# Nonlinear convective stability of a critical pulled front undergoing a
Turing bifurcation at its back: a case study

Louis Garénaux  Institut de Mathématiques de Toulouse, UMR 5219, Université de
Toulouse CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France.
<EMAIL_ADDRESS>
###### Abstract.

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.

###### Key words and phrases:

Reaction-diffusion, Propagating monostable front, Asymptotic stability, Turing
instability, Weighted spaces, Point-wise resolvent estimate, Mode filters,
Amplitude equation

###### 1991 Mathematics Subject Classification:

35K57, 35C07, 35B35, 35B36, 35Q56, 35K46

###### Contents

  1. 1 Introduction
  2. 2 Main result
    1. 2.1 Notations
    2. 2.2 Sketch of the proof
  3. 3 Decay of perturbations in fully weighted space
    1. 3.1 Essential and point spectrum
    2. 3.2 Construction of decaying ODE solutions
    3. 3.3 Construction and estimations of the kernel for the resolvent operator
    4. 3.4 Linear dynamic
    5. 3.5 Nonlinear dynamic
  4. 4 Perturbations in partially weighted space are bounded in time
    1. 4.1 Mode filters
    2. 4.2 Linear dynamic
    3. 4.3 Nonlinear dynamic: shadowing the global attractor of the Ginzburg-Landau equation
  5. A Ginzburg-Landau equation
  6. B Proof of proposition 3.8
  7. C Three equilibrium points
  8. D Proof of lemma 3.4
  9. E Standard Lemmas

##  1\. 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
$x\in\mathbb{R}$ and $t>0$:

(1) $\left\\{\begin{array}[]{l}\partial_{t}u=d\partial_{xx}u+\alpha
u(1-u^{2})+\beta v,\\\\[5.0pt]
\partial_{t}v=-(\partial_{xx}+1)^{2}v+v(\mu-\sigma v^{2})-\gamma
v(1-u),\end{array}\right.$

with parameters $d$, $\alpha$, $\mu$, $\sigma$, $\gamma$ positive, and
$\beta\in\mathbb{R}$ nonzero. The scalar KPP equation is a typical model for
front propagation from a stable to an unstable state [KPP37, Fis37]:

(2) $\partial_{t}u=d\partial_{xx}u+f(u).$

The diffusion coefficient $d$ is positive and the reaction term
$f\in\mathcal{C}^{2}$ is of KPP type: it admits two equilibrium states
$f(0)=0=f(1)$ with distinct stability w.r.t. time
$f^{\prime}(0)>0>f^{\prime}(1)$. Generally, a concavity hypothesis is added:
$f^{\prime\prime}(u)<0$ for $u\in(0,1)$. Although $f(u)=\alpha u(1-u)$ is the
canonical choice, we will work with $f(u)=\alpha u(1-u^{2})$. All statements
and proofs below adapt to the $u(1-u)$ 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
($q_{c}\to 1$ when $x\to-\infty$) to the unstable one ($q_{c}\to 0$ when
$x\to+\infty$) [AW78]. Supercritical fronts
$c>c_{*}:=2\sqrt{d\,f^{\prime}(0)}=2\sqrt{d\alpha}$ 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 $\lambda=0$, creating a spectral gap at linear level
[Sat76]. 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
$\left\\{-\xi^{2}:\xi\in\mathbb{R}\right\\}\subset\mathbb{C}$ up to the
origin. Last advances in this direction [Gal94, FH18, AS21] 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 non-monotonic, 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) $\partial_{t}v=-(\partial_{xx}+1)^{2}v+\mu v-\sigma v^{3}.$

When the small parameter $\mu$ becomes positive, (3) undergoes a supercritical
Turing bifurcation: a curve of essential spectrum crosses the imaginary axis
twice, for nonzero Fourier frequencies $\pm 1$. This causes the constant
equilibrium state $u=0$ to bifurcate into periodic profile of amplitude
$\sqrt{\mu}$. It can be obtained through center manifold theory [EW91].

For all values of $\mu$, the coupled system (1) admits
$Q=(q_{*},0)^{\mathrm{T}}$ as a front with propagation speed $c_{*}$. The
linear coupling term $\beta v$ feeds the front dynamic with oscillatory
pattern, while the nonlinear coupling term $\gamma v(1-u)$ stabilize at linear
level ahead of the front. Thus oscillating profile are expected to appear only
behind the front.

Multiple works about bistable front – _i.e._ a front that connects two stable
states: $f(-1)=0=f(1)$ and $f^{\prime}(-1)<0$, $f^{\prime}(1)<0$ – undergoing
a Turing bifurcation have been done. When bifurcation occurs behind the front,
[SS01] showed that no modulated front appears: the Turing pattern travels at
slow speed $\mathcal{O}(\sqrt{\mu})$, hence is left behind the front. However,
the front survive after bifurcation, although through a non-modulated form.
Its nonlinear stability is showed in [BGS09] for a general second-order
setting.

When the state ahead of a bistable front bifurcates, [SS01] showed existence
and linear stability of modulated fronts – _i.e._ coherent structures that
link stable state behind to Turing pattern ahead – in a general second-order
setting. For a system that ressembles (1), [GSU04] obtain nonlinear stability
of such structure. We also mention [GS07] for a quadratic coupling $\beta
v^{2}$ 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 [SS00] and [SS04], while nonlinear stability of a traveling pulse for a
precise system is showed in [GSU04].

For monostable KPP fronts, previous works rather investigated the Turing
bifurcation in a non-local KPP context. [BNPR09] 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 [FH15], while global properties are obtained in [HR14]: 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 [NPT11, HR14]. 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 $0\in\mathbb{C}$ in
the essential spectrum does not contribute to a zero of the so-called Evans
function, due to the corresponding eigenfunction $q_{*}^{\prime}$ having only
weak exponential decay: $q_{*}^{\prime}(x)\sim(ax+b)e^{-\frac{c_{*}}{2}x}$
when $x\to+\infty$. This removes the technical aspect of stability up to a
shift in space, also referred to as orbital stability or modulation.

Figure 1. Illustration of a typical solution $(u,v)$, for $\mu$ 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 $\mathcal{O}(\sqrt{\mu})$, 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.

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

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 [FHSS21],
where this exact situation is investigated. Using a precise decomposition of
perturbations, they conclude on nonlinear stability of the front $Q_{*}$ for
$\mu<0$. Although we could have followed this direction in our situation, we
prefer to set $\gamma$ 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 [Sat76]. 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 [BGS09]. 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 $\gamma v(1-u)$ by $\gamma uv$. We point out
that a modulated front for such system is necessarily non-positive, which
prevent the use of a KPP reaction term $u(1-u)$. To ensure existence of
solutions globally in time, it is possible to rather study a monostable front
emanating from a bistable reaction term $u(1-u^{2})$.

##  2\. Main result

###  2.1. Notations

We investigate now the stability of the traveling front
$Q_{*}=(q_{*},0)^{\mathrm{T}}$. Insert the following ansatz in (1):

$\begin{pmatrix}u(t,x)\\\ v(t,x)\end{pmatrix}=Q(\tilde{x})+P(t,\tilde{x}),$

with $\tilde{x}=x-c_{*}t$. Then the perturbation
$P=(p_{1},p_{2})^{\mathrm{T}}$ satisfies

$\partial_{t}P=\mathcal{A}P+N(P):=\begin{pmatrix}\mathcal{A}^{\text{kpp}}&\beta\\\
0&\mathcal{A}^{\text{sh}}\end{pmatrix}P+\begin{pmatrix}N_{1}(P)\\\
N_{2}(P)\end{pmatrix},$

where the linear operator $\mathcal{A}:L^{2}(\mathbb{R})^{2}\to
L^{2}(\mathbb{R})^{2}$ is closed, has dense domain $H^{2}(\mathbb{R})\times
H^{4}(\mathbb{R})$ and is defined by:

(4)
$\mathcal{A}^{\text{kpp}}=d\partial_{xx}+c_{*}\partial_{x}+\alpha(1-3q_{*}(\tilde{x})),\hskip
20.00003pt\mathcal{A}^{\text{sh}}=-(1+\partial_{xx})^{2}+c_{*}\partial_{x}+\mu-\gamma(1-q_{*}(\tilde{x})),$

while remaining nonlinear terms express as:

