N + M } such that r i = -r i . Now, for X = (X, X ′ ) with X ′ = (X N +1 , ..., X N +M ), set F ( X) = F (X).

Thus F satisfies (10.3) with N replaced by Ñ = N + M and if moreover φ solves cφ ′ (z) = F ((φ(z + r i )) i=0,...,N ), then it solves cφ ′ (z) = F ((φ(z + r i )) i=0,..., Ñ ).

In addition, if F is C 1 in a neighborhood of {0} N +1 , then F is C 1 in a neighborhood of {0} with I = i ∈ {1, ..., N } such that there exists i ∈ {1, ..., N } with r i = -r i .

Proof of Proposition 10.4

The proof is very similar to the proof of Proposition 10.1, so we give only a few details. Let X ∈ [-1, 1] N +1 , then define the extension function G by

(10.6) G(X) = F (P roj | Q (X)) -F (-P roj | Q ′ (X)),
where we recall that X i = X i with r i = -r i .

Step 1: G(-X) = -G(X)

We have G(-X) = F (P roj | Q (-X)) -F (-P roj | Q ′ (-X))

= F (-P roj | Q ′ (X)) -F (-P roj | Q ′ (-X)) (using Lemma 10.3 ii) and -X = -X)

= F (-P roj | Q ′ (X)) -F (P roj | Q (X)) (using again Lemma 10.3 ii))

= -G(X).

Step 2: G satisfies (A Lip ) on [-1, 1] N +1 This step is an analogous of Step 2 in the proof of Proposition 10.1.

Step 3: checking (10.5)

We have G(X) = F (X + ) -F (-(X) -)

Let Σ = (σ 0 , σ 1 , ..., σ N ) and define Σ = (σ 0 , σ 1 , ..., σ N ) such that σ i = σ i for all i = 0, ...., N ; and then recall Q Σ = X = (X 0 , ..., X N ) ∈ [-1, 1] N +1 such that σ i X i ∈ [0, 1] for i = 0, ..., N .

We have

X ∈ Q Σ ⇐⇒ X ∈ Q Σ .
Let X, X + aE close enough to {0} N +1 with a > 0 small and E = (1, ..., 1). where aΘ = (X + aE) + -X + with Θ = (θ i ) i=0,...,N , where

θ i = 1 if σ i = 1 0 if σ i = -1
and aΓ = (X + aE) --(X) -with Γ = (γ i ) i=0,...,N , where

γ i = 0 if σ i := σ i = 1 1 if σ i := σ i = -1.
Hence, using the fact that F is C 1 , we get where we have used in the fourth line the fact that θ i + γ i = 1 for all i = 0, ..., N, which follows from the definition of θ i and γ i and the fact that σ i = σ i .

Case 2: X ∈ Q Σ and X + aE ∈ Q Σ This case is exactly the same as Case 2 of Step 3 in the proof of Proposition 10.1. However, in this case, we can choose η = a 2 (Θ + Γ).∇F (0) > 0.

Here, we recall two comparison principle results on half lines that we will also use to prove that c + ≥ 0.

1

 1 General motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Main results in the monostable case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Main result on the velocity function . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Notations of our assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 Introduction 1.

Figure 1 :

 1 Figure 1: Positive degenerate monostable non-linearity f

Figure 4 :Remark 1 . 9 (

 419 Figure 4: Typical graph of the velocity function c(σ) with vertical branches at σ = σ ± .

u

  * (y) = lim sup x→y u(x) and u * (y) = lim inf x→y u(x).

Definition 2 . 1 (

 21 Viscosity solution)Let I = I ′ = R (or I = (-r * , +∞) and I ′ = (0, +∞)) and u : I → R be a locally bounded function, c ∈ R and F defined on R N +1 .

Proposition 2 . 3 (

 23 Perron's method ([16, Proposition 2.8])) Let I = (-r * , +∞) and I ′ = (0, +∞) and F be a function satisfying ( ÃLip ) (introduced in Subsection 7.1). Let u and v defined on I satisfying u ≤ v on I,

  Now, since 0 < m p → 0 and k i → b for all i (see [1, Proposition 2.3, Steps 4.1, 4.2 and 5]), then c i → b for all i ∈ {0, ..., N }.

