Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic environment

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Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic

environment

Benjamin Contri

To cite this version:

Benjamin Contri. Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic environment. 2015. �hal-01178306�

Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic

environment

Benjamin CONTRI

Aix Marseille Université, CNRS, Centrale Marseille

Institut de Mathématiques de Marseille, UMR 7373, 13453 Marseille, France

Abstract

This paper is devoted to reaction-diffusion equations with bistable nonlinearities depending periodically on time. These equations admit two linearly stable states.

However, the reaction terms may not be bistable at every time. These may well be a periodic combination of standard bistable and monostable nonlinearities. We are interested in a particular class of solutions, namely pulsating fronts. We prove the existence of such solutions in the case of small time periods of the nonlinearity and in the case of small perturbations of a nonlinearity for which we know there exist pulsating fronts. We also study uniqueness, monotonicity and stability of pulsating fronts.

Keywords: Bistable reaction diffusion equation; Pulsating front; Time periodicity.

1 Introduction and main results

In this paper we investigate equations of the type

u

− u

= f

(t, u), t ∈

, x ∈

, (1) where

f

(t + T, u) = f

(t, u), ∀t ∈

, ∀u ∈ [0, 1], and

f

(t, 0) = f

(t, 1) = 0, ∀t ∈

. (2) Throughout this article, we assume the function f

:

× [0, 1] →

is of class C

with respect to t uniformly for u ∈ [0, 1], and C

with respect to x uniformly for t ∈

. The main hypotheses imposed on the function f

are the following

1 T

Z T

0

f

(s, 0)ds < 0 and 1 T

Z T

0

f

(s, 1)ds < 0. (3) We say the function f

is bistable on average if it satisfies hypotheses (2) and (3).

∗Address correspondence to Benjamin Contri: benjamin.contri@univ-amu.fr

We begin by recalling the definition of monostable and bistable homogeneous nonlinear- ities. A function f : [0, 1] →

is said monostable if it satisfies f (0) = f (1) = 0 and f > 0 on (0,1). If in addition to this we have f (u) ≤ f

(0)u on (0, 1), we say that f is of KPP type. A function f : [0, 1] →

is called bistable if there exists θ ∈ (0, 1) such that f (0) = f (θ) = f (1) = 0, f < 0 on (0, θ) and f > 0 on (θ, 1).

We give now two examples of bistable on average functions f

:

× [0, 1] →

. The first example is a bistable homogeneous function balanced by a periodic function depending only on time. Namely, if g : [0, 1] →

is a bistable function and m :

→

is a T -periodic function, then the function f

defined by f

(t, u) = m(t)g(u) is bistable on average if and only if the quantity

m(s)ds is positive and both g

(0) and g

(1) are negative. The second example of bistable on average function is a combination of a bistable homogeneous function and a monostable homogeneous function, both balanced by periodic functions (with the same period) depending only on time. Namely, if g

: [0, 1] →

is a monostable function, g

: [0, 1] →

is a bistable function, and m

, m

:

→

are two T -periodic functions, then the function f

defined by f

(t, u) = m

(t)g

(u) + m

(t)g

(u) is bistable on average if and only if we have µ

g

(0) + µ

g

(0) < 0 and µ

g

(1) + µ

g

(1) < 0, where µ

:=

m

(s)ds. It is important to note that for a bistable on average func- tion, there can very well exist times t for which the homogeneous function f

(t, ·) is not a bistable function in the sense of homogeneous nonlinearities. Indeed, if we set in the previous case g

(u) = u(1 − u), g

(u) = u(1 − u)(u − θ) with 0 < θ < 1, m

(t) = sin(2πt) and m

(t) = 1 − sin(2πt), we can notice that although the function f

(t, u) = m

(t)g

(u) + m

(t)g

(u) is bistable on average, the homogeneous function f

(1/4, ·) is of KPP type.

Context

The study of reaction-diffusion equations began in the 1930’s. Fisher [12] and Kolmogorov, Petrovsky and Piskunov [17] were interested in the equation

u

− u

= f (u), t ∈

, x ∈

, (4)

with a nonlinearity f of KPP type. They proved existence and uniqueness (up to trans- lation) of planar fronts U

of speed c, for all speeds c ≥ c

:= 2

f

(0), that is, for all c ≥ c

, there exists a function u

satisfying (4) and which can be written u

(t, x) = U

(x − ct), with 0 < U

< 1, U

(−∞) = 1 and U

(+∞) = 0. Numerous articles have been dedicated to the study of existence, uniqueness, stability, and other properties of planar fronts for various nonlinearities, see e.g. [2, 11, 16, 18, 24]. In particular, for bistable nonlinearities, there exists a unique (up to translation) planar front U(x − ct) and a unique speed c solution of (4). When the nonlinearity is not homogeneous, there are no planar front solutions of (1) anymore. For equations with coefficients depending on the space variable, Shigesada, Kawasaki and Teramoto [28] defined in 1986 a notion more general than the planar fronts, namely the pulsating fronts. This notion can be extended for time dependent equations as follows.