(5) $N_{1}(P)=-\alpha(3q_{*}(\tilde{x}){p_{1}}^{2}+{p_{1}}^{3}),\hskip
40.00006ptN_{2}(P)=\gamma p_{1}p_{2}-\sigma{p_{2}}^{3}.$

The essential spectrum of $\mathcal{A}$ is unstable, see fig. 2, we do not
hope for an asymptotic stability result against general perturbation $P\in
H^{2}(\mathbb{R})\times H^{4}(\mathbb{R})$. Instead, we restrict to weighted
perturbations $P(t,\tilde{x})=\omega(\tilde{x})U(t,\tilde{x})$, with $U\in
H^{2}(\mathbb{R})\times H^{4}(\mathbb{R})$. As we shall see in further remark
3.2, the coupling term $\beta v$ imposes us to use scalar weight. Namely
$\omega_{*}:=\omega_{\text{kpp}}\omega_{\text{sh}}$, with smooth positive
functions

(6) $\omega_{\text{kpp}}(x):=\begin{cases}1&\text{if }x\leq-1,\\\
e^{-\frac{c_{*}}{2d}x}&\text{if }x\geq 1,\end{cases}\hskip
40.00006pt\omega_{\text{sh}}(x):=\begin{cases}e^{\theta x}&\text{if
}x\leq-1,\\\ 1&\text{if }x\geq 1,\end{cases}$

where the exponent $\theta<0$ is small, see proposition 3.1. The weight
$\omega_{\text{kpp}}$ acts only at $+\infty$, and allows to marginally
stabilize the KPP spectrum, while $\omega_{\text{sh}}$ acts at $-\infty$ and
stabilizes the SH spectrum. Hence, we insert the new ansatz

(7) $\begin{pmatrix}u(t,x)\\\
v(t,x)\end{pmatrix}=Q(\tilde{x})+\omega_{*}(\tilde{x})U(t,\tilde{x})$

in (1) to obtain that $U=(u_{1},u_{2})^{\mathrm{T}}$ satisfies:

(8)
$\partial_{t}U=\mathcal{L}U+\mathcal{N}(U):=\begin{pmatrix}\mathcal{L}^{\text{kpp}}&\beta\\\
0&\mathcal{L}^{\text{sh}}\end{pmatrix}U+\begin{pmatrix}\mathcal{N}_{1}(U)\\\
\mathcal{N}_{2}(U)\end{pmatrix},$

where all linear terms are closed, densely defined operators. They are
obtained as conjugation of the unweighted linear operators,

$\mathcal{L}^{i}=\omega_{*}^{-1}\mathcal{A}^{i}\omega_{*}$

when $i\in\\{\text{kpp},\text{sh}\\}$ and nonlinear terms express according to
$\mathcal{N}_{i}(U)=\omega_{*}^{-1}N_{i}(\omega_{*}U)$.

Figure 2. Essential spectrum of linear operators. Fredholm borders
corresponding to KPP (respectively SH) component are plain (respectively
dashed). Curves in red with single arrow (respectively blue with double arrow)
correspond to $+\infty$ (respectively $-\infty$) borders. Left:
$\Sigma_{\text{ess}}(\mathcal{A})$ with two unstable curves,
$\Sigma_{\text{}}(\mathcal{A}^{\text{sh},+})$ is out of picture to the left.
Center: $\Sigma_{\text{ess}}(\mathcal{L})$ with only marginally stable
spectrum. Right: $\Sigma_{\text{ess}}(\mathcal{T}^{-}+c_{*}\partial_{x})$ with
one unstable curve, which corresponds to Turing instability. The essential
spectrum of $\mathcal{T}^{-}$ is real and obtained from the one represented by
application of $z\mapsto\operatorname*{\mathrm{Re}}{z}$. Notice that the
spectrum of $\mathcal{T}$ can not be marginally stabilized, due to the lack of
a first order derivative.

Since the translation eigenmode $q_{*}^{\prime}$ in weighted spaces is
unbounded at $x\to+\infty$, we expect $U$ to decay in time with rate
$t^{-3/2}$ if the following assumption of a marginally stable essential
spectrum holds, see [AS21, FH18]:

($H_{1}$)
$\Sigma_{\text{ess}}(\mathcal{L})\subset\left\\{\operatorname*{\mathrm{Re}}{\lambda}\leq-\eta\right\\}\cup\left\\{-\xi^{2}:\xi\in\mathbb{R}\right\\}.$

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 [BGS09]. More details are
given in section 2.2. We do so with the second ansatz

(9) $\begin{pmatrix}u(t,x)\\\
v(t,x)\end{pmatrix}=Q(\tilde{x})+\rho_{*}(\tilde{x})\omega_{\text{kpp}}(\tilde{x})\,V(t,x).$

Here again $\tilde{x}=x-c_{*}t$, and the polynomial weight $\rho_{*}$ is a
smooth positive function defined by

(10) $\rho_{*}(x):=\begin{cases}1&\text{if }x\leq-1,\\\ \langle
x\rangle:=\sqrt{1+x^{2}}&\text{if }x\geq 1.\end{cases}$

We note $\varpi:=\rho_{*}\omega_{\text{kpp}}$, and insert (9) into (1) to
obtain that

$\varpi\partial_{t}V-c_{*}(\partial_{x}\varpi)V=\mathcal{A}(\varpi
V)-c_{*}\partial_{x}(\varpi V)+N(\varpi V),$

which rewrites as

$\displaystyle\partial_{t}V$ $\displaystyle=\mathcal{T}V+\mathcal{Q}(V),$ (11)
$\displaystyle=\mathcal{T}^{-}V+\mathcal{Q}^{-}(V)+\mathcal{S}(\tilde{x},V),$

where linear operators express as:

$\mathcal{T}:=\varpi(\tilde{x})^{-1}(\mathcal{A}-c_{*}\partial_{x})\varpi(\tilde{x})+c_{*}\frac{\partial_{x}\varpi}{\varpi}(\tilde{x})\hskip
40.00006pt\mathcal{T}^{-}=\lim_{\tilde{x}\to-\infty}\mathcal{T}(\tilde{x}),$

while nonlinear terms are defined by
$\mathcal{Q}(V)=\varpi(\tilde{x})^{-1}N(\varpi(\tilde{x})V)$, and
$\mathcal{Q}^{-}(V)=\lim_{\tilde{x}\to-\infty}\mathcal{Q}(V)$. We are left
with an error term

$\mathcal{S}(V):=\mathcal{T}V-\mathcal{T}^{-}V+\mathcal{Q}(V)-\mathcal{Q}^{-}(V).$

Since $\varpi(\tilde{x})=1$ for $\tilde{x}\leq-1$, we decompose
$\mathcal{T}^{-}$ as:

(12) $\mathcal{T}^{-}=\begin{pmatrix}d\partial_{xx}-2\alpha&\beta\\\
0&-(1+\partial_{xx})^{2}\end{pmatrix}+\begin{pmatrix}0&0\\\
0&\mu\end{pmatrix}=:\mathcal{T}_{0}+\mathcal{T}_{\mu}.$

and $\mathcal{Q}^{-}$ as

$\mathcal{Q}^{-}(V)=\begin{pmatrix}-3\alpha{v_{1}}^{2}\\\
\gamma{v_{1}}{v_{2}}\end{pmatrix}+\begin{pmatrix}-\alpha{v_{1}}^{3}\\\
-\sigma{v_{2}}^{3}\end{pmatrix}=:\mathcal{Q}_{2}+\mathcal{Q}_{3}.$

Thus we get the following expression for the source term:

(13) $\displaystyle\mathcal{S}(\cdot,V)={}$
$\displaystyle\left[\varpi^{-1}(\mathcal{A}-c_{*}\partial_{x})\varpi-(\mathcal{A}-c_{*}\partial_{x})+c_{*}\frac{\partial_{x}\varpi}{\varpi}+\begin{pmatrix}3\alpha(1-q_{*}^{2})&0\\\
0&-\gamma(1-q_{*})\end{pmatrix}\right]V$
$\displaystyle{}+(\varpi-1)\begin{pmatrix}-3\alpha q_{*}{v_{1}}^{2}\\\
\gamma{v_{1}}{v_{2}}\end{pmatrix}+(\varpi^{2}-1)\begin{pmatrix}-\alpha{v_{1}}^{3}\\\
-\sigma{v_{2}}^{3}\end{pmatrix}.$

To control perturbation $V$, we will compute an amplitude equation, which
reveals to be a Ginzburg-Landau equation with real coefficients

(GL) $\partial_{T}A=4\partial_{XX}A+A+bA\lvert A\rvert^{2}.$

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
$(u,v)^{\mathrm{T}}=Q(\tilde{x})+\rho_{*}(\tilde{x})\omega_{\text{kpp}}(\tilde{x})V(t,\tilde{x})$.
The actual choice lighten the computation to obtain the GL equation.
Conjugation by $\varpi:=\rho_{*}\omega_{\text{kpp}}$ is needed so that
proposition 4.5 holds: this weight is natural for KPP equation, in the sense
that its spatial decay at $+\infty$ copy the one of the translation
eigenfunction $q_{*}^{\prime}$ [FH18].