  Moreover, either φ satisfies m = φ(-∞) and φ(+∞) = m + 1 or there exists two solutions φ a and φ b such that m = φ a (-∞) and φ a (+∞) = b and b = φ b (-∞) and φ b (+∞) = m + 1.

2 :

 2 limits of the profile Since φ(±∞) solves f = 0, then φ(±∞) ∈ {m, b, m + 1}. Therefore, either φ satisfies m = φ(-∞) and φ(+∞) = m + 1 or there exists two solutions φ a and φ b such that φ a satisfies m = φ a (-∞) and φ a (+∞) = b ) * (0) ≤ m + b 2 (φ a ) * (0) ≥ m + b 2 and φ b satisfies b = φ b (-∞) and φ b (+∞) = m + 1 and (4.11)

  φ a n and φ b n satisfies resp. (4.10) and (4.11).

Theorem 4 . 4 (

 44 Identification of the limits of the profile) We work under the assumptions of Lemma 4.2 with F n = F + σ n , m n = m σn , b n = b σn and F satisfying ( ÃC 1

Proposition 4 . 5 (

 45 Existence and uniqueness of c = c(σ) for σ ∈ (σ -, σ + ))

Lemma 4 . 6 (

 46 Continuity and monotonicity of m σ , b σ over [σ -, σ + ])

(4. 15 )

 15 |∇F (X + t(a, ..., a)) -∇F (m σ , ..., m σ )| ≤ δ for all t ∈ [0, 1]. Hence using (4.15), we get

Proposition 5 . 1 (

 51 Monotonicity of the velocity)Assume ( ÃC 1 ) and let σ ∈ [σ -, σ + ]. Let (c 1 , φ 1 ) and (c 2 , φ 2 ) be respectively a sub and a supersolution of (1.19) such that(5.1)

Lemma 5 . 5 ( 1 < m σ 2 . 2 ( 5 . 5 )

 5512255 Strict monotonicity)Under the assumptions of Proposition 5.1, there exists a constant K > 0 such that c(σ) satisfies(5.4) dc dσ ≥ K|c| on (σ -, σ + )in the viscosity sense.Proof of Lemma 5.5Clearly, if c = 0 then (5.4) holds true. Let σ 1 , σ 2 ∈ (σ -, σ + ) with σ 1 < σ 2 and, as in the proof of Proposition 5.1, let us call c 1 ≤ c 2 the associated velocities and φ 1 , φ 2 the corresponding profiles with φ i (-∞) = m σ i for i = 1, 2 and m σ Recall also that (c, φ) = (c i , φ i ) solves for σ = σ i and i = 1, cφ ′ = F ((φ(x + r i )) i=0,...,N ) + σ.

Proposition 6 . 1 (

 61 Because φ(±∞) solves f (φ(±∞)) + σ + = 0, the solution has to satisfy φ(-∞) = m σ + = 0 and φ(+∞) = 1. Therefore φ = φ + solves (1.20). 6 Filling the gaps: traveling waves for c ≥ c + and c ≤ c - We prove, in this section, for each c ≥ c + (resp. c ≤ c -) the existence of a solution of (1.20) (resp. (1.21)). We also prove that (1.20) (resp. (1.21)) admits no solution for any c < c + (resp. c > c -). Existence of solution for vertical branches of velocities) Let F be a given function satisfying assumptions ( ÃC 1 ) and ( BC 1 ). Let c + < +∞ and c -> -∞ be given by Corollary 5.3. Then for every c > c + (resp. c < c -), there exists a solution φ of (1.20) (resp. (1.21)). For c = c + or c >> 1 (resp. c = c -or c << 1) we already have the existence of a solution of φ of (1.20) (resp. (1.21)). Proposition 6.1 fills the gap for all c ≥ c + (resp. c ≤ c -).

Lemma 6 . 2 (

 62 Existence of a hull function ([16, Theorem 1.5 and Theorem 1.6 a1]))

Lemma 6 . 4 (

 64 Non-existence of solution for c < c + and c > c -) Consider a function F and assume ( ÃC 1 ) and ( BC 1 ). Let σ = σ + (resp. σ = σ -) and c + < +∞ (resp. c -> -∞) be given by Corollary 5.3. Let (c, φ) be a solution of (1.20) (resp. (1.21)), then c ≥ c + (resp. c ≤ c -).

F

  is globally Lipschitz continuous over R N +1 .

