Louis Gar Énaux
email: louis.garenaux@math.univ-toulouse.fr

NONLINEAR CONVECTIVE STABILITY OF A CRITICAL PULLED FRONT UNDERGOING A TURING BIFURCATION AT ITS BACK: A CASE STUDY

Keywords: Mathematics Subject Classification. 35K57, 35C07, 35B35, 35B36, 35Q56, 35K46 Reaction-diffusion, Propagating monostable front, Asymptotic stability, Turing instability, Weighted spaces, Point-wise resolvent estimate, Mode filters, Amplitude equation

We investigate a specific reaction-diffusion system that admits a monostable pulled front propagating at constant critical speed. When a small parameter changes sign, the stable equilibrium behind the front destabilizes, due to essential spectrum crossing the imaginary axis, causing a Turing bifurcation. Despite both equilibrium states are unstable, the front continues to exist, and is shown to be asymptotically stable, against suitably-localized perturbations, with algebraic temporal decay rate t -3/2 . To obtain such decay, we rely on point-wise semigroup estimates, and show that the Turing pattern behind the front remains bounded in time, by use of mode-filters.

Introduction

Analysis of evolution problems in PDE often reveals a large variety of nontrivial dynamics. Among parabolic problems, the class of reaction-diffusion equations furnish a wide class of stable -and thus observable -solutions: planar waves such as propagating fronts or periodic patterns, rotating spirals, etc.

We are interested here in a system coupling a Kolmogorov-Petrovski-Piskunov (KPP) equation together with a Swift-Hohenberg (SH) equation, We work with x ∈ R and t > 0:

(1)

∂ t u = d∂ xx u + αu(1 -u 2 ) + βv, ∂ t v = -(∂ xx + 1) 2 v + v(µ -σv 2 ) -γv(1 -u),
with parameters d, α, µ, σ, γ positive, and β ∈ R nonzero. The scalar KPP equation is a typical model for front propagation from a stable to an unstable state [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF][START_REF] Fisher | The wave of advance of advantageous genes[END_REF]:

(2)

∂ t u = d∂ xx u + f (u).
The diffusion coefficient d is positive and the reaction term f ∈ C 2 is of KPP type: it admits two equilibrium states f (0) = 0 = f (1) with distinct stability w.r.t. time f (0) > 0 > f (1). Generally, a concavity hypothesis is added: f (u) < 0 for u ∈ (0, 1). Although f (u) = αu(1u) is the canonical choice, we will work with f (u) = αu(1u 2 ). All statements and proofs below adapt to the u(1u) case. Equation (2) admits a family of fronts (q c ) c>0 -with q c propagating at constant speed c -that connect the stable equilibrium point (q c → 1 when x → -∞) to the unstable one (q c → 0 when x → +∞) [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. Supercritical fronts c > c * := 2 d f (0) = 2 √ dα are convectively stable with exponential decay in time, against sufficiently localized perturbations. Indeed, when set in a moving frame, conjugating the problem with spatial exponential weights allow to both stabilize the essential spectrum and erase the eigenvalue at λ = 0, creating a spectral gap at linear level [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF]. Stability of the critical front q * := q c * is more involved, since the linear essential spectrum can only be marginally stabilized : in the optimal exponentially weighted space, essential spectrum lies at left of the imaginary axis but touches it, due to the presence of absolute spectrum -ξ 2 : ξ ∈ R ⊂ C up to the origin. Last advances in this direction [START_REF] Gallay | Local stability of critical fronts in nonlinear parabolic partial differential equations[END_REF][START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF][START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF] state that q * is asymptotically stable with decay rate t -3/2 , and that this rate is optimal. Gallay used renormalization group method; Faye and Holzer relied on point-wise estimates for the resolvent of the full problem; Avery and Scheel reduced the problem to its asymptotic properties through far-field decomposition, and use resolvent estimates on each side. For subcritical speeds 0 < c < c * , there still exists nonmonotonic, homoclinic trajectories q c from 1 to 0. However, Sturm Liouville theory ensures the existence of unstable point spectrum, preventing stability.

The scalar SH equation is a simple model for the formation of periodic pattern [SH77, CH93]:

(3)

∂ t v = -(∂ xx + 1) 2 v + µv -σv 3 .
When the small parameter µ becomes positive, (3) undergoes a supercritical Turing bifurcation: a curve of essential spectrum crosses the imaginary axis twice, for nonzero Fourier frequencies ±1. This causes the constant equilibrium state u = 0 to bifurcate into periodic profile of amplitude √ µ.

It can be obtained through center manifold theory [START_REF] Eckmann | Propagating fronts and the center manifold theorem[END_REF]. For all values of µ, the coupled system (1) admits Q = (q * , 0) T as a front with propagation speed c * . The linear coupling term βv feeds the front dynamic with oscillatory pattern, while the nonlinear coupling term γv(1u) stabilize at linear level ahead of the front. Thus oscillating profile are expected to appear only behind the front.

Multiple works about bistable fronti.e. a front that connects two stable states: f (-1) = 0 = f (1) and f (-1) < 0, f (1) < 0 -undergoing a Turing bifurcation have been done. When bifurcation occurs behind the front, [START_REF] Sandstede | Essential instabilities of fronts: bifurcation, and bifurcation failure[END_REF] showed that no modulated front appears: the Turing pattern travels at slow speed O( √ µ), hence is left behind the front. However, the front survive after bifurcation, although through a non-modulated form. Its nonlinear stability is showed in [START_REF] Beck | Nonlinear convective stability of travelling fronts near turing and hopf instabilities[END_REF] for a general second-order setting.

When the state ahead of a bistable front bifurcates, [START_REF] Sandstede | Essential instabilities of fronts: bifurcation, and bifurcation failure[END_REF] showed existence and linear stability of modulated frontsi.e. coherent structures that link stable state behind to Turing pattern ahead -in a general second-order setting. For a system that ressembles (1), [START_REF] Gallay | Stable transport of information near essentially unstable localized structures[END_REF] obtain nonlinear stability of such structure. We also mention [START_REF] Ghazaryan | Nonlinear convective instability of turing unstable fronts near onset: A case study[END_REF] for a quadratic coupling βv 2 instead of a linear one. The signed term therein allows to obtain a priori estimates by applying comparison principle to simplify the system. When bifurcation occurs both ahead and behind a bistable front, spectral stability of either traveling or standing pulses is showed for a general system [START_REF] Sandstede | Spectral stability of modulated travelling waves bifurcating near essential instabilities[END_REF] and [START_REF] Sandstede | Absolute instabilities of standing pulses[END_REF], while nonlinear stability of a traveling pulse for a precise system is showed in [START_REF] Gallay | Stable transport of information near essentially unstable localized structures[END_REF].

For monostable KPP fronts, previous works rather investigated the Turing bifurcation in a nonlocal KPP context. [START_REF] Berestycki | The non-local Fisher-KPP equation: travelling waves and steady states[END_REF] first investigate the existence of steady states and propagating fronts, together with the monotonicity of this fronts. A precise construction of modulated fronts is made in [START_REF] Faye | Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach[END_REF], while global properties are obtained in [START_REF] Hamel | On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds[END_REF]: boundedness of solutions when initial condition is non-positive, together with bounds on propagation speeds. It seems that the question of the stability of such structure is open, as remarked in [START_REF] Nadin | Can a traveling wave connect two unstable states? the case of the nonlocal Fisher equation[END_REF][START_REF] Hamel | On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds[END_REF]. Our conclusion is that Turing bifurcation behave the same when it occurs behind a monostable or a bistable front: see fig. 1 for a view on a typical solution. However, the spectrum in the present situation is quite different. On one hand, the essential spectrum is unstable, which require to work in exponentially weighted spaces. Even so, the optimal exponential weight only allows to marginally stabilize the spectrum. On the other hand, the translation eigenvalue 0 ∈ C in the essential spectrum does not contribute to a zero of the so-called Evans function, due to the corresponding eigenfunction q * having only weak exponential decay: q * (x) ∼ (ax + b)e -c * 2 x when x → +∞. This removes the technical aspect of stability up to a shift in space, also referred to as orbital stability or modulation.

Let us emphasize that the coupling at linear level βv forces us to work with the same weights on the KPP and SH components, see further remark 3.2. This typically happens at +∞: conjugation by the critical KPP weight, although necessary to stabilize the KPP spectrum, may destabilize the SH part of the spectrum, if general parameters α, d, γ are set. We counter this effect by imposing a lower bound on γ, which allows to stabilize the SH spectrum at +∞.

When no such parameter is available, one has to work with unstable spectrum in any weighted space. This is referred to as remnant instability in [START_REF] Faye | Invasion into remnant instability: a case study of front dynamics[END_REF], where this exact situation is investigated. Using a precise decomposition of perturbations, they conclude on nonlinear stability of the front Q * for µ < 0. Although we could have followed this direction in our situation, we prefer to set γ large enough so that computations are lightened.

In the supercritical case c > c * , it is possible to obtain spectral gap for the essential spectrum, and to remove the translation eigenvalue, by use of a strong enough weight [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF]. In such case, the situation is simpler than bistable case: exponential stability without a phase can be obtained by adapting the current proofs or the ones of [START_REF] Beck | Nonlinear convective stability of travelling fronts near turing and hopf instabilities[END_REF]. If one rather use the optimal supercritical weight, the situation has to be discussed.

In the monostable critical case, instability ahead of the front is still an open problem, and would be an interesting direction to follow. For example, one can replace the coupling term γv(1u) by γuv. We point out that a modulated front for such system is necessarily non-positive, which prevent the use of a KPP reaction term u(1u). To ensure existence of solutions globally in time, it is possible to rather study a monostable front emanating from a bistable reaction term u(1u 2 ).

x u(t, •) v(t, •) O( √ µ) O( √ µ) c *

Main result

2.1. Notations. We investigate now the stability of the traveling front Q * = (q * , 0) T . Insert the following ansatz in (1):

u(t, x) v(t, x) = Q(x) + P (t, x),
with x = xc * t. Then the perturbation P = (p 1 , p 2 ) T satisfies

∂ t P = AP + N (P ) := A kpp β 0 A sh P + N 1 (P ) N 2 (P ) ,
where the linear operator

A : L 2 (R) 2 → L 2 (R) 2 is closed, has dense domain H 2 (R) × H 4 (R)
and is defined by: (4)

A kpp = d∂ xx + c * ∂ x + α(1 -3q * (x)), A sh = -(1 + ∂ xx ) 2 + c * ∂ x + µ -γ(1 -q * (x)),
while remaining nonlinear terms express as:

(5)

N 1 (P ) = -α(3q * (x)p 1 2 + p 1 3 ), N 2 (P ) = γp 1 p 2 -σp 2 3 .
The essential spectrum of A is unstable, see fig. 2, we do not hope for an asymptotic stability result against general perturbation P ∈ H 2 (R) × H 4 (R). Instead, we restrict to weighted perturbations

P (t, x) = ω(x)U (t, x), with U ∈ H 2 (R)×H 4 (R).
As we shall see in further remark 3.2, the coupling term βv imposes us to use scalar weight. Namely ω * := ω kpp ω sh , with smooth positive functions (6) where the exponent θ < 0 is small, see proposition 3.1. The weight ω kpp acts only at +∞, and allows to marginally stabilize the KPP spectrum, while ω sh acts at -∞ and stabilizes the SH spectrum. Hence, we insert the new ansatz

ω kpp (x) := 1 if x ≤ -1, e -c * 2d x if x ≥ 1, ω sh (x) := e θx if x ≤ -1, 1 if x ≥ 1, Re λ Im λ Σ(A kpp,+ ) Σ(A kpp,-) Σ(A sh,-) Re λ Im λ Σ(L kpp,+ ) Σ(L sh,+ ) Σ(L kpp,-) Σ(L sh,-) Re λ Im λ Σ(T -+ c * ∂x) Figure 2.
(7) u(t, x) v(t, x) = Q(x) + ω * (x)U (t, x)
in (1) to obtain that U = (u 1 , u 2 ) T satisfies:

(8)

∂ t U = LU + N (U ) := L kpp β 0 L sh U + N 1 (U ) N 2 (U ) ,
where all linear terms are closed, densely defined operators. They are obtained as conjugation of the unweighted linear operators,

L i = ω -1 * A i ω * when i ∈ {kpp, sh} and nonlinear terms express according to N i (U ) = ω -1 * N i (ω * U
). Since the translation eigenmode q * in weighted spaces is unbounded at x → +∞, we expect U to decay in time with rate t -3/2 if the following assumption of a marginally stable essential spectrum holds, see [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF][START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF]:

(H 1 ) Σ ess (L) ⊂ {Re λ ≤ -η} ∪ -ξ 2 : ξ ∈ R .
However at nonlinear level, it appears we also need to study the dynamic near the steady state (u, v) = (1, 0) undergoing a Turing bifurcation. This is classic for such problem and was already done in [START_REF] Beck | Nonlinear convective stability of travelling fronts near turing and hopf instabilities[END_REF]. More details are given in section 2.2. We do so with the second ansatz

(9) u(t, x) v(t, x) = Q(x) + ρ * (x)ω kpp (x) V (t, x).
Here again x = xc * t, and the polynomial weight ρ * is a smooth positive function defined by

(10) ρ * (x) := 1 if x ≤ -1, x := √ 1 + x 2 if x ≥ 1.
We note := ρ * ω kpp , and insert (9) into (1) to obtain that

∂ t V -c * (∂ x )V = A( V ) -c * ∂ x ( V ) + N ( V ),
which rewrites as

∂ t V = T V + Q(V ), = T -V + Q -(V ) + S(x, V ), ( 11
)
where linear operators express as:

T := (x) -1 (A -c * ∂ x ) (x) + c * ∂ x (x) T -= lim x→-∞ T (x),
while nonlinear terms are defined by

Q(V ) = (x) -1 N ( (x)V ), and Q -(V ) = lim x→-∞ Q(V ).
We are left with an error term

S(V ) := T V -T -V + Q(V ) -Q -(V ).
Since (x) = 1 for x ≤ -1, we decompose T -as:

(12) T -= d∂ xx -2α β 0 -(1 + ∂ xx ) 2 + 0 0 0 µ =: T 0 + T µ .
and Q -as

Q -(V ) = -3αv 1 2 γv 1 v 2 + -αv 1 3 -σv 2 3 =: Q 2 + Q 3 .
Thus we get the following expression for the source term:

S(•, V ) = -1 (A -c * ∂ x ) -(A -c * ∂ x ) + c * ∂ x + 3α(1 -q 2 * ) 0 0 -γ(1 -q * ) V (13) + ( -1) -3αq * v 1 2 γv 1 v 2 + ( 2 -1) -αv 1 3 -σv 2 3 .
To control perturbation V , we will compute an amplitude equation, which reveals to be a Ginzburg-Landau equation with real coefficients (GL)

∂ T A = 4∂ XX A + A + bA|A| 2 .
An important property for our result to hold is that the Turing bifurcation is supercritical :

(H 2 ) In (GL), the coefficient b is negative.
We emphasize that the perturbation V is not traveling in the laboratory frame x, but that we measure it using a weight that follows the front. Despite V is measured in the steady frame x, it would be possible to replace ansatz (9) by (u, v) T = Q(x) + ρ * (x)ω kpp (x)V (t, x). The actual choice lighten the computation to obtain the GL equation. Conjugation by := ρ * ω kpp is needed so that proposition 4.5 holds: this weight is natural for KPP equation, in the sense that its spatial decay at +∞ copy the one of the translation eigenfunction q * [START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF].