The perturbations $U$ and $V$ are linked through

(14) $V(t,x)=\frac{\omega_{\text{sh}}(\tilde{x})}{\rho_{*}(\tilde{x})}\
U(t,\tilde{x}).$

To measure space localization and regularity, we use Sobolev spaces, and
introduce the following norm:

$\lVert U\rVert_{X}:=\lVert U\rVert_{W^{2,\infty}(\mathbb{R})\times
W^{4,\infty}(\mathbb{R})}+\lVert\rho_{*}^{3}U\rVert_{L^{\infty}(\mathbb{R})}.$

We can now state our main result.

######  Theorem 2.1.

There exists an open, nonempty set of parameters $\Omega$ such that for
$(\alpha,\beta,d,\sigma,\gamma)$ in $\Omega$, both hypothesis ($H_{1}$) and
($H_{2}$) holds. For such a choice of parameters, the following holds. There
exists positive constants $C$, $\mu_{0}$, and $\delta$ such that for all
$0<\mu<\mu_{0}$, if $U_{0}$, $V_{0}$ satisfies

$K_{0}:=\frac{1}{\sqrt{\mu}}\left(\lVert
V_{0}\rVert_{W^{1,\infty}(\mathbb{R})}+\lVert
U_{0}\rVert_{X}\right)\leq\delta,$

then the solution of (1) with initial condition
$Q+\omega_{\text{kpp}}\omega_{\text{sh}}U_{0}=Q+\rho_{*}\omega_{\text{kpp}}V_{0}$
exists for all time, and writes as
$Q(\tilde{x})+\omega_{\text{kpp}}(\tilde{x})\omega_{\text{sh}}(\tilde{x})U(t,\tilde{x})=Q(\tilde{x})+\rho_{*}(\tilde{x})\omega_{\text{kpp}}(\tilde{x})V(t,x)$.
For all $t>0$, it satisfies:

$\left\lVert\frac{U(t,\cdot)}{\rho_{*}}\right\rVert_{L^{\infty}(\mathbb{R})}\leq
C\frac{\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{\infty}(\mathbb{R})}}{(1+t)^{3/2}},\hskip
40.00006pt\left\lVert V(t,\cdot)\right\rVert_{L^{\infty}(\mathbb{R})}\leq
C\sqrt{\mu}\,(1+K_{0}e^{-t/\mu}).$

It happens we can refine the decay of perturbation $U$.

######  Corollary 2.2.

Let $1<p\leq+\infty$ and $(\alpha,\beta,d,\sigma,\gamma)\in\Omega$, with
$\Omega$ as in theorem 2.1. There exists positive constants $C$, $\mu_{0}$,
$\delta$ and $\eta$ such that if

$K_{0}\leq\delta,\hskip 40.00006pt\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{p}(\mathbb{R})}\leq\delta,$

then the solution $U=(u_{1},u_{2})^{\mathrm{T}}$ of (1) with initial condition
$U_{0}$ satisfies:

$\left\lVert\frac{u_{1}(t,\cdot)}{\rho_{*}}\right\rVert_{L^{p}(\mathbb{R})}\leq
C\frac{\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{p}(\mathbb{R})}}{(1+t)^{\frac{3}{2}-\frac{1}{2p}}},\hskip
40.00006pt\lVert u_{2}(t,\cdot)\rVert_{L^{p}(\mathbb{R})}\leq Ce^{-\eta
t}\lVert U_{0}\rVert_{L^{p}(\mathbb{R})}.$

###  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:

$\partial_{t}U=\mathcal{L}U+\mathcal{N}(U).$

Where the linear operator $\mathcal{L}$ has marginally stable spectrum,
similarly to the KPP equation, and the nonlinear part $\mathcal{N}$ is
unbounded w.r.t. to $x\in\mathbb{R}$. At linear level, we expect an algebraic
decay: $t^{-3/2}$. We follow the approach from [FH18]: we first look at the
solution to the linear Cauchy problem

(15) $\partial_{t}p=\mathcal{L}p,\hskip 40.00006ptp(0,\cdot)=p_{0},$

and assume that it is expressed through a kernel:
$p(t,x)=\int_{\mathbb{R}}\mathcal{G}_{t}(x,y)p_{0}(y)\mathrm{d}y$. The matrix
valued function $\mathcal{G}_{\cdot}(\cdot,y)$ has to satisfy the linear
problem (15), with a Dirac delta initial condition:
$\mathcal{G}_{0}(x,y)=\delta_{y}(x)\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$.
Remark that $\mathcal{G}$ is an upper-triangular matrix, due to the triangular
structure of $\mathcal{L}$ and $\mathcal{G}_{0}$. We note it

$\mathcal{G}=\begin{pmatrix}\mathcal{G}^{\text{kpp}}&\mathcal{G}^{\text{co}}\\\
0&\mathcal{G}^{\text{sh}}\end{pmatrix},$

where $\mathcal{G}^{i}$ solves
$\partial_{t}\mathcal{G}_{t}=\mathcal{L}^{i}\mathcal{G}_{t}$ when
$i\in\left\\{\text{kpp},\text{sh}\right\\}$, and $\mathcal{G}_{t}^{\text{co}}$
accounts for the coupling terms. We then apply Laplace transform to obtain the
spectral Green kernel $G_{\lambda}(x,y)=\int_{0}^{+\infty}e^{-\lambda
t}\mathcal{G}_{t}(x,y)\mathrm{d}t$, that satisfies the _fundamental
eigenproblem_

$(\lambda-\mathcal{L})G_{\lambda}=\delta_{y},$

for suitable $\lambda\in\mathbb{C}$. As above, it is an upper-triangular
matrix that writes

$G=\begin{pmatrix}G^{\text{kpp}}&G^{\text{co}}\\\
0&G^{\text{sh}}\end{pmatrix},$

where each $G^{i}$ is the Laplace transform of $\mathcal{G}^{i}$ for
$i\in\left\\{\text{kpp},\text{sh},\text{co}\right\\}$. Hence, they satisfy

$(\lambda-\mathcal{L}^{\text{kpp}})G_{\lambda}^{\text{kpp}}(\cdot,y)=\delta_{y},\hskip
40.00006pt(\lambda-\mathcal{L}^{\text{sh}})G_{\lambda}^{\text{sh}}(\cdot,y)=\delta_{y},$
$(\lambda-\mathcal{L}^{\text{kpp}})G_{\lambda}^{\text{co}}(\cdot,y)=\beta
G_{\lambda}^{\text{sh}}(\cdot,y).$

The homogeneous eigenproblems $(\lambda-\mathcal{L}^{i})\phi=0$ with
$i\in\left\\{\text{kpp},\text{sh}\right\\}$ are ODEs, their solutions admit
exponential behaviors at $\pm\infty$:

$\phi(x)=e^{\nu(\lambda)x}(1+e^{-r\lvert x\rvert}\kappa(x)),$

where $\nu\in\mathbb{C}$ is a solution of the dispersion relations
$\lambda-\mathcal{L}^{i,\pm}(\nu)=0$; the real $r>0$ does not depend on $\nu$
nor $\lambda$; $\kappa$ is a bounded function on the half-line
$\mathbb{R}_{\pm}$. Such solutions can be concatenated to construct
$G_{\lambda}^{i}(\cdot,y)\in L^{2}(\mathbb{R})$. Then, the coupled spectral
green function expresses as the $L^{2}(\mathbb{R})$-inner product