Definition 1.1. A pulsating front connecting 0 and 1 for equation (1) is a solution u :

R

×

→ [0, 1] such that there exists a real number c and a function U :

×

→ [0, 1]

verifying











u(t, x) = U (t, x − ct), ∀t ∈

, ∀x ∈

,

U(·, −∞) = 1, U (·, +∞) = 0, uniformly on

, U(t + T, x) = U (t, x), ∀t ∈

, ∀x ∈

.

So, a pulsating front connecting 0 and 1 for equation (1) is a solution couple (c, U (t, ξ)) of the problem











U

− cU

− U

− f

(t, U ) = 0, ∀(t, ξ) ∈

×

, U(·, −∞) = 1, U (·, +∞) = 0, uniformly on

, U(t + T, ξ) = U(t, ξ), ∀(t, ξ) ∈

×

.

(5)

For an environment depending on space only, we can refer to [4, 5, 6, 7, 8, 10, 15, 19, 29, 30, 31] for some existence, uniqueness and stability results. As far as environments depending on time (and possibly on space), Nolen, Rudd and Xin in [22] were interested in equations with a homogenous nonlinearity and an advection coefficient depending periodically on space and on time. Frejacques in [13] proved the existence of pulsating fronts in the case of a time periodic environment with positive and combustion nonlinearities. Nadin in [21]

proved the existence of pulsating fronts in an environment depending on space and time with KPP type nonlinearity. If we consider Nadin’s results in the context of our equation, he imposes in his existence results that the steady state 0 is unstable in the sense that the principal eigenvalue associated with the equilibrium 0 is negative. Yet, we shall see in section 2 that hypotheses (2) and (3) in our paper are equivalent to the fact that 0 and 1 are stable steady states, that is, the principal eigenvalues associated with equilibria 0 and 1 are positive. Shen in [25] and [26] defined and proved the existence of pulsating fronts in the case of an almost-periodic environment with a bistable nonlinearity, that is, for functions f which satisfy f(·, 0) = f (·, 1) = 0 and are negative near the equilibrium 0 and positive near the equilibrium 1 for any time. More exactly, it is assumed that there exists γ > 0 and δ ∈ (0, 1/2) such that







f(t, u) ≤ −γu, ∀(t, u) ∈

× [0, δ], f(t, u) ≥ γ(1 − u), ∀(t, u) ∈

× [δ, 1].

Alikakos, Bates and Chen in [1] (in the case of a periodic nonlinearity) and Shen in [27]

(in the case of an almost periodic nonlinearity) consider as in our paper the equation (1).

They impose the Poincaré map associated with the function f

has exactly two stable fixed points and one unstable fixed point in between. They prove under this hypothesis there exists a unique pulsating front (U, c) solution of the problem (5). They show that for each t, the function U (t, ·) is monotonic and that U , U

and U

exponentially approach their limits as ξ → ±∞. They also prove a global exponential stability result. In section 3, we will see that hypotheses (2) and (3) are equivalent to the fact that 0 and 1 are two stable fixed points of the Poincaré map associated with f

. Let us note that in this paper, we do not impose the uniqueness of intermediate fixed points between 0 and 1.

We now give the main results of the paper.

Uniqueness and monotonicity of pulsating fronts

In this paper, we begin by showing the monotonicity of pulsating front solving (5). Then

we use this result to prove the uniqueness (up to translation) of the pair (U, c) solving

(5). For that purpose, we use some new comparison principles adapted to hypotheses (2) and (3), that we show using sliding methods. We have the following theorem.

Theorem 1.1. There exists at most one couple (c, U ) solution of problem (5), the function U (t, ξ) being unique up to shifts in ξ. In this case, we have that

U

(t, ξ) < 0, ∀t ∈

, ∀ξ ∈

. Asymptotic stability of pulsating fronts

We now investigate global stability of pulsating fronts. In case of homogeneous bistable nonlinearities, Fife and McLeod in [11] proved the global stability of planar fronts. For heterogeneous bistable nonlinearities, Alikakos, Bates and Chen [1] (in case depending on time variable) and Ding, Hamel and Zhao [8] (in case depending on space variable) also proved a global exponential stability result. In our case with assumptions (2) and (3), the following stability result is proved.