The perturbations U and V are linked through

(14) V (t, x) = ω sh (x) ρ * (x) U (t, x).
To measure space localization and regularity, we use Sobolev spaces, and introduce the following norm:

U X := U W 2,∞ (R)×W 4,∞ (R) + ρ 3 * U L ∞ (R) .
We can now state our main result.

Theorem 2.1. There exists an open, nonempty set of parameters Ω such that for (α, β, d, σ, γ) in Ω, both hypothesis (H 1 ) and (H 2 ) holds. For such a choice of parameters, the following holds. There exists positive constants C, µ 0 , and δ such that for all 0 < µ < µ 0 , if U 0 , V 0 satisfies

K 0 := 1 √ µ V 0 W 1,∞ (R) + U 0 X ≤ δ,
then the solution of (1) with initial condition Q + ω kpp ω sh U 0 = Q + ρ * ω kpp V 0 exists for all time, and writes as

Q(x) + ω kpp (x)ω sh (x)U (t, x) = Q(x) + ρ * (x)ω kpp (x)V (t, x)
. For all t > 0, it satisfies:

U (t, •) ρ * L ∞ (R) ≤ C U 0 ρ 3 * L ∞ (R) (1 + t) 3/2 , V (t, •) L ∞ (R) ≤ C √ µ (1 + K 0 e -t/µ ).
It happens we can refine the decay of perturbation U .

Corollary 2.2. Let 1 < p ≤ +∞ and (α, β, d, σ, γ) ∈ Ω, with Ω as in theorem 2.1. There exists positive constants C, µ 0 , δ and η such that if

K 0 ≤ δ, U 0 ρ 3 * L p (R)
≤ δ, then the solution U = (u 1 , u 2 ) T of (1) with initial condition U 0 satisfies:

u 1 (t, •) ρ * L p (R) ≤ C U 0 ρ 3 * L p (R) (1 + t) 3 2 -1 2p , u 2 (t, •) L p (R) ≤ Ce -ηt U 0 L p (R) .
2.2. Sketch of the proof. We informally present the main ideas of our proof. First, the dynamic for fully weighted perturbation U writes as:

∂ t U = LU + N (U ).
Where the linear operator L has marginally stable spectrum, similarly to the KPP equation, and the nonlinear part N is unbounded w.r.t. to x ∈ R. At linear level, we expect an algebraic decay: t -3/2 . We follow the approach from [START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF]: we first look at the solution to the linear Cauchy problem (15)

∂ t p = Lp, p(0, •) = p 0 ,
and assume that it is expressed through a kernel: p(t, x) = ´R G t (x, y)p 0 (y)dy. The matrix valued function G • (•, y) has to satisfy the linear problem (15), with a Dirac delta initial condition: G 0 (x, y) = δ y (x) 1 0 0 1 . Remark that G is an upper-triangular matrix, due to the triangular structure of L and G 0 . We note it

G = G kpp G co 0 G sh ,
where G i solves ∂ t G t = L i G t when i ∈ {kpp, sh}, and G co t accounts for the coupling terms. We then apply Laplace transform to obtain the spectral Green kernel G λ (x, y) = ´+∞ 0 e -λt G t (x, y)dt, that satisfies the fundamental eigenproblem

(λ -L)G λ = δ y ,
for suitable λ ∈ C. As above, it is an upper-triangular matrix that writes

G = G kpp G co 0 G sh ,
where each G i is the Laplace transform of G i for i ∈ {kpp, sh, co}. Hence, they satisfy

(λ -L kpp )G kpp λ (•, y) = δ y , (λ -L sh )G sh λ (•, y) = δ y , (λ -L kpp )G co λ (•, y) = βG sh λ (•, y).
The homogeneous eigenproblems (λ -L i )φ = 0 with i ∈ {kpp, sh} are ODEs, their solutions admit exponential behaviors at ±∞:

φ(x) = e ν(λ)x (1 + e -r|x| κ(x)),
where ν ∈ C is a solution of the dispersion relations λ -L i,± (ν) = 0; the real r > 0 does not depend on ν nor λ; κ is a bounded function on the half-line R ± . Such solutions can be concatenated to construct G i λ (•, y) ∈ L 2 (R). Then, the coupled spectral green function expresses as the

L 2 (R)-inner product G co λ (x, y) = G kpp λ (x, •), βG sh λ (•, y)
. This leads to spatial localization of G λ , which is converted into temporal decay for G t through the inverse Laplace transform:

G t (x, y) = 1 2iπ ˆΛ e λt G λ (x, y)dλ,
where Λ is a contour in the resolvent set C\Σ(L), that can be chosen as a continuous deformation of a sectorial contour, see [Dav02, section 3]. To control the full non-linear dynamic, we then use Duhamel's formula:

U (t, x) = ˆR G t (x, y)U 0 (y)dy + ˆt 0 ˆR G t-τ (x, y)N (U (τ, y))dydτ.
The fact that N (U ) is unbounded 1 is a major issue. In [START_REF] Beck | Nonlinear convective stability of travelling fronts near turing and hopf instabilities[END_REF], it is absorbed by transforming nonlinear terms into linear ones: writing ω -1 sh (ω sh u) p = (ω sh u) p-1 u and showing that ω sh u L ∞ (R) is bounded w.r.t. time. We follow the same line. The material presented above is detailed in section 3, and leads to the following proposition, the proof of which can be found in section 3.5.

Proposition 2.3 (V bounded implies decay of U ). Assume hypothesis (H 1 ) holds. There exists positive constants δ, µ 0 , C stab and η such that for all 0 < µ < µ 0 , the following holds. Fix C and t V positive constants, an initial condition U 0 such that U 0 ρ 3 * ∈ L p (R), and assume that for all

0 ≤ t ≤ t V , V (t) L ∞ (R) ≤ C ≤ δ. 1 Due to |ω sh (x)| → ∞ when x → -∞
Then the solution U for the Cauchy problem (8) with initial condition U 0 is defined for all 0 ≤ t ≤ T , and satisfies

u 1 (t) ρ * L p (R) ≤ C stab U 0 ρ 3 * L p (R) (1 + t) 3 2 -1 2p , u 2 (t) L p (R) ≤ C stab e -ηt U 0 L p (R) .
Furthermore, C stab depends neither on C or t V .

Second, we turn to ω sh ρ * U = V , and show it is bounded in time. 2 Remind that

∂ t V = T V + Q(V ) = T -V + Q -(V ) + S(x, V ).
We show that S is sufficiently localized in space so that we can extract a ω sh from it. This allows to control S(V ) by U/ρ * -see ( 14) -which ensures decay in time as done in [GS07, below Proposition 3.3] and [BGS09, Lemma 4.1]. Hence after a suitable time, V is driven by the dynamic at -∞, which allows to get rid of the marginally stable curve of essential spectrum at +∞. Since T - undergoes a Turing bifurcation, we follow the steps of [START_REF] Schneider | Global existence via Ginzburg-Landau formalism and pseudo-orbits of Ginzburg-Landau approximations[END_REF][START_REF] Beck | Nonlinear convective stability of travelling fronts near turing and hopf instabilities[END_REF], that mostly relies on the use of mode-filters, see [START_REF] Schneider | A new estimate for the Ginzburg-Landau approximation on the real axis[END_REF].

We first show that periodic patterns are naturally selected: after a time T µ , perturbation V has at first order an oscillating profile. This is commonly referred to as the approximation property. For such profiles, using multi-scale analysis, dynamic of the whole system reduces to an amplitude equation, which in our case appears to be the Ginzburg-Landau (GL) equation:

setting ε = √ µ, if V = ψ(A) + O(ε 2 ), with ψ(ε, A)(t, x) := εA(ε 2 t, εx)e ix c + c.c, 3 with a suitable c ∈ R 2 , then A(T, X) ∈ C satisfies ∂ T A = 4∂ XX A + A + bA|A| 2 .
We refer to [START_REF] Mielke | The Ginzburg-Landau equation in its role as a modulation equation[END_REF][START_REF] Mielke | Attractors for modulation equations on unbounded domains-existence and comparison[END_REF] for the derivation of amplitude equation. For suitable values of the parameter γ > 0, coefficient b ∈ R is negative, has shown in appendix A. Hence the Turing bifurcation is supercritical, and (GL) is known to have a bounded global attractor -see [MS95, Theorem 3.4] -which ensures that A is bounded w.r.t. time. In case where the bifurcation is subcritical, a quintic term is often add to recover a precise behavior, we do not explore this line.

To conclude that V stays small for all time, we use the approximation property: if V is close to ψ(A) at time t 0 > 0, then the solution of the whole system emanating from V 0 is defined upon time t 0 + T /µ, and remains close to ψ(A). This last step can be applied as many time as needed, without deterioration of constants. All these arguments are made precise in section 4. They lead to the next proposition, which is proved on page 31.

Proposition 2.4 (Decay of U implies V bounded). Assume hypothesis (H 2 ) holds. There exists positive constants δ, µ 0 such that for all 0 < µ < µ 0 , the following holds. Assume that

K 0 := 1 √ µ V 0 W 1,∞ (R) + U 0 X ≤ δ.
Then V (t) is defined for all times t ≥ 0. Assume further that C 1 and t U are positive constants, such that for all 0 ≤ t ≤ t U :

U (t) ρ * L ∞ (R) ≤ C 1 U 0 ρ 3 * L ∞ (R) (1 + t) 3/2 .
2 The presence of an extra 1 ρ * is necessary so that source term S(x, V ) decay in time, see proposition 4.5. 3 Here and in the following, we note c.c for the complex conjugate: z + c.c := z + z = 2 Re(z).

Then there exists C 2 -that may depend on C 1 , but neither on t U , δ or µ 0 -such that for all

0 ≤ t ≤ t U , V (t) L ∞ (R) ≤ C 2 √ µ (1 + K 0 e -t/µ ).
Finally, we combine propositions 2.3 and 2.4 to prove theorem 2.1. It may seems unclear how to use jointly those two propositions. The important point is that C stab is independent of the bound on V . It reads as follows.

Proof. Theorem 2.1.

First apply lemma A.1 to obtain the existence of Ω that allows to fulfill both hypothesis (H 1 ) and (H 2 ). Take µ 0 small enough, and fix ρ * V 0 = ω sh U 0 such that:

U 0 ρ 3 * ∈ L ∞ (R), K 0 ≤ δ.
Since T is sectorial, and 11) is uniquely defined on a open, nonempty, maximal set I -see [START_REF] Henry | Geometric Theory of Semilinear Parabolic Equations[END_REF]. Hence there exists t 0 and C positive constants such that

V → Q(V ) is locally Lipschitz, the solution t ∈ I → V (t) ∈ L ∞ (R) to equation (
V (t) L ∞ (R) ≤ C
for all 0 ≤ t ≤ t 0 . Applying proposition 2.3, this ensures that for all 0 ≤ t ≤ t 0 , the perturbation U/ρ * is uniquely defined in L ∞ (R), and satisfies ( 16)

N (t) := 1 (1 + t) 3/2 U (t) ρ * L ∞ (R) ≤ C stab U 0 ρ 3 * L ∞ (R) .
Consider t 1 the first time where this inequality may fail:

t 1 := inf I, I := t > 0 : N (t) > C stab U 0 ρ 3 * L ∞ (R) .
Assuming by contradiction that I is nonempty, remark that 0 < t 0 ≤ t 1 < +∞. From proposition 2.4, V is defined for all times 0 ≤ t ≤ t 1 , with bound

V (t) L ∞ (R) ≤ C 2 √ µ (1 + K 0 ), 0 ≤ t ≤ t 1
where C 2 depends on C stab . Reasoning as above, there exists t 2 > t 1 such that V is defined up to time t 2 , with bound

V (t) L ∞ (R) ≤ 2C 2 √ µ (1 + K 0 ), 0 ≤ t ≤ t 2 .
Assuming that µ 0 (1+K 0 ) is small enough, we can assume 2C 2 √ µ (1+K 0 ) ≤ δ so that proposition 2.3 applies again, and we recover (16) for times 0 ≤ t ≤ t 2 . This is a contradiction with the definition of t 1 , hence we conclude that I is empty, and that (16) holds for all times. Applying proposition 2.4, we recover the claimed bound on V . Assuming now that ρ 3 * U 0 L p ≤ δ, we can finally apply proposition 2.3 to recover L p (R) estimates on U for all times.

Remark 2.5. To understand how V stays bounded in time, we separate critical from stable frequencies, using mode-filters in Fourier space. This imposes us to work with uniformly localized spaces, which are difficult to combine with Green's kernel approach. An alternative strategy would be to describe Turing patterns as solution of the eigenvalue ODE problem -hoping they will reflect on the Green kernel -and separate critical from stable mode in Laplace space. It would allow to work both in Sobolev spaces, and with the full dynamic T rather than the asymptotic one T -. This last point allow to remove the source term S, and as so to break the U -V interaction in our proof.

Decay of perturbations in fully weighted space

3.1. Essential and point spectrum. We say that a scalar differential operator T (x) = n j=0 a j (x)∂ j x is exponentially asymptotic if it converges with uniform exponential rate at ±∞: there exists r > 0 such that for all 0 ≤ j ≤ n,

lim x→±∞ e r|x| |a j (x) -a ± j | = 0.
We note Σ(T ) the spectrum of T , i.e. the set of complex numbers λ such that λ -T :

H n (R) ⊂ L 2 (R) → L 2 (R) is not bounded invertible.
To study the spectrum, we decompose it into two distinct parts. We say that λ -T is Fredholm if its Fredholm index

Fred(λ -T ) := dim ker(λ -T ) -codim im(λ -T )
is defined and finite. We define the point spectrum as

Σ pt (T ) := {λ ∈ Σ(T ) : Fred(λ -T ) = 0} ,
it corresponds to the subset in which the rank-nullity theorem holds. Then, the essential spectrum is the complementary set:

Σ ess (T ) := {λ ∈ Σ(T ) : Fred(λ -T ) is non defined or nonzero} .
With such definitions, the asymptotic operator with piece-wise constant coefficients

T ∞ (x) := T + = n j=0 a + j ∂ j x if x > 0, T -= n j=0 a - j ∂ j x if x < 0,
share the same essential spectrum as T , see [START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF]p 40]. If both T + and T -are elliptic -in one dimensional space, it comes down to both (-1) n/2 a + n and (-1) n/2 a - n being positive -then Σ ess (T ) = Σ ess (T ∞ ) is located to the left of the asymptotic Fredholm borders

Σ(T + ) ∪ Σ(T -) =    n j=0 a + j (iξ) j : ξ ∈ R    ∪    n j=0 a - j (iξ) j : ξ ∈ R    ⊂ C, see again [KP13, Theorem 3.1.13]. 4
For a matrix operator:

T = T 1,1 T 1,2 T 2,1 T 2,2
, the asymptotic spectra Σ(T ± ) are still obtained through Fourier transform. The operator λ -T + is invertible iff the matrix λ -T + (ξ) is invertible for all ξ ∈ R -since Fourier transform is an isometry -which is equivalent to ξ → det(λ -T + (ξ)) never vanishing. In our special case, L + is triangular, hence the determinant is nonzero if both diagonal coefficients maintain away from zero. We conclude that

Σ(L + ) = Σ(L kpp,+ ) ∪ Σ(L sh,+ ),
and the same goes at -∞. Once again, the essential spectrum of L is located to the left of Σ(L + ) ∪ Σ(L -).