$G_{\lambda}^{\text{co}}(x,y)=\langle G_{\lambda}^{\text{kpp}}(x,\cdot),\beta
G_{\lambda}^{\text{sh}}(\cdot,y)\rangle.$

This leads to spatial localization of $G_{\lambda}$, which is converted into
temporal decay for $\mathcal{G}_{t}$ through the inverse Laplace transform:

$\mathcal{G}_{t}(x,y)=\frac{1}{2i\pi}\int_{\Lambda}e^{\lambda
t}G_{\lambda}(x,y)\mathrm{d}\lambda,$

where $\Lambda$ is a contour in the resolvent set
$\mathbb{C}\backslash\Sigma_{\text{}}(\mathcal{L})$, 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:

$U(t,x)=\int_{\mathbb{R}}\mathcal{G}_{t}(x,y)U_{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{R}}\mathcal{G}_{t-\tau}(x,y)\mathcal{N}(U(\tau,y))\mathrm{d}y\mathrm{d}\tau.$

The fact that $\mathcal{N}(U)$ is unbounded111Due to
$\lvert\omega_{\text{sh}}(x)\rvert\to\infty$ when $x\to-\infty$ is a major
issue. In [BGS09], it is absorbed by transforming nonlinear terms into linear
ones: writing
$\omega_{\text{sh}}^{-1}\left(\omega_{\text{sh}}u\right)^{p}=(\omega_{\text{sh}}u)^{p-1}u$
and showing that $\lVert\omega_{\text{sh}}u\rVert_{L^{\infty}(\mathbb{R})}$ 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 $\delta$,
$\mu_{0}$, $C_{\text{stab}}$ and $\eta$ such that for all $0<\mu<\mu_{0}$, the
following holds. Fix $C$ and $t_{V}$ positive constants, an initial condition
$U_{0}$ such that $U_{0}\rho_{*}^{3}\in L^{p}(\mathbb{R})$, and assume that
for all $0\leq t\leq t_{V}$,

$\lVert V(t)\rVert_{L^{\infty}(\mathbb{R})}\leq C\leq\delta.$

Then the solution $U$ for the Cauchy problem (8) with initial condition
$U_{0}$ is defined for all $0\leq t\leq T$, and satisfies

$\left\lVert\frac{u_{1}(t)}{\rho_{*}}\right\rVert_{L^{p}(\mathbb{R})}\leq
C_{\text{stab}}\frac{\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{p}(\mathbb{R})}}{(1+t)^{\frac{3}{2}-\frac{1}{2p}}},\hskip
40.00006pt\lVert u_{2}(t)\rVert_{L^{p}(\mathbb{R})}\leq
C_{\text{stab}}e^{-\eta t}\lVert U_{0}\rVert_{L^{p}(\mathbb{R})}.$

Furthermore, $C_{\text{stab}}$ depends neither on $C$ or $t_{V}$.

Second, we turn to $\frac{\omega_{\text{sh}}}{\rho_{*}}U=V$, and show it is
bounded in time.222The presence of an extra $\frac{1}{\rho_{*}}$ is necessary
so that source term $\mathcal{S}(\tilde{x},V)$ decay in time, see proposition
4.5. Remind that

$\partial_{t}V=\mathcal{T}V+\mathcal{Q}(V)=\mathcal{T}^{-}V+\mathcal{Q}^{-}(V)+\mathcal{S}(\tilde{x},V).$

We show that $\mathcal{S}$ is sufficiently localized in space so that we can
extract a $\omega_{\text{sh}}$ from it. This allows to control
$\mathcal{S}(V)$ by $U/\rho_{*}$ – 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 $-\infty$, which allows to get
rid of the marginally stable curve of essential spectrum at $+\infty$. Since
$\mathcal{T}^{-}$ undergoes a Turing bifurcation, we follow the steps of
[Sch94b, BGS09], that mostly relies on the use of _mode-filters_ , see
[Sch94c].

We first show that periodic patterns are naturally selected: after a time
$\frac{T}{\mu}$, 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 $\varepsilon=\sqrt{\mu}$, if
$V=\psi(A)+\mathcal{O}(\varepsilon^{2})$, with
$\psi(\varepsilon,A)(t,x):=\varepsilon A(\varepsilon^{2}t,\varepsilon
x)e^{ix}\varrho_{\text{c}}+c.c$,333Here and in the following, we note $c.c$
for the complex conjugate: $z+c.c:=z+\bar{z}=2\operatorname*{\mathrm{Re}}(z)$.
with a suitable $\varrho_{\text{c}}\in\mathbb{R}^{2}$, then
$A(T,X)\in\mathbb{C}$ satisfies

$\partial_{T}A=4\partial_{XX}A+A+bA\lvert A\rvert^{2}.$

We refer to [Mie02, MS95] for the derivation of amplitude equation. For
suitable values of the parameter $\gamma>0$, coefficient $b\in\mathbb{R}$ 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
$\psi(A)$ at time $t_{0}>0$, then the solution of the whole system emanating
from $V_{0}$ is defined upon time $t_{0}+T/\mu$, and remains close to
$\psi(A)$. 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 section 4.3.

######  Proposition 2.4 (Decay of $U$ implies $V$ bounded).

Assume hypothesis ($H_{2}$) holds. There exists positive constants $\delta$,
$\mu_{0}$ such that for all $0<\mu<\mu_{0}$, the following holds. Assume that

$K_{0}:=\frac{1}{\sqrt{\mu}}\left(\lVert
V_{0}\rVert_{W^{1,\infty}(\mathbb{R})}+\lVert
U_{0}\rVert_{X}\right)\leq\delta.$

Then $V(t)$ is defined for all times $t\geq 0$. Assume further that $C_{1}$
and $t_{U}$ are positive constants, such that for all $0\leq t\leq t_{U}$:

$\left\lVert\frac{U(t)}{\rho_{*}}\right\rVert_{L^{\infty}(\mathbb{R})}\leq
C_{1}\frac{\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{\infty}(\mathbb{R})}}{(1+t)^{3/2}}.$

Then there exists $C_{2}$ – that may depend on $C_{1}$, but neither on
$t_{U}$, $\delta$ or $\mu_{0}$ – such that for all $0\leq t\leq t_{U}$,

$\lVert V(t)\rVert_{L^{\infty}(\mathbb{R})}\leq
C_{2}\sqrt{\mu}\,(1+K_{0}e^{-t/\mu}).$

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 $C_{\text{stab}}$ 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 $\Omega$ that
allows to fulfill both hypothesis ($H_{1}$) and ($H_{2}$). Take $\mu_{0}$
small enough, and fix $\rho_{*}V_{0}=\omega_{\text{sh}}U_{0}$ such that:

$U_{0}\rho_{*}^{3}\in L^{\infty}(\mathbb{R}),\hskip
40.00006ptK_{0}\leq\delta.$

Since $\mathcal{T}$ is sectorial, and $V\mapsto\mathcal{Q}(V)$ is locally
Lipschitz, the solution $t\in I\mapsto V(t)\in L^{\infty}(\mathbb{R})$ to
equation $\eqref{e:system_V}$ is uniquely defined on a open, nonempty, maximal
set $I$ – see [Hen81]. Hence there exists $t_{0}$ and $C$ positive constants
such that

$\lVert V(t)\rVert_{L^{\infty}(\mathbb{R})}\leq C$

for all $0\leq t\leq t_{0}$. Applying proposition 2.3, this ensures that for
all $0\leq t\leq t_{0}$, the perturbation $U/\rho_{*}$ is uniquely defined in
$L^{\infty}(\mathbb{R})$, and satisfies

(16)
$N(t):=\frac{1}{(1+t)^{3/2}}\left\lVert\frac{U(t)}{\rho_{*}}\right\rVert_{L^{\infty}(\mathbb{R})}\leq
C_{\text{stab}}\lVert U_{0}\rho_{*}^{3}\rVert_{L^{\infty}(\mathbb{R})}.$

Consider $t_{1}$ the first time where this inequality may fail:

$t_{1}:=\inf I,\hskip 40.00006ptI:=\left\\{t>0:N(t)>C_{\text{stab}}\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{\infty}(\mathbb{R})}\right\\}.$