Theorem 1.2. Assume there exists a pulsating front U with speed c solution of (5). We consider a solution u of the Cauchy problem







u

− u

= f

(t, u) ∀t > 0, ∀x ∈

, u(0, x) = h(x), ∀x ∈

,

where the initial condition h :

→ [0, 1] is uniformly continuous. We denote v (t, ξ) = u(t, x) = u(t, ξ + ct). There exists a constant γ ∈ (0, 1) depending only on f

such that if

lim inf

ξ→−∞

h(ξ) > 1 − γ and lim sup

ξ→+∞

h(ξ) < γ, then, there exists ξ

∈

such that

t→

lim

v(t, ξ) − U (t, ξ + ξ

)

= 0, uniformly for ξ ∈

.

Roughly speaking, if the initial condition h "looks like" a front, then v converges to a front as t → +∞. To prove this theorem we use the method of sub- and supersolution. We adapt here some ideas used in [11] in case of a bistable homogeneous nonlinearity to our equation (1) with assumptions (2) and (3).

Existence and convergence of pulsating fronts for small periods

We are interested here in understanding the role of the period T of the function f

in the limit of small periods. We consider nonlinearities of the form

f

(t, u) = f ( t

T , u), ∀t ∈

, ∀u ∈ [0, 1].

The function f is 1-periodic in time, and hypothesis (3) becomes

Z ₁

0

f

(s, 0)ds < 0 and

Z ₁

0

f

(s, 1)ds < 0.

Consequently, the sign of the quantities (3) do not depend on the period T . In order to understand the homogeneization limit as T → 0

, we define the averaged nonlinearity

g : [0, 1] →

u 7→

Z ₁

0

f(s, u)ds (6)

We assume that the function g is a bistable function, that is, there exists θ

∈ (0, 1) such

that





g(0) = g(θ

) = g(1) = 0,

g(u) < 0, ∀u ∈ (0, θ

), g(u) < 0, ∀u ∈ (θ

, 1).

We also assume that

g

(θ

) > 0. (7)

Let us noticing that according to (3), one gets g

(0) < 0 and g

(1) < 0. We have the following existence theorem.

Theorem 1.3. Under the above assumptions, there exists T

> 0 such that for all T ∈ (0, T

), there exists a unique pulsating front (U

(t, ξ), c

) solving















(U

)

− c

(U

)

− (U

)

= f

(t, U

), on

×

, U

(·, −∞) = 1, U

(·, +∞) = 0, uniformly on

, U

(t + T, ξ ) = U

(t, ξ), ∀(t, ξ) ∈

×

,

U

(0, 0) = θ

.

We are then interested in the convergence of the couple (c

, U

) as T → 0. We recall from [2] that for the bistable nonlinearity g, there exists a unique planar fronts (c

, U

) solving











U

+ c

U

+ g(U

) = 0, on

, U

(−∞) = 1, U

(+∞) = 0, U

(0) = θ

.

For any 1 ≤ p ≤ +∞, we define

W

(

) =

U ∈ L

(

) | ∂

U, ∂

U, ∂

U ∈ L

(

)

.

For any k ∈

and any α ∈ (0, 1), we also define C

(

) the space of functions of class C

(

) with the k

partial derivatives α-Hölder.

In terms of convergence, we extend the function U

on

by U

(t, ξ) = U

(ξ) for any (t, ξ) ∈

, and we have the following result

Theorem 1.4. As T → 0, the speed c

converges to c

and U

converges to U

in W

(

) weakly and in C

(

), for any 1 < p < +∞ and for any α ∈ (0, 1).

Let us mention that for nonlinearities depending only on space variable, such convergence results were proved by Ding, Hamel and Zhao [8] in the bistable case and by El Smaily [9] in the KPP case.