Proposition 3.1 (Marginally stable essential spectrum). Fix α, d positive. Then there exists µ 0 > 0 such that for all 0 < µ < µ 0 , there exists θ < 0 such that the monotonic weight ω * = ω kpp ω sh , with

ω kpp (x) = 1 if x ≤ -1, e -c * 2d x if x ≥ 1, ω sh (x) = e θx if x ≤ -1, 1 if x ≥ 1,
satisfies the following. For all γ > γ rem , with γ rem as in [START_REF] Faye | Invasion into remnant instability: a case study of front dynamics[END_REF]:

γ rem := 8 α d 2 + 4 α d -2α + µ 0 ,
the operator L = ω -1 * Aω * has marginally stable essential spectrum. More precisely, the Fredholm curve corresponding to the essential KPP spectrum touches the imaginary axis only at λ = 0:

Σ(L kpp,+ ) = -ξ 2 : ξ ∈ R ,
while the three other Fredholm borders have spectral gap: there exists η > 0 depending only on µ 0 , d, α and γ such that

(17) Σ(L kpp,-) ∪ Σ(L sh,+ ) ∪ Σ(L sh,-) ⊂ {λ ∈ C : Re λ ≤ -3η} .
In particular, hypothesis (H 1 ) is fulfilled. Furthermore, θ → 0 when µ 0 → 0.

Proof. The asymptotic operators L kpp,± and L sh,± are obtained as conjugation of A kpp,± and A sh,± with pure exponential weight e θx or e -c * 2d x . Direct computations (see lemma

E.1 below) show that L kpp,+ = A kpp,+ (-c * 2d + ∂ x )
, where we identify the operator A kpp,+ with its symbol A kpp,+ (X) = dX 2 + c * X + α evaluated at X = ∂ x . Hence, the Fredholm border is given by

Σ(L kpp,+ ) = A kpp,+ - c * 2d + iξ : ξ ∈ R = -ξ 2 : ξ ∈ R .
Using similar notations for the three other curves, we get

Re A kpp,-(θ + iξ) = Re d(θ + iξ) 2 + c * (θ + iξ) -2α ≤ -2α + dθ 2 + c * θ < -α
for θ ∈ (-θ 0 , θ 0 ), where θ 0 > 0 only depends on α and d.

Similarly with A sh,-(X) = -(X 2 + 1) 2 + cX + µ, a direct computation shows that Re A sh,-(θ + iξ) is maximal at ±ξ 0 = ± √ 1 + 3θ 2 , hence: Re A sh,-(θ + iξ) ≤ -θ 4 -2θ 2 + c * θ + µ + ξ 4 0 -1 = µ + c * θ + 4θ 2 + 8θ 4 Fix η > 0 small,
it is easily seen that the right hand side is less than -3η for some θ < 0 that goes to 0 when µ 0 → 0.

Finally for A sh,+ (X) = A sh,-(X)γ, the same calculation with

ζ 0 := 1 + 3( c * 2d ) 2 shows that Re A sh,+ - c * 2d + iξ ≤ - c * 2d 4 -2 c * 2d 2 - c 2 * 2d + µ 0 -γ + ζ 4 0 -1 = γ rem -γ.
Hence for γ > γ rem , there exists η > 0 such that Re A sh,+ -c * 2 + iξ < -3η as claimed. Remark 3.2. We examine the case of distinct weights on KPP and SH component. Note Ω(x) = Diag(ω 1 (x), ω 2 (x)), and replace ansatz (7) by (u, v) T (t, x) = Q(x) + Ω(x)U (t, x), such that perturbation U is driven at linear level by

∂ t U = ω 1 -1 A kpp ω 1 ω2 ω1 β 0 ω 2 -1 A sh ω 2 U.
For our study to be relevant, we need ω2(x) ω1(x) to be bounded w.r.t. x. The computations done in proposition 3.3 shows that both ω 1 (x) ≤ e -c * 2d x for x ≥ 1 and ω 2 (x) ≥ e θx for x ≤ -1 are necessary to obtain marginal spectral stability. With the condition ω 2 (x) ≤ Cω 1 (x) for x ∈ R, we are left with ω 1 = ω 2 .

Proposition 3.3 (Stable point spectrum).

With the same assumptions on µ 0 , µ, α, d, γ, η as in previous proposition 3.1, the operator L has no eigenvalue in the following set:

Ω eig := {λ ∈ C : Re λ > -3η} \(-∞, 0).
Proof. We assume by contradiction that there exists λ ∈ Ω eig and a nonzero φ

= (φ kpp , φ sh ) ∈ H 2 (R, R) × H 4 (R, R) such that (λ -L)φ = 0.
In particular, the second eigenproblem is decoupled: (λ -L sh )φ sh = 0. We show it has no other solution than φ sh = 0 in H 4 (R), which will imply that the first eigenproblem (λ -L kpp )φ kpp = βφ sh admits no other solution than φ kpp = 0. This is a contradiction. Both L sh and A sh are not easy to work with, because they respectively have non-constant coefficients or unstable essential spectrum. Instead, we use

B sh := e -θx A sh e θx = e -θx ω * L sh ω -1 * e θx = b 0 (x) + 4 j=1 b j ∂ j x ,
with θ and ω * (x) defined in the above proposition 3.1. Then ϕ := e -θx ω * φ sh lies in H 4 (R) due to e -θx ω * (x) ≤ 1, 5 and satisfies (λ -B sh )ϕ = 0. Take the L 2 (R, C)-inner product with ϕ:

λ ϕ 2 L 2 = b 0 (x)ϕ, ϕ + 4 j=1 b i ∂ j x ϕ, ϕ .
Then ∂ j x ϕ, ϕ = (-1) j ∂ j x ϕ, ϕ , so that the real part of the above equality writes:

(18) Re(λ) ϕ 2 L 2 = ˆR b 0 |ϕ| 2 dx -b 2 ∂ x ϕ 2 L 2 + b 4 ∂ 2 x ϕ 2 L 2 ≤ -3η ϕ 2 L 2 .
The inequality is obtained using that b 2 = 2(1 + 3θ 2 ) > 0, that b 4 = -1 < 0 and finally that

b 0 (x) = µ -γ(1 -q * (x)) + c * θ -(θ 2 + 1) 2 ≤ µ + c * θ ≤ µ + c * θ + 4θ 2 + 8θ 4 ≤ -3η.
Recall we have chosen Re λ > -3η, hence (18) implies ϕ = 0 and then φ sh = 0 as claimed. Now the first eigenproblem writes (λ -L kpp )φ kpp = 0, hence ω * φ kpp ∈ H 2 (R) is an eigenfunction for A kpp . Using Sturm-Liouville theory -see e.g. [KP13, p 33.] -eigenvalues of A kpp are real and non-positive, since q * ∈ H 2 (R) does not vanish and satisfies A kpp q * = 0. Hence A kpp has no eigenvalues outside of (-∞, 0], and so does L kpp . Since 1 ω * (x) q * (x) ∼ ax + b when x → +∞, the derivative of the front does not contribute to an eigenfunction for L kpp , hence it has no eigenvalue oustide (-∞, 0). The first eigenproblem imposes either λ ∈ (-∞, 0) or φ kpp = 0, which is a contradiction, and complete the proof.

5 Recall that -c * 2 ≤ θ when µ is small enough.

Construction of decaying ODE solutions.

Here we solve the linear non-autonomous ODE (λ -L i )φ = 0, with unknown φ and i ∈ {kpp, sh}. To keep notations simple, we concentrate on i = sh. Since L sh is exponentially asymptotic, φ expresses at first order using the solutions of the asymptotic ODE, as described in the incoming Lemma. After vectorialization, (λ -L sh )φ = 0 writes ∂ x Φ = A sh (λ, x)Φ, with a matrix A sh (λ, x) that exponentially converges towards matrices A sh,± (λ) when x → ±∞.

Lemma 3.4. Let A : t ∈ R → A(t) ∈ M n (C) be a continuous, matrix-valued function, that converges at exponential speed towards A ∞ when t → +∞. Let α > 0 that satisfies A(t) -A ∞ ≤ Ce -αt for t ≥ 0. 6 Let (v i ) 1≤i≤n be a basis of C n . Then there exists (y i ) 1≤i≤n a basis of solutions for the ODE

(19) y (t) = A(t)y(t),
and κ i ∈ L ∞ (0, +∞) n such that for t ≥ 0 we have

y i (t) = e tA∞ (v i + e -αt κ i (t)).
Furthermore, if λ → A(t, λ), A ∞ and v are holomorphic with respect to an extra parameter λ, then λ → κ(λ, •) is also holomorphic.

We delay the proof to the later appendix D, and apply this Lemma to the matrix A sh (λ, x), for λ to the right of Σ(L sh ). There, eigenvalues of A sh,± are distinct and holomorphic. Since asymptotic matrices are companion, they admit a basis v sh,± i (λ) of holomorphic eigenvectors. This allows to define two basis of solutions (φ sh,+ i (λ, •)) 1≤i≤4 and (φ sh,- i (λ, •)) 1≤i≤4 for the ODE (λ -L sh )φ = 0, with exponential behavior at +∞ or -∞:

φ sh,± i = e ν sh,± i x (v sh,± i + e -r|x| κ sh,± i (x)),
where ν sh,+ i is the eigenvalue associated to v sh,+ i , and κ sh,+ i ∈ L ∞ (R + ). We use similar notations for L kpp . Lemma 3.5. The dispersion relation λ -L kpp,± (ν) = 0 and λ -L sh,± (ν) = 0 are respectively second order and fourth order polynomial in ν. Their respective roots ν kpp,± i (λ) and ν sh,± i (λ) have the following localization:

(1) (spectral gap away from the spectrum) There exists κ 2 > 0 such that for all λ ∈ C with Re λ ≥ -2η, |Re ν| ≥ κ 2 , where ν stands for ν sh,± i (λ) or ν kpp,- i (λ).

(2) (pinched double root at the origin) Let I = (-∞, 0) ⊂ C, and V be a neighborhood of 0 ∈ C.

There exists C positive such that for λ ∈ V \I, and λ → 0,

ν kpp,+ 1 (λ) ∼ -C √ λ, ν kpp,+ 2 (λ) ∼ C √ λ.
(3) (elliptic operators) There exists R, C, θ positive constant such that for λ ∈ C with |λ| ≥ R and Re λ ≤ -θ|Im λ|,

|Re ν kpp,± i (λ)| ≥ C|λ| 1/2 , |Re ν sh,± i (λ)| ≥ C|λ| 1/4 .
6 Here . is any operator norm.

Proof. Items (1) and (2) are easily obtained. Item (3) relies on a scaling argument, see [FH18, Lemma 3.1] for ν kpp,± i , the case of ν sh,± i adapts as follow. First set ψ(x) := φ(x|λ| 1/4 ), and apply lemma 3.4 to construct solutions that are close to solutions of λ + |λ|∂ 4

x ψ = 0. In particular, the asymptotic matrix eigenvalues for ψ are ν(λ) := z 4 √ λ/|λ|, where z ∈ e i π 4 , e -i π 4 , e i 3π 4 , e -i 3π 4 .

Notice that |Re z| = 1/ √ 2, hence we get |Re ν| ≥ C, by restricting to λ of the form re iθ , with |θ| ≤ 3π/4. Reverting back to the original variable, the asymptotic matrix eigenvalues for φ satisfy |Re ν| ≥ C|λ| 1/4 .

In the following, we will compute determinant of several matrices, whose columns depend on φ sh,± i or φ kpp,± i . Suppose we are given n scalar functions φ 1 , . . . , φ n depending on the space variable x. Then, we write

(20) Det(φ 1 , . . . , φ n ) := det    φ 1 • • • φ n . . . . . . ∂ n-1 x φ 1 • • • ∂ n-1 x φ n    = det ∂ j-1 x φ i 1≤i,j≤n .
where "det" stands for the determinant. We also introduce the wronskian function composed of all decaying solutions:

W kpp := Det(φ kpp,+ 1 , φ kpp,- 2 ), W sh := Det(φ sh,+ 1 , φ sh,+ 2 , φ sh,- 3 , φ sh,- 4
).

For each λ ∈ C, the function y → W kpp (λ, y) is either identically zero or does not vanish, see lemma E.2. The particular value W kpp (λ, 0) is often called the Evans function. When λ is such that φ kpp,+ 1 and φ kpp,- 2 are decaying at x → ±∞, 7 W kpp (λ, 0) vanishes exactly when λ is an eigenvalue for L kpp , with same multiplicity. The same goes for W sh .

3.3.

Construction and estimations of the kernel for the resolvent operator. First, we construct and control the Green function G kpp λ (x, y), which corresponds to the pure KPP equation. Remark that such a result was already proved in [FH18, Lemma 3.2]. We rewrite it in a more condensed form that separates the behaviors at +∞ and -∞. See the further remark 3.12. Proposition 3.6. Let K ⊂ C be a compact set to the right of Σ(L kpp,-), and note I = (-∞, 0) the real negative axis that corresponds to the absolute spectrum of L kpp . Then λ → G kpp λ (x, y) is holomorphic from K\I to R. Furthermore, there exists positive constants κ 1 , C such that for all λ ∈ K\I and 0 ≤ j ≤ 2, we have the following. If x ≥ 0 or y ≥ 0,

G kpp λ (x, y) ≤ Ce -Re √ λ|x-y| ω κ1,0 (x) ω κ1,0 (y).
If on the contrary, x ≤ 0 and y ≤ 0, then

G kpp λ (x, y) ≤ Ce -κ1|x-y| .
Proof. We use the decaying ODE solution φ kpp,- 2 , φ kpp,+ 1 constructed above. To keep notation as light as possible, we write them φ -and φ + respectively. Similarly, we note ψ -= φ kpp,- 1 and ψ + = φ kpp,+ 2 the growing ODE solutions. Remind that both (φ -, ψ -) and (φ + , ψ + ) are basis for solution of the ODE (λ -L kpp )ϕ = 0. The Green function expresses as

G kpp λ (x, y) = 1 W kpp (λ, y) φ -(y)φ + (x) if y < x, φ -(x)φ + (y) if x < y.
7 When λ lies to the right of Σess(L kpp ), this decay property hold.

L. GAR ÉNAUX

We begin by the case x ≥ 0 or y ≥ 0. For λ to the right of Σ(L kpp,-), the spatial eigenvalues are such that the exponential obtained in [FH18, Lemma 3.2] are smaller than e -|x-y| Re √ λ . Then each O leads to our weights ω κ1,0 (x)ω κ1,0 (y). The case x ≤ 0 and y ≤ 0 is in [FH18, Lemma 3.2].