Assuming by contradiction that $I$ is nonempty, remark that $0<t_{0}\leq
t_{1}<+\infty$. From proposition 2.4, $V$ is defined for all times $0\leq
t\leq t_{1}$, with bound

$\lVert V(t)\rVert_{L^{\infty}(\mathbb{R})}\leq
C_{2}\sqrt{\mu}\,(1+K_{0}),\hskip 40.00006pt0\leq t\leq t_{1}$

where $C_{2}$ depends on $C_{\text{stab}}$. Reasoning as above, there exists
$t_{2}>t_{1}$ such that $V$ is defined up to time $t_{2}$, with bound

$\lVert V(t)\rVert_{L^{\infty}(\mathbb{R})}\leq
2C_{2}\sqrt{\mu}\,(1+K_{0}),\hskip 40.00006pt0\leq t\leq t_{2}.$

Assuming that $\mu_{0}(1+K_{0})$ is small enough, we can assume
$2C_{2}\sqrt{\mu}\,(1+K_{0})\leq\delta$ so that proposition 2.3 applies again,
and we recover (16) for times $0\leq t\leq t_{2}$. 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
$\lVert\rho_{*}^{3}U_{0}\rVert_{L^{p}}\leq\delta$, we can finally apply
proposition 2.3 to recover $L^{p}(\mathbb{R})$ 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 $\mathcal{T}$
rather than the asymptotic one $\mathcal{T}^{-}$. This last point allow to
remove the source term $\mathcal{S}$, and as so to break the $U$ \- $V$
interaction in our proof.

##  3\. Decay of perturbations in fully weighted space

###  3.1. Essential and point spectrum

We say that a scalar differential operator
$\mathcal{T}(x)=\sum_{j=0}^{n}a_{j}(x)\partial_{x}^{j}$ is _exponentially
asymptotic_ if it converges with uniform exponential rate at $\pm\infty$:
there exists $r>0$ such that for all $0\leq j\leq n$,

$\lim_{x\to\pm\infty}e^{r\lvert x\rvert}\lvert a_{j}(x)-a_{j}^{\pm}\rvert=0.$

We note $\Sigma_{\text{}}(\mathcal{T})$ the spectrum of $\mathcal{T}$, _i.e._
the set of complex numbers $\lambda$ such that
$\lambda-\mathcal{T}:H^{n}(\mathbb{R})\subset L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})$ is not bounded invertible. To study the spectrum, we
decompose it into two distinct parts. We say that $\lambda-\mathcal{T}$ is
Fredholm if its Fredholm index

$\operatorname*{Fred}(\lambda-\mathcal{T}):=\dim\ker(\lambda-\mathcal{T})-\operatorname*{codim}\operatorname{im}(\lambda-\mathcal{T})$

is defined and finite. We define the point spectrum as

$\Sigma_{\text{pt}}(\mathcal{T}):=\left\\{\lambda\in\Sigma_{\text{}}(\mathcal{T}):\operatorname*{Fred}(\lambda-\mathcal{T})=0\right\\},$

it corresponds to the subset in which the rank-nullity theorem holds. Then,
the essential spectrum is the complementary set:

$\Sigma_{\text{ess}}(\mathcal{T}):=\left\\{\lambda\in\Sigma_{\text{}}(\mathcal{T}):\operatorname*{Fred}(\lambda-\mathcal{T})\text{
is non defined or nonzero}\right\\}.$

With such definitions, the asymptotic operator with piece-wise constant
coefficients

$\mathcal{T}^{\infty}(x):=\begin{cases}\mathcal{T}^{+}=\sum_{j=0}^{n}a_{j}^{+}\partial_{x}^{j}&\text{if
}x>0,\\\ \mathcal{T}^{-}=\sum_{j=0}^{n}a_{j}^{-}\partial_{x}^{j}&\text{if
}x<0,\end{cases}$

share the same essential spectrum as $\mathcal{T}$, see [KP13, p 40]. If both
$\mathcal{T}^{+}$ and $\mathcal{T}^{-}$ 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
$\Sigma_{\text{ess}}(\mathcal{T})=\Sigma_{\text{ess}}(\mathcal{T}^{\infty})$
is located to the left of the asymptotic Fredholm borders

$\Sigma_{\text{}}(\mathcal{T}^{+})\cup\Sigma_{\text{}}(\mathcal{T}^{-})=\left\\{\sum_{j=0}^{n}a_{j}^{+}(i\xi)^{j}:\xi\in\mathbb{R}\right\\}\cup\left\\{\sum_{j=0}^{n}a_{j}^{-}(i\xi)^{j}:\xi\in\mathbb{R}\right\\}\subset\mathbb{C},$

see again [KP13, Theorem 3.1.13].444If $\mathcal{T}^{\pm}$ are elliptic, then
both $\lambda-\mathcal{T}^{\pm}$ are invertible for
$\operatorname*{\mathrm{Re}}{\lambda}$ large enough. This is equivalent to
have $\frac{n}{2}$ stable (respectively unstable) spatial eigenvalues for
$\lambda-\mathcal{T}^{+}$ (respectively $\lambda-\mathcal{T}^{-}$) and ensures
that the region $S_{1}$ that contains $(\eta,+\infty)$ – for
$\eta\in\mathbb{R}$ large enough – does not belong to
$\Sigma_{\text{ess}}(\mathcal{T}^{\infty})$.

For a matrix operator:
$\mathcal{T}=\begin{pmatrix}\mathcal{T}_{1,1}&\mathcal{T}_{1,2}\\\
\mathcal{T}_{2,1}&\mathcal{T}_{2,2}\end{pmatrix}$, the asymptotic spectra
$\Sigma_{\text{}}(\mathcal{T}^{\pm})$ are still obtained through Fourier
transform. The operator $\lambda-\mathcal{T}^{+}$ is invertible _iff_ the
matrix $\lambda-\hat{\mathcal{T}}^{+}(\xi)$ is invertible for all
$\xi\in\mathbb{R}$ – since Fourier transform is an isometry – which is
equivalent to $\xi\mapsto\det(\lambda-\hat{\mathcal{T}}^{+}(\xi))$ never
vanishing. In our special case, $\mathcal{L}^{+}$ is triangular, hence the
determinant is nonzero if both diagonal coefficients maintain away from zero.
We conclude that

$\Sigma_{\text{}}(\mathcal{L}^{+})=\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},+})\cup\Sigma_{\text{}}(\mathcal{L}^{\text{sh},+}),$

and the same goes at $-\infty$. Once again, the essential spectrum of
$\mathcal{L}$ is located to the left of
$\Sigma_{\text{}}(\mathcal{L}^{+})\cup\Sigma_{\text{}}(\mathcal{L}^{-})$.

######  Proposition 3.1 (Marginally stable essential spectrum).

Fix $\alpha$, $d$ positive. Then there exists $\mu_{0}>0$ such that for all
$0<\mu<\mu_{0}$, there exists $\theta<0$ such that the monotonic weight
$\omega_{*}=\omega_{\text{kpp}}\,\omega_{\text{sh}}$, with

$\omega_{\text{kpp}}(x)=\begin{cases}1&\text{if }x\leq-1,\\\
e^{-\frac{c_{*}}{2d}x}&\text{if }x\geq 1,\end{cases}\hskip
40.00006pt\omega_{\text{sh}}(x)=\begin{cases}e^{\theta x}&\text{if
}x\leq-1,\\\ 1&\text{if }x\geq 1,\end{cases}$

satisfies the following. For all $\gamma>\gamma_{\mathrm{rem}}$, with
$\gamma_{\mathrm{rem}}$ as in [FHSS21]:

$\gamma_{\mathrm{rem}}:=8\left(\frac{\alpha}{d}\right)^{2}+4\frac{\alpha}{d}-2\alpha+\mu_{0},$

the operator $\mathcal{L}=\omega_{*}^{-1}\mathcal{A}\omega_{*}$ has marginally
stable essential spectrum. More precisely, the Fredholm curve corresponding to
the essential KPP spectrum touches the imaginary axis only at $\lambda=0$:

$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},+})=\left\\{-\xi^{2}:\xi\in\mathbb{R}\right\\},$

while the three other Fredholm borders have spectral gap: there exists
$\eta>0$ depending only on $\mu_{0}$, $d$, $\alpha$ and $\gamma$ such that

(17)
$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},-})\cup\Sigma_{\text{}}(\mathcal{L}^{\text{sh},+})\cup\Sigma_{\text{}}(\mathcal{L}^{\text{sh},-})\subset\left\\{\lambda\in\mathbb{C}:\operatorname*{\mathrm{Re}}{\lambda}\leq-3\eta\right\\}.$

In particular, hypothesis ($H_{1}$) is fulfilled. Furthermore, $\theta\to 0$
when $\mu_{0}\to 0$.

