Montie Avery
email: avery142@umn.edu

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

Spectral stability of the critical front in the extended Fisher-KPP equation

Keywords: AMS subject classifications: 35B35, 35B25 traveling waves, pulled fronts, singular perturbations, embedded eigenvalues

 which establish stability of fronts without identifying a precise decay rate. We verify that the front is marginally spectrally stable: while the essential spectrum touches the imaginary axis at the origin, there are no unstable eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum at the origin. Together with the recent work of Avery and Scheel [3], this implies nonlinear stability of the critical front with sharp t -3/2 decay rate, as previously obtained in the classical Fisher-KPP equation. The main challenges are to regularize the singular perturbation in the extended Fisher-KPP equation and to track eigenvalues near the essential spectrum, and we overcome these difficulties with functional analytic methods.

Introduction

Background and main results

We consider the extended Fisher-KPP equation

u t = -δ 2 u xxxx + u xx + f (u), (1.1)
for small values of the parameter δ. Notice that the sign of δ ∈ R will play no role in what follows. Here we assume f is of Fisher-KPP type: f (0) = f (1) = 0, f (0) > 0, f (1) < 0, and f (u) < 0 for all u ∈ (0, 1); see Section 1.2 for comments on this last assumption. Equation (1.1) may be derived as an amplitude equation near certain co-dimension 2 bifurcations in reaction-diffusion systems [START_REF] Rottschäfer | On the transition from the Ginzburg-Landau equation to the extended Fisher-Kolmogorov equation[END_REF], and for δ = 0, it reduces to the classical Fisher-KPP equation, which is a fundamental model for front propagation into unstable states. The extended Fisher-KPP equation is of interest in the physics literature as a concrete model in which one can study the transition from monotone to pattern-forming invasion fronts [START_REF] Dee | Bistable systems with propagating fronts leading to pattern formation[END_REF]. In both the classical and extended Fisher-KPP equations, much of the interesting dynamics are driven by traveling fronts, which solve 0 = -δ 2 q + q + cq + f (q), q(-∞) = 1, q(∞) = 0.

(1.2)

In the classical Fisher-KPP equation, δ = 0, the front equation (1.2) admits monotone solutions for all speeds c ≥ c * (0) = 2 f (0); see [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF]. Using comparison principles [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF][START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Nolen | Convergence to a single wave in the Fisher-KPP equation[END_REF][START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov[END_REF] or probabilistic methods [START_REF] Bramson | Maximal displacement of branching Brownian motion[END_REF][START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to traveling waves[END_REF] it can be shown that the critical front with speed c * (0) governs the dynamics for large classes of initial data, in the sense that solutions typically converge to the critical front, with some higher order corrections to the position. Such results are not available for the extended Fisher-KPP equation due to the lack of a comparison principle.

1 Much work has also been done on the stability of fronts in the Fisher-KPP equation with respect to sufficiently localized perturbations. For supercritical fronts, with speeds c > c * (0), stability with an exponential decay rate e -Λt can be achieved by using exponential weights to push the essential spectrum entirely into the left half-plane and suppress the translational eigenvalue at the origin. Stability of the critical front is more difficult, since with the optimal choice of exponential weight, the essential spectrum still touches the imaginary axis due to the presence of absolute spectrum [START_REF] Sandstede | Absolute and convective instabilities of waves on unbounded and large bounded domains[END_REF] at the origin. Local stability of the critical front was established by Kirchgässner [START_REF] Kirchgässner | On the nonlinear dynamics of travelling fronts[END_REF] and later refined in [START_REF] Eckmann | The nonlinear stability of front solutions for parabolic partial differential equations[END_REF][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]. The sharpest results [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] establish that sufficiently localized perturbations to the critical front decay with algebraic rate t -3/2 . This result was first obtained by Gallay [START_REF] Gallay | Local stability of critical fronts in nonlinear parabolic partial differential equations[END_REF] using renormalization group theory, and it was later observed in [START_REF] Sandstede | Evans function and blow-up methods in critical eigenvalue problems[END_REF] that this improved decay rate, compared to the diffusive decay rate t -1/2 , can be explained by the lack of an eigenvalue embedded in the essential spectrum at the origin, a perspective directly exploited to obtain nonlinear stability in [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].

One typically expects that the linearization about a traveling wave in a translation invariant system will have an eigenvalue at the origin, with the derivative of the wave giving the eigenfunction. However, in the classical Fisher-KPP equation, the critical front q * (•; 0) has weak exponential decay q * (x; 0) ∼ xe -η * (0)x , as x → ∞, where η * (0) = c * (0)/2 = f (0). Since ones uses exponential weight to stabilize the essential spectrum, the derivative of the front is unbounded in this weighted space, with e η * (0)x q * (x; 0) ∼ η * (0)x as x → ∞. Hence, in the appropriately weighted space, there is not an eigenvalue at the origin even corresponding to a bounded eigenfunction. Such an embedded eigenvalue to a bounded eigenfunction is sometimes called a resonance. This generalized notion of an eigenvalue is the appropriate, dynamically relevant approach to considering eigenvalues embedded in the essential spectrum -for instance, resonances correspond to zeros of the Evans function, once the Evans function is extended into the essential spectrum with the gap lemma [START_REF] Kapitula | Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations[END_REF][START_REF] Gardner | The gap lemma and geometric criteria for instability of viscous shock profiles[END_REF]. The lack of a zero of the Evans function at the origin was central to the proof of stability of the critical front via pointwise semigroup methods in [START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF].

For δ small, the extended Fisher-KPP equation is a singular perturbation of the classical Fisher-KPP equation. The presence of fourth order dissipation, however small, immediately breaks the comparison principle, rendering stability questions more difficult. Using geometric singular perturbation theory, Rottschäfer and Wayne showed [START_REF] Rottschäfer | Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation[END_REF] that the picture for existence of fronts in the classical Fisher-KPP equation extends to δ small but nonzero: there is a critical speed c * (δ) such that there exist monotone front solutions to (1.2) if and only if c ≥ c * (δ). The authors also proved asymptotic stability of all of these fronts in the extended Fisher-KPP equation against sufficiently localized perturbations. However, their approach, using control of certain energy functionals, does not identify a precise decay rate for the perturbations. We also mention that the existence of both invasion fronts and fronts connecting two stable states in fourth order parabolic equations, including the extended Fisher-KPP equation with δ not necessarily small, was established in [START_REF] Bouwe Van Den Berg | Travelling waves for fourth order parabolic equations[END_REF] using topological arguments.

Here, we revisit the stability of the critical front in the extended Fisher-KPP equation, putting this equation in the general framework of stability of pulled fronts recently presented in [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF]. There, Scheel and the first author obtain the t -3/2 decay rate for perturbations to critical fronts in general higher order parabolic equations, under assumptions on the existence and spectral stability of critical fronts. The main difficulty in verifying these assumptions for the extended Fisher-KPP equation is to simultaneously handle the singular nature of the perturbation while also tracking potential eigenvalues embedded in the essential spectrum. The main contributions of the present paper are as follows:

1. We verify the spectral stability assumptions of [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF] for (1.2), thereby obtaining the t -3/2 decay rate as well as an asymptotic description of the perturbation as immediate corollaries.

2. Our approach illustrates conceptually that these spectral stability assumptions as well as the existence of critical fronts are robust within the class of equations considered in [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF]: these features persist under small perturbations to the equation considered, even in our singularly perturbed setting.

3. We do not rely on geometric singular perturbation theory or the machinery of normally hyperbolic invariant manifolds, instead further developing self-contained, functional analytic approaches to singular perturbations and the study of eigenvalues embedded in the essential spectrum previously explored in Re λ

Im λ σ(A -(δ)) σ(A + (δ)) σ ess (A(δ)) Re λ Im λ σ ess (L(δ))
Re λ

Im λ

Figure 1: Left: the essential spectrum of the unweighted operator A(δ) is bounded by the Fredholm borders in blue and red. Middle and right: overview and zoom near the origin of the essential spectrum of the weighted operator L(δ).

[17] and [START_REF] Pogan | Instability of spikes in the presence of conservation laws[END_REF]. We also reprove the existence of the critical front using a far-field/core decomposition relying only on functional analysis.

The threshold speed c * (δ) identified in [START_REF] Rottschäfer | Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation[END_REF] is the linear spreading speed for the extended Fisher-KPP equation, in the sense that solutions with compactly supported initial data to the linearization of (1.2) about u ≡ 0 in a co-moving frame,

u t = -δ 2 u xxxx + u xx + cu x + f (0)u
generically grow pointwise for c < c * (δ) and decay exponentially pointwise for c > c * (δ). In this case, the linear spreading speed is determined by the presence of a simple pinched double root of the dispersion relation

d + c (λ, ν) = -δ 2 ν 4 + ν 2 + cν + f (0) -λ, ( 1.3)
at (λ, ν) = (0, -η * (δ)) for c = c * (δ); see Lemma 1.1 for details, and [START_REF] Holzer | Criteria for pointwise growth and their role in invasion processes[END_REF] for background on linear spreading speed theory. We first reprove the existence of the critical front using a functional analytic approach to regularize the perturbation.

Theorem 1 (Existence of the critical front). For δ sufficiently small and for c = c * (δ), there exists a smooth traveling front q * solving (1.2), such that

q * (x; δ) = (µ(δ) + x)e -η * (δ)x + O(e -(η * (δ)+η)x ), x → ∞
for some η > 0. Moreover, q * (•; δ) → q * (•; 0) uniformly in space as δ → 0.