In the following, we will also need to bound the derivatives of the spectral Green kernel. Since our problem is parabolic, to differentiate the semigroup must gains us extra temporal decay. At spectral level, it converts into extra power of √ λ. Due to G kpp λ being a kernel, Dirac delta correction terms appear when differentiating more than the order of L kpp . However, they will be easily absorbed at temporal level. Proposition 3.7. Let K ⊂ C be a compact set to the right of Σ(L kpp,-), and note I = (-∞, 0) the real negative axis that corresponds to the absolute spectrum of L kpp . Fix an integer j ≥ 1. There exists a sum of Dirac delta derivatives:

P j (δ x=y ) = j-2 k=0 a j k (λ, y)δ (k)
x=y with coefficients a k holomorphic w.r.t. λ ∈ K\I and bounded with respect to y ∈ R, such that λ → ∂ j x G kpp λ (x, y) -P j (δ x=y ) is holomorphic from K\I to R. Furthermore, there exists positive constants κ 1 , C such that for all λ ∈ K\I, we have the following. If x ≥ 0 or y ≥ 0,

∂ j x G kpp λ (x, y) -P j (δ x=y ) ≤ C|λ| j/2 e -Re √ λ|x-y| ω κ1,0 (x) ω κ1,0 (y).
If on the contrary, x ≤ 0 and y ≤ 0, then

∂ j x G kpp λ (x, y) -P j (δ x=y ) ≤ C|λ| j/2 e -κ1|x-y| .
Proof. For the case j = 1 no correction terms appear, and the proof is close to [START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF]. We compute

∂ x G kpp λ (•, y) = 1 W kpp λ (y) φ -(y)∂ x φ + on (y, +∞) φ + (y)∂ x φ -on (-∞, y)
Begin first with case x > y. When x ≥ 0, we use

∂ x φ + (x) = - √ λe - √ λ (1 + κ(x)) rather than φ + (x) = e - √ λ (1 + κ(x))
, which gains us the claimed |λ| 1/2 in comparison with the case j = 0. When x ≤ 0, we project onto the (φ -, ψ -) basis:

∂ x φ + (x) = Det(∂ x φ + , ψ -) Det(φ -, ψ -) φ -(x) + Det(φ -, ∂ x φ + ) Det(φ -, ψ -) ψ -(x),
we refer to the above mentioned reference for more details, or to the proof of proposition 3.8, see appendix B. Since both φ -and ψ -are bounded by e κ1x , and using that Det(φ -, ψ -) > 0 uniformly w.r.t. λ, we get to

|∂ x φ + (x)| ≤ C|λ| 1/2 e κ1x
. This conclude the case j = 1. When j = 2, we compute

∂ 2 x G kpp λ (•, y) = δ y + 1 W kpp λ (y) φ -(y)∂ 2 x φ + on (y, +∞), φ + (y)∂ 2 x φ -on (-∞, y),
and similar computations as above show the claimed estimate for ∂ 2 x G kpp λ (x, y)δ y . For j ≥ 2, the correction terms can be computed recursively:

P j+1 (δ x=y ) = δ x=y φ -∂ j+1 y φ + -φ + ∂ j+1 y φ - W λ (y) + j-1 k=1 a j k (y)δ (k)
x=y Then, ∂ j x G kpp λ (x, y) -P j (δ x=y ) is handled as above. We state a similar result for G sh λ . Since L sh is a fourth order operator, the proof slightly differs. Due to its length, we present the proof in the later appendix B. Note that [START_REF] Howard | Pointwise Green's function estimates toward stability for multidimensional fourth-order viscous shock fronts[END_REF] presents similar computations in a viscous shock front context. Proposition 3.8. Let K ⊂ C be a compact set to the right of Σ(L sh,- θ ) ∪ Σ(L sh,+ ), for example K ⊂ {Re(λ) ≥ -2η}. Then λ → G sh λ (•, y) is holomorphic from K to H 4 (R\{y}), and there exists C, κ 2 > 0 such that the following holds. For all x, y ∈ R 2 , and λ ∈ K: G sh λ (x, y) ≤ Ce -κ2|x-y| . Furthermore, if x ≥ 0 or y ≥ 0, then we have:

(21)
G sh λ (x, y) ≤ Ce -κ2|x-y| ω κ2,0 (y). We can now obtain a statement on G co λ , similar to the above proposition 3.6. We use the structure of the equation satisfied by G co λ to directly transfer the spatial decay from G kpp λ to G co λ . Proposition 3.9. Let K ⊂ C be a compact set to the right of Σ(L kpp,-) ∪ Σ(L sh,- θ ) ∪ Σ(L sh,+ ), and note I = (-∞, 0). Then λ → G co λ (•, y) is holomorphic from K\I to H 2 (R\{y}). Furthermore, there exists κ 3 , C > 0 such that for all λ ∈ K\I, we have the following. If x ≥ 0 or y ≥ 0, then:

|G co λ (x, y)| ≤ Ce -Re √ λ|x-y| ω κ3,0 (x) ω κ3,0 (y).
If on the contrary x ≤ 0 and y ≤ 0,

|G co λ (x, y)| ≤ Ce -κ3|x-y| . Proof. Recall that (22) (λ -L kpp )G co λ = βG sh λ .
Since λ is located to the right of Σ(L kpp ), the operator λ -L kpp is invertible, and we get

G co λ (x, y) = β ˆR G kpp λ (x, τ )G sh λ (τ, y)dτ.
Indeed, we have constructed G kpp λ (x, y) as the solution of the fundamental problem associated with (22), i.e. replace the right hand side by a Dirac delta. When the Dirac delta is replaced by any source term, it give birth to a convolution-like solution as above. We first assume x ≥ 0 or y ≥ 0. Then from proposition 3.6, where we neglect the ω κ1,0 (τ ) terms, and from the refined estimate (21) in proposition 3.8, we obtain that

|G co λ (x, y)| ≤ C ˆR e -Re √ λ|x-τ | ω κ1,0 (x) e -κ2|τ -y| ω κ2,0 (y) dτ.
Using triangular inequality |xτ | + |τ -y| ≥ |x -y|, we obtain

|G co λ (x, y)| ≤ Ce -Re √ λ|x-y| ω κ1,0 (x) ω κ2,0 (y) ˆR e -(κ2-Re √ λ)|τ -y| dτ, ≤ Ce -Re √ λ|x-y| ω κ1,0 (x) ω κ2,0 (y),
which is the first claimed estimate. Now when x ≤ 0 and y ≤ 0, we first assume that κ 1 ≤ κ 2 . We introduce κ 3 := 1 2 κ 1 , and obtain -again using triangular inequality -that

|G co λ (x, y)| ≤ C ˆR e -κ1|x-τ | e -κ2|τ -y| dτ ≤ Ce -κ3|x-y| ˆR e -κ3|x-τ | e -(κ2-κ3)|τ -y| ≤ Ce -κ3|x-y| .
If on the contrary we have κ 1 > κ 2 , we set κ 3 := 1 2 min(κ 1 , κ 2 ) and obtain the same inequality. To finish this section, we show bounds on G λ for large λ. This is needed to transfer spectral behavior into temporal decay. Indeed, G λ will be integrated along a spectral contour that surround the spectrum and hence goes to infinity. Proposition 3.10. There exists R, C > 0 such that: for all λ to the right of Σ(L), with |λ| ≥ R, and all x, y ∈ R 2 , we have

(23) G kpp λ (x, y) ≤ C |λ| 1/2 e -|λ| 1/2 |x-y| , G sh λ (x, y) ≤ C |λ| 3/4 e -|λ| 1/4 |x-y| .
The first estimate still holds if G kpp is replaced by G co . In particular, for i ∈ {kpp, sh, co}

G i λ L ∞ (Rx,L 1 (Ry)) ≤ C |λ| , G i λ L ∞ (Ry,L 1 (Rx)) ≤ C |λ| .
Proof. The last claimed inequalities are easily obtained from the first ones, and implies that

G i λ • v L p (R) ≤ C
|λ| v L p (R) using complex interpolation. The first estimates are obtained through the scaling argument began in the proof of lemma 3.5. The second ones are necessary to obtain small time estimates on ´Γ e λt G λ dλ, with Γ being a ray that extends at infinity.

Linear dynamic.

We now convert spatial localization of spectral Green functions G λ into temporal decay in appropriate space using the inverse Laplace transform and adequate integration contours.

Proposition 3.11. There exists positive constants C, η such that:

|G sh t (x, y)| ≤ Ce -ηt e -κ2|x-y| .
In particular, for any 1 ≤ p ≤ +∞, we have

(24) G sh t • w L p (R) ≤ Ce -ηt w L p (R)
, where we have noted (G sh t • w)(x) := ´R G sh t (x, y)w(y)dy.

Proof. Since L sh is sectorial with spectral gap η from propositions 3.1 and 3.3, this proof easily follows from the spatial localization of G sh λ , see propositions 3.8 and 3.10. It is enough to use inverse Laplace transform with a spectral contour in λ ∈ C : Re(λ) ≤ -η 2 for small λ, while restrict to a contour in {λ ∈ C : Arg(λ) ≤ 3π/4} in the region of large λ. Up to a change of notation η = η/2, the proof is complete. Remark 3.12. We now state a similar result for G kpp t and G co t . Remark that in [FH18, Proposition 4.1], the situation is similar except that the weight ω * was exponentially decaying both at +∞ and -∞. In our setting, since ω * (x) → +∞ when x → -∞, we can not absorb any supplementary polynomial during the nonlinear argument. Hence, we need to refine the aforementioned result with respect to the polynomial weight. We use the spatial decay ω κ1,0 (y) obtained for both G kpp λ and G co λ in the above propositions 3.6 and 3.9. This allows to bypass the weight at -∞ during nonlinear argument. Remark 3.13. Depending on one aim, the following pointwise bounds for the derivatives may not be adapted. We allow ourselves to keep a single power of |x -y|, which transfers into t -3/2 decay. To show stronger decay, it is necessary to keep more powers of ρ * , thus weakening the weight. In our situation, more powers of ρ * would result in the translation eigenvalue λ = 0 appearing again, due to q * ω kpp ρ 2 * ∼ 1

x at x → +∞. To guess what bound may be obtained, one can differentiate 1 t 1/2 e -x 2 t (resp. x t 3/2 e -x 2 t ) in the case where |x -y| ≥ Kt (resp. |x -y| ≤ Kt) Proposition 3.14. Recall that ρ * is defined by (10). Fix 0 ≤ j ≤ 1 and restrict to t > 1. Then there exists positive constants C, K and κ 1 such that the following two pointwise estimate hold. If |x -y| ≥ Kt, then

∂ j x G kpp t (x, y) ≤ C t j+1 2 1 + |x -y| √ t j e -κ1 |x-y| 2 t h(x, y).
If on the contrary |x -y| ≤ Kt, then

∂ j x G kpp t (x, y) ≤ 1 + |x -y| 1+ j/2 t 3/2+ j/2 e -κ1 |x-y| 2 t h(x, y).
In the above expressions, x, y → h(x, y)/ρ * (y) 2 is integrable in y uniformly in x:

0 ≤ h(x, y) ≤ C ω κ1,0 (x) ω κ1,0 (y) if x ≥ 0 or y ≥ 0, e -κ1|x-y| if x ≤ 0 and y ≤ 0.
In particular, for 0 ≤ j ≤ 1, for all 1 ≤ q ≤ p ≤ +∞, all x, y ∈ R and t ≥ 1, (25)

∂ j x G kpp t • w ρ * L p (R) ≤ C t 3 2 -1 2p ρ 3 * w L q (R) .
All the above still hold when G kpp t and κ 1 are replaced by G co t and κ 3 respectively. Proposition 3.15. There exists positive constants C, κ 1 and a constant c = 0 such that for t > 1, the two point-wise estimate of the above proposition hold with j = 2 for ∂ 2

x G kpp t (x, y)ce -t δ x=y .

Hence, for all 1 ≤ q ≤ p ≤ +∞, all x, y ∈ R and t ≥ 1,

(26) ∂ 2 x G kpp t • w ρ * L p (R) ≤ C t 3 2 -1 2p ρ 3 * w L q (R) + Ce -t w L p (R) .
All the above still hold when G kpp t and κ 1 are replaced by G co t and κ 3 respectively. Proof. Propositions 3.14 and 3.15.

We first investigate the case j = 0. We recall that [FH18, Proposition 4.1] proves there exists K > 0 such that:

• if |x -y| ≥ Kt, then G kpp t (x, y) ≤ C √ t e -κ |x-y| 2 t , • else, |x -y| ≤ Kt implies G kpp t (x, y) ≤ C 1+|x-y| t 3/2 e -κ |x-y| 2 t .
Having kept the extra exponential localization -ω κ1,0 (x)ω κ1,0 (y) when x ≥ 0 or y ≥ 0 -in our estimates of G kpp λ (x, y), the proof of [START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF] straightforwardly adapt with this extra decay. The same goes for the other case x ≤ 0 and y ≤ 0.

In case where j > 0, derivatives pass through the inverse Laplace transform formula, and we can choose a contour Λ that is sectorial before differentiation:

∂ j x G kpp t (•, y) = 1 2iπ ˆΛ e λt ∂ j x G kpp λ (•, y)dλ,
We first treat the long time scenario: |x-y| ≤ Kt. Using estimate from proposition 3.7, there exists a function H holomorphic w.r.t. λ outside I := (-∞, 0), such that H(λ, •, •) L ∞ (Rx,L 1 (Ry)) < +∞ and

∂ j x G kpp λ (x, y) -P j (δ x=y ) = √ λ j e - √
λ|x-y| H(λ, x, y).

Setting ρ = κ |x-y| t with κ > 0 to be fixed later. Close to the origin, we use a parabolic contour:

Λ 1 (ρ) = λ(ξ) = (ρ + iξ) 2 : ξ ∈ (-ξ 0 , ξ 0 ) ,
and obtain with H(ξ, x, y, t)

:= 1 π e i(2ρξt-ξ|x-y|) H λ(ξ) (x, y) that 1 2iπ ˆΛ1 e λt (∂ j x G kpp λ (•, y) -P j (δ x=y ))dλ = e ρ 2 t-ρ|x-y| ˆξ0 -ξ0
e -ξ 2 t H(ξ, x, y, t)(ρ + iξ) j+1 dξ, Due to L kpp having real coefficients, we have G kpp λ = G kpp λ , and the same goes for H. Since λ(-ξ) = λ(ξ) we deduce that the term below the integral in the right hand side of the above equation satisfies z(-ξ) = z(ξ). As a consequence, we compute

1 2iπ ˆΛ1 e λt (∂ j x G kpp λ (•, y) -P j (δ x=y ))dλ = e (κ 2 -κ) |x-y| 2 t ˆξ0 -ξ0 e -ξ 2 t Re H(ξ, x, y, t)(ρ + iξ) j+1 dξ, = e (κ 2 -κ) |x-y| 2 t ˆξ0 -ξ0
e -ξ 2 t Re H(ξ, x, y, t) Re (ρ + iξ) j+1

-Im H(ξ, x, y, t) Im (ρ + iξ) j+1 dξ.