###### Proof.

The asymptotic operators $\mathcal{L}^{\text{kpp},\pm}$ and
$\mathcal{L}^{\text{sh},\pm}$ are obtained as conjugation of
$\mathcal{A}^{\text{kpp},\pm}$ and $\mathcal{A}^{\text{sh},\pm}$ with pure
exponential weight $e^{\theta x}$ or $e^{-\frac{c_{*}}{2d}x}$. Direct
computations (see lemma E.1 below) show that
$\mathcal{L}^{\text{kpp},+}=\mathcal{A}^{\text{kpp},+}(-\frac{c_{*}}{2d}+\partial_{x})$,
where we identify the operator $\mathcal{A}^{\text{kpp},+}$ with its symbol
$\mathcal{A}^{\text{kpp},+}(X)=dX^{2}+c_{*}X+\alpha$ evaluated at
$X=\partial_{x}$. Hence, the Fredholm border is given by

$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},+})=\left\\{\mathcal{A}^{\text{kpp},+}\left(-\frac{c_{*}}{2d}+i\xi\right):\xi\in\mathbb{R}\right\\}=\left\\{-\xi^{2}:\xi\in\mathbb{R}\right\\}.$

Using similar notations for the three other curves, we get

$\operatorname*{\mathrm{Re}}\left(\mathcal{A}^{\text{kpp},-}(\theta+i\xi)\right)=\operatorname*{\mathrm{Re}}\left(d(\theta+i\xi)^{2}+c_{*}(\theta+i\xi)-2\alpha\right)\leq-2\alpha+d\theta^{2}+c_{*}\theta<-\alpha$

for $\theta\in(-\theta_{0},\theta_{0})$, where $\theta_{0}>0$ only depends on
$\alpha$ and $d$. Similarly with
$\mathcal{A}^{\text{sh},-}(X)=-(X^{2}+1)^{2}+cX+\mu$, a direct computation
shows that
$\operatorname*{\mathrm{Re}}\left(\mathcal{A}^{\text{sh},-}(\theta+i\xi)\right)$
is maximal at $\pm\xi_{0}=\pm\sqrt{1+3\theta^{2}}$, hence:

$\operatorname*{\mathrm{Re}}\left(\mathcal{A}^{\text{sh},-}(\theta+i\xi)\right)\leq-\theta^{4}-2\theta^{2}+c_{*}\theta+\mu+\xi_{0}^{4}-1=\mu+c_{*}\theta+4\theta^{2}+8\theta^{4}$

Fix $\eta>0$ small, it is easily seen that the right hand side is less than
$-3\eta$ for some $\theta<0$ that goes to $0$ when $\mu_{0}\to 0$.

Finally for
$\mathcal{A}^{\text{sh},+}(X)=\mathcal{A}^{\text{sh},-}(X)-\gamma$, the same
calculation with $\zeta_{0}:=\sqrt{1+3(\frac{c_{*}}{2d})^{2}}$ shows that

$\operatorname*{\mathrm{Re}}\left(\mathcal{A}^{\text{sh},+}\left(-\frac{c_{*}}{2d}+i\xi\right)\right)\leq-\left(\frac{c_{*}}{2d}\right)^{4}-2\left(\frac{c_{*}}{2d}\right)^{2}-\frac{c_{*}^{2}}{2d}+\mu_{0}-\gamma+\zeta_{0}^{4}-1=\gamma_{\mathrm{rem}}-\gamma.$

Hence for $\gamma>\gamma_{\mathrm{rem}}$, there exists $\eta>0$ such that
$\operatorname*{\mathrm{Re}}\left(\mathcal{A}^{\text{sh},+}\left(-\frac{c_{*}}{2}+i\xi\right)\right)<-3\eta$
as claimed. ∎

######  Remark 3.2.

We examine the case of distinct weights on KPP and SH component. Note
$\Omega(x)=\mathrm{Diag}(\omega_{1}(x),\omega_{2}(x))$, and replace ansatz (7)
by $(u,v)^{\mathrm{T}}(t,x)=Q(\tilde{x})+\Omega(\tilde{x})U(t,\tilde{x})$,
such that perturbation $U$ is driven at linear level by

$\partial_{t}U=\begin{pmatrix}{\omega_{1}}^{-1}\mathcal{A}^{\text{kpp}}\omega_{1}&\frac{\omega_{2}}{\omega_{1}}\beta\\\
0&{\omega_{2}}^{-1}\mathcal{A}^{\text{sh}}\omega_{2}\end{pmatrix}U.$

For our study to be relevant, we need $\frac{\omega_{2}(x)}{\omega_{1}(x)}$ to
be bounded w.r.t. $x$. The computations done in proposition 3.3 shows that
both $\omega_{1}(x)\leq e^{-\frac{c_{*}}{2d}x}$ for $x\geq 1$ and
$\omega_{2}(x)\geq e^{\theta x}$ for $x\leq-1$ are necessary to obtain
marginal spectral stability. With the condition $\omega_{2}(x)\leq
C\omega_{1}(x)$ for $x\in\mathbb{R}$, we are left with
$\omega_{1}=\omega_{2}$.

######  Proposition 3.3 (Stable point spectrum).

With the same assumptions on $\mu_{0}$, $\mu$, $\alpha$, $d$, $\gamma$, $\eta$
as in previous proposition 3.1, the operator $\mathcal{L}$ has no eigenvalue
in the following set:

$\Omega_{\mathrm{eig}}:=\left\\{\lambda\in\mathbb{C}:\operatorname*{\mathrm{Re}}{\lambda}>-3\eta\right\\}\backslash(-\infty,0).$

###### Proof.

We assume by contradiction that there exists $\lambda\in\Omega_{\mathrm{eig}}$
and a nonzero $\phi=(\phi^{\text{kpp}},\phi^{\text{sh}})\in
H^{2}(\mathbb{R},\mathbb{R})\times H^{4}(\mathbb{R},\mathbb{R})$ such that

$(\lambda-\mathcal{L})\phi=0.$

In particular, the second eigenproblem is decoupled:
$(\lambda-\mathcal{L}^{\text{sh}})\phi^{\text{sh}}=0$. We show it has no other
solution than $\phi^{\text{sh}}=0$ in $H^{4}(\mathbb{R})$, which will imply
that the first eigenproblem
$(\lambda-\mathcal{L}^{\text{kpp}})\phi^{\text{kpp}}=\beta\phi^{\text{sh}}$
admits no other solution than $\phi^{\text{kpp}}=0$. This is a contradiction.

Both $\mathcal{L}^{\text{sh}}$ and $\mathcal{A}^{\text{sh}}$ are not easy to
work with, because they respectively have non-constant coefficients or
unstable essential spectrum. Instead, we use

$\mathcal{B}^{\text{sh}}:=e^{-\theta x}\mathcal{A}^{\text{sh}}e^{\theta
x}=e^{-\theta x}\omega_{*}\mathcal{L}^{\text{sh}}\omega_{*}^{-1}e^{\theta
x}=b_{0}(x)+\sum_{j=1}^{4}b_{j}\partial_{x}^{j},$

with $\theta$ and $\omega_{*}(x)$ defined in the above proposition 3.1. Then
$\varphi:=e^{-\theta x}\omega_{*}\phi^{\text{sh}}$ lies in $H^{4}(\mathbb{R})$
due to $e^{-\theta x}\omega_{*}(x)\leq 1$,555Recall that
$-\frac{c_{*}}{2}\leq\theta$ when $\mu$ is small enough. and satisfies
$(\lambda-\mathcal{B}^{\text{sh}})\varphi=0$. Take the
$L^{2}(\mathbb{R},\mathbb{C})$-inner product with $\varphi$:

$\lambda\lVert\varphi\rVert_{L^{2}}^{2}=\langle
b_{0}(x)\varphi,\varphi\rangle+\sum_{j=1}^{4}b_{i}\langle\partial_{x}^{j}\varphi,\varphi\rangle.$