  0 (because of the condition on L 0 ). Since F δ +σ satisfies (A Lip ) and (B Lip ) with [0, 1] N +1 replaced by [m δ,σ -1, m δ,σ ] N +1 and b replaced by b δ,σ (see (7.7) and (7.9)), then applying the result of Proposition 4.1 (but now on [m δ,σ -1, m δ,σ ] N +1

Substep 6 . 1 . 4 :

 614 construction of a solution (c, φ) associated to some σ Since the map(7.22) σ → λ(σ, p) := pc δ,σ,p is continuous with λ(±∞, p) = ±∞ (see Lemma 6.2 applied to F δ instead of F ), then for every 0 < δ ≤ δ 0 , 0 < -σ ≤ σ δ and 0 < p ≤ p δ,σ such that (7.[START_REF] Guo | Front propagation for discrete periodic monostable equations[END_REF]) and (7.20) hold true, there exists σ = σ δ,p ∈ R and a function φ = φ δ,σ,p : R → R (see Corollary 6.3) such that c δ,σ,p = c and (c, φ) solves (7.23)

( 8 .

 8 10) ψ * (•, 0) = εH * and ψ * (•, 0) = εH * , where H = 1 [0,+∞) is the Heaviside function.

  ), where r * = max i=0,...,N |r i |. Since F is Lipschitz (with constant Lipschitz L), then F (φ(x), φ(x + r 1 ), ..., φ(x + r N )) ≤ L φ(x) φ(x + r * ) . . . φ(x + r * ) ≤ L 1 φ(x + r * ) with L 1 = L

Remark 9 . 1 (

 91 About the assumption (1.14)) It is possible to show that Proposition 1.5 still holds true if we replace (1.14) by ∃ i 1 ∈ {0, ..., N } such that r i 1 < 0 and ∂F ∂X i 1 (0, ..., 0) > 0 if c + < 0.

( 9 .Lemma 9 . 3 (

 993 26) c + ≤ c for all c > c * . This implies that c + ≤ c * . Now, we give an example of non-linearity where we have c + > c * . Example with c + > c * ) Consider the function F

  δ was denoted by b δ in the proof of Theorem 1.1). Since F δ satisfies (A Lip ) and (P Lip ) with [0, 1] N +1 replaced by [0 δ , 1] N +1 , then applying the result of Theorem 1.1, we deduce that there exists a minimal velocity c + F δ and a profile φ solution of (9.46) δ φ ′ (z) = F δ (φ(z + r 0 ), φ(z + r 1 ), ..., φ(z + r N )) on R φ is non-decreasing over R φ(-∞) = 0 δ and φ(+∞) = 1.

= ( 1 -

 1 0 δ )c + F δ and F δ ((X i ) i=0,...,N ) = F δ (((1 -0 δ )X i + 0 δ ) i=0,...,N ). ′ (z) = F δ (( φ(z + r i )) i=0,...,N ) on R φ is non-decreasing over R φ(-∞) = 0 and φ(+∞) = 1 and c + F δis the critical velocity associated to F δ which is defined on [0, 1] N +1 . Moreover, we still have | F δ -F | → 0 as δ → 0 and F δ satisfies (A Lip ) and (P Lip ) on [0, 1] N +1 . In addition, since 0 δ → 0 as δ → 0, then from (9.47) we still have

Lemma 10 . 3 (

 103 Some properties of orthogonal projection)Let X = (X i ) i=0,...,N ∈ [-1, 1] N +1 and call P roj | Q (X) the orthogonal projection of X on Q = [0, 1] N +1 . Then P roj | Q (X) = (P roj | [0,1] (X i )) i=0,...,N .Moreover, we have i) Order preservationLet Y = (Y i ) i=0,...,N ∈ [-1, 1] N +1 and assume that X ≥ Y in sense that X i ≥ Y i for all i ∈ {0, ..., N }, then P roj | Q (X) ≥ P roj | Q (Y ).