First develop (ρ + iξ) j+1 = j+1 k=0 j+1 k (iξ) k ρ j+1-k , and notice that ρ ≤ κK. Then for k even, we bound

ρ j+1-k ≤ κ 1+j-k K j-j/2 -k/2 |x -y| t 1+ j/2 -k/2
, with jj/2k/2 ≥ 0. Finally, each power of ξ provide a power of 1/ √ t when integrated:

0 ≤ ˆξ0 -ξ 0 ξ k e -ξ 2 t dξ ≤ 1 t (k+1)/2 ˆR ξ √ t k e -ξ 2 t √ tdξ ≤ C k t (k+1)/2 .
Hence, using that Re H is bounded, we compute that ˆξ0

-ξ0 e -ξ 2 t Re H(ξ, x, y, t) Re (ρ + iξ) j+1 dξ ≤ C j 1 + |x -y| 1+ j/2 t 3/2+ j/2 .

The terms with k odd, corresponding to the imaginary part of (ρ + iξ) j+1 , are bounded similarly:

ρ j+1-k ≤ κ j+1-k K j-j/2 -k-1 2 |x -y| t 1+ j/2 -k+1 2 .
Since ξ → Im( H(ξ)) is odd, it provide an extra power of ξ: |Im H(ξ)| ≤ ξC. Hence we compute that ˆξ0

-ξ0

e -ξ 2 t Im H(ξ, x, y, t)

Im (ρ + iξ) j+1 dξ ≤ C j 1 + |x -y| 1+ j/2 t 3/2+ j/2 .
In the short time scenario: |x -y| ≥ Kt, we assume K is large enough so that the parabolic contour Λ 1 is far enough from the origin, which in turn guarantees that large λ estimates from proposition 3.10 applies. The computations are then similar to above, with (ρ + iξ) j+1 replaced by

(ρ + iξ) j = j k=0 j k (iξ) k ρ j-k .
We now use ρ = κ |x-y| t , to obtain that the k-th term in the above some contribute as

j k (iξ) k ρ j-k = C √ t ξ √ t k × 1 t j+1 2 |x -y| √ t j-k
, which leads to the claimed pointwise estimate. We now express the correction terms. None appear for j = 1, while for j = 2, there is only one coefficient a 0,2 = 1. It is holomorphic on λ ∈ C, thus we can move the integration contour to the left of the imaginary axis, and obtain:

1 2iπ
ˆΛ e λt δ y=x dλ = ce -t δ x=y .

Turning now to the L q -L p estimate ( 25), we use the extra exponential localization we have gained in h. First when |x -y| ≤ Kt, we use both that (1 + |x -y|)h(x, y) ≤ ρ * (x)ρ * (y)h(x, y) and that ( |x-y| t ) j/2 ≤ C to obtain that ˆ|x-y|≤Kt

∂ j x G kpp t (x, y)w(y) ρ * (x) dy ≤ ρ 3 * w L ∞ (R) t 3/2 ˆ|x-y|≤Kt h(x, y) ρ * (y) 2 dy.
The integral on the right hand side is finite. In the other case, we rather use that K ≤ 1 √ t |x-y| √ t , and change variables to gain as much decay as needed:

ˆ|x-y|≥Kt

∂ j x G kpp t (x, y)w(y) ρ * (x) dy ≤ ρ * w L ∞ (R) t j+3 2 ˆ|x-y|≥Kt 1 + |x -y| √ t j+2 e -κ1 |x-y| 2 t dy, ≤ ρ 3 * w L ∞ (R) t 3/2 ˆR(1 + z) j+2 e -κ1z 2 dz.
We have shown (p, q) = (+∞, +∞). Then (p, q) = (1, 1) reads ¨|x-y|≤Kt

G kpp t (x, y)w(y) ρ * (x) dydx ≤ C t ˆR x → 1 √ t e -κ |x-y| 2 t L 1 (R)
|ρ * (y)w(y)|dy,

≤ C t ρ * w L 1 (R) , together with ρ * w L p (R) ≤ ρ 3 * w L p (R)
. The other term ˜|x-y|≥Kt bound similarly and decay as fast as needed. The case (p, q) = (+∞, 1) proves similarly. Interpolation between those three cases leads to the 1 ≤ q ≤ p ≤ +∞ estimate. It only remains to show (26). The part which is absolutely continuous with respect to Lebesgue measure leads to the rational decay similarly as above. Then the Dirac delta provide the exponentially decaying term. The polynomial weight ρ * (x) 2 ≥ 1 is removed.

Remark 3.16. Using standard parabolic regularity, we obtain for 0 < t ≤ 1 and 1 ≤ p ≤ +∞ that:

G kpp t • w W j,p (R) ≤ C w W j,p (R) .
Then from inf(1, 1 t ) ≤ 2 1+t , estimates (25)-( 26) hold true for t ≥ 0, p = q, and 1 t replaced by 1 1+t .

Remark 3.17. When p = +∞, inequality (25) is the optimal t -3/2 decay. It does not seem possible to obtain estimates with p < q, since G t (x, y) can not absorb both integral in x and y. Remark that ρ 3 * is arbitrary, and could be replaced by ρ * h with ´+∞ 0 dy h(y) < +∞.

Nonlinear dynamic.

Here we finally prove proposition 2.3, i.e. that U = (u 1 , u 2 ) T decays in time provided V = (v 1 , v 2 ) T is bounded. Throughout this section, we will assume that there exists

C µ > 0 such that for all 0 ≤ t ≤ t V , (27) ω sh ρ * U (t) L ∞ (R) = V (t) L ∞ (R) ≤ C µ .
To lighten notations, we may note C µ instead of CC µ when C is a constant not depending on C µ . Remind that V (t) and ω sh ρ * U (t) differs only by a change of space variable, so that the first equality is automatically satisfied. Proposition 3.18. For all 1 < p ≤ +∞, there exists positive constants C stab , η, δ, such that if C µ ≤ δ, then the solution U of (8) emanating from initial condition U 0 -with U 0 ρ 3 * ∈ L p (R) -is defined for 0 ≤ t ≤ t V , and satisfies

u 1 (t, •) ρ * L p (R) ≤ C stab U 0 ρ 3 * L p (R) (1 + t) 3 2 -1 2p , u 2 (t, •) L p (R) ≤ C stab e -η 2 t U 0 L p (R) .
Furthermore, C stab is independent of C µ .

Proof. In this proof, we abbreviate K p := U 0 ρ 3 * L p (R) . We follow the same argument as in [BGS09, Lemma 3.2], which is adapted to the nonlinear system after conjugation by an unbounded weight.

The key ingredient is to use the nonlinear term to absorb ω sh . Then nonlinear terms are seen as linear ones.

We fix 0 < η < η where η is given by the exponential decay of u 2 at linear level: see (24) in proposition 3.11. For 1 < p ≤ +∞, we note σ(p) := 3 2 -1 2p > 1. Then we note Θ 1 (t) := sup s∈(0,t)

(1 + s) σ(p) u 1 (s) ρ * L p (R) Θ 2 (t) := sup s∈(0,t) e ηs u 2 (s) L p (R) ,
together with Θ(t) = max(Θ 1 (t), Θ 2 (t)), and show that Θ is bounded in time. Using (27), we can absorb the unbounded ω sh that comes from the nonlinear terms. We have both

|N 1 (U )| ≤ |3αq * ||ω sh u 1 ||ω kpp u 1 | + |αω kpp ||ω sh u 1 | 2 |ω kpp u 1 | ≤ C µ |ω kpp u 1 | and |N 2 (U )| ≤ C|ω sh u 1 ||ω kpp u 2 | + C|u 2 ω sh | 2 |ω kpp u 2 | ≤ C µ |ω kpp u 2 |.
In particular, U → N (U ) is globally Lipschitz, hence the solution U is defined in L p (R) for times 0 ≤ t ≤ T , see [START_REF] Henry | Geometric Theory of Semilinear Parabolic Equations[END_REF]. We can now use the ω kpp factor to gain a polynomial weight -remind that to apply the linear estimates (25), we need one ρ * to control the Green kernel, and two extra ones to make the integral w.r.t. y converge:

ω kpp ≤ C ρ 4 * -which leads to |N 1 (s, y)| ≤ C µ ρ * (y) 3 (1 + s) σ(p) (1 + s) σ(p) ρ * (y) u 1 (s, y) , hence ρ 3 * N 1 (s) L p (R) ≤ C µ (1 + s) σ(p) Θ 1 (t).
Similarly we get to

ρ 3 * N 2 (U (s)) L p ≤ C µ e -ηs e ηs u 2 (s, •) L p ≤ C µ e -ηs Θ 2 (t). The Duhamel's formula decomposes into u 2 (t, x) = ˆR G sh t (x, y)u 0 2 (y)dy + ˆt 0 ˆR G sh t-s (x, y)N 2 (s, y)dyds. u 1 (t, x) = ˆR G kpp t u 0 1 + G co t u 0 2 dy + ˆt 0 ˆR G kpp t-s N 1 + G co t-s N 2 dyds.
We use Minkowski inequality as well as the injection L 1 ((0, t), L p (R)) ⊂ L p (R, L 1 (0, t)) -see lemma E.3 -onto Duhamel's formula on u 2 to obtain that

u 2 (t) L p (R) ≤ G sh t • u 0 2 L p + ˆt 0 G sh t-s • N (s, •) L p ds.
Now using the linear estimate (24) from proposition 3.11 and the above nonlinear estimate, we obtain:

e ηt u 2 (t) L p (R) ≤ C u 0 2 L p (R) + C µ Θ 2 (t) ˆt 0 e -(η-η)(t-s) ds,
which after integration and taking the supremum on t ∈ (0, τ ) reads

Θ 2 (τ ) ≤ C u 0 2 L p (R) 1 - Cµ η-η ,
assuming that the denominator is positive. By imposing C µ ≤ δ ≤ η-η 2 , this condition is fulfilled and we recover Θ 2 (t) ≤ 2C u 0 2 L p (R) for 0 ≤ t ≤ T . Remind that C does not depend on C µ . This is the claimed exponential temporal decay for u 2 . Turning now towards Duhamel's formula for u 1 , we apply linear estimate (25) with p = q -see also remark 3.16 -together with both above nonlinear estimates, to obtain that:

(1 + t) σ(p) u 1 (t, •) ρ * L p ≤ C U 0 ρ 3 * L p (R) + ˆt 0 (1 + t) σ(p) (1 + t -s) σ(p) ρ 3 * N 1 (s, •) L p (R) + ρ 3 * N 2 (s, •) L p (R) ds, ≤ CK p + C µ Θ(t) ˆt 0 (1 + t) (1 + t -s)(1 + s) σ(p) ds.
Standard computations on integral -see lemma E.4 and remark that σ(p) > 1 -lead to

Θ 1 (t) ≤ CK p + C µ Θ(t).
Summing the above Θ 2 (t) ≤ 2CK p , we get to Θ(t) ≤ CKp 1-Cµ as soon as 1 -C µ > 0. By assuming that C µ ≤ δ ≤ 1 2 , we get to Θ(t) ≤ CK p , which implies the claimed temporal decay for u 1 . Remark that C stab := C does not depends on C µ . 4. Perturbations in partially weighted space are bounded in time 4.1. Mode filters. Since part of the spectrum of T -is unstable, the dynamic for V is unstable at linear level. We count on the nonlinear terms to control V for large bounded times t ≤ T µ . To do so, we mostly follow the approach of Guido Schneider, see [START_REF] Schneider | Global existence via Ginzburg-Landau formalism and pseudo-orbits of Ginzburg-Landau approximations[END_REF]. We separate the critical from the stable modes in our solution: the first ones grow or are bounded at linear level, while the second ones decay exponentially, uniformly in µ. Let us use a smooth, positive cut-off function χ:

(28) χ c (x) := 1 if x ∈ I c := [-9 8 , -7 8 ] ∪ [ 7 8 , 9 8 ], 0 if x / ∈ I c + B(0, 1 8 ),
with B(x, r) the ball centered at x with radius r. We then work in Fourier space: the matrix T -(ξ) has two eigenvalues

λ c (ξ) = -(1 -ξ 2 ) + µ, λ s (ξ) = -dξ 2 -2α,
with associated eigenvectors i , for i ∈ {s, c}. We note Π i (ξ) V = V , * i (ξ) i (ξ) the respective parallel projections onto each of the eigenspaces, and we separate critical from stable frequencies:

Π c V = F -1 ξ → χ c (ξ) V (ξ), * c (ξ) c (ξ) , Π s V = V -Π c V
Remark that χ 2 c = χ c , so that neither Π c or Π s are projections. However, using ), in a similar way as (28). Then with

I h c := [-5 4 , -3 4 ] ∪ [ 3 4 , 5 4 ] and I h s := [-
Π h c V := F -1 (χ h c Π 1 V ), Π h s V := F -1 ((1 -χ h s )Π 1 V ) + F -1 (Π 2 V ), we have Π h i Π i = Π i for i ∈ {c, s}.
Ultimately, we will use a scalar "projection" onto half of the critical Fourier modes, to select only frequencies close to ξ = 1:

(29) π h 1 V = F -1 ξ → χ h c (ξ)1 ξ>0 1 ρ c (1), ρ * c (ξ) V (ξ), * c (ξ) .
Such decompositions are well-behaved with Fourier transform. However, we need to measure objects with L ∞ (R) norms, since the pattern we want to study behave as the solution of a bistable equation.

To combine those two constraint, we use the so-called uniformly-localized space, that were introduce by Guido Schneider, see for example [START_REF] Schneider | A new estimate for the Ginzburg-Landau approximation on the real axis[END_REF]. We first define a weight

ρ u,l (x) := 1 1 + x 2 . Then, we say that u ∈ L 2 u,l (R) if u L 2 u,l (R) := sup y∈R ρ u,l (• -y)u L 2 (R) < +∞.
Similarly, we define H s u,l (R) and its norm by u H s u,l (R) := sup y∈R ρ u,l (•y)u H s (R) when s ≥ 0. Then, we use the following injections to link with Sobolev spaces, that were used to estimate Green's kernel.

Lemma 4.1. Let s ∈ R with s ≥ 1. Then the following injections are continuous:

W s,∞ (R) ⊂ H s u,l (R) ⊂ L ∞ (R) ⊂ L 2 u,l (R).
Proof. Let first recall the definition of .

H s u,l (R) that we use. Let ρ u,l (x) := 1 1+x 2 , then u H s u,l (R) := sup y∈R u(• -y)ρ u,l H s (R) .
From one hand the injection

H s (R) ⊂ L ∞ (R) leads to u H s u,l (R) ≥ sup y∈R u(• -y)ρ u,l L ∞ (-1,1) ≥ 1 2 sup y∈R u(• -y) L ∞ (-1,1) = 1 2 u L ∞ (R) ,
It corresponds to the second injection. From the other hand, uρ u,l

H s (R) ≤ u W s,∞ (R) ρ u,l H s (R)
leads to the first and third injection.

As announced, we will need to estimate operators in Fourier space. We will use the following Lemma, see [Sch94a, Section 3.1] and [Sch94b, Lemma 5] for proofs.