Then
$\langle\partial_{x}^{j}\varphi,\varphi\rangle=(-1)^{j}\,\overline{\langle\partial_{x}^{j}\varphi,\varphi\rangle}$,
so that the real part of the above equality writes:

(18)
$\operatorname*{\mathrm{Re}}(\lambda)\,\lVert\varphi\rVert_{L^{2}}^{2}=\int_{\mathbb{R}}b_{0}\lvert\varphi\rvert^{2}\mathrm{d}x-b_{2}\lVert\partial_{x}\varphi\rVert_{L^{2}}^{2}+b_{4}\lVert\partial_{x}^{2}\varphi\rVert_{L^{2}}^{2}\leq-3\eta\lVert\varphi\rVert_{L^{2}}^{2}.$

The inequality is obtained using that $b_{2}=2(1+3\theta^{2})>0$, that
$b_{4}=-1<0$ and finally that

$b_{0}(x)=\mu-\gamma(1-q_{*}(x))+c_{*}\theta-(\theta^{2}+1)^{2}\leq\mu+c_{*}\theta\leq\mu+c_{*}\theta+4\theta^{2}+8\theta^{4}\leq-3\eta.$

Recall we have chosen $\operatorname*{\mathrm{Re}}{\lambda}>-3\eta$, hence
(18) implies $\varphi=0$ and then $\phi^{\text{sh}}=0$ as claimed.

Now the first eigenproblem writes
$(\lambda-\mathcal{L}^{\text{kpp}})\phi^{\text{kpp}}=0$, hence
$\omega_{*}\phi^{\text{kpp}}\in H^{2}(\mathbb{R})$ is an eigenfunction for
$\mathcal{A}^{\text{kpp}}$. Using Sturm-Liouville theory – see _e.g._ [KP13, p
33.] – eigenvalues of $\mathcal{A}^{\text{kpp}}$ are real and non-positive,
since $q_{*}^{\prime}\in H^{2}(\mathbb{R})$ does not vanish and satisfies
$\mathcal{A}^{\text{kpp}}q_{*}^{\prime}=0$. Hence $\mathcal{A}^{\text{kpp}}$
has no eigenvalues outside of $(-\infty,0]$, and so does
$\mathcal{L}^{\text{kpp}}$. Since
$\frac{1}{\omega_{*}(x)}q_{*}^{\prime}(x)\sim ax+b$ when $x\to+\infty$, the
derivative of the front does not contribute to an eigenfunction for
$\mathcal{L}^{\text{kpp}}$, hence it has no eigenvalue oustide $(-\infty,0)$.
The first eigenproblem imposes either $\lambda\in(-\infty,0)$ or
$\phi^{\text{kpp}}=0$, which is a contradiction, and complete the proof. ∎

###  3.2. Construction of decaying ODE solutions

Here we solve the linear non-autonomous ODE $(\lambda-\mathcal{L}^{i})\phi=0$,
with unknown $\phi$ and $i\in\left\\{\text{kpp},\text{sh}\right\\}$. To keep
notations simple, we concentrate on $i=\text{sh}$. Since
$\mathcal{L}^{\text{sh}}$ is exponentially asymptotic, $\phi$ expresses at
first order using the solutions of the asymptotic ODE, as described in the
incoming Lemma. After vectorialization,
$(\lambda-\mathcal{L}^{\text{sh}})\phi=0$ writes
$\partial_{x}\Phi=A^{\text{sh}}(\lambda,x)\Phi$, with a matrix
$A^{\text{sh}}(\lambda,x)$ that exponentially converges towards matrices
$A^{\text{sh},\pm}(\lambda)$ when $x\to\pm\infty$.

######  Lemma 3.4.

Let $A:t\in\mathbb{R}\mapsto A(t)\in\mathcal{M}_{n}(\mathbb{C})$ be a
continuous, matrix-valued function, that converges at exponential speed
towards $A_{\infty}$ when $t\to+\infty$. Let $\alpha>0$ that satisfies $\lVert
A(t)-A_{\infty}\rVert\leq Ce^{-\alpha t}$ for $t\geq 0$.666Here
$\lVert.\rVert$ is any operator norm. Let $(v_{i})_{1\leq i\leq n}$ be a basis
of $\mathbb{C}^{n}$. Then there exists $(y_{i})_{1\leq i\leq n}$ a basis of
solutions for the ODE

(19) $y^{\prime}(t)=A(t)y(t),$

and $\kappa_{i}\in L^{\infty}(0,+\infty)^{n}$ such that for $t\geq 0$ we have

$y_{i}(t)=e^{tA_{\infty}}(v_{i}+e^{-\alpha t}\kappa_{i}(t)).$

Furthermore, if $\lambda\mapsto A(t,\lambda)$, $A_{\infty}$ and $v$ are
holomorphic with respect to an extra parameter $\lambda$, then
$\lambda\mapsto\kappa(\lambda,\cdot)$ is also holomorphic.

We delay the proof to the later appendix D, and apply this Lemma to the matrix
$A^{\text{sh}}(\lambda,x)$, for $\lambda$ to the right of
$\Sigma_{\text{}}(\mathcal{L}^{\text{sh}})$. There, eigenvalues of
$A^{\text{sh},\pm}$ are distinct and holomorphic. Since asymptotic matrices
are companion, they admit a basis
$\left(v_{i}^{\text{sh},\pm}(\lambda)\right)$ of holomorphic eigenvectors.
This allows to define two basis of solutions
$(\phi^{\text{sh},+}_{i}(\lambda,\cdot))_{1\leq i\leq 4}$ and
$(\phi^{\text{sh},-}_{i}(\lambda,\cdot))_{1\leq i\leq 4}$ for the ODE
$(\lambda-\mathcal{L}^{\text{sh}})\phi=0$, with exponential behavior at
$+\infty$ or $-\infty$:

$\phi_{i}^{\text{sh},\pm}=e^{\nu_{i}^{\text{sh},\pm}x}(v_{i}^{\text{sh},\pm}+e^{-r\lvert
x\rvert}\kappa_{i}^{\text{sh},\pm}(x)),$

where $\nu_{i}^{\text{sh},+}$ is the eigenvalue associated to
$v_{i}^{\text{sh},+}$, and $\kappa_{i}^{\text{sh},+}\in
L^{\infty}(\mathbb{R}_{+})$. We use similar notations for
$\mathcal{L}^{\text{kpp}}$.

######  Lemma 3.5.

The dispersion relation $\lambda-\mathcal{L}^{\text{kpp},\pm}(\nu)=0$ and
$\lambda-\mathcal{L}^{\text{sh},\pm}(\nu)=0$ are respectively second order and
fourth order polynomial in $\nu$. Their respective roots
$\nu_{i}^{\text{kpp},\pm}(\lambda)$ and $\nu_{i}^{\text{sh},\pm}(\lambda)$
have the following localization:

  1. (1)

(spectral gap away from the spectrum) There exists $\kappa_{2}>0$ such that
for all $\lambda\in\mathbb{C}$ with
$\operatorname*{\mathrm{Re}}{\lambda}\geq-2\eta$,

$\lvert\operatorname*{\mathrm{Re}}{\nu}\rvert\geq\kappa_{2},$

where $\nu$ stands for $\nu_{i}^{\text{sh},\pm}(\lambda)$ or
$\nu_{i}^{\text{kpp},-}(\lambda)$.