We then turn to the stability of this critical front under a reasonable class of perturbations. Writing the solution to (1.1)

as u(t, x) = q * (x -c * (δ)t; δ) + v(t, x -c * (δ)), the perturbation v solves v t = A(δ)v + f (q * + v) -f (q * ) -f (q * )v, (1.4)
where A(δ) :

H 4 (R) ⊂ L 2 (R) -→ L 2 (R)
is the linearization about the critical front, defined through

A(δ) := -δ 2 ∂ 4 x + ∂ 2 x + c * ∂ x + f (q * (x; δ)). (1.5)
This linear operator has unstable essential spectrum at +∞, due to the fact that f (0) > 0; see Figure 1. Hence, the critical front q * (•; δ) is unstable to perturbations v 0 ∈ H 4 (R), and we thereby restrict ourselves to exponentially localized perturbations. For each δ small we define a smooth positive exponential weight

ω * (•; δ) satisfying ω * (x; δ) = e η * (δ)x , x ≥ 1, 1, x ≤ -1, (1.6)
and assume that the perturbation has the form v(t, •) = p(t, •)ω -1 * , with p(t, •) ∈ H 4 (R). We sometimes suppress the dependence of ω * on δ if it is clear from context. The linear dynamics for p are then driven by

L(δ) : H 4 (R) ⊂ L 2 (R) -→ L 2 (R), given by L(δ) := ω * A(δ)ω -1 * = -δ 2 ∂ 4 x + δ 2 a 3 ∂ 3 x + 1 + δ 2 a 2 ∂ 2 x + a 1 ∂ x + a 0 ,
where the coefficients a i (x; δ) converge to limits a ± i (δ) exponentially quickly when x → ±∞, and are defined using the local notation (x) := 1/ω * (x) by the following expressions:

a 3 = -4 , a 2 = -6 , a 1 = c * + 2 -4δ 2 , a 0 = f (q * ) + c * + -δ 2 . (1.7) We note that (k) (x)/ (x) = (-η * ) k for x ≥ 1.
For such a linear operator, the essential spectrum is delimited by the two Fredholm borders, which are defined using the asymptotic dispersion relations. Lemma 1.1 (Linear spreading speed). There exists δ 0 > 0 such that for δ ∈ (-δ 0 , δ 0 ), there exists a critical speed c * = c * (δ), and an exponent η = η * (δ) > 0 for the critical weight such that the right dispersion relation (1.3) satisfies the following properties.

(i) Simple pinched double root: for λ, ν near 0 ∈ C:

d + c * (λ, -η * + ν) = ν 2 1 -12δ 2 f (0) -λ + O(ν 3 ), (1.8
)

with 1 -12δ 2 f (0) > 0. (ii) Minimal critical spectrum: If d + c * (iκ, -η * + ik) = 0 for some κ, k ∈ R, then κ = k = 0. (iii) No unstable essential spectrum: If d + c * (λ, -η * + ik) = 0 for some k ∈ R and λ ∈ C, then Re λ ≤ 0.
We prove this lemma in Section 2.1. The left dispersion relation, given by

d - c * (λ, ν) = -δ 2 ν 4 + ν 2 + c * (δ)ν + f (1)
λ is also relevant for determining the essential spectrum of L(δ); intuitively, it determines what happens to perturbations in the wake of the front. Since the invading state u ≡ 1 is locally stable, with f (1) < 0, one immediately obtains the following lemma.

Lemma 1.2 (Stability on the left). If d -

c * (λ, ik) = 0 for some k ∈ R, then Re λ < 0. These two lemmas together with Palmer's theorem imply that the essential spectrum of L(δ) is marginally stable, touching the imaginary axis only at the origin [START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF][START_REF] Fiedler | Spatio-temporal dynamics of reaction-diffusion patterns[END_REF]; see Figure 1. Our main result is the following theorem, which states that L(δ) has no unstable eigenvalues and no resonance at λ = 0.

Theorem 2 (Spectral stability).

There exists a δ 0 > 0 such that for all δ ∈ (-δ 0 , δ 0 ) the operator L(δ) has no eigenvalues λ with Re λ ≥ 0, and there does not exist a bounded solution to L(δ)u = 0.

Together, Lemma 1.1, Lemma 1.2, and Theorem 2 imply that for δ small, the critical front in the extended Fisher-KPP equation satisfies the spectral stability assumptions of [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF], and so as a corollary we obtain two results on nonlinear stability of the critical front. To state these, we first define for r ∈ R a smooth positive one-sided algebraic weight ρ r which satisfies 3 2 . There exist constants ε > 0 and C > 0 such that if

ρ r (x) = 1, x ≤ -1, (1 + x 2 ) r/2 , x ≥ 1.

Corollary 1.3 (Nonlinear stability). Let r >

ω * ρ r v 0 H 1 < ε, then ω * ρ -r v(t, •) H 1 ≤ Cε (1 + t) 3/2 ,
where v is the solution to (1.4) with initial data v 0 . Corollary 1.4 (Nonlinear stability with asymptotics). Let r > 5 2 . There exist constants ε > 0 and C > 0 such that if ω * ρ r v 0 || H 1 < ε, then there exists a real number α * = α * (ω * ρ r v 0 ), depending smoothly on ω * ρ r v 0 in H 1 (R), such that for t > 1,

ρ -r ω * (v(t, •) -α * t -3/2 q * (•; δ)) H 1 ≤ Cε (1 + t) 2 ,
where v is the solution to (1.4) with initial data v 0 .

Remarks

Assumptions on f . Since we prove our results by perturbing from the classical Fisher-KPP equation, our results hold for any smooth nonlinearity f which satisfies f (0) = f (1) = 0, f (0) > 0, f (1) < 0, and for which existence and spectral stability of the critical front hold for the classical Fisher-KPP equation with this reaction term. In particular, this is implied by the assumption f (u) < 0 for u ∈ (0, 1) [32, Theorem 5.5], which we state in the introduction. This can be weakened, for instance, to the assumption that 0 < f (u) ≤ f (0)u for u ∈ (0, 1); see e.g. [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF].

General approach -preconditioning. Our approach to regularizing the singular perturbation is based on preconditioning with an appropriately chosen operator. To illustrate the main idea, briefly consider the eigenvalue problem for the unweighted linearization, (A(δ

) -λ)u = 0. Applying (1 -δ 2 ∂ 2 x ) -1 to A(δ) -λ, we obtain (1 -δ 2 ∂ 2 x ) -1 (A(δ) -λ) = (1 -δ 2 ∂ 2 x ) -1 [(1 -δ 2 ∂ 2 x )∂ 2 x + c * (δ)∂ x + f (q * ) -λ] = ∂ 2 x + (1 -δ 2 ∂ 2 x ) -1 (c * (δ)∂ x + f (q * ) -λ) = ∂ 2 x + c * (δ)∂ x + f (q * (•; δ)) -λ + T (δ) c * (δ)∂ x + f (q * (•; δ)) -λ , where T (δ) = (1 -δ 2 ∂ 2 x ) -1 -1.
Once we prove that the terms involving T (δ) are continuous in δ, the eigenvalue problem becomes essentially a regular perturbation of the classical Fisher-KPP linearization, at δ = 0. We prove the necessary estimates on the preconditioners using direct Fourier analysis in Section 2.3. This approach is inspired by that used to construct oblique stripe solutions in a quenched Swift-Hohenberg equation in [START_REF] Goh | Pattern-forming fronts in a Swift-Hohenberg equation with directional quenching -parallel and oblique stripes[END_REF].

Stability to less localized perturbations. We note that under the spectral stability conditions we prove here, in addition to Corollaries 1.3 and 1.4, one also immediately obtains from the results of [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF] stability under less localized perturbations, with a prescribed decay rate which is slower than t -3/2 . See [3, Theorems 3 and 4] for details.

Geometric vs. functional analytic point of view.

We remark here that one should also be able to prove the spectral stability results obtained here using geometric dynamical systems methods, in particular geometric singular perturbation theory in the sense of Fenichel [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF] together with the gap lemma [START_REF] Gardner | The gap lemma and geometric criteria for instability of viscous shock profiles[END_REF][START_REF] Kapitula | Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations[END_REF], which is used to extend the Evans function into the essential spectrum. An attractive feature of our approach here is that it is quite self contained, ultimately relying mostly on basic Fredholm theory and Fourier analysis. We also remark that in principle the functional analytic methods could be adapted, together with the approach to linear stability through obtaining resolvent estimates via far-field/core decompositions in [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF], to problems in stability of critical fronts in nonlocal equations, since these methods do not rely as heavily on the presence of an underlying phase space. Some of the relevant Fredholm theory for nonlocal operators has been developed in [START_REF] Faye | Fredholm properties of nonlocal differential equations via spectral flow[END_REF][START_REF] Faye | Existence of pulses in excitable media with nonlocal coupling[END_REF].