Existence and convergence of pulsating fronts for small perturbations

In this part, we consider some families of functions f

:

× [0, 1] →

having the same regularity as f

and such that







f

(t, 0) = f

(t, 1) = 0, ∀t ∈

,

f

(t, u) = f

(t + T, u), ∀(t, u) ∈

× [0, 1]. (8)

We also suppose that there exists a bounded function ω(ε) : (0, +∞) →

satisfying ω(ε) −−→

0 and such that

|f

(t, u) − f

(t, u)| ≤ ω(ε), ∀(t, u) ∈ [0, T ] × [0, 1]. (9) We will first show that if f

satisfies the hypotheses of existence and uniqueness theorem of Alikakos, Bates and Chen [1], then for ε > 0 small enough, the Poincaré map associated with f

also verifies it. As consequence, the following theorem holds.

Theorem 1.5. We suppose the Poincaré map associated to f

has exactly two stable fixed points 0 and 1 and one unstable fixed point α

between both.

Then, there exists ε

> 0 such that for all ε ∈ (0, ε

), there exists a unique pulsating front (U

(t, ξ), c

) solving















(U

)

− c

(U

)

− (U

)

− f

(t, U

) = 0, on ∈

×

, U

(·, −∞) = 1, U

(·, +∞) = 0, uniformly on

,

U

(t + T, ξ) = U

(t, ξ), ∀(t, ξ) ∈

×

, U

(0, 0) = α

.

(10)

Let us note that by hypothesis on f

, there exists a unique pulsating front (U

, c

) solving problem (5) with U

(0, 0) = α

. We have then the following convergence result

Theorem 1.6. As ε → 0, the speed c

converges to c

and U

converges to U

in W

(

) weakly and in C

(

), for any 1 < p < +∞ and for any α ∈ (0, 1).

Outline

Section 2 of this paper is devoted to some equivalent formulations of hypotheses (2) and (3), in particular using Poincaré map and principal eigenvalue. In the following two sections, we prove uniqueness, monotonicity and uniform stability of pulsating front solution of (1) that is, Theorems 1.1 and 1.2. In Section 5, we prove Theorems 1.3 and 1.4 on the homogenization limit. In Section 6, we prove Theorems 1.5 and 1.6 on a small perturbation of a given pulsating front.

2 Preliminaries on the characterization of the asymp- totic stability of equilibrium state

This part is devoted to the study of various characterizations of bistable on average functions. As we mentioned it previously, it is necessary to know its various points of view to be able to place the results of our paper in the literature already existing. We begin by defining the notion of equilibrium state.

Definition 2.1. Consider the problem







∂

U − c∂

U − ∂

U = f

(t, U ) on

,

U(t, ξ) = U (t + T, ξ), ∀t ∈

, ∀ξ ∈

. (11)

The T -periodic solutions of the equation y

= f

(t, y) are called equilibrium states of (11).

Indeed, if U :

→ [0, 1] is a solution of (11) and if there exists a function t 7→ θ(t) such that |U(t, ξ) − θ(t)| −−−→

0 for all t ∈

, then, by standard parabolic estimates, the function θ is a T -periodic solution of the equation y

= f

(t, y) on

. We recall the notion of uniformly asymptotic stability of such an equilibrium state.

Definition 2.2. Let t 7→ θ(t) be an equilibrium state of (11). Let (t, ξ) 7→ U (t, ξ) be another solution of the same equation. We say that θ is a uniformly asymptotic stable equilibrium if there exists η > 0 such that

∀ξ ∈

, |U(0, ξ) − θ(0)| < η

= ⇒ lim

t→+∞

|U (t, ·) − θ(t)| = 0 unif. on

.

If θ is not a uniformly asymptotic stable equilibrium, we say it is a uniformly asymptotic unstable equilibrium.

We approach now the notion of principal eigenvalue.

Proposition 2.1. [14],[20] Let t 7→ θ(t) an equilibrium state of (11). There exists a constant λ

and a function Φ

∈ C

(

×

,

) such that















∂

Φ

− c∂

Φ

− ∂

Φ

= f

(t, θ(t))Φ

+ λ

Φ

on

, Φ

> 0 on

,

Φ

(·, ξ) is T − periodic, ∀ξ ∈

, Φ

(t, ·) is 1 − periodic, ∀t ∈

.

(12)

The real number λ

is unique. It is called the principal eigenvalue associated with the function f

and the equilibrium state θ. The function Φ

is unique up to multiplication by a positive constant. It is called the principal eigenfunction associated with the function f

and the equilibrium state θ.

We give an explicit formulation of the principal eigenvalue.

Proposition 2.2. The function Φ

does not depend on the variable ξ, and the constant λ

is given by

λ

= − 1 T

Z T

0

f

(s, θ(s))ds.