Proof of Proposition 10. 1

 1 Let X = (X i ) i=0,...,N ∈ [-1, 1] N +1 , then define the extension function G by:(10.2) G(X) = F (P roj | Q (X)) -F (-P roj | Q ′ (X)),

(10. 5 )

 5 G(X + (a, ..., a)) -G(X) ≥ ηa.Remark 10.5 (On the reflection condition (10.3)) Notice that we can always assume that the reflection condition (10.3) is satisfied up to modify the function F. Indeed, if F does not satisfy the reflection condition (10.3), i.e. we have {i 1 , ..., i M } = {i ∈ {1, ..., N }, such thatr i / ∈ {r j } j=1,...,N } with M ≥ 1, then let us define r N +j = -r i j for j = 1, ..., M.

Case 1 :

 1 X, X + aE ∈ Q Σ Since F is C 1 over a neighborhood of {0} N +1 , then (as in the proof of Proposition 10.1, Step 3) we haveG(X + aE) -G(X) = F ((X + aE) + ) -F (X + ) -F (-(X + aE) -) -F (-(X) -) = F ((X + aE) + ) -F (X + ) -F (-(X + aE) -) -F (-(X) -) = aΘ.∇F (X + ) + o(|aΘ|) + aΓ.∇F (-(X) -) + o(|aΓ|),

G

  (X + aE) -G(X) = a (Θ + Γ).∇F (0) + Θ.(∇F (X + ) -∇F (0)) + o(1) + Γ.(∇F (-(X) -) -∇F (0))

Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, 6-8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex

2, France

ACKNOWLEDGMENTS

The first author would like to thank the Lebanese National Council for Scientific Research (CNRS-L) and the Campus France (EGIDE earlier) for supporting him. He also want to thank professor R. Talhouk and the Lebanese university. The last author was also partially supported by the contract ERC ReaDi 321186. Finally, this work was partially supported by ANR HJNet (ANR-12-BS01-0008-01) and by ANR-12-BLAN-WKBHJ: Weak KAM beyond Hamilton-Jacobi.

Finite threshold velocities (c + < +∞ and c -> -∞)

In this subsection, we show that c + < +∞ (resp. c -> -∞) and we prove the existence of a solution for c = c + (resp. c = c -) of (1.20) (resp. (1.21)).

In order to prove that c + < +∞ and c -> -∞, we need to start with the following useful lemma.

Preliminary for the critical velocity: Harnack inequality

We prove in this subsection a Harnack inequality (Proposition 8.4) for the profile that we use in Subsection 9.1 to show that c + ≥ c * . Our approach is inspired by Hamel [START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF]. The proof will use a strong maximum principle for a linear evolution problem that we also prove in this subsection. Proposition 8.1 (Strong maximum principle for a linear evolution problem) Let F be a function satisfying (A Lip ) and differentiable at {0} N +1 . Assume that (8.1) ∃ i 0 ∈ {0, ..., N } such that r i 0 > 0 and ∂F ∂X i 0 (0, ..., 0) > 0.

Let T > 0 and u : R×[0, T ) → [0, +∞) be a lower semi-continuous function which is a supersolution of (8.2)

∂F ∂X i (0, ..., 0)u(x + r i , t) for (x, t) ∈ R × (0, T ).

If u(x 0 , t 0 ) = 0 for some (x 0 , t 0 ) ∈ R × (0, T ), then u(x 0 + kr i 0 , t) = 0 for all k ∈ N and 0 ≤ t ≤ t 0 .

Proof of Proposition 8.1 Let u be a supersolution of (8.2) such that u ≥ 0 and assume that there exists some (x 0 , t 0 ) ∈ R × (0, T ) such that u(x 0 , t 0 ) = 0.

Step 1: u(x 0 , s) = 0 for all 0 ≤ s ≤ t 0

Step 1.1: preliminary Since u is a supersolution of (8.2) on R × (0, T ), then u satisfies in the viscosity sense

∂F ∂X i (0, ..., 0)u(x + r i , t) for all (x, t) ∈ R × (0, T ).

Because (8.3) ∂F ∂X i (0, ..., 0) ≥ 0 for all i = 0 and ∂F ∂X 0 (0, ..., 0) ≤ L, where L is the Lipschitz constant of F, we get in the viscosity sense (using u ≥ 0):

Step 1.2: u(x 0 , •) is a viscosity supersolution of (8.4) on (0, T ) We now set v(t) = u(x 0 , t). We claim that v satisfies in the viscosity sense (8.5) v t ≥ -Lv on (0, T ).