Natural range for δ. In this paper, we have restricted to small δ. However, we believe that similar results should hold true for larger values of this parameter. While the existence of fronts is established in [START_REF] Bouwe Van Den Berg | Travelling waves for fourth order parabolic equations[END_REF] for all speeds c > 0 and δ ∈ R, we do not have access to explicit decay at +∞ for this fronts, which seems necessary to establish precise stability. Monotonicity of the front would imply such a precise decay by use of Ikehara's theorem [START_REF] Carr | Uniqueness of traveling waves for nonlocal monostable equations[END_REF]. An important value is δ = 1/ 12f (0), at which the dispersion relation admits a triple root, and the essential spectrum of the linearized operator becomes tangent to the imaginary axis. Stability at or above this value of δ is therefore fundamentally outside the scope of [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF]. Supercritical and subcritical fronts. If we consider a supercritical front, traveling with speed c > c * (δ) and constructed in [START_REF] Rottschäfer | Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation[END_REF], one can simplify the argument of Theorem 2 to prove that the linearization about such a front has no unstable point spectrum. For these fronts, one can use an exponential weight to push the essential spectrum entirely into the left half plane, and thereby with the analogue of Theorem 2 obtain stability of supercritical fronts with an exponential decay rate using standard semigroup methods (see e.g. [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF]). Subcritical fronts, with c < c * (δ), have unstable absolute spectrum, meaning in particular that the essential spectrum of the linearization about any of these fronts is unstable in any exponentially weighted space. A modified version of our proof of Theorem 1 should also give existence of these supercritical and subcritical fronts using functional analytic methods, although we do not give the details here.

Outline. The remainder of this paper is organized as follows. In Section 2, we compute some preliminary information needed for our analysis (the linear spreading speed in (1.2) and the cokernel of L(0)) and prove some necessary estimates on our preconditioner. In Section 3, we use our preconditioning and a far-field/core decomposition to prove Theorem 1, establishing existence of the critical front. In Section 4, we define a functional analytic analogue of the Evans function near λ = 0, and use it together with knowledge of the spectrum of L(0) to prove that L(δ) has no resonance at the origin or unstable eigenvalues for δ small. In Section 5, we complete the proof of Theorem 2 by showing that there are also no unstable eigenvalues away from the origin.

Acknowledgements. The authors are grateful to Arnd Scheel and Grégory Faye for helpful comments. MA was supported by the National Science Foundation through the Graduate Research Fellowship Program under Grant No. 00074041. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Preliminaries

Exponential weights

In addition to the critical weight (1.6) which we use to shift the essential spectrum out of the right half plane, we will further need exponential weights to recover Fredholm properties of L(δ) and related operators for our far-field/core analysis. For η ± ∈ R, we define a smooth positive weight function ω η-,η+ satisfying

ω η-,η+ = e η-x , x ≤ -1, e η+x , x ≥ 1.
If η -= 0 and η + = η, then we write ω η-,η+ = ω η . If η -= η + = η, we choose ω η,η (x) = e ηx . Given an integer m, we define the exponentially weighted Sobolev space

H m η-,η+ (R) through the norm ||f || H m η -,η + = ||ω η-,η+ f || H m . We note that for η > 0 we have H m 0,η (R) = H m (R) ∩ H m η,η (R) as well as the following equivalence of norms ||f || H m 0,η ∼ ||f || H m + ||f || H m η,η . (2.1)
This characterization of the one-sided weighted spaces is useful in obtaining estimates on operators defined by Fourier multipliers on these spaces, and we make use of this below in Section 2.3.

Essential spectrum

The boundaries of the essential spectrum of L are determined by the essential spectrum of the limiting operators L ± , obtained by sending x → ±∞ [START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF][START_REF] Fiedler | Spatio-temporal dynamics of reaction-diffusion patterns[END_REF]. From the construction of c * , η * (see the proof of Lemma 1.1 below), we have at +∞:

L + (δ) = -δ 2 ∂ 4 x + 4η * δ 2 ∂ 3 x + (1 -6δ 2 η 2 * )∂ 2 x . (2.2)
The spectrum of this constant coefficient operator is, via Fourier transform, readily seen to be marginally stable, see the red curves of Figure 1. Notice that η = η * is the only reasonable value for which L + has a non positive zeroth order term. Any other uniformly bounded η(δ) will lead to spectral instability for L(δ). At -∞, there is no contribution from ω * , hence

L -= A -= -δ 2 ∂ 4 x + ∂ 2 x + c * ∂ x + f (1)
has a stable spectrum, with spectral gap f (1) < 0, by Lemma 1.2.

Proof of Lemma 1.1. We first look for c * , η * > 0 which satisfy (1.8). The polynomial ν → d c (λ, ν) at λ = 0 admits -η as a double root if and only if

0 = d + c (0, -η) = -δ 2 η 4 + η 2 -c η + f (0), 0 = ∂ ν d + c (0, -η) = 4δ 2 η 3 -2 η + c.
We remove c from the first equation by using the second one, and find a quadratic equation satisfied by η 2 , which has roots ±η 1 , ±η 2 where

η 1 := 1 |δ| √ 6 1 + 1 -12δ 2 f (0) ∼ 1 |δ| √ 3 , η 2 := 1 |δ| √ 6 1 -1 -12δ 2 f (0) ∼ f (0). (2.3)
where the asymptotics hold for δ → 0. The choice

η * = η 2 and c * = 2η * -4δ 2 η 3 * leads to c * (δ) = 2 f (0) -δ 2 f (0) 3/2 + O(δ 4 ).
The other double roots do not determine linear spreading speeds, as they are not pinched; see [START_REF] Holzer | Criteria for pointwise growth and their role in invasion processes[END_REF] for details. We now fix δ 0 = 1/ 12f (0). Then for |δ| < δ 0 , and using the expression of η * = η 2 , we obtain:

∂ 2 ν d + c (0, -η * ) 2! = 1 -6δ 2 η 2 * = 1 -12δ 2 f (0) > 0.
Hence (λ, ν) = (0, -η * ) is a simple double root and (1.8) is proved. Such an expansion together with the lack of unstable essential spectrum ensures that this root is pinched; see [START_REF] Holzer | Criteria for pointwise growth and their role in invasion processes[END_REF]Lemma 4.4]. Alternatively, Lemma 4.2 below directly proves that the root is pinched. We now check the two remaining conditions in Lemma 1.1. We equate the polynomial P (X) = d + * (λ, X) with its Taylor series centered at the double root X = -η * to obtain

Re d + * (λ, -η * + ik) = -Re λ + Re 4 j=0 (ik) j P (j) (-η * ) j! = -Re λ -δ 2 k 4 -(1 -6δ 2 η 2 * )k 2 ≤ 0 (2.4)
if Re λ > 0, since from (2.3), we have 1 -6δ 2 η 2 * = 1 -12δ 2 f (0) ≥ 0. This proves hypothesis (iii). Furthermore, the inequality in (2.4) is an equality if and only if k = 0 and λ = 0, for which we have d + * (0, -η * ) = 0. Hence, hypothesis (ii) is proved.

Preconditioner estimates

Here we prove the estimates we will need on our preconditioner (1δ 2 ∂ 2 x ) -1 , by directly examining its Fourier symbol.

Lemma 2.1. Fix η > 0 sufficiently small, and fix an integer m. Then there exist constants δ 0 > 0 and

C = C(δ 0 , η) such that if |δ| < δ 0 , ||(1 -δ 2 ∂ 2 x ) -1 || L 2 0,η →L 2 0,η ≤ C, (2.5) ||(1 -δ 2 ∂ 2 x ) -1 || H m 0,η →H m+1 0,η ≤ C |δ| . (2.6)
Proof. By (2.1), it suffices to prove the estimates separately for L 2 and for L 2 η,η with η > 0 small. Since multiplication by e η• is an isomorphism from L 2 η,η (R) to L 2 (R), to prove estimates for (1δ 2 ∂ 2 x ) -1 on L 2 η,η , it suffices to consider the inverse of the conjugate operator

e η• (1 -δ 2 ∂ 2 x )e -η• = 1 -δ 2 (∂ x -η) 2 acting on L 2 (R)
. This is the advantage of using (2.1) to separate estimates on L 2 0,η (R) into estimates on L 2 (R) and L 2 η,η (R): the conjugate operator arising from studying (1δ 2 ∂ 2 x ) on L 2 η,η (R) has constant coefficients since the weight is a fixed exponential function, and so we can directly estimate its inverse using the Fourier transform.

Fix η ≥ 0. By Plancherel's theorem,

(1 -δ 2 (∂ x -η) 2 ) -1 f || L 2 = 1 1 -δ 2 (i • -η) 2 f (•) L 2 ≤ sup k∈R 1 1 -δ 2 (ik -η) 2 || f || L 2 . Let δ 0 = 1/( √ 2η), so that δ 2 0 η 2 = 1/2, and hence if |δ| < δ 0 , 1 + δ 2 (k 2 -η 2 ) = 1 -δ 2 η 2 + δ 2 k 2 ≥ 1 2 + δ 2 k 2 . (2.7)
Then for any δ with |δ| < δ 0 , we have

1 1 -δ 2 (ik -η) 2 2 = 1 (1 + δ 2 (k 2 -η 2 )) 2 + 4k 2 δ 4 η 2 ≤ 1 (1 + δ 2 (k 2 -η 2 )) 2 ≤ 1 1 2 + δ 2 k 2 ≤ C
, with C depending only on δ 0 and η, and so

(1 -δ 2 ∂ 2 x ) -1 L 2 η,η →L 2
η,η ≤ C. Since this holds for any fixed 0 ≤ η < 1, in particular also for η = 0, we obtain (2.5) by combining these estimates with (2.1). Now we prove (2.6), again by obtaining bounds on the Fourier symbol of the inverse of the conjugate operator for η > 0 and η = 0. By Plancherel's theorem, for any fixed 0 ≤ η < 1, we have

||(1 -δ 2 (∂ 2 x -η)) -1 f || H m+1 = 1 1 -δ 2 (i • -η) 2 • m+1 f (•) ≤ sup k∈R 1 1 -δ 2 (ik -η) 2 k f H m .
Again, let δ 0 = 1/( √ 2η). Then, by (2.7), we have

1 1 -δ 2 (ik -η) 2 k 2 = δ 2 + δ 2 k 2 (1 + δ 2 (k 2 -η 2 )) 2 + 4k 2 δ 4 η 2 1 δ 2 ≤ δ 2 + δ 2 k 2 1 2 + δ 2 k 2 2 1 δ 2 ≤ C δ 2 ,
from which we obtain

||(1 -δ 2 (∂ 2 x -η)) -1 f || H m+1 ≤ C |δ| ||f || L 2 .
Since this holds for η ≥ 0, we obtain (2.6) from the equivalence of norms (2.1).