Proof. We know that Φ

is unique up to multiplication by a positive constant. Let us suppose for example that kΦ

k

= 1. Let ξ

∈

. The function Φ

(t, ξ + ξ

) is also a positive solution of the problem (12). Yet kΦ

(·, · + ξ

)k

= 1. So, by uniqueness Φ

(t, ξ + ξ

) = Φ

(t, ξ) for any (t, ξ) in

. Since ξ

is arbitrary, it follows that Φ

does not depend on the variable ξ. Furthermore, the first equation in (12) becomes

∂

Φ

= f

(t, θ(t))Φ

+ λ

Φ

, ∀t ∈

.

We divide this equation by Φ

, then we integrate between 0 and T . According to the fact that Φ

is a T -periodic function, we obtain the expression of λ

given in Proposition 2.2.

We give a characterization of the uniformly asymptotic stability of the equilibrium state θ from the principal eigenvalue λ

.

Proposition 2.3. If λ

> 0 (resp. < 0), then the equilibrium state θ is uniformly

asymptotically stable (resp. unstable).

Proof. We are going to handle the case where λ

> 0. We consider a solution U (t, ξ) of (11). We saw that the function Φ

= Φ

(t) satisfies











∂

Φ

= f

(t, θ(t))Φ

+ λ

Φ

on

, Φ

> 0,

Φ

is T − periodic.

There exists ε

> 0 small enough such that for any t ≥ 0 we have

− λ

4 ε

e

λθ,f T 2 t

Φ

(t) ≤ f

(t, θ(t) + ε

e

λθ,f T 2 t

Φ

(t))−

f

(t, θ(t)) − f

(t, θ(t))ε

e

λθ,f T

2 t

Φ

(t) ≤ λ

4 ε

e

λθ,f T

2 t

Φ

(t).

We note χ(t) = θ(t) + ε

e

λθ,f T

2 t

Φ

(t). For any t ≥ 0, we have χ

(t) − cχ

(t) − χ

(t) − f

(t, χ(t)) ≥ ε

e

λθ,f T 2 t

Φ

(t)[λ

− λ

2 − λ

4 ] > 0.

If we suppose that for any ξ in

, we have U (0, ξ) ≤ θ(0) + ε

Φ

(0), then, applying the maximum principle, we have that

U(t, ξ) ≤ θ(t) + ε

e

λθ,f T 2 t

Φ

(t), ∀t ≥ 0, ∀ξ ∈

.

In the same way, possibly reducing ε

, one can show that if we suppose for any ξ ∈

that we have θ(0) − ε

Φ

(0) ≤ U (0, ξ), then

θ(t) − ε

e

λθ,f T

2 t

Φ

(t) ≤ U (t, ξ), ∀t ≥ 0, ∀ξ ∈

. Consequently, lim

t→+∞

|U (t, ξ) − θ(t)| = 0 uniformly on

.

We are now interested in case where λ

< 0. Let η > 0. There exists ε

> 0 small enough such that ε

≤

θ,f T(0)

and, for any t ∈ [0, T ] and any ε ∈ (0, ε

) we have λ

4 εe

λθ,f T 2 t

Φ

(t) ≤ f

(t, θ(t) + εe

λθ,f T 2 t

Φ

(t))−

f

(t, θ(t)) − f

(t, θ(t))εe

λθ,f T

2 t

Φ

(t) ≤ − λ

4 εe

λθ,f T 2 t

Φ

(t).

We note χ

(t) = θ(t) + εe

λθ,f T 2 t

Φ

(t). The same calculations as previously give us that for any t ∈ [0, T ] we have

(χ

)

(t) − c(χ

)

(t) − (χ

)

(t) − f

(t, χ

(t)) ≤ 0.

We define U

:

×

→

solution of the Cauchy problem







(U

)

− c(U

)

− (U

)

− f

(t, U

) on

×

, U

(0, ·) = χ

(0) on

.

Applying the maximum principle on [0, T ] ×

to the function U

and χ

, we obtain that U

≥ χ

on [0, T ] ×

. In particular, since χ

is a nondecreasing function on [0, T ], we have

U

(T, ·) ≥ χ

(0) on

. (13)

The function U

(· + T, ·) satisfies the same equation as U

. According to (13), we can apply the maximum principle on [0, T ] ×

to the function U

(· + T, ·) and χ

. We obtain for t = T that

U

(2T, ·) ≥ χ

(T ) ≥ χ

(0) on

. For any n ∈

, we can show by induction that

U

(nT, ·) ≥ χ

(0) on

.