Lemma 4.2. Let d, n be positive integers, and M an operator that acts in Fourier space as a point-wise linear application:

M (ξ) ∈ L(R d , R n ) with M u := F -1 (ξ → M (ξ)û(ξ)). Then for q, s ≥ 0, M u H s u,l (R) d ≤ C(s, q) ξ → ξ s-q 2 M (ξ) C 2 b (R,L(R d ,R n )) u H q u,l (R) n
, with a positive constant C(s, q) independent of u and M . Furthermore if α and ξ 0 are reals, then

x → e iξ0x M u(αx) H s u,l (R) d ≤ C x → e iξ0x u(αx) H q u,l (R) n , where the above constant satisfies C ≤ C(s, q) ξ → ξ s-q 2 M ξ α -ξ 0 C 2 b (R,L(R d ,R n ))
.

4.2. Linear dynamic.

Lemma 4.3. Let T and µ 0 be positive constants. Then there exists C > 0 and κ > 0 such that for all 0 < µ < µ 0 , the following holds. For s ≥ 0, and 0 ≤ t ≤ T µ ,

e tT -Π h c V H s u,l (R) ≤ C V H s u,l (R) , while for any t ≥ 0, e tT -Π h s V H s u,l (R) ≤ Ce -κt V H s u,l (R)
. Both estimates still holds when Π h i are replaced by Π i . Proof. Since T -has constant coefficients -see (12) -it acts in Fourier space through multiplication. Hence we rely on multiplier theory, see [START_REF] Mielke | Attractors for modulation equations on unbounded domains-existence and comparison[END_REF]. Since at fixed Fourier parameter ξ ∈ R, the eigenvalues of matrix T -(ξ) satisfy Re λ i (ξ) ≤ µ, we obtain using lemma 4.2 that for t ≤ T µ :

e tT -Π h c V H s u,l (R) ≤ exp(t T -) Πh c C 2 (R,M2(R)) V H s u,l (R) ≤ Ce 2tµ V H s u,l (R) ,
with M 2 (R) the set of 2 × 2 matrices. The case of e tT -Π h s adapts easily: for t ≥ 0 and ξ in the support of χ s , the eigenvalues of T -(ξ) satisfy Re(λ j (ξ)) ≤ -2κ. To see that Π h i may be replaced by Π i , simply use that e tT -Π i = e tT -Π h i Π i and that

Π i V H s u,l (R) ≤ V H s u,l (R)
to obtain the result.

In the following, we can combine lemmas 4.1 and 4.3 to glue the mode filters techniques with our usual Sobolev spaces -to the cost of one derivativevia estimates of the form:

V (t) L ∞ (R) ≤ C V (t) H 1 u,l (R) ≤ Cϑ(t) V 0 H 1 u,l (R) ≤ Cϑ(t) V 0 W 1,∞ (R) . 4.3.
Nonlinear dynamic: shadowing the global attractor of the Ginzburg-Landau equation. Following [START_REF] Ghazaryan | Nonlinear convective instability of turing unstable fronts near onset: A case study[END_REF], we drive the perturbation using the dynamic at -∞, to the cost of an extra source term S. We first show that this term defined by ( 13) is sufficiently localized in space so that we can extract a

V ω sh (•-c * t) L ∞ = U ρ * L ∞ from it.
In the rest of this section, we assume that there exists C 1 and t U positive constants such that for all 0 ≤ t ≤ t U , (30)

U (t) ρ * L ∞ (R) ≤ C 1 U 0 X (1 + t) 3/2 .
As above, we will note C 1 instead of CC 1 when C is a constant not depending on C 1 . All arguments below rely on the fact that U does not blow up in finite time. Hence in the rest of this section, we always restrict -even when it is not clearly stated -to times 0 ≤ t ≤ t U . Furthermore, we recall the notation µ = ε 2 , with ε > 0.

We first obtain decay of derivatives of U , and then control the source term. Recall that

U X := U W 2,∞ (R)×W 4,∞ (R) + ρ 3 * U L ∞ (R) .
Proposition 4.4. There exist a positive constant C such that for 0 ≤ t ≤ t U :

U (t) ρ * W 2,∞ (R)×W 4,∞ (R) ≤ C U 0 X (1 + t) 3 2
.

Proof. First write the Duhamel formula for u 1 , and then differentiate it:

∂ j x u 1 (t, x) = ˆR ∂ j x G kpp t u 0 1 + ∂ j x G co t u 0 2 dy + ˆt 0 ˆR ∂ j x G kpp t-s N 1 + ∂ j x G co t-s N 2 dyds.
We can then bound the right hand side using linear decay from proposition 3.15 -see also remark 3.16 -and the decay (30) of U (t). The localized weight ω kpp in nonlinear terms allows to gain as many powers of ρ * (y) as needed. We use the same approach for ∂ k x u 2 , the linear estimate

G sh t u 2 W k,∞ (R) ≤ Ce -ηt u 2 W k,∞ (
R) comes from standard parabolic regularity: L sh is sectorial with spectral gap. Proposition 4.5. There exists positive constants δ and C such that for all 0 ≤ t ≤ t U ,

S(• -c * t, V (t, •)) H 1 u,l (R) ≤ C V (t, •) ω sh (• -c * t) W 2,∞ (R)×W 4,∞ (R) ≤ C U 0 X (1 + t) 3/2 . Proof. From lemma 4.1, it is enough to bound S(•, V (t)) W 1,∞ (R) . For x ≤ -1, we have (x) = 1, hence S(x, V ) = (1 -q * (x)) 3α(1 + q * (x))v 1 -γv 2 .
We show that ω sh (1q * ) W 1,∞ (-∞,0) is finite. The equilibrium point (q, q ) = (1, 0) for the front equation is a saddle, with one positive eigenvalue κ = (-1 + √ 3) α d . Basic ODE dynamic then ensures that sup

x≤0 |q * (x) -1|e -κx < +∞, sup x≤0 |q * (x)|e -κx < +∞.
For x ≤ -1 and |θ| small enough, ω sh (x) = e θ x ≤ e -κx , which leads to

S(• -c * t, V (t, •)) W 1,∞ (-∞,0) ≤ C V (t) ω sh (• -c * t) W 1,∞ (-∞,0)
. Now for x ≥ 1, recall the expression of the source term (13). For linear terms, we use the commutator to gain one derivative: the operator

-1 (A -c * ∂ x ) -(A -c * ∂ x ) = -1 [A -c * ∂ x , ]
exhibit at most third order derivatives, so that it maps

W 2,∞ (R) × W 4,∞ (R) onto W 1,∞ (R) × W 1,∞ (R). Furthermore -1 ∂ x W 1,∞ (0,+∞) < ∞. Altogether, it leads to S(•, V ) W 1,∞ (0,+∞) ≤ C V W 2,∞ ×W 4,∞ (0,+∞) + C V 2 W 1,∞ (0,+∞) + C V 3 W 1,∞ (0,+∞) .
Remark that ω kpp (x) = 1, hence the first claimed estimate is shown. Then V (t,x) ω sh (x) = U (t,x) ρ * (x) , and proposition 4.4 ensures the second claimed estimate. The proof is complete.

Remark 4.6. Since t → (1 + t) -3/2 is integrable at 0, the above control of the source term implies that solutions to (11) with initial condition at t = 0 are defined and continuous on an open interval.

We now state nonlinear estimate, which relies on mode filters, see section 4.1.

Lemma 4.7 (Non-linear estimates). There exists C > 0 such that for all V ∈ H 1 u,l (R),

Π s Q -(εΠ c V + ε 2 Π s V ) H 1 u,l (R) ≤ Cε 2 ( Π c V H 1 u,l (R) + Π s V H 1 u,l (R) ) 2 , while Π c Q -(εΠ c V + ε 2 Π s V ) H 1 u,l (R) ≤ Cε 3 ( Π c V H 1 u,l (R) + Π s V H 1 u,l (R) ) 2 . Proof.
The first estimate is immediate. The second one comes from [START_REF] Schneider | A new estimate for the Ginzburg-Landau approximation on the real axis[END_REF], and proves as follows. In Fourier space, multiplication becomes convolution. Since Supp(χ c ) = [-5/4, -3/4] ∪ [3/4, 5/4] and Supp(χ c * χ c ) = [-5/2, -3/2] ∪ [-1/2, 1/2] ∪ [3/2, 5/2] do not intersect, we deduce that

Π c (Π c V 1 × Π c V 2 ) = F -1 (χ c × (χ c V1 * χ c V2 )) = 0.
Hence lowest order quadratic terms vanish when Π c is applied, leaving ε 3 terms at leading order.

We now begin the proof of proposition 2.4 by decomposing T -in Fourier space. This allows to show that after a time T att /ε 2 , the perturbation V is at leading order a critical oscillating mode. Lemma 4.8 (Attractivity). Let T att > 0 be fixed. There exists a constant C > 0 -depending on C 1 -and a positive constant δ such that for all V 0 satisfying (31)

K 1 := 1 ε V 0 H 1 u,l (
R) + U 0 X ≤ δ, the solution V to (11) with initial condition V 0 exists for all time 0 ≤ t ≤ Tatt ε 2 , and decomposes as

V = V c + V s , with Π h i V i = V i . When 0 ≤ t ≤ Tatt ε , it satisfies V i (t) H 1 u,l (R) ≤ CK 1 ε, i ∈ {c, s} , while for times Tatt ε 2/3 ≤ t ≤ Tatt ε 2 , we have V c (t) H 1 u,l (R) ≤ CK 1 ε, V s (t) H 1 u,l (R) ≤ CK 1 ε 2 .
Finally, the initial condition A 0 for (GL) is bounded uniformly in ε: A 0 H 1 u,l (R) ≤ CK 1 , with

(32) A 0 (X) := 1 ε e -i X ε π h 1 V c T att ε 2 , X ε . Proof. Since T -is sectorial, V → Q -(V )
is locally Lipschitz and t → S(x, V (t, x)) is integrable at 0, solution V to (11) with initial condition V 0 is uniquely defined as long as it does not blow up. Hence the estimates that follow ensure that V is defined up to time T att /ε 2 .

To construct V , it is enough to solve the following system, with initial condition V i (0) = Π i V (0):

(33)

∂ t V c = T -V c + Π c Q -(V c + V s ) + Π c S(x, V c + V s ), ∂ t V s = T -V s + Π s Q -(V c + V s ) + Π s S(x, V c + V s ),
and then set V = V c + V s . A solution (V c , V s ) of the decoupled system (33) is not guaranteed to write as V i = Π i V , however the critical-stable separation still holds, so that lemma 4.3 applies:

e tT -V s H 1 u,l (R) = e tT -Π h s V s H 1 u,l (R) ≤ e -κt V s H 1 u,l (R)
. We introduce the local notations W i := Vi ε and Θ i (t) := sup 0≤τ ≤t W i (τ ) H 1 u,l (R) . Using Duhamel formula and decay of S, we see that

W c (t) H 1 u,l (R) ≤ C W c (0) H 1 u,l (R) + Cε ˆt 0 (Θ c (t) + Θ s (t)) 2 dτ + C 1 U 0 X ε ˆt 0 dτ (1 + τ ) 3/2 , ≤ C 1 K 1 + Cεt(Θ c (t) + Θ s (t)) 2 .
In a similar way, standard integral computations -see lemma E.4 -ensures that

W s (t) H 1 u,l (R) ≤ Ce -κt W s (0) H 1 u,l (R) + Cε(Θ c (t) + Θ s (t)) 2 ˆt 0 e -κ(t-τ ) dτ + C 1 U 0 X ε ˆt 0 e -κ(t-τ ) dτ (1 + τ ) 3/2 , ≤ C 1 K 1 (1 + t) 3/2 + Cε(Θ c (t) + Θ s (t)) 2 .
For 0 ≤ t ≤ Tatt ε , we take the sup in the two above equations to obtain -with Θ := max(Θ c , Θ s ) -that Θ(t) ≤ C 1 K 1 + CΘ(t) 2 . Applying a standard nonlinear argument -see lemma E.5 -with K 1 ≤ δ small enough, we recover Θ(T att /ε) ≤ C 1 K 1 , which is the first claimed estimate. Now for T att /ε 2/3 ≤ t ≤ T att /ε, we have

1 (1+t) 3/2 ≤ Cε, hence W s (t) H 1 u,l (R) ≤ C 1 K 1 ε.
This little improvement will allow us to propagate estimates until time T att /ε 2 . For t ≥ 0, we now use the new local notation

V (t + T att /ε 2/3 ) =: εW c (t) + ε 2 W s (t), together with Θ i (t) := sup 0≤τ ≤t W i (τ ) H 1 u,l (R) .
Remark that from the above, we know that

W c (0) H 1 u,l (R) + W s (0) H 1 u,l (R) ≤ C 1 K 1 . Then nonlinear estimate from lemma 4.7 ensures that for t ≤ T att /ε 2 , W c (t) H 1 u,l (R) ≤ C W c (0) H 1 u,l (R) + Cε 2 t (Θ c (t) + Θ s (t)) 2 + C 1 U 0 X ε ˆt 0 dτ (1 + T att /ε 2/3 + τ ) 3/2 , ≤ C 1 K 1 + C(Θ c (t) + Θ s (t)) 2 .
Since (1 + T att /ε 2/3 + τ ) -3/2 ≤ ε when τ ≥ 0, we deduce as above that

W s (t) H 1 u,l (R) ≤ Ce -κt W s (0) H 1 u,l (R) + C ˆt 0 e -κ(t-τ ) dτ (Θ c (t) + Θ s (t)) 2 + C 1 U 0 X ε 2 ˆt 0 e -κ(t-τ ) dτ (1 + T att /ε 2/3 + τ ) 3/2 , ≤ C 1 K 1 + C(Θ c (t) + Θ s (t)) 2 .
Taking the sup for 0 ≤ t ≤ T att /ε 2 , we obtain as previously that Θ(T att /ε 2 ) ≤ C 1 K 1 for K 1 small enough. This is the second claimed estimate. To bound A 0 , we use the scaled estimate in lemma 4.2

X → e -i X ε M u X ε H 1 u,l (R) ≤ ξ → ξ 1/2 M (εξ + 1) C 2 b (R,L(R 2 ,R)) X → e -i X ε u X ε L 2 u,l (R)
, with M = π h 1 and u = V c ( Tapr ε 2 ). To estimate M (εξ + 1), we can restrict to the C 0 b norm, since derivative gains us power of ε. Recall that

M (ξ)û = χ h c (ξ) c (1), * c (ξ) -1 û, * c (ξ) so that Cauchy-Schwartz leads to M (ξ) L(R 2 ,R 2 ) ≤ |χ h c (ξ) * c (ξ) c (1), * c (ξ) -1 | ≤ C|χ h c (ξ)|.
Here we assume that the support of χ c is so small that c (1), * c (ξ) does not vanish by continuity. We emphasize that this support is still independent of µ, so that the separation of frequencies comes with spectral gap. Using the support of χ h c , we obtain