We now state and prove the estimates we will need on the difference between the preconditioner and the identity,

T (δ) = (1 -δ 2 ∂ 2 x ) -1 -1. Lemma 2.2.
Fix η > 0 sufficiently small. There exists a constant δ 0 such that the mapping δ → T (δ) is continuous from (-δ 0 , δ 0 ) to B(H 1 0,η , L 2 0,η ), the space of bounded linear operators from H 1 0,η (R) to L 2 0,η (R) with the operator norm topology.

Proof. As in the proof of Lemma 2.1, it suffices to establish continuity in δ of the conjugate operator

T η (δ) := (1 -δ 2 (∂ x -η) 2 ) -1 -1 on L 2 (R) for η ≥ 0 sufficiently small. For δ nonzero, we write 1 -δ 2 (∂ x -η) 2 = δ 2 1 δ 2 -(∂ x -η) 2 .
By standard spectral theory, we therefore see that T η (δ) is continuous in δ provided δ is nonzero and 1/δ 2 is in the resolvent set of the operator (∂ xη) 2 . Computing the spectrum of this operator with the Fourier transform, one readily finds that there exists a δ 1 depending on η such that this continuity holds for 0 < δ < δ 1 .

We now establish continuity at δ = 0 via direct estimates on the Fourier multiplier.

Tη (δ, k) = δ 2 (ik -η) 2 1 -δ 2 (ik -η) 2 .
Since we are proving continuity of T η (δ) from H 1 to L 2 , we gain a helpful factor of k -that is, it suffices to estimate | Tη (δ, k)|/ k . By (2.7), for |δ| < δ 0 := min{δ 1 , 1/ √ 2η} we have Tη (δ, k)

1 k = δ 2 (ik -η) 2 1 -δ 2 (ik -η) 2 1 k = δ 2 (η 2 -k 2 ) 2 + 4k 2 η 2 (1 -δ 2 (η 2 -k 2 )) 2 + 4δ 4 k 2 η 2 |δ| (δ 2 + δ 2 k 2 ) 1/2 ≤ |δ|   δ 4 η 4 + δ 4 k 4 + 2δ 4 k 2 η 2 1 2 + δ 2 k 2 2 + 4δ 4 k 2 η 2 (δ 2 + δ 2 k 2 )   1/2
, using (2.7) in the denominator. We now split the factor in the parenthesis, first estimating

δ 4 η 4 + 2δ 4 k 2 η 2 1 2 + δ 2 k 2 2 + 4δ 4 k 2 η 2 (δ 2 + δ 2 k 2 ) ≤ δ 4 η 4 + 2δ 4 k 2 η 2 1 4 (δ 2 + δ 2 k 2 ) = δ 2 η 4 + 2δ 2 k 2 η 2 1 4 (1 + k 2 ) ≤ C,
where C depends only on δ 0 and η. For the remaining term, we have

δ 4 k 4 1 2 + δ 2 k 2 2 + 4δ 4 k 2 η 2 (δ 2 + δ 2 k 2 ) ≤ δ 4 k 4 1 2 + δ 2 k 2 2 (δ 2 k 2 ) = δ 2 k 2 1 2 + δ 2 k 2 2 ≤ C,
again with constant C only depending on η and δ 0 . From this estimate on the Fourier symbol together with Plancherel's theorem, we obtain

||T η (δ)|| H 1 →L 2 ≤ C|δ|,
for |δ| < δ 0 , and so in particular δ → T η (δ) is continuous at δ = 0, which completes the proof of the lemma.

Fredholm properties at δ = 0

We will further need the Fredholm properties of L(0), which is the linearization in the weighted space of the classical FKPP problem δ = 0. The classical Fisher-KPP front, at δ = 0, may be constructed via simple phase plane methods (see [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF]), and we denote this front by q 0 . In the following two lemmas, we describe the kernel, the cokernel and the range of L(0). They will both be needed for the existence of the critical front q * (•; δ) in Section 3, and for the control of small eigenvalues in Section 4.

Re λ

Im λ

Re λ

Im λ

Re λ

Im λ

Figure 2: Essential spectrum of L η (0) for η < 0 at left, η = 0 at middle, and η > 0 at right. For η > 0, the positive Fredholm border has reverse orientation so that fred(L η ) = -1, while fred(L η ) = 1 for η < 0.

Lemma 2.3. For η > 0, the operator L(0) : H 2 0,η (R) → L 2 0,η (R) is Fredholm with index -1, with trivial kernel and with cokernel spanned by ϕ(x) = (ω * (x; 0)) -1 e c * (0)x q 0 (x).

Proof. Recall that the asymptotic operators are given by L + (0) = ∂ 2

x and L -(0) = ∂ 2 x + c * (0)∂ x + f (1). For η > 0, define the conjugate operator:

L η (0) = ω 0,η L(0) ω -1 0,η : H 2 (R) -→ L 2 (R), with asymptotic operators L η,+ = (∂ x -η) 2 and L η,-= L -. Since the multiplication ω 0,η • : L 2 0,η (R) -→ L 2 (R) is an isomorphism, the Fredholm indexes satisfy fred L η (0) = 0 + fred L(0) + 0 = fred L(0).
Then the conjugate operator is defined on a unweighted space, and its Fredholm borders are the two oriented curves σ(L η,+ ) = {-k 2 + i2ηk + η 2 : k ∈ R} and σ(L η,-) = {-k 2 + ic * (0)k + f (1) : k ∈ R}, which are away from 0 ∈ C; see Figure 2. This ensures that L η (0) is Fredholm, we now compute its index fred(L η (0)λ) at λ = 0. For λ to the right of the essential spectrum, we use Palmer's theorem to compute the Fredholm index from the Morse indices (see e.g. [START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF][START_REF] Fiedler | Spatio-temporal dynamics of reaction-diffusion patterns[END_REF]),

fred(L η (0) -λ) = dim E u -(λ) -dim E u + (λ) = 2 -2 = 0.
where E u ± are the unstable eigenspaces at ±∞. To prove that this spaces share the same dimension, one can |λ| large enough and use a standard normalization; see [10, proof of Lemma 3.1]. Then the index decreases to -1 when λ crosses σ(L η,+ ), since the latter curve has reverse orientation, see [START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF]. Hence at λ = 0, we have shown that fred L(0) = fred L η (0) = -1.

To compute the kernel, we note that

u ∈ ker L(0) if and only if u ∈ H 2 0,η (R) and A(0)ω -1 * u = 0, (2.8)
with

A(0) = ∂ 2 x + c * (0)∂ x + f (q 0 (x)).
Studying the asymptotic growth of the ODE A(0)u = 0, one can construct a basis of solutions {q 0 , φ}, with exponential behavior at -∞: φ(x) ∼ exp((f (0)α)x) and q 0 (x) ∼ exp((-

f (0) + α)x) with α = f (0) -f (1) > f (0) > 0. See [10, proof of Lemma 2.2]
for a similar construction. Furthermore, the derivative of the front has weak exponential decay at +∞: q 0 (x) ∼ xω * (x) -1 . Hence, neither φ nor q 0 are sufficiently localized to satisfy the right hand condition in (2.8), so that ker L(0) = {0}.

Finally, it is easily computed that Ã(0

) := exp( c * 2 •)A(0) exp(-c * 2 •) is self-adjoint, so that for v ∈ H 2 0,-η (R) and u ∈ H 2 0,η (R): u, L * v = Lu, v = A(ω -1 * u), ω * v = Ã(e c * 2 • ω -1 * u), e -c * 2 ω * v = e c * • ω -1 * u, A(e -c * • ω * v) , which ensures that v ∈ ker(L(0) * ) if and only if v ∈ H 2 0,-η (R) and A(0)e -c * • ω * v = 0. For x → -∞, ω * (x) -1 e c * x φ(x) ∼ exp(( f (0) -α)x) is not bounded, hence ker(L(0) * ) = Span(ω -1 * e c * • q 0 ).
Lemma 2.4. For η > 0 small enough, the range of L(0) :

H 2 0,η (R) → L 2 0,η (R) is im(L(0)) = {u ∈ L 2 0,η (R) : u, ϕ = 0},
where ϕ is defined in the above Lemma 2.3. We let P : L 2 0,η (R) -→ im(L(0)) denote the orthogonal projection onto im L(0) with respect to the L 2 0,η (R)-inner product.