According to the fact that χ

(0) > θ(0), and that θ(nT ) = θ(0) we have that lim

t→+∞

(U

(t, ·)−

θ(t)) 6= 0 for any ξ ∈

, although we have |U

(0, ξ) − θ(0)| ≤ η, for any ξ ∈

Finally, we connect now the previous notions with the notion of Poincaré map associated with the function f

.

Definition 2.3. For any α ∈ [0, 1], let w(α, ·) be the solution of the Cauchy problem







y

= f

(t, y),

y(0) = α. (14)

The Poincaré map associated with f

is the function P : [0, 1] → [0, 1] such that P (α) := w(α, T ).

Let α

be a fixed point of P . We say that α

is stable (resp. unstable) if P

(α

) < 1 (resp. P

(α

) > 1).

We give the link between equilibrium states of (11) and fixed points of the Poincaré map associated with f

. First of all, it follows from the definition of P that a real number α ∈ [0, 1] is a fixed point of P if and only if w(α, ·) is an equilibrium state of (11).

Proposition 2.4. Let α be a fixed point of P . We have P

(α) = e

. Proof. We have

∂

w(α, t) = f

(t, w(α, t)), ∀t ∈

.

Differentiating with respect to α, we obtain that ∂

w(α, ·) solves the linear ODE y

= yf

(s, w(α, s)). It follows that

∂

w(α, t) = ∂

w(α, 0)e

t

0f_u^T(s,w(α,s))ds

, ∀t ∈

. If we take t = T , we have that ∂

w(α, T ) = ∂

w(α, 0)e

T

0 f_u^T(s,w(α,s))ds

, and since ∂

w(α, 0) = 1, we infer that ∂

w(α, T ) = e

T

0 f_u^T(s,w(α,s))ds

. In other words, by Proposition 2.2, P

(α) = e

.

Consequently, the fact that a fixed point α of the Poincaré map associated with f

is stable (resp. unstable) in the sense of Definition 2.3 is equivalent to the fact that the principal eigenvalue associated with w(α, ·) and f

is positive (resp. negative), that is, by Proposition (2.3), the solution w(α, ·) of (14) is a uniformly asymptotic stable (resp.

unstable) equilibrium of (11).

In particular, in our paper, the hypothetis (2) implies that 0 and 1 are two fixed points

of the Poincaré map associated with f

, and the condition (3) is a condition of positivity

of the principal eigenvalues associated with 0 and 1. In this way, the equilibria 0 and 1

are uniformly asymptotically stable for the equation (1).

3 Uniqueness and monotonicity of pulsating front

This section is devoted to the proof of Theorem 1.1.

3.1 Two comparison principles

Lemma 3.1. Let us fix c ∈

, R

∈

and α ∈ (0, 1). We consider two functions g and g of class C

(

× [0, 1],

), T-periodic and such that

g (t, u) ≤ g(t, u), ∀t ∈

, ∀u ∈ [0, 1]. (15) We assume g satisfies the hypothesis (2) and the first inequality of (3).

Suppose there exist two functions v :

× [R

, +∞) → [0, 1], (t, ξ) 7→ v(t, ξ) and v :

R

× [R

, +∞) → [0, 1], (t, ξ) 7→ v (t, ξ) of class C

(

) in t uniformly for ξ ∈ [R

, +∞) and of class C

([R

, +∞)) in ξ uniformly for t ∈

, and such that

∂

v − c∂

v − ∂

v ≥ g(t, v) on

× [R

, +∞), (16)

∂

v − c∂

v − ∂

v ≤ g(t, v) on

× [R

, +∞), (17) v (t, R

) ≤ v(t, R

), ∀t ∈

, (18)

v(·, +∞) = 0 uniformly on

. (19)

There exists δ

∈ (0, 1) depending only on g such that if we have

v (t, ξ) ≤ δ

, ∀t ∈

, ∀ξ ∈ [R

, +∞), (20) then

v(t, x) ≤ v(t, x), ∀t ∈

, ∀ξ ∈ [R

, +∞).