ξ → ξ 1/2 M (εξ + 1) C 2 b (R,L(R 2 ,R)) ≤ C √ ε . Now rescaling the L 2 u,l (R) norm, we get X → e -i X ε u(X/ε) L 2 u,l (R) ≤ √ ε e -ix u L 2 u,l (R) ≤ √ ε u L 2 u,l (R) . Hence we have shown that A 0 H 1 u,l (R) ≤ Cε -1 V c (ε -2 T apr ) L 2 u,l (R) ≤ C 1 K 1 .
Now that V is at leading order a critical oscillating mode, we can approximate it by ψ(ε, A) := εψ c (ε, A) + ε 2 ψ s (ε, A), with A the solution to the GL equation with initial condition A 0 , see (32), with the critical part of ψ being:

(34) ψ c (ε, A)(t, x) = (e ix A(ε 2 t, εx) + c.c) c ,
and the stable part:

ψ s (ε, A)(t, x) = |A(ε 2 t, εx)| 2 0 + e ix ∂ X A(ε 2 t, εx) 1 + e 2ix A(ε 2 t, εx) 2 2 + c.c
. All i and c are two-dimensional vectors with explicit expressions, see appendix A. The choice of ψ s is made clearer therein, it ensures that the error of approximation R = Vψ(ε, A) stays small when time evolves. Lemma 4.9 (Global attractor for Ginzburg-Landau). Let T, X → A(T, X) be a solution of the Ginzburg-Landau equation (GL). Then there exists a constant C GL > 0 such that

A(T ) W 1,∞ (R) ≤ C GL + e -T /2 A(0) W 1,∞ (R) .
Proof. Since the coefficients of GL are real, the dynamic is close to that of a bistable equation. We follow [Sch94b, Theorem 7]. From computations made in [CE90, Theorem 4.1], we recover that

A(T )h L 2 (R) ≤ C GL + e -2T A(0)h L 2 (R)
where h a (X) = 1 1+(X-a) 2 . The same estimate holds for ∂ x A. It leads to

|A(T, X)| ≤ 1 h(X) 2 C GL + e -T A(0)h H 1 (R) , which after taking the supremum over a ∈ R writes A(T ) L ∞ (R) ≤ C GL + e -T A(0) H 1 u,l (
R) . The bound on ∂ X A(T ) is then obtained from standard parabolic regularity: we differentiate the Duhamel formula and use the above bound of A(T ). To handle the term that is constant in time, which appears in the non-linear term, we simply deteriorate the exponential decay.

We now show that when t ≤ ε -7/4 T apr , the error R stays small. We then apply this argument again to reach any finite time. For similar problems, this approximation property is usually shown on a time scale ε -2 T apr , which is optimal due to the linear growth of critical modes e

tT -Π c V 0 H 1 u,l (R) ≥ Ce ε 2 t Π c V 0 H 1 u,l (
R) . In our case, the slowly decaying inhomogeneous term S prevents us to go beyond ε -7/4 T apr .

Lemma 4.10 (Approximation). For T apr > 0 fixed, there exists a positive constant C apr such that the following holds. Let t 0 ≥ Tatt ε 2 , A 0 ∈ H 1 u,l (R) and a solution V of (11) such that

R(t 0 ) H 1 u,l (R) = V (t 0 ) -ψ(ε, A 0 ) H 1 u,l (
R) ≤ ε 5/4 . Note A the solution of (GL) with initial condition A 0 at t = t 0 , and R(t) = V (t)ψ(ε, A(t)). Then for t ≥ t 0 , the error term decomposes as

R(t) = R c (t) + R s (t), with Π h i R i (t) = R i (t). Furthermore, for 0 ≤ t -t 0 ≤ Tapr ε 1/4 , R i (t) H 1 u,l (R) ≤ C apr ε 5/4 , i ∈ {c, s} , while for Tapr ε 1/4 ≤ t -t 0 ≤ Tapr ε 7/4 , R c (t) H 1 u,l (R) ≤ C apr ε 5/4 , R s (t) H 1 u,l (R) ≤ C apr ε 9/4 . Proof. Insert R(t) = V (t) -ψ(ε, A(t)) in (11) to obtain that for t ≥ t 0 , R satisfies ∂ t R = T -R + 2B(ψ, R) + Q -(R) + S(t) + Res(ψ),
where B is a symmetric bilinear function that satisfies Q -(V ) = B(V, V ), and the residual term is

Res(ψ) = -∂ t ψ + T -ψ + Q -(ψ).
We decompose Duhamel formula into a critical and stable part:

Π i R(t) = e tT -Π i R(t 0 ) + ˆt t0 e (t-s)T -Π i 2B(ψ, R) + Q -(R) + S + Res(ψ) ds,
with i ∈ {c, s}. To close a nonlinear argument, we follow [START_REF] Schneider | Global existence via Ginzburg-Landau formalism and pseudo-orbits of Ginzburg-Landau approximations[END_REF]. Since A satisfies (GL), the leading order terms in the residual Res(ψ(ε, A)) vanish, see appendix A. Hence we can control Res(ψ(ε, A)) for times smaller than ε -2 T apr , as done in [Sch94b, Lemma 14]. To handle the source term -which is not present in the above reference -we use proposition 4.5 above, to conclude that S(t, •) H 1 u,l (R) ≤ Cε 3 when t ≥ t 0 ≥ Tatt ε 2 . This implies that ˆt t0 e (t-τ )T -Π c S(τ )dτ

H 1 u,l (R) ≤ C(t -t 0 )ε 3 , while ˆt t0 e (t-τ )T -Π s S(τ )dτ H 1 u,l (R) ≤ Cε 3 ˆt t0 e -κ(t-τ ) dτ ≤ ε 3 .
Hence when t-t 0 ≤ ε -1/4 T apr , the estimates on S can be sub-summed into the one on Res(ψ(ε, A)). Now for ε -1/4 T apr ≤ tt 0 ≤ ε -7/4 T apr ≤, we can once again sub-summed S into Res(ψ(ε, A)). The remaining terms are handled as in the proof of [Sch94b, Lemma 11], see Lemma 13 therein.

We are now ready to prove that V remains bounded in time.

Proof. Proposition 2.4.

Beginning with V (0) H 1 u,l (R) ≤ C V 0 W 1,∞ ≤ εK 1 ≤ εδ,
we wait a time t 0 := ε -2 T att , see lemma 4.8. Now V decomposes as V = V c + V s , with the leading order V c writing as:

Vc (t 0 , ξ) = F(x → εe ix A 0 (εx))(ξ) c (ξ) + c.c,
i.e. V c (t 0 ) is obtained as the modulation of a large profile x → A 0 (εx), with A 0 H 1 u,l (R) ≤ CK 1 from lemma 4.8. We propagate this ansatz by decomposing V (t, x) = ψ(ε, A)(t, x) + R(t, x), see (34). Applying lemma 4.9 and

A 0 L ∞ (R) ≤ C A 0 H 1 u,l (R) from lemma E.3, we conclude that A(ε 2 t) W 1,∞ (R) ≤ C GL + CK 1
for all times t ≥ 0. However, A and ψ(ε, A) does not depend on the same space variable, so that rescaling H 1 u,l (R) norms loses us a ε -1/2 . Instead, we use lemma E.3 and its injections to rescale in W 1,∞ (R) space:

ψ c (ε, A)(t) H 1 u,l (R) ≤ C x → A(ε 2 t, εx) H 1 u,l (R) , ≤ C x → A(ε 2 t, εx) W 1,∞ (R) , ≤ C A(ε 2 t) W 1,∞ (R) , ≤ C GL + CK 1 e -t/ε 2 Recall the expression of K 1 (31). Similar estimates lead to ψ s (ε, A)(t) H 1 u,l (R) ≤ C GL + CK 1 e -t ε 2 . Now for t ≥ t 0 we can estimate V (t) H 1 u,l (R) ≤ ψ(ε, A)(t) H 1 u,l (R) + V (t) -ψ(ε, A)(t) H 1 u,l (R) , ≤ Cε(1 + K 1 e -t/ε 2 ) + R(t) H 1 u,l (
R) , and it only remains to bound the error R(t). We initialy decompose it as R c (t

0 ) = V c (t 0 ) -εψ c (A 0 ) and R s (t 0 ) = V s (t 0 ) -ε 2 ψ s (A 0 ). From the definition of A 0 , π h 1 V c (t 0 , x) = εe ix A 0 (εx), while definitions of ψ c (34) and π h 1 (29) ensure that π h 1 (εψ c (A 0 ))(x) = F -1 χ h c 1 ξ>0 F(εe iy A 0 (εy) = εe ix F -1 (χ 0 F(A 0 (εy))). where χ 0 (ξ) = χ h c (ξ -1)1 ξ>1 is supported near ξ = 0. Hence π h 1 (V c (t 0 ) -εψ c (A 0 ))(x) = εe ix F -1 ((1 -χ 0 )F(A 0 (ε•))) .
Since R c (t 0 ) has compact support in Fourier space, we can gain one derivative, and treat separately the behavior close to 1 and -1:

R c (t 0 ) H 1 u,l (R) ≤ C π h 1 R c (t 0 ) L 2 u,l (R) + π h -1 R c (t 0 ) L 2 u,l (
R) . We first rescale space, then apply the scaled estimate from lemma 4.2, and finally use the localization of χ 0 together with the bound on A 0 -see lemma 4.8 -to obtain that

π h 1 R c (t 0 ) L 2 u,l (R) ≤ 1 √ ε x → π h 1 R c t 0 , x ε L 2 u,l (R) , ≤ √ ε (1 -χ 0 (εξ -1)) ξ -1/2 C 2 b (R) X → e iX A 0 (X) H 1 u,l (R) , ≤ ε 3/2 CK 1 . The conjugated part π h -1 R c (t 0 ) bound identically. Furthermore: R s (t 0 ) H 1 u,l (R) ≤ V s (t 0 ) H 1 u,l (R) + ε 2 ψ s (A 0 ) H 1 u,l (R) ≤ Cε 2 + Cε 2 x → A 0 (εx) H 2 u,l (
R) . Using that x → A 0 (εx) has compact support in Fourier space, we can gain two derivatives, inject into L ∞ (R), scale, and finally inject back into H 1 u,l (R). It reads:

x → A 0 (εx) H 2 u,l (R) ≤ C A 0 H 1 u,l (R) . Hence R(t 0 ) H 1 u,l (R) ≤ ε 5/4
, and lemma 4.10 ensures that R(t) H 1 u,l (R) ≤ Cε 5/4 for times t 0 ≤ t ≤ t 1 = t 0 + ε -7/4 T apr . Applying the same Lemma as many time as needed, we conclude that the above bound on R(t) holds for times t 0 ≤ t ≤ t n = t 0 + nε -7/4 T apr , without deterioration of the constant. Hence V (t) H 1 u,l (R) ≤ Cε(1 + K 1 e -t/ε 2 ) + Cε 5/4 ≤ C √ µ(1 + K 1 e -t/µ ), and the proof is complete.

Appendix A. Ginzburg-Landau equation

Here we derive the Ginzburg-Landau amplitude equation from the dynamic at -∞. We study the following system:

(35) ∂ t V = T 0 V + T µ V + N 2 (V ) + N 3 (V ),
where the linear terms are given by:

T 0 = d∂ 2 x -2α β 0 -(1 + ∂ 2 x ) 2 , T µ = 0 0 0 µ ,
while the quadratic and cubic terms write as:

N 2 (V ) = -3αv 1 2 γv 1 v 2 N 3 (V ) = - αv 1 3 σv 2 3 .
In the following, we note M (X) the symbol of the constant coefficient operator T 0 , such that T 0 = M (∂ x ). We also note ε = √ µ. Assume that the solution to such system writes according to the following ansatz:

V (t, x) = ε(e ix A(ε 2 t, εx) + c.c) c + ε 2 (A 0 0 + e ix A 1 1 + e 2ix A 2 2 + c.c).

Here and in the following c.c is the complex conjugate: z + c.c stands for z + z = 2 Re z. The amplitude A ∈ C is the main unknown, while each A i (ε 2 t, εx) is an amplitude we will fix later.

The vector c is chosen to be an eigenvector for matrix M (i), associated to eigenvalue 0, while the vectors i will be fixed later. We also introduce * c , in the kernel of M (i) * , normalized so that c , * c = 1. We propose We inject the ansatz in system (35), and then identify terms of order ε l e ikx for l ∈ N * and k ∈ Z. As usual for such computation, the ε order will be trivially satisfied. Then ε 2 e ikx terms will determine amplitudes A k and vectors k . Finally, the ε 3 e ix equation will lead to Ginzburg-Landau equation. Hence, we may neglect all terms of order ε 3 e 2ix and ε 3 e 0 . We easily obtain:

∂ t V = ε 3 (e ix ∂ T A c + c.c)
with slow time variable T = ε 2 t. Similarly, we note X = εx the large space variable. To compute T 0 u, we use a formal Taylor expansion, see lemma E.1:

T 0 (εe ix A c ) = εe ix M (i + ∂ x )A c , = εe ix A M (i) c + ε∂ X A M (i) c + 1 2 ε 2 ∂ XX A M (i) c + O(ε 4 ).
The computation can be easily adapt for T 0 (ε 2 e 2ikx A k k ), for k ∈ {0, 1, 2}. The T µ (V ) term is easily developed, and we are left with the nonlinear terms. We note B : R 2 × R 2 -→ R 2 a symmetric bilinear application that satisfies B(V, V ) = N 2 (V ). It is given by:

B(V, W ) = -3αv 1 w 1 γ 2 (v 1 w 2 + v 2 w 1 )
.

Hence, we obtain: We now collect previous calculus, and identify terms of same order ε l e ikx . The ε 2 e ikx equations write ∂ X A M (i) c + A 1 M (i) c = 0,

N 2 (V ) = ε 2 e 2ix
A 0 M (0) 0 + 2|A| 2 N 2 ( c ) = 0, A 2 M (2i) 2 + A 2 N 2 ( c ) = 0.
Both matrix M (0) and M (2i) are invertible, so that we can impose:

A 1 = ∂ X A, A 0 = |A| 2 , A 2 = A 2 , 0 = -2M (0) -1 N 2 ( c ), 2 = -M (2i) -1 N 2 ( c
). We also fix 1 such that M (i) c + M (i) 1 = 0, which is possible due to M (i) c ∈ im(M (i)). The ε 3 e ix equation then writes

∂ T A c = ∂ XX A (M (i) c + M (i) 1 ) + A 0 0 0 1 c + A|A| 2 (2B( c , 0 ) + 2B( c , 2 ) + 3N 3 ( c )) .
Taking the scalar product with the normalized left eigenvector * c leads to the scalar Ginzburg-Landau equation. In the following, we make explicit its coefficients. It is easily seen that M (i) c + M (i) 1 , * c = 4, and that 0 0 0 1 c , * c = 1. We are left with the coefficient in front of the cubic terms. We explicitly compute .

M (0) -1 = -1 2α 1 β 0 2α M (
We can finally compute that:

2B( c , 0 ) + 2B( c , 2 ) + 3N 3 ( c ), * c = γ 2 β 2 19 9 + (d + 2α) 1 α + 1 9(4d + 2α) -3γβ 2 1 + α 4d + 2α -3σ(d + 2α) 2 .
The right hand side reads as a degree 2 polynomial in γ which we note P (γ). Altogether, we obtain the following Ginzburg-Landau equation:

(GL) ∂ T A = 4∂ XX A + A + P (γ)A|A| 2 .