Proof. Assume that u ∈ im(L(0)), so that u = L(0)ũ with ũ ∈ H 2 0,η (R). Then u, ϕ = ũ, L(0) * ϕ = 0. To prove the reverse inclusion, write u ∈ L 2 0,η (R) as u = P u + (1 -P )u. From Lemma 2.3, L(0) is Fredholm, hence its range is closed and P is well defined. Furthermore, fred L(0) = -1 and ker L(0) = {0}, so that 1 -P has a one dimensional range:

(1 -P )u = α(u)ψ, with ψ ∈ L 2 0,η (R) fixed, and α : H 2 0,η (R) -→ R linear. Assuming that u, ϕ = 0, we obtain 0 = P u, ϕ + (1 -P )u, ϕ = ũ, L(0

) * ϕ + α(u) ψ, ϕ = α(u) ψ, ϕ , (2.9)
for some ũ ∈ H 2 η (R). Hence either α(u) = 0 or ψ, ϕ = 0. If ψ, ϕ = 0, then for all v ∈ H 2 0,η (R), we would have v, ϕ = ṽ, L(0) * ϕ + α(v) ψ, ϕ = 0, which is to say that ϕ = 0 and is a contradiction. Hence from (2.9) we conclude α(u) = 0, so that u = P u ∈ im(L(0)).

Existence of the critical front -proof of Theorem 1

Our approach is to capture the weak exponential decay at +∞ implied by the pinched double root by solving (1.2) with an ansatz q(x; δ) = χ -(x) + w(x) + χ + (x)(µ + x)e -η * (δ)x , (

where χ + is a smooth positive cutoff function satisfying

χ + (x) = 1, x ≥ 3, 0, x ≤ 2, (3.2)
and χ -(x) = χ + (-x). For brevity, we denote by ψ(µ, δ) the function ψ(x; µ, δ) = (µ + x)e -η * (δ)x .

We will require w to be exponentially localized, with a decay rate faster than e -η * (δ)x -this localized piece is the core of the solution, while χ -and χ + ψ capture the far-field behavior. Similar far-field/core decompositions have been used to construct heteroclinic solutions to pattern-forming systems in [START_REF] Avery | Growing stripes, with and without wrinkles[END_REF][START_REF] Goh | Pattern-forming fronts in a Swift-Hohenberg equation with directional quenching -parallel and oblique stripes[END_REF]. Inserting the ansatz (3.1) into the traveling wave equation (1.2), we get an equation

A + (δ)(χ -+ w + χ + ψ(µ, δ)) + N (χ -+ w + χ + ψ(µ, δ)) = 0, (3.3)
where

A + (δ) = -δ 2 ∂ 4 x + ∂ 2 x + c * (δ)∂ x + f (0)
, and N (q) = f (q)f (0)q. Since we want to require w to decay faster than the front itself, we first let v = ω * w, so that (3.3) becomes

0 = F (v; µ, δ) := S(δ)v + ω * A + (δ)(χ -+ χ + ψ) + ω * N (χ -+ ω -1 * v + χ + ψ), where S(δ) = ω * A + (δ)ω -1
* is the conjugate operator

S(δ) = -δ 2 ∂ 4 x + δ 2 a 3 (x; δ)∂ 3 x + (1 + δ 2 a 2 (x; δ))∂ 2 x + a 1 (x; δ)∂ x + ã0 (x; δ), ( 3.4)
where the coefficients a i are given in (1.7) for i = 1, 2 or 3 while

ã0 = f (0) + ω * c * ∂ x + ∂ 2 x -δ 2 ∂ 4
x ω -1 * , since we are linearizing about the unstable state u ≡ 0 rather than the front itself, which we are in the process of constructing.

Since ω * (x; δ) = 1 on the support of χ -and ω * (x; δ) = e η * (δ)x on the support of χ + , we simplify F to

F (v; µ, δ) = S(δ)v + A + (δ)χ -+ S(δ)[(µ + •)χ + ] + ω * N (χ -+ ω -1 * v + χ + ψ).
Then, we extract from N terms that are linear in v, together with residual terms that are v-independent. We write

ω * N (χ -+ ω -1 * v + χ + ψ) = N (v; µ, δ) + Q(µ, δ)v + R(µ, δ) where N (v; µ, δ) = ω * f (χ + + ω -1 * v + χ + ψ) -f (χ -+ χ + ψ) -f (χ -+ χ + ψ)ω -1 * v , ( 3.5)
and

Q(µ, δ)v = (f (χ -+ χ + ψ) -f (0))v, R(µ, δ) = ω * [f (χ -+ χ + ψ) -f (0)(χ -+ χ + ψ)].
Altogether, F decomposes as the sum of a linear term, a residual term, and a nonlinear term:

F (v; µ, δ) = [S(δ) + Q(µ, δ)]v + R(µ, δ) + N (v; µ, δ), ( 3.6)
where N (v; µ, δ) is given by (3.5), and

R(µ, δ) = R(µ, δ) + A + (δ)χ -+ S(δ)[(µ + •)χ + ].
At δ = 0, the equation F (v; µ, 0) = 0 is the traveling wave equation for the Fisher-KPP equation, and so we have a solution F (v 0 ; µ 0 , 0) = 0 where

v 0 = ω * (•; 0)q 0 -χ --χ + ω * (•; 0)ψ(µ 0 , 0),
and q 0 is the translate of the critical Fisher-KPP front for which q 0 (x) = (µ 0 + x)e -η * (0)x + O(x 2 e -2η * (0)x ), x → ∞, so that v 0 is exponentially localized (see e.g. [START_REF] Gallay | Local stability of critical fronts in nonlinear parabolic partial differential equations[END_REF] for asymptotics of the critical Fisher-KPP front).

To regularize the singular perturbation and enforce exponential localization of v, we consider

G(v; µ, δ) = (1 -δ 2 ∂ 2 x ) -1 F (v; µ, δ), as a nonlinear function G : H 2 0,η (R) × R × R → L 2 0,η (
R), for η > 0 sufficiently small. Lemma 3.1. Fix η > 0 sufficiently small. There exists Proof. We use (3.6) to write G as

δ 0 > 0 such that (v, µ, δ) → G(v, µ, δ) : H 2 0,η (R) × R × (-δ 0 , δ 0 ) → L 2 0,η (R) is well-defined, smooth in v,
G(v; µ, δ) = (1 -δ 2 ∂ 2 x ) -1 S(δ)v + (1 -δ 2 ∂ 2 x ) -1 [Q(µ, δ)v + R(µ, δ) + N (v; µ, δ)]. (3.7)
Using the fact that f is smooth and that H 2 0,η (R) is a Banach algebra, one readily finds by Taylor expanding f where it appears in N and R that if v ∈ H 2 0,η (R), then

||Q(µ, δ)v + R(µ, δ) + N (v; µ, δ)|| L 2 0,η < ∞.
The remaining terms A + (δ)χ -and S(δ)[(µ + •)χ + ] in R(µ, δ) are strongly localized by the choice of the far-field ansatz: χ -(x) is identically zero for x large, and for x large every term in S(δ) has at least two derivatives in it, so S(δ)(µ + •) ≡ 0 on the support of χ + , and the only terms that remain are compactly supported commutator terms. Hence we also obtain R(µ, δ) L 2 0,η < ∞. Together with (2.5) of Lemma (2.1), this implies that the second term of (3.7) is in L 2 0,η (R), and so to check that G is well-defined, it only remains to estimate the first term in (3.7). For this term, we use the specific form of S(δ), given in (3.4), to write

(1 -δ 2 ∂ 2 x ) -1 S(δ) = ∂ 2 x + δ 2 (1 -δ 2 ∂ 2 x ) -1 [a 3 ∂ 3 x + a 2 ∂ 2 x ] + (1 -δ 2 ∂ 2 x ) -1 (a 1 ∂ x + ã0 ). (3.8)
Since a 3 and a 2 are smooth, constant outside of fixed compact set, and bounded uniformly in δ, we have

||a 3 ∂ 3 x + a 2 ∂ 2 x || H 2 0,η →H -1 0,η ≤ C.
Combining this with estimate (2.6) of Lemma 2.1, we obtain

||δ 2 (1 -δ 2 ∂ 2 x ) -1 (a 3 ∂ 3 x + a 2 ∂ 2 x )|| H 2 0,η →L 2 0,η ≤ C|δ|. (3.9)
The other terms in (3.8) are readily seen to be uniformly bounded in δ as operators from H 2 0,η (R) to L 2 0,η (R) for δ sufficiently small, from which we conclude that G is well-defined.

Since f is smooth, smoothness in v follows readily from the fact that H 2 0,η (R) is a Banach algebra whose norm controls the L ∞ norm. Smoothness in µ is also readily attainable from smoothness of f and the exponential localization of our ansatz. The preconditioner plays little role in these arguments -when treating the residual terms or the nonlinearity, we do not need to use the preconditioner at all to obtain smoothness in v and µ.

The residual terms as well as the nonlinearity are also readily seen to be continuous in δ. The main subtlety is to handle the term (1δ 2 ∂ 2

x ) -1 S(δ), which we write as

(1 -δ 2 ) -1 S(δ) = (∂ 2 x + a 1 ∂ x + ã0 ) + δ 2 (1 -δ 2 ∂ 2 x ) -1 (a 3 ∂ 3 x + a 2 ∂ 2 x ) + T (δ)(a 1 ∂ x + ã0 ), where T (δ) = (1 -δ 2 ∂ 2 x ) -1 -1. The operator ∂ 2 x + a 1 (x, δ)∂ x + ã0 (x, δ) is continuous in δ from H 2 0,η to L 2 0,η
, since the coefficients are smooth and uniformly bounded in δ. The second term is continuous in δ by (3.9), and the last term is continuous in δ by Lemma 2.2. Continuity in δ of ∂ v G and ∂ µ G proceeds analogously.