Proof. Let us first introduce a few notations. We denote λ

and Φ

the principal eigen- value and the principal eigenfunction associated with the function g and the equilibrium 0. We saw in Section 2 that Φ

depends only on t (and not on ξ). Consequently, we have

Φ

(t) = (λ

+ g

(t, 0))Φ

(t), ∀t ∈

, (21) We also saw in Section 2 that λ

> 0. Since g is of class C

(

× [0, 1],

) and periodic in t, there exists δ

> 0 such that

∀t ∈

, ∀(u, u

) ∈ [0, 1], |u − u

| ≤ δ

⇒ |g

(t, u) − g

(t, u

)| < λ

2 . (22) The general strategy to prove Lemma 3.1 consists in using a sliding method. To do so, we define

ε

= inf

ε ≥ 0 | v − v Φ

+ ε ≥ 0 on

× [R

, +∞)

.

We note that ε

is a real number since v and v are bounded and min

Φ

> 0. Further- more, by continuity

v − v Φ

+ ε

≥ 0 on

× [R

, +∞). (23)

We are going to show bwoc that ε

= 0. Thus let us suppose that ε

> 0. We consider a sequence (ε

)

satisfying ε

n→+∞

−−−−→ ε

and 0 < ε

< ε

. There exists (t

, ξ

) ∈

R

× [R

, +∞) such that

v(t

, ξ

) − v(t

, ξ

)

Φ

(t

) + ε

< 0. (24)

We write t

= k

T + t

, with k

∈

and |t

| < T . Since (t

)

is bounded, we thus have up to extraction of a subsequence that t

−−−−→

t

. Furthermore, the sequence (ξ

)

is also bounded. Indeed, let us suppose it is not the case. So, according to (19), there exists R ∈

such that

∀n ∈

, ∀ξ ≥ R, 0 ≤ v(t

, ξ) ≤ ε

4 min

R

Φ

.

Now, if (ξ

)

is not bounded, there exists N ∈

such that ξ

≥ R and ε

>

. So, v(t

, ξ

) − ε

Φ

(t

) ≤ v(t

, ξ

) − ε

2 min

R

Φ

≤ ε

4 min

R

Φ

− ε

2 min

R

Φ

< 0.

Now, by (24) we have v(t

, ξ

) ≤ v (t

, ξ

) − ε

Φ

(t

). Hence v(t

, ξ

) < 0, which contradicts the fact that v ∈ [0, 1]. We thus have up to extraction of a subsequence that ξ

n→+∞

−−−−→ ξ

∈ [R

, +∞). We define v

(t, ξ) = v(t + k

T, ξ), and v

(t, ξ) = v(t + k

T, ξ).

As g and g are T −periodic, v

and v

satisfy respectively (16) and (17). Furthermore, v

and v

converge up to extraction of a subsequence respectively to v

∈ [0, 1] and v

∈ [0, 1]

in C

(

× [R

, +∞)). Passing to the limit in (23) we obtain v

− v

Φ

+ ε

≥ 0 on

× [R

, +∞).

According to (24), we have

v

(t

, ξ

) − v

(t

, ξ

)

Φ

(t

) + ε

< 0.

So, passing to the limit, we have

v

(t

, ξ

) − v

(t

, ξ

)

Φ

(t

) + ε

= 0. (25)

We define the open set

Ω =

(t, ξ) ∈

× [R

, +∞) | 0 ≤ v

(t, ξ) < v

(t, ξ)

.

This set is open by continuity and by (18). We note that (t

, ξ

) ∈ Ω because v

(t

, ξ

) − v

(t

, ξ

) = −ε

Φ

(t

) < 0. Furthermore, according to (18), we have v

(t

, R

) − v

(t

, R

) ≥ 0. So, ξ

> R

. We will apply a strong maximum principle to the non- negative function

z = v

− v

Φ

+ ε

.

There exists θ :

× [R

, +∞) → [0, 1], (t, ξ) 7→ θ(t, ξ), with θ(t, ξ) between v

(t, ξ) and v

(t, ξ) such that we have on

× [R

, +∞)

∂

z − c∂

z − ∂

z ≥ g(t, v

) − g(t, v

)

Φ

− ∂

Φ

(v

− v

)

Φ

((16) and (17))

≥ g(t, v

) − g(t, v

) Φ

− (g

(t, 0) + λ

)(v

− v

) Φ

((21) and (15))

= 1

Φ

(g

(t, θ) − g

(t, 0) − λ

)(v

− v

),

= (g

(t, θ) − g

(t, 0) − λ

(z − ε

)g

(t, θ) − g

(t, 0) − λ

).

We define the bounded function α(t, ξ) := g

(t, θ(t, ξ)) − g

(t, 0) − λ

on

× [R

, +∞).