In order to prove our claim, let φ be a test function such that (8.6)

Proposition 8.4 (Harnack inequality)

Let F be a function satisfying (A Lip ), (P Lip ) and assume that F is C 1 over a neighborhood of {0} N +1 in [0, 1] N +1 . Assume moreover that (8.25) ∃ i 0 ∈ {0, ..., N } such that r i 0 > 0 and ∂F ∂X i 0 (0, ..., 0) > 0.

Let (c, u) with c = 0 be a solution of (8.26)

Then for every ρ > 0 there exists a constant κ 1 = κ 1 (ρ) > 1 such that for every x ∈ R, we have

u.

Moreover, there exists κ 0 > 1 such that

where r * = max i=0,...,N

We refer the reader to Remark 9.1 for comments on assumption (8.25).

Proof of Proposition 8.4

Let F be the extension of F on R N +1 given by Lemma 7.1. Then define the function

where u ∈ C 1 , because c = 0. Thus u satisfies (8.29) u t (x, t) = F ((u(x + r i , t)) i=0,...,N ) for all (x, t) ∈ R × (0, +∞) and (8.30) u(x, 0) = u(x).

Let x 0 ∈ R such that 1 ≥ u(x 0 ) > 0. Since u is non-decreasing, then for all x ∈ R we have

where H = 1 [0,+∞) is the Heaviside function.

For ε ∈ (0, 1] that will be fixed later, let ψ ε = ψ be the solution given by Lemma 8.2 with initial conditions (8.10) and let v(x, t) = ψ ε (xx 0 , t). Now, using Proposition 8.3, we deduce that there exists some ε 0 ∈ (0, 1] and T 0 such that for all δ ∈ (0, T 0 ) and R > 0 there exists a constant

We now choose ε = min(ε 0 , u(x 0 )).

with a constant L 0 > 2Lip(F ) > 0 large enough. Let δ ∈ (0, 1 2 ) and set

(where 1 δ was denoted by m δ in the proof of Theorem 1.1).

Part I: antisymmetric extension of F δ and proof for ii) Using Proposition 10.1, there exists an antisymmetric extension

(because of (P C 1 ) and (9.29)) and f ′ δ (0) = f ′ (0) > 0, then there exists η > 0 such that for every X, X + (a, ..., a) ∈ [-1, 1] N +1 close to {0} N +1 with a > 0 small, we have

In addition, the function

Step replaced by [-1 δ , 1 δ ] N +1 and b replaced by 0, we deduce that there exists a real c 0 δ and a function φ 0 δ solution of (9.32)

Step I.2: c 0 δ ≥ 0 We show in this step that c 0 δ is non-negative under ii), i.e. assuming r i ≥ 0 for all i ∈ {0, ..., N }.

) is a subsolution of (9.32). Using an argument similar to the computation of (7.9) for L 0 large enough (here

, that is G δ satisfies (10.7) and (10.9) (for s = -1 δ and s ′ = 1 δ ).

Applying the comparison principle results (Proposition 10.6 and Proposition 10.7) and the ideas of the proof of Proposition 5.1, we deduce that

Step I.3: comparing c and c 0 δ Recall that (c 0 δ , φ 0 δ ) is a solution of (9.32). Moreover, since G δ = F δ ≤ F over [0, 1] N +1 , then (c, φ) is a supersolution for (1.9), with F replaced by G δ .

Since φ(-∞) = 0 and φ(+∞) = 1

and -1 δ < 0 < 1 δ < 1, that is φ 0 δ (±∞) < φ(±∞) and φ 0 δ (+∞) > φ(-∞), then using the proof of Proposition 5.1 (which still true for sub and supersolutions), we deduce that

Part II: extension of F δ by antisymmetry-reflection and proof for iii)

In this part, we assume that F (and then F δ ) satisfies the strict monotonicity condition (1.12). Using Remark 10.5, we can assume that the set I defined in (1.11) satisfies

i.e. for all i ∈ {1, ..., N }, there exists i ∈ {1, ..., N } such that r i = -r i . Using now Proposition 10.4, there exists an extension

and satisfying (A

, then (using Proposition 10.4) there exists η > 0 such that for every X, X + (a, ..., a) ∈ [-1, 1] N +1 close to {0} N +1 with a > 0 small, we have (9.34) G δ (X + (a, ..., a)) -G δ (X) ≥ ηa.