With the appropriate regularity of G in hand, we now aim to solve near (v 0 , µ 0 , 0) using the implicit function theorem. The linearization about this solution in v is given by

∂ v G(v 0 ; µ 0 , 0) = S(0) + Q(µ 0 , 0) + ∂ v N (v 0 ; µ 0 , 0) = S(0) + f (q 0 ) -f (0) = L(0). From Lemma 2.3, ∂ v G(v 0 ; µ 0 , 0) is Fredholm with index -1, so that the joint linearization ∂ (v,µ) G(v 0 ; µ 0 , 0)
is Fredholm index 0 by the Fredholm bordering lemma [START_REF] Sandstede | Relative morse indices, Fredholm indices, and group velocities[END_REF]Lemma 4.4]. We show that in fact the joint linearization has full range, and hence is invertible.

Lemma 3.2. The joint linearization ∂ (v,µ) G(v 0 ; µ 0 , 0) : H 2 0,η (R) × R → L 2 0,η (R) is invertible. Proof. To show that ∂ (v,µ) G(v 0 ; µ 0 , 0) is invertible, we show that ∂ µ G(v 0 ; µ 0 , 0) is linearly independent from the range of L(0). From Lemma 2.4, it is enough to obtain ∂ µ G(v 0 ; µ 0 , 0), ϕ = 0. After a short computation, one finds ∂ µ G(v 0 ; µ 0 , 0) = S(0)χ + + (f (q 0 ) -f (0))χ + = L(0)χ + .
We compute L(0)χ + , ϕ via integration by parts, with the goal being to move L(0) onto the other side of the inner product as its adjoint and exploit the fact that L(0) * ϕ = 0. However, we must be careful since ϕ and χ + are not localized at ∞, and in fact there is one boundary term from integration by parts which does not vanish. We see this by writing

R χ + ϕ dx = - R χ + ϕ dx = R χ + ϕ -[χ + ϕ ] ∞ -∞ = χ + , ϕ -ϕ (∞) = χ + , ϕ + η * (0),
where we have observed from Lemma 2.3 that ϕ (∞) = -η * (0). Recalling that L(0) = ∂ 2 x + f (q * ) for x ≥ 1, we obtain

L(0)χ + , ϕ = χ + , L(0) * ϕ + η * (0) = η * (0) = c * (0) 2 > 0,
which concludes the proof.

Proof of Theorem 1. Since G(v 0 ; µ 0 , 0) = 0, G is smooth in v and µ and continuous in δ near (v 0 ; µ 0 , 0), ∂ (v,µ) G(v 0 ; µ 0 , 0) is invertible, and ∂ (v,µ) G(v; µ, δ) is continuous in δ, the implicit function theorem implies that for δ small, there exist v(δ) ∈ H 2 0,η (R) and µ(δ) ∈ R depending continuously on δ near δ = 0 such that G(v(δ); µ(δ), δ) = 0. By construction of G, this implies that

q * (x; δ) := χ -(x) + ω * (x; δ) -1 v(x; δ) + χ + (x)(µ(δ) + x)e -η * (δ)x
solves (1.2). The claim that q * (•, δ) → q * (•; 0) = q 0 uniformly in space follows from the form of this ansatz, together with the fact that

H 2 0,η (R) is continuously embedded in L ∞ (R).

Small eigenvalues

Having established existence of the critical front, we are now ready to study the point spectrum of the linearization about the front. Here we show that there is no eigenvalue in a neighborhood of the origin, and in particular no resonance embedded in the essential spectrum at the origin. For this, we follow [START_REF] Pogan | Instability of spikes in the presence of conservation laws[END_REF]: apply a Lyapunov-Schmidt reduction to construct a scalar function which vanishes at the eigenvalues, in a similar manner to the Evans function.

Throughout this section, we set Ω(δ) := {0} ∪ (C\σ ess (L δ )), and restrict to λ ∈ Ω(δ). Then λ is off the negative real axis, so that the principal value of γ := √ λ is defined by Re γ ≥ 0.

Proposition 4.1.

There exists δ 0 , γ 0 > 0 and a function E : (-δ 0 , δ 0 ) × B(0, γ 0 ) -→ C, continuous in δ and analytic in γ such that for all γ ∈ Ω(δ), the eigenvalue problem

(L(δ) -γ 2 )u = 0 (4.1)
admits a bounded solution u if and only if E(δ, γ) = 0. Furthermore, E(0, 0) = 0. In particular, there exists γ 1 , δ 1 > 0 such that for all δ ∈ (-δ 1 , δ 1 ), L(δ) has no eigenvalues on B(0, γ 1 2 ) ∩ Ω(δ 1 ).

For any fixed δ = 0, notice that (4.1) is a linear, non degenerate ODE with smooth coefficients, so that any solution u is smooth. Furthermore, such a solution admits exponential expansions at ±∞ (see the proof of Lemma 4.2 hereafter), so that when γ 2 is to the right of the essential spectrum, u is bounded if and only if it lies in H 4 (R), which is to say it is an eigenfunction. We will therefore consider bounded solutions from this point forward: for γ 2 to the right of the essential spectrum, they correspond with eigenfunctions, while at γ = 0 they capture resonances of L(δ).

We first show that a bounded solution of (4.1) decomposes into two parts: a uniformly localized part, and a slowly decaying part, whose rate is γ-close to 0. Lemma 4.2. Near (δ, γ) = (0, 0), the roots of the polynomial ν → d + c * (γ 2 , -η * + ν) satisfy:

ν 1 = - 1 |δ| + O(1), ν 2 = -γ + O(δγ + γ 2 ), ν 3 = γ + O(δγ + γ 2 ), ν 4 = 1 |δ| + O(1),
where each O is taken as δ and γ goes to 0.

In particular, there exists δ 0 > 0, γ 0 > 0 and η > 0 such that for all δ ∈ (-δ 0 , δ 0 ), and γ ∈ B(0, γ 0 ) with γ 2 ∈ Ω(δ), a bounded and smooth solution u of (4.1) decompose as

u(x) = w(x) + βχ + (x)e ν2x , ( 4.2)
where w ∈ H 2 0,η (R) and β ∈ C. In this decomposition, χ + is the cutoff function (3.2).

Proof. The claimed expansions of the four roots is purely technical and is postponed to the end of the proof. Rewrite (4.1) as a first order ODE in R 4 :

∂ x U = M (x; δ, γ).
where U = (u, u , u , u (3) ) T . The matrix M converges towards M ± (δ, γ) when x → ±∞, with an exponential rate which is independent of δ and γ. The eigenvalues of this asymptotic matrices M ± are the roots of the dispersion relations d ± c * (γ 2 , -η * + •). It is standard that with such a convergence rate, these eigenvalues determine the behavior of U at ±∞; see for example [10, proof of Lemma 2.2].

More precisely, the behavior at +∞ is the following. For γ = 0, the four roots are distinct, so that the exponential behavior is ensured: U (x) ∼ 4 i=1 c i (U )e νix when x → +∞, with c i (U, δ, γ) are vectors that does not depend on x. As γ 2 / ∈ σ ess (L(δ)), the two small roots satisfy Re ν 2 (δ, γ) < 0 < Re ν 3 (δ, γ), so that a bounded U has exactly the claimed form. At γ = 0, the two small roots merge to form a Jordan block. The proof in the above reference adapts, and we have the following expansion: U (x) ∼ c 1 (U )e ν1x + c 2 (U ) + c 3 (U )x + c 4 (U )e ν4x when x → +∞. Once again the claimed decomposition is satisfied.

At -∞, the four roots of d - c * (γ 2 , -η * + •) are distinct, and bounded away from 0 with spectral gap uniform in (δ, γ). Then the expansion U (x) ∼ 4 i=1 c - i (U )e ν - i x holds at x → -∞, so that any bounded U lies in H 2 (R -). Hence the claimed decomposition holds. For an alternative argument not relying on the dynamical systems view of exponential expansions, see Remark 4.7.

We now establish the expansions of the roots by applying the implicit function theorem to d + c * . From the choice of η * (see also (2.2)) we have

g 0 (δ, γ, ν) := d + c * (γ 2 , -η * + ν) = -δ 2 ν 4 + 4η * δ 2 ν 3 + (1 -6δ 2 η 2 * )ν 2 -γ 2 .
To avoid any δ singularity, we get rid of the δ 2 in the dominant term by changing variables µ := ν|δ|:

g 1 (δ, γ, µ) := δ 2 g 0 γ, δ, µ |δ| = -µ 4 + 4η * |δ|µ 3 + (1 -6δ 2 η 2 * )µ 2 -γ 2 δ 2 .
At (δ, γ) = (0, 0), this reduces to g 1 (0, 0, µ) = -µ 2 (µ -1)(µ + 1). Applying implicit function theorem to the simple root -1, we construct a root µ 1 (δ, γ) for g 1 (δ, γ, •) whose derivatives can be computed iteratively by differentiating the relation g 1 (δ, γ, µ 1 (δ, γ)) = 0. One can show by induction that any pure derivative in γ is null: ∂ k γ µ 1 (0, 0) = 0 for k ∈ N * . This ensures that the Taylor expansion has the form

µ 1 (δ, γ) = -1 -δ ∂ δ g 1 (0, 0, -1) ∂ µ g 1 (0, 0, -1) -γ ∂ γ g 1 (0, 0, -1) ∂ µ g 1 (0, 0, -1) + O(δ 2 + δγ) = -1 + O(δ).
Coming back to the original variable, we define ν 1 (δ, γ) = µ 1 (δ, γ)/|δ|, which satisfies the claimed expansion.