According to (20) we have v

≤ δ

on

× [R

, +∞). Furthermore, we have 0 ≤ v

< v

on Ω, whence |θ| ≤ δ

on Ω. So, by (22) we have on Ω

∂

z − c∂

z − ∂

z − α(t, ξ)z ≥ −ε

(g

(t, θ) − g

(t, 0) − λ

) ≥ λ

ε

2 > 0.

Furthermore, according to (23), we have z ≥ 0 on Ω and even on

× [R

, +∞). As z(t

, ξ

) = 0 and (t

, ξ

) ∈ Ω, it follows from [23, Th2 p168] that

z ≡ 0 on Ω

, (26)

where Ω

is the set of point (t, ξ) ∈ Ω such that there exists a continuous path γ : [0, 1] → Ω such that γ(0) = (t, ξ) and γ(1) = (t

, ξ

), and the time component of γ(s) is nondecreasing with respect to s ∈ [0, 1]. Now, we define

ξ = inf

ξ ≥ R

| ∀ ξ ˜ ∈ (ξ, ξ

], (t

, ξ) ˜ ∈ Ω

o

. The equality (26) and the continuity of z imply that

z(t

, ξ) = 0.

So, according to (18), we have ξ > R

(because v

(t

, R

) ≥ v

(t

, R

)). Consequently, the definition of ξ and the continuity of v

and v

imply that

v

(t

, ξ) = v

(t

, ξ).

Finally, the previous two displayed equalities imply that ε

= 0, which contradicts the fact that ε

is positive. Consequently ε

= 0 and the lemma is proved.

With a similar proof, we can obtain the following lemma.

Lemma 3.2. Let us fix c ∈

, R

∈

and α ∈ (0, 1). We consider two functions g and g of class C

(

× [0, 1],

), T-periodic and such that

g(t, u) ≤ g(t, u), ∀t ∈ [0, T ], ∀u ∈ [0, 1].

We assume g satisfies the hypothesis (2) and the second inequality of (3).

Suppose there exist two functions v :

× (−∞, R

] → [0, 1], (t, ξ) 7→ v(t, ξ) and v :

R

× (−∞, R

] → [0, 1], (t, ξ) 7→ v(t, ξ) of class C

(

) in t uniformly for ξ ∈ (−∞, R

] and of class C

((−∞, R

]) in ξ uniformly for t ∈

, and such that

∂

v − c∂

v − ∂

v ≥ g(t, v) on

× (−∞, R

],

∂

v − c∂

v − ∂

v ≤ g(t, v) on

× (−∞, R

], v(t, R

) ≤ v(t, R

), ∀t ∈

,

v(t, −∞) = 1 uniformly on

. There exists δ

∈ (0, 1) depending only on g such that, if we have

v(t, ξ) ≥ 1 − δ

, ∀t ∈

, ∀ξ ∈ (−∞, R

], then

v(t, x) ≤ v(t, x) ∀t ∈

, ∀ξ ∈ (−∞, R

].

3.2 Monotonicity of the front

Let us consider (U, c) a solution of (5), with U :

×

→ [0, 1]. We want to prove that for all τ ≥ 0, we have

U (t, ξ) ≤ U(t, ξ − τ), ∀t ∈

, ∀ξ ∈

. We start with the following lemma.

Lemma 3.3. There exists τ

≥ 0 such that for any τ ≥ τ

, we have U (t, ξ) ≤ U (t, ξ − τ), ∀t ∈

, ∀ξ ∈

.

Proof. As U (·, +∞) = 0 uniformly on

, there exists a real R

such that

U (t, ξ) ≤ δ

, ∀t ∈

, ∀ξ ∈ [R

, +∞), (27) with δ

∈ (0, 1) defined in Lemma 3.1 with g = g = f.

As U (·, −∞) = 1 uniformly on

, there exists τ

≥ 0 such that

U (t, ξ − τ ) ≥ 1 − δ

, ∀τ ≥ τ

, ∀t ∈

, ∀ξ ∈ (−∞, R

], where δ

∈ (0, 1) is defined in Lemma 3.2 with g = g = f .

Without loss of generality, we can assume that max {δ

, δ

} < 1/2, whence U (t, R

) ≤ U (t, R

− τ ) for all t ∈

and τ ≥ τ

. We can apply Lemma 3.1 on

× [R

, +∞) and Lemma 3.2 on

× (−∞, R

] to the functions v = U and v = U (·, · − τ ) for any τ ≥ τ

, and the lemma is proved.

Proposition 3.4. We have

∂

U (t, ξ) < 0, ∀t ∈

, ∀ξ ∈

.