In addition, the function

Step II.1: existence of traveling waves for G δ This step is a variant of Step I.1 with G δ replaced G δ . Thus we deduce that there exists a real c 0 δ and a function φ 0 δ solution of (9.36)

thus condition (1.12) is equivalent to f ′ (0) > 0. Therefore, we can apply iii) which shows that c + ≥ 0.

Complement: another proof for i) and ii)

We show in this complement the result of Proposition 1.3 i) and ii) using a different approach.

The proof is done by contradiction. Assume to the contrary that c + < 0.

Using Proposition 1.5, we deduce that c + ≥ c * .

Since F satisfies (P C 1 ), then

Moreover, we have that P is convex and that

Getting a contradiction

Clearly, if F satisfies the reflection symmetry condition i), then we get that P ′ (0) = 0. Similarly, if r i ≥ 0 for all i ∈ {1, ..., N }, then P ′ (0) ≥ 0. But P is convex, hence we deduce that P (λ) > 0 for every λ > 0.

Therefore, we get that c * ≥ 0, which is a contradiction with 0 > c + ≥ c * . This implies that c + > 0 under the conditions i) and ii).

Lemma 9.4 (Sign of c + and c -for (FK) model (1.17))

Consider the Frenkel-Kontorova model with β > 0

. Let c ± be the critical velocity associated to σ ± . Then

Proof of Lemma 9.4 Let σ = σ + = β and let us show that c + > 0. Let φ be non-decreasing with φ(-∞) = 0 and φ(+∞) = 1. Integrating over the real line the equation

where

Since f > 0 on (0, 1), if c + = 0, then φ(z) = 0 or 1 almost everywhere.

This implies that ∆ 1 φ(z) := φ(z + 1) + φ(z -1) -2φ(z) = 0 almost everywhere.

Consider now the set

which has measure zero. Thus the set A + Z has also measure zero. Hence for a fixed a ∈ R\(A + Z) = ∅, we have ∆ 1 φ(a + k) = 0 for every k ∈ Z.

This implies that there exists λ, b ∈ R (that may depend on a) such that

But φ is bounded, then λ = 0 and hence φ(a

Similarly, for σ = σ -= -β, we show that c -< 0, since f -2β < 0 on -1 2 , 1 2 .

Instability of critical velocity

In this section, we show that the critical velocity c + given in Theorem 1.1 is unstable in the sense of Proposition 1.2, which we prove in this section. Before proving Proposition 1.2, we give an example of a non-linearity F for which the associated critical velocity is negative. This example will be the proof of Proposition 1.4.

Proof of Proposition 1.4

The aim is to construct a function F satisfying (A Lip ) and (P C 1 ) such that the associated critical and let us show that G(X) ≥ G(Y ). In fact, since the orthogonal projection preserve the ordering (see Lemma 10.3) and since F is non-decreasing w.r.t. X i for all i ∈ {1, ..., N }, we conclude that G is non-decreasing w.r.t. X i for all i ∈ {1, ..., N } over [-1, 1] N +1 .

Step 3: checking (10.1) We first give some notations for the projection function. Consider X = (X 0 , ..., X N ) ∈ [-1, 1] N +1 , then from Lemma 10.3, we have

Similarly, we have (with

We also define

where Σ = (σ 0 , ..., σ N ) and σ i = ±1. Now, we go back to the proof of (10.1) which is splitted in two cases. Let X, X + (a, ..., a) close to {0} N +1 with a > 0 small: Case 1: X, X + (a, ..., a) ∈ Q Σ From the definition of G (see (10.2)) and the notations introduced at the beginning of this step, we have

where E = (1, ..., 1). Thus, we get G(X + (a, ..., a)) -G(X) = aΘ.∇F (X + ) + o(|aΘ|) + aΓ.∇F (-X -) + o(|aΓ|), where aΘ = (X + aE) + -X + with Θ = (θ i ) i=0,...,N , where

and aΓ = (X + aE) --X -with Γ = (γ i ) i=0,...,N , where

Hence, we obtain there exists η 0 > 0 such that if X = (X 0 , ..., X N ), X + (α, ..., α) ∈ [s ′η 0 , s ′ ] N +1 then F (X + (α, ..., α)) < F (X) if α > 0.

Let u, v : [-r * , +∞) → [s, s ′ ] be respectively a sub and a supersolution of