The same steps can be applied to define µ 4 (δ, γ) = 1 + O(δ), which in turn leads to ν 4 (δ, γ) as claimed.

To unfold the double root at µ = 0, we change variables once again to ν = γσ:

g 2 (δ, γ, σ) = g 0 (δ, γ, γσ) γ 2 = -δ 2 γ 2 σ 4 + 4η * δ 2 γσ 3 + (1 -6δ 2 η 2 * )σ 2 -1.
At (δ, γ) = (0, 0), this reduces to g 2 (0, 0, σ) = (σ -1)(σ + 1). Applying the implicit function theorem once again gives rise to

σ 2 (δ, γ) = -1 + O(δ + γ), σ 3 (δ, γ) = 1 + O(δ + γ)
, which in turns leads to the claimed estimates on ν 2 (δ, γ) = γσ 2 (δ, γ) and ν 3 (δ, γ) = γσ 3 (δ, γ).

As in the existence of the critical front, our problem is singular at δ = 0. Hence, we apply the same preconditioner: when δ is small, (4.1) is equivalent to

(1 -δ 2 ∂ 2 x ) -1 (L(δ) -γ 2
)u = 0. We now use the decomposition of Lemma 4.2 to separate out the localized part of our problem from the far-field behavior, which will allow us to make use of the Fredholm properties on weighted spaces of Section 2.4. In the following, for δ ∈ (-δ 0 , δ 0 ) and γ ∈ B(0, γ 0 ) we let A(δ, γ, η) = {w + βχ + e ν2(δ,γ)• : w ∈ H 2 0,η (R), β ∈ R}, denote the set where the ansatz obtained above holds.

Lemma 4.3.

There exist positive constants δ 0 , γ 0 and η such that if δ ∈ (-δ 0 , δ 0 ), γ ∈ B(0, γ 0 ) and

u ∈ A(δ, γ, η), then (1 -δ 2 ∂ 2 x ) -1 (L(δ) -γ 2 )u ∈ L 2 0,η (R).
Proof. First, (1δ 2 ∂ 2 x ) -1 (L(δ)γ 2 )w belongs to L 2 0,η (R) by the choice of the preconditioner, using the same regularization effect we observed in (3.8). Then, as χ + is smooth, vanishes on (-∞, 2) and is constant on (3, +∞), it only remains to show that (L(δ)γ 2 )e ν2• ∈ L 2 0,η (R + ). For x ≥ 1, almost all coefficients of L are constants, see (1.7), hence we compute

(L(δ) -γ 2 )e ν2x = (L(δ) -L + (δ))e ν2x + (L + (δ) -γ 2 )e ν2x = (f (q * (x; δ)) -f (0)) e ν2x + P (ν 2 , δ, γ)e ν2x ,
where the polynomial P (X, δ, γ) is the symbol defined by: L + (δ)γ 2 = P (∂ x , δ, γ), and L + is the asymptotic operator (2.2). From the definition of ν 2 (δ, γ), P (X, δ, γ) vanishes at X = ν 2 (δ, γ), hence for x ≥ 1:

(L(δ) -γ 2 )e ν2x = (f (q * (x)) -f (0))e ν2x = f (0)q * (x; δ)e ν2x + O e ν2x q * (x; δ) 2 .
The right hand side belongs to L 2 0,η (R) as long as η satisfies

-η * + Re ν 2 (δ, γ) < -η. (4.3)
We can take a smaller γ 0 than in Lemma 4.2, so that sup δ,γ {-η * (δ) + Re ν 2 (δ, γ)} < 0, which then allows to fix η > 0 so that (4.3) is satisfied for all δ ∈ (-δ 0 , δ 0 ) and γ ∈ B(0, γ 0 ). This concludes the proof.

We can now use Lemma 2.4 to decompose our problem into a part which belongs to im L(0) and a complementary part. Recall that P : L 2 0,η (R) -→ im(L(0)) and that ϕ allows to describe im(L(0)). Fix δ ∈ (-δ 0 , δ 0 ), and γ ∈ B(0, γ 0 ) ∩ Ω(δ). If u is a bounded solution of (4.1) then (w, β) ∈ H 2 0,η (R) × C defined in Lemma 4.2 solves:

P (1 -δ 2 ∂ 2 x ) -1 (L(δ) -γ 2 )(w + βh) = 0, (1 -δ 2 ∂ 2 x ) -1 (L(δ) -γ 2 )(w + βh), ϕ = 0, (4.4)
where h(x) = χ + (x)e ν2(γ)x . Reciprocally, if (w, β) ∈ H 2 0,η (R) × C satisfies (4.4), then u = w + βh is bounded and satisfies (4.1). We write the first equation as 0 = F(w, β; γ, δ)

:= P (1 -δ 2 ∂ 2 x ) -1 (L(δ) -γ 2 )(w + βh). (4.5)
and solve it with the implicit function theorem. We will then use the second equation to define E(δ, γ).

Lemma 4.4. For η > 0 sufficiently small, the map F :

H 2 0,η (R) × C × B(0, γ 0 ) × (-δ 0 , δ 0 ) → L 2 0,η (
R) is smooth in w and β, analytic in γ, and continuous in δ. Moreover, ∂ w F(w, β; γ, δ) is continuous in β, γ, and δ.

Proof. Note that F is linear in w and β, so smoothness is automatic provided the linear part in w is well defined, which is guaranteed here by Lemma 4.3. For the continuity of ∂ w F(w, β; γ, δ), we write

(1 -δ 2 ∂ 2 x ) -1 L(δ) = ∂ 2 x + a 1 ∂ x + a 0 + δ 2 (1 -δ 2 ∂ 2 x ) -1 (a 3 ∂ 3 x + a 2 ∂ 2 x ) + T (δ)(a 1 ∂ x + a 0
). We see by Lemmas 2.1 and 2.2 that (1δ 2 ∂ 2

x ) -1 L(δ) is a a well-defined family of bounded operators from H 2 0,η to L 2 0,η , depending continuously on δ. This is of course preserved when we compose with the projection P . We write the other term in the linearization in w as

γ 2 (1 -δ 2 ∂ 2 x ) -1 w = γ 2 w + γ 2 T (δ)
w, which is again continuous in γ and δ as a bounded linear operator from H 2 0,η to L 2 0,η by Lemma 2.2. Hence ∂ w F is continuous in its three last variables. Analyticity of F in γ follows as in [START_REF] Pogan | Instability of spikes in the presence of conservation laws[END_REF]Proposition 5.11]. For the continuity of F with respect to δ, it only remains to look at the terms associated to h. We rewrite (L(δ)γ 2 )h = [L + (δ), χ + ]e ν2(δ,γ)• + (L(δ) -L + (δ))e ν2(δ,γ)• , using the fact that (L + (δ)-γ 2 )e ν2(δ,γ)• = 0, and where [L + (δ), χ + ] = L + (δ)(χ + •)-χ + L + (δ) is the commutator between these operators. In this form, we recognize that [L + (δ), χ + ] and (L(δ) -L + (δ)) are both differential operators with exponentially localized coefficients, with rate uniform in δ for δ small. By Lemma 4.2, e ν2(δ,γ)x is continuous in γ and δ for each fixed x, and the uniform localization of [L + (δ), χ + ] and (L(δ) -L + (δ)) guarantees that these terms are continuous in δ in L 2 0,η for η small. In fact, since h is a smooth function, we see that δ → (L(δ)γ 2 )h is in particular continuous from (-δ 1 , δ 1 ) to H 1 0,η . Taking into account the preconditioner, we write Moreover, any solution to (4.5) with γ, δ small has this form.

(1 -δ 2 ∂ 2 x ) -1 (L(δ) -γ 2 )h = (L(δ) -γ 2 )h + T (δ)(L(δ) -γ 2 )h. By Lemma 2.
Proof. We begin with the trivial solution F(0, 0; 0, 0) = 0. The linearization in w about this trivial solution is ∂ w F(0, 0; 0, 0) = P L(0), which is invertible by Lemmas 2.3 and 2.4. Together with Lemma 4.4, this implies that we can solve near this trivial solution with the implicit function theorem, obtaining a unique solution w(β; γ, δ) in a neighborhood U of (0, 0; 0, 0). Since (4.5) is linear in w and β, by uniqueness any solution in this neighborhood can be written as w(β; γ, δ) = β w(γ, δ) Proposition 5.1. There exists a compact set K ⊂ C such that for all δ small, any eigenvalue λ of L(δ) with Re λ ≥ 0 lies in K. More precisely, an eigenvalue λ satisfies:

Re λ ≤ b(•; δ) ∞ , |Im λ| ≤ b(•; δ) ∞ + c * b(•; δ) ∞ -Re λ, where b(• ; δ) = f (q * (• ; δ)) is uniformly bounded.
Proof. We work with δ ∈ (-δ 0 , δ 0 ), with δ 0 small enough so that Theorem 1 apply. Assume that λ ∈ C and ψ ∈ H where Fu = û denotes the Fourier transform of a function u. Using the facts that φ = φ(-•), that the Fourier transform preserves the inner product, we can rewrite the 0-th order term as: We now conclude the proof of Theorem 2 by excluding the possibility of any eigenvalues in the intermediate region; see Figure 3. Proposition 5.2. For each δ 0 > 0 sufficiently small, there exists r(δ 0 ) > 0 with r(δ 0 ) → 0 as δ 0 → 0 such that for all δ with |δ| < δ 0 , the operator L(δ) has no eigenvalues in {Re λ ≥ 0} \ B(0, r(δ 0 )).