Since P admits two roots with distincts sign, there exists γ GL > 0 such that for all γ ∈ (0, γ GL ), we have P (γ) < 0, i.e. hypothesis (H 2 ) is fulfilled. Recall that α, d, σ > 0. We compute: Lemma A.1. There exists an open, nonempty set of parameters Ω such that for all (α, β, d, σ) ∈ Ω, the ordering γ rem < γ GL holds. In particular for γ ∈ (γ rem , γ GL ), both hypothesis (H 1 ) and (H 2 ) holds true.

γ GL = 3(
Proof. The fact that Ω is open comes from the continuity of both γ rem and γ GL with respect to α, β, d, σ > 0. To see that Ω = ∅, remark that γ GL → +∞ when either β → 0 or σ → +∞.

Remark A.2. The ansatz we propose here only develop up to order ε 2 , while the information we extract is held by ε 3 terms. If ones try to push the ansatz one order further:

V (t, x) = ε(e ix A c + c.c) + ε 2 (e ix A 1,1 1,1 + e 2ix A 1,2 1,2 + c.c + A 1,0 1,0 ) + ε 3 (e ix A 2,1 2,1 + e 2ix A 2,2 2,2 + e 3ix A 2,3 2,3 + c.c + A 2,0 2,0 ), the only changes is the presence of an extra A 2,1 L(i) 2,1 , * c term in (GL). By definition of * c , this term vanishes anyway.

Lemma E.4. Let α, β > 1 and γ > 0 be reals. Then there exists C > 0 such that for all t ≥ 0:

(44) ˆt 0 ds (1 + s) α (1 + ts) β ≤ C (1 + t) min(α,β) , ˆt 0 e -γ(t-τ ) (1 + τ ) α dτ ≤ C (1 + t) α .

Proof. One can slightly adapt this statement when α = 1 or β = 1, see [START_REF] Xin | Multidimensional stability of traveling waves in a bistable reaction-diffusion equation[END_REF]. First, remark that

I α,β (t) := ˆt/2 0 ds (1 + s) α (1 + t -s) β ≤ 1 (1 + t/2) β 1 -α (1 + s) α-1 t/2 0 ≤ α -1 (1 + t/2) β .
Hence, we cut the integral in (44) at t/2, and the change of variable z = ts leads to ˆt 0 ds (1 + s) α (1 + ts) β = I α,β (t) + I β,α (t) ≤ C (1 + t/2) min(α,β) ≤ C 2 min(α,β) (1 + t) min(α,β) .

Which is the first claimed estimate. The second one follows from e -γ(t-τ ) ≤ C (1+t-τ ) α . Lemma E.5. Let b, ε be positive constants, and note a 0 (b, ε) = 1 2 (2b) -1 ε > 0. Then for all 0 < a < a 0 , the following holds. If t → x(t) is a positive continuous function that satisfy x(t) ≤ a + bx(t) 1+ε and such that x(0) is small enough, then x(t) ≤ 2a for all t ≥ 0.

Proof. Introduce f (x) = a + bx 1+εx. Then f (2a) = a(2 1+ε ba ε -1) < 0 when a is smaller than a 0 , while f (0) = a > 0. Assume x(0) ≤ 2a. Then the connected component of f -1 (R + ) that contains x(0) is include in (-∞, 2a). Since f (x(t)) ≥ 0 by construction, the continuity of x implies x(t) ≤ 2a for all t > 0. This conclude the proof.

Figure 1 .

 1 Figure 1. Illustration of a typical solution (u, v), for µ positive and small. The front -right brace -has fix shape and moves to the right at constant speed c * . The Turing pattern -left brace -continuously expands at slow speed O( √ µ), towards left and right. Both states (0, 0) and (1, 0) -respectively ahead and behind the front -are pointwise unstable, but convectively stable in the critical weighted space.

  A 2 N 2 ( c ) + c.c + 2ε 2 |A| 2 N 2 ( c ) + ε 3 (e ix 2AA 0 B( c , 0 ) + c.c) + ε 3 (e ix 2 ĀA 2 B( c , 2 ) + c.c) + O |k| =1 (ε 3 e ikx + ε 4 ).Similarly, the cubic term develop as:N 3 (V ) = ε 3 (e i x3A|A| 2 N 3 ( c ) + c.c) + O |k| =1 (ε 3 e ikx + ε 4 ).

  + 2α) β 2 (γ(d + 2α) -27α) γβ(d + 2α)(4d + 2α).Then, it followsN 3 ( ) = -αβ 3 σ(d + 2α) 3 , B( c , 0 ) = -3αβ 3 (γ d+2α α + 2α) -3αβ 3 (γ(d + 2α) -27α) β 2 γ 2 (d + 2α)(γ(5d + 4α) -27α)

If T ± are elliptic, then both λ -T ± are invertible for Re λ large enough. This is equivalent to have n 2 stable (respectively unstable) spatial eigenvalues for λ -T + (respectively λ -T -) and ensures that the region S 1 that contains (η, +∞) -for η ∈ R large enough -does not belong to Σess(T ∞ ).

Acknowledgments

The author thanks Grégory Faye for multiple fruitful discussions. This works was partially supported by Labex CIMI under grant agreement ANR-11-LABX-0040.

Proof. Proposition 3.8. Here, we write a(x, λ) b(x, λ) to stand for a(x, λ) ≤ Cb(x, λ) where C is a constant. Recall we have construct ODE solutions with exponential behavior. For 0 ≤ j ≤ 3, we have (36) ∂ j x φ(λ, x) = e νx ν j + O R ± ×K (e -α|x| ) ,

where φ stands for φ sh,± i and ν for ν sh,± i (λ). Here, we noted g(λ, x) = O R + (f (x)) if |g(λ, x)| f (x) when x ≥ 0, and if λ → g(λ, x) is holomorphic on K for almost all x ≥ 0. The according notation holds for O R -. In particular, |φ sh,± i (x)| e x Re(ν ± i ) when x ∈ R ± . In the following, we drop the "sh" exponent. Then, the Green function expresses using the decaying solutions:

i=3 b i (λ, y)φ - i (λ, x) if x < y, where the coefficients b i (λ, y) are determined by the jump condition:

Recall notation (20) for determinants. We use Cramer's rule to compute the coefficients b i (λ, y):

, and

It may happens that ν + 2ν + 1 or ν - 4ν - 3 vanish, causing a singularity in (b 1 , b 2 ) or (b 3 , b 4 ). However, this singularity can be erased in the expression of G sh λ since at first order b 1 (λ, y)

. Recall that we assume the ν ± i are sorted by real part: Re ν + 1 ≤ Re ν + 2 ≤ Re ν + 3 ≤ Re ν + 4 (the same ordering holds for the ν - i ), and that from lemma 3.5item (1), there exists κ 2 > 0 such that:

We now prove the claimed result, depending on the ordering between x, y and 0.

(i ) y < 0 < x. We need to decompose φ + 2 (y) into the (φ - i ) 1≤i≤4 basis: φ + 2 = i=1 4 c i φ - i . We can differentiate three times this decomposition to obtain a Cramer system. Solving it leads to

Once again, the y dependence in each of the fraction can be dropped thanks to lemma E.2. The numerator is holomorphic with respect to λ, hence bounded when λ ∈ K, and the denominator does not vanish. 8 In the following, we may simply write

) to note the first coefficient of ϕ when decomposed in the (φ - i ) 1≤i≤4 basis. We extend this notation to other coefficients: c - i (ϕ), and to the decomposition into (φ + i ) 1≤i≤4 basis: c + i (ϕ). Using the above decomposition into the expression of G sh λ , we obtain:

)y W sh (λ, 0). Using proposition 3.3, we see that W sh (λ, 0) 1 when λ lies in K. Collecting the above estimates (39), ( 36) and (38), we now see that

e -κ2(x-y) .

The exact same approach for the second term leads to:

e -κ2(x-y) .

The first case is now done. For the following ones, same tools are used. We nevertheless give the general plan of the proofs to clarify some points.

(ii ) 0 < y < x. We decompose both φ - 3 (y) and φ - 4 (y) into the (φ + i ) 1≤i≤4 basis, and use the ordering (38) of the spatial eigenvalues to obtain

, e -y Re(ν + 1 ) + e -y Re(ν + 3 ) + e -y Re(ν + 4 ) .

The second and third term are treated directly: -y Re(ν + 3,4 ) ≤ -yκ 2 ≤ yκ 2 . For the first term, we need to use the ordering of x and y. Since xy ≥ 0, we have Re(ν + 1 )(xy) ≤ -κ 2 (xy). Hence:

A similar argument leads to:

e -κ2(x-y) .

8 A vanishing determinant would implies that λ is an eigenvalue, which can not hold due to Re λ ≥ -2η.

(iii ) y < x < 0. Here we need to decompose both φ + 1 (x) and φ + 2 (y). Taken independently, both term φ + 1 b 1 and φ + 2 b 2 do not decay as claimed, due to the following asymmetric growing rates, which correspond to terms where the two projections are done on the same element of (φ - i ) 1≤i≤4 :

and

However, this terms appear both in φ + 1 b 1 and φ + 2 b 2 , so that they cancel out in the expression of G sh λ , recall from (37) that b 1 and b 2 are obtained with opposite sign in their expressions. We compute

+ e x Re(ν - 2 ) + e x Re(ν - 3 ) + e x Re(ν - 4 ) e -y Re(ν - 2 ) , e -κ2(x-y) .

For the second inequality we have used the same method as in the previous case. From one hand (xy) Re(ν + 1 ) ≤ -κ 2 (xy) due to the ordering of x and y. From the other hand, x Re(ν 3 ) ≤ xκ 2 ≤ -xκ 2 coupled with -y Re(ν - 1 ) ≤ -yκ 2 . Similar computations leads to

We conclude using triangular inequality: |φ + 1 (x)b 1 (y) + φ + 2 (x)b 2 (y)| e -κ2(x-y) . We are now done with all cases where y < x. The x < y cases are mirror versions of the three above cases. Computations can be adapted easily, we omit them. This complete the proof of the first stated inequality. We now treat the second one (21). Remark that when y ≥ 1, (21) adds no information to the first inequality, since ω κ2,0 (y) = 1. Hence we are only left with the case y ≤ 0 ≤ x, in which we have:

2 y e -κ 2 2 x ≤ e -κ 2 2 |x-y| ω κ2/2,0 (y).

Up to a change of notation κ2 = 1 2 κ 2 , this is the claimed (21). The proof is now complete.

Appendix C. Three equilibrium points

It appears that for certain sets of parameters (d, α, β, γ, σ), system (1) admits nontrivial equilibrium (u eq , v eq ) with v eq ∼ √ µ. To keep dynamic as simple as possible, for example when working with numerical simulations, we can restrict to parameters that satisfy the following statement. This is by no mean necessary to our study, since the main result is perturbative.

Proposition C.1 (Three equilibrium points). Set µ 0 < 1 and γ > 0. Then for positive d, α, β, there exists σ 0 > 0 such that for all σ > σ 0 , and 0 ≤ µ < µ 0 , system (1) admits only the steady states (u, v) = (±1, 0) and (u, v) = (0, 0).

L. GAR ÉNAUX

Proof. Assume that (u, v) ∈ R 2 is a constant solution of (1). If v = 0, the only solutions are (±1, 0) and (0, 0). Else, the point (u, v) lies in the intersection of two curves:

For v ≤ 0, they do not intersect since they respectively ensures u ≤ 1 and u > 1. 9 For v > 0, we use the tangent line to the first curve at (u, v) = (1, 0):

Hence the two curves do not intersect for positive v provided the second curve do not intersect T , which reads:

It holds true for sufficiently large σ > 0. This complete the proof.

Appendix D. Proof of lemma 3.4

Proof. Lemma 3.4. We make the change of variable z(t) = e -tA∞ y(t), so that

which writes as

)ds. Hence for z bounded, equation (40) together with condition z(t) -→ v when t → +∞ is equivalent to (41). We now make the change of variable x(t) = e α 2 t (z(t)v), that satisfies:

The operator K is a contraction from L ∞ (T, +∞) n to itself, provided T is large enough. Indeed, for t ≥ T we have

The Picard fixed point theorem ensures the existence of a unique κ ∈ L ∞ (T, +∞) solution of (42).

Reverting back all changes of variable, we obtain a solution of (19) with y(t) = e tA∞ (v + e -α 2 t κ(t)) as claimed. By Cauchy-Lipschitz theorem, we flow this solution backward to define it on R.

We now assume that (v i ) 1≤i≤n is a basis of R n . Since the family (e -tA∞ y i (t)) i converges to (v i ) i when t → +∞, we obtain that det(e -tA∞ y 1 (t), . . . , e -tA∞ y n (t)) = det(e -tA∞ ) det(y 1 (t), . . . , y n (t)) is nonzero for t large enough, by continuity of the determinant. This ensures that (y i ) i is a basis for the solutions of equation ( 19).

Finally, we suppose that A(t), A ∞ and v are holomorphic with respect to λ. Then the operator K λ : L ∞ (T, +∞) n → L ∞ (T, +∞) n defined by ( 42) is holomorphic with respect to λ, and so is κ. This conclude the proof.

Appendix E. Standard Lemmas

Lemma E.1. Let M be a matrix with polynomial coefficients: M (X) = (P i,j (X)) 1≤i,j≤n , we note d := max i,j deg(P i,j ) the greatest degree in M , and L the associated differential operator: L = M (∂ x ). Then the conjugation of L by the weight e ϑx writes as:

where M (k) is the matrix whose coefficients are the polynomial's derivatives: (P

Proof. We first show the scalar case n = 1. By linearity, it is enough to consider the case L = ∂ d x , which corresponds to P (X) = X d . It can be easily computed that e -ϑx ∂ x e ϑx • = (ϑ + ∂ x ), rising it to the power d reads: e -ϑx L e ϑx • = (ϑ + ∂ x ) d = P (ϑ + ∂ x ). To finish the scalar case, we do a Taylor expansion at order d. Such expansion is exact due to P being polynomial:

We then formally replace X by ∂ x . The case of a matrix of differential operator is straightforward, is suffice to apply the above to each component of the matrix.

Lemma E.2. Let L = n j=0 a j (x)∂ j x be a differential operator with coefficients x → a j (x). Let φ 1 , . . . , φ n be solutions of the ODE Proof. For the proof, we will note Φ the n × n matrix whose i-th column is Φ i . Remark that W(λ, x) := det(Φ)(λ, x) satisfy:

where A(λ, x) is the n×n matrix obtained if one vectorise the ODE (43). Since the trace is invariant under circular permutation, we recognize

This scalar ODE on W is easily solved. Noticing that Tr(A(λ, x)) = -a n-1 (x) conclude the proof.

Lemma E.3. Let p, q ∈ R ∪ {+∞} such that 1 ≤ q ≤ p ≤ +∞, and X, Y be two measurable spaces.

Then the following injection is continuous:

Proof. This is the well-known Minkowsky's integral inequality.