I 0 := R F (
Proof. Suppose to the contrary that there exists a sequence δ n → 0 with corresponding eigenvalues λ n bounded away from the origin, with Re λ n ≥ 0, and with eigenfunctions u n . We normalize the eigenfunctions so that ||u n || H 2 = 1 for all n. By Proposition 5.1, these eigenvalues all belong to the compact set K. By compactness, we extract a subsequence along which λ n → λ ∞ for some λ ∞ with λ ∞ ∈ K, and λ ∞ = 0, since the sequence was bounded away from the origin. We now show that in this limit, λ ∞ is an eigenvalue for L(0) with Re λ ≥ 0, contradicting the spectral stability of this operator.

These eigenfunctions solve (L(δ n )λ n )u n = 0. We precondition by applying (1δ 2 n ∂ 2 x ) -1 to both sides of this equation, obtaining where

∂ 2 x + a 1 (•, δ n )∂ x + a 0 (•, δ n ) + E 1 (δ n ) + E 2 (δ n ) -λ n + λ n T (δ n ) u n = 0, ( 5
E 1 (δ n ) = δ 2 n (1 -δ 2 n ∂ 2 x ) -1 (a 3 (•, δ n )∂ 3 x + a 2 (•, δ n )∂ 2 x ), E 2 (δ n ) = T (δ n )(a 1 (•, δ n )∂ x + a 0 (•, δ n )).
We relate this to the KPP linearization L(0) by rewriting (5.5) as

(L(0) -λ ∞ )u n = -E 1 (δ n )u n -E 2 (δ n )u n + E 3 (δ n )u n + (λ n -λ ∞ )u n + λ n T (δ n )u n =: f n where E 3 (δ n ) = (a 1 (•, 0) -a 1 (•, δ n ))∂ x + (a 0 (•, 0) -a 0 (•, δ n )).
It follows from Lemma 2.1 and the fact that the coefficients a j (•; δ) are uniformly bounded in δ that

||E 1 (δ n )u n || L 2 ≤ Cδ n ||u n || H 2 = Cδ n .
Similarly, by Lemma 2.2 we see that E 2 (δ n )u n → 0 and λ n T (δ n )u n → 0 in L 2 as n → ∞, since λ n and u n are uniformly bounded in n. Lastly, by the construction of the exponential weights, the fact that q * (•; δ) converges uniformly to q * (•; 0) as δ → 0 by Theorem 1, and the assumption that ||u n || H 2 is uniformly bounded, we see that also E 3 (δ n )u n → 0 in L 2 as n → ∞. Hence f n converges to zero in L 2 as n → ∞. Since λ ∞ is not in the spectrum of L(0), we can can invert (L(0)λ ∞ ) to write

u n = (L(0) -λ ∞ ) -1 f n ,
from which we observe that u n → 0 in H 2 (R) as n → ∞ by boundedness of the resolvent operator. This is a contradiction since we have normalized u n so that ||u n || H 2 = 1.

Proof of Theorem 2. By Proposition 4.1, there exist γ 1 , δ 1 > 0 so that for all δ ∈ (-δ 1 , δ 1 ), L(δ) has no eigenvalues in B(0, γ 2 1 ), and also has no resonance at λ = 0. By Proposition 5.2, there exists a δ 0 > 0 so for all δ ∈ (-δ 0 , δ 0 ), L(δ) has no eigenvalues in {Re λ ≥ 0} \ B(0, γ 2 1 2 ). Hence for all δ ∈ (-δ 0 , δ 0 ), L(δ) has no eigenvalues in {Re λ ≥ 0}, as desired.

  and continuous in µ and δ. Moreover, ∂ v G and ∂ µ G are continuous in δ.

Corollary 4 . 5 .

 45 2, this term is continuous in δ, as desired. For γ, δ sufficiently small, and for β ∈ C, there is a family of solutions w to (4.5) which have the form w(β; γ, δ) = β w(γ, δ).(4.6)

  4 (R) satisfy L(δ)ψ = λψ. Coming back to the unweighted operator A(δ) = ω -1 * L(δ)ω * , defined by (1.5), we obtainA(δ)φ = λφ (5.1) with φ = ω -1 * ψ ∈ H 4 (R).Up to a scalar multiplication, we can assume that φ L 2 (R) = 1. Now we take the Fourier transform in (5.1), multiply by φ(ξ) and finally integrate to obtain:R (-δ 2 ξ 4ξ 2 + ic * ξ)| φ(ξ)| 2 dξ + R F(f (q * )φ) φ dξ = λ, (5.2)

2 ≤ c 2 * φ 2 L 2 Rξ 2 |

 2222 bφ) φ dξ = R b(x)φ(x)φ(-x) dx.(5.3)The real and imaginary parts of equation (5.2) giveRe λ -ReI 0 = -R (δ 2 ξ 4 + ξ 2 )| φ| 2 dξ, Im λ -Im I 0 = c * R ξ| φ| 2 dξ.Hence by the Cauchy-Schwartz inequality:0 ≤ (Im λ -Im I 0 ) φ| 2 dξ ≤ c 2 * R (δ 2 ξ 4 + ξ 2 )| φ| 2 dξ = c 2 * (Re I 0 -Re λ) .(5.4)Note that b(• ; δ) is uniformly bounded with respect to δ ∈ (-δ 0 , δ 0 ), since this holds for q * (• ; δ) from Theorem 1, and since f is continuous.Applying the Cauchy-Schwartz inequality in(5.3) leads to |I 0 | ≤ b ∞ φ 2 L 2 .Inserting this into (5.4) leads to the claimed bound on Re λ. Then, we take the square root of (5.4) to obtain:|Im λ| -|Im I 0 | ≤ |Im λ -Im I 0 | ≤ c * b ∞ -Re λ,which is the claimed bound on Im λ. These bounds together with the requirement Re λ ≥ 0 define a compact set K.

Figure 3 :

 3 Figure 3: Three regions for the study of the point spectrum. The function E(δ, γ) from Section 4 rules out point spectrum in the dashed ball centered at the origin, together with a potential eigenvalue at the origin. Proposition 5.1 excludes point spectrum to the right of the dashed curve. Finally, the green region to the right contains no point spectrum provided δ is small enough, see Proposition 5.2.

for some function w(γ, δ) ∈ H 2 0,η (R). If for some fixed γ, δ small we have another solution (w 0 , β 0 ) to (4.5) which does not a priori have this form, by dividing by a sufficiently large constant K(||w 0 || H 2 0,η , β) we get another solution which belongs to the neighborhood U where we have solved with the implicit function theorem, and so we conclude that

and hence the solution (w 0 , β 0 ) in fact has the form (4.6), as claimed.

Having solved the first equation in (4.4) with the implicit function theorem, we now insert this solution w(β; γ, δ) = β w(γ, δ) into the second equation, so that (4.4) has a solution if and only if

Note that we have been able to eliminate the β dependence in this equation, since all terms in this equation are linear in β by Corollary 4.5. Since the projection P played no role in the proof of Lemma 4.4, the same argument shows that E is continuous in both of its arguments.

Lemma 4.6. The function E : (-δ 0 , δ 0 ) × B(0, γ 0 ) → C is continuous in both arguments, and analytic in γ for fixed δ.

Proof of Proposition 4.1. It only remains to prove that E(0, 0) = 0. From (4.7), we see that

Since w(0, 0) ∈ H 2 0,η (R), we have

Hence we obtain

by the computation in the proof of Lemma 3.2.

Remark 4.7. Rather than using the spatial dynamics approach to exponential expansions outlined in the proof of Lemma 4.2 to show directly that eigenfunctions have the form (4.2), one can instead show that for

and γ 2 is to the right of the essential spectrum of L(δ), using an argument adapted from [START_REF] Pogan | Instability of spikes in the presence of conservation laws[END_REF]. Indeed, if γ 2 is to the right of the essential spectrum, then L(δ)γ 2 is Fredholm index 0, and in particular has closed range, so to invert this operator on L 2 (R), it suffices to solve (L(δ)γ 2 )u = g for g in the dense subspace L 2 0,η (R). The fact that the range of L(δ)γ 2 is closed then implies L(δ)γ 2 is surjective, and hence invertible since it is Fredholm of index 0. The open mapping theorem then implies the inverse is bounded, so γ 2 will be in the resolvent set of L(δ). To solve (L(δ)γ 2 )u = g for g ∈ L 2 0,η (R), one looks for solutions in the form (4.2), and finds that (w, β) solve the system (4.4) but with (0, 0) T on the right hand side replaced by (P g, g, φ ) T . We can always solve the first equation with the implicit function theorem, and we can solve the second equation precisely when E(δ, γ) = 0, as claimed. At γ = 0 we lose Fredholm properties on L 2 (R), but the fact that E(δ, 0) implies there is no solution to L(δ)u = 0 of the form u = w + χ + for w exponentially localized, and this is actually all that is needed in [START_REF] Avery | Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum[END_REF] to prove nonlinear stability. One could additionally use a modified far-field/core decomposition at γ = 0 to prove that all bounded solutions to L(δ)u = 0 have the form u = w + χ + .

Large and intermediate eigenvalues -proof of Theorem 2

Here, we conclude the study of the point spectrum. We first exclude any large unstable point spectrum, using mostly that the operator is sectorial.