  2. (2)

(pinched double root at the origin) Let $I=(-\infty,0)\subset\mathbb{C}$, and
$V$ be a neighborhood of $0\in\mathbb{C}$. There exists $C$ positive such that
for $\lambda\in V\backslash I$, and $\lambda\to 0$,

$\nu_{1}^{\text{kpp},+}(\lambda)\sim-C\sqrt{\lambda},\hskip
40.00006pt\nu_{2}^{\text{kpp},+}(\lambda)\sim C\sqrt{\lambda}.$

  3. (3)

(elliptic operators) There exists $R$, $C$, $\theta$ positive constant such
that for $\lambda\in\mathbb{C}$ with $\lvert\lambda\rvert\geq R$ and
$\operatorname*{\mathrm{Re}}{\lambda}\leq-\theta\lvert\operatorname*{\mathrm{Im}}{\lambda}\rvert$,

$\lvert\operatorname*{\mathrm{Re}}{\nu_{i}^{\text{kpp},\pm}(\lambda)}\rvert\geq
C\lvert\lambda\rvert^{1/2},\hskip
40.00006pt\lvert\operatorname*{\mathrm{Re}}{\nu_{i}^{\text{sh},\pm}(\lambda)}\rvert\geq
C\lvert\lambda\rvert^{1/4}.$

###### Proof.

Items (1) and (2) are easily obtained. Item (3) relies on a scaling argument,
see [FH18, Lemma 3.1] for $\nu_{i}^{\text{kpp},\pm}$, the case of
$\nu_{i}^{\text{sh},\pm}$ adapts as follow. First set
$\psi(x):=\phi(x\lvert\lambda\rvert^{1/4})$, and apply lemma 3.4 to construct
solutions that are close to solutions of
$\lambda+\lvert\lambda\rvert\partial_{x}^{4}\psi=0$. In particular, the
asymptotic matrix eigenvalues for $\psi$ are
$\tilde{\nu}(\lambda):=z\sqrt[4]{\lambda}/\lvert\lambda\rvert$, where
$z\in\left\\{e^{i\frac{\pi}{4}},e^{-i\frac{\pi}{4}},e^{i\frac{3\pi}{4}},e^{-i\frac{3\pi}{4}}\right\\}$.
Notice that $\lvert\operatorname*{\mathrm{Re}}{z}\rvert=1/\sqrt{2}$, hence we
get $\lvert\operatorname*{\mathrm{Re}}{\tilde{\nu}}\rvert\geq C$, by
restricting to $\lambda$ of the form $re^{i\theta}$, with
$\lvert\theta\rvert\leq 3\pi/4$. Reverting back to the original variable, the
asymptotic matrix eigenvalues for $\phi$ satisfy
$\lvert\operatorname*{\mathrm{Re}}{\nu}\rvert\geq C\lvert\lambda\rvert^{1/4}$.
∎

In the following, we will compute determinant of several matrices, whose
columns depend on $\phi_{i}^{\text{sh},\pm}$ or $\phi_{i}^{\text{kpp},\pm}$.
Suppose we are given $n$ scalar functions $\phi_{1},\dots,\phi_{n}$ depending
on the space variable $x$. Then, we write

(20)
$\operatorname*{Det}(\phi_{1},\dots,\phi_{n}):=\det\begin{pmatrix}\phi_{1}&\cdots&\phi_{n}\\\
\vdots&&\vdots\\\
\partial_{x}^{n-1}\phi_{1}&\cdots&\partial_{x}^{n-1}\phi_{n}\end{pmatrix}=\det\left(\left(\partial_{x}^{j-1}\phi_{i}\right)_{1\leq
i,j\leq n}\right).$

where “$\det$” stands for the determinant. We also introduce the wronskian
function composed of all decaying solutions:

$\mathcal{W}^{\text{kpp}}:=\operatorname*{Det}(\phi_{1}^{\text{kpp},+},\phi_{2}^{\text{kpp},-}),\hskip
40.00006pt\mathcal{W}^{\text{sh}}:=\operatorname*{Det}(\phi_{1}^{\text{sh},+},\phi_{2}^{\text{sh},+},\phi_{3}^{\text{sh},-},\phi_{4}^{\text{sh},-}).$

For each $\lambda\in\mathbb{C}$, the function
$y\mapsto\mathcal{W}^{\text{kpp}}(\lambda,y)$ is either identically zero or
does not vanish, see lemma E.2. The particular value
$\mathcal{W}^{\text{kpp}}(\lambda,0)$ is often called the Evans function. When
$\lambda$ is such that $\phi_{1}^{\text{kpp},+}$ and $\phi_{2}^{\text{kpp},-}$
are decaying at $x\to\pm\infty$,777When $\lambda$ lies to the right of
$\Sigma_{\text{ess}}(\mathcal{L}^{\text{kpp}})$, this decay property hold.
$\mathcal{W}^{\text{kpp}}(\lambda,0)$ vanishes exactly when $\lambda$ is an
eigenvalue for $\mathcal{L}^{\text{kpp}}$, with same multiplicity. The same
goes for $\mathcal{W}^{\text{sh}}$.

###  3.3. Construction and estimations of the kernel for the resolvent
operator

First, we construct and control the Green function
$G_{\lambda}^{\text{kpp}}(x,y)$, 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 $+\infty$ and
$-\infty$. See the further remark 3.12.

######  Proposition 3.6.

Let $K\subset\mathbb{C}$ be a compact set to the right of
$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},-})$, and note $I=(-\infty,0)$ the
real negative axis that corresponds to the absolute spectrum of
$\mathcal{L}^{\text{kpp}}$. Then $\lambda\mapsto
G_{\lambda}^{\text{kpp}}(x,y)$ is holomorphic from $K\backslash I$ to
$\mathbb{R}$. Furthermore, there exists positive constants $\kappa_{1}$, $C$
such that for all $\lambda\in K\backslash I$ and $0\leq j\leq 2$, we have the
following. If $x\geq 0$ or $y\geq 0$,

$\left\lvert G_{\lambda}^{\text{kpp}}(x,y)\right\rvert\leq
Ce^{-\operatorname*{\mathrm{Re}}{\sqrt{\lambda}}\lvert x-y\rvert}\
\omega_{\kappa_{1},0}(x)\ \omega_{\kappa_{1},0}(y).$

If on the contrary, $x\leq 0$ and $y\leq 0$, then

$\left\lvert G_{\lambda}^{\text{kpp}}(x,y)\right\rvert\leq
Ce^{-\kappa_{1}\lvert x-y\rvert}.$

###### Proof.

We use the decaying ODE solution $\phi_{2}^{\text{kpp},-}$,
$\phi_{1}^{\text{kpp},+}$ constructed above. To keep notation as light as
possible, we write them $\phi^{-}$ and $\phi^{+}$ respectively. Similarly, we
note $\psi^{-}=\phi^{\text{kpp},-}_{1}$ and $\psi^{+}=\phi^{\text{kpp},+}_{2}$
the growing ODE solutions. Remind that both $(\phi^{-},\psi^{-})$ and
$(\phi^{+},\psi^{+})$ are basis for solution of the ODE
$(\lambda-\mathcal{L}^{\text{kpp}})\varphi=0$. The Green function expresses as

$G_{\lambda}^{\text{kpp}}(x,y)=\frac{1}{\mathcal{W}^{\text{kpp}}(\lambda,y)}\begin{cases}\phi^{-}(y)\phi^{+}(x)&\text{
if }y<x,\\\ \phi^{-}(x)\phi^{+}(y)&\text{ if }x<y.\end{cases}$

We begin by the case $x\geq 0$ or $y\geq 0$. For $\lambda$ to the right of
$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},-})$, the spatial eigenvalues are
such that the exponential obtained in [FH18, Lemma 3.2] are smaller than
$e^{-\lvert x-y\rvert\operatorname*{\mathrm{Re}}{\sqrt{\lambda}}}$. Then each
$\mathcal{O}$ leads to our weights
$\omega_{\kappa_{1},0}(x)\omega_{\kappa_{1},0}(y)$. The case $x\leq 0$ and
$y\leq 0$ 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 $\sqrt{\lambda}$. Due to $G_{\lambda}^{\text{kpp}}$ being a kernel,
Dirac delta correction terms appear when differentiating more than the order
of $\mathcal{L}^{\text{kpp}}$. However, they will be easily absorbed at
temporal level.

######  Proposition 3.7.

Let $K\subset\mathbb{C}$ be a compact set to the right of
$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},-})$, and note $I=(-\infty,0)$ the
real negative axis that corresponds to the absolute spectrum of
$\mathcal{L}^{\text{kpp}}$. Fix an integer $j\geq 1$. There exists a sum of
Dirac delta derivatives:

$P_{j}(\delta_{x=y})=\sum_{k=0}^{j-2}a_{k}^{j}(\lambda,y)\delta_{x=y}^{(k)}$

with coefficients $a_{k}$ holomorphic w.r.t. $\lambda\in K\backslash I$ and
bounded with respect to $y\in\mathbb{R}$, such that
$\lambda\mapsto\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(x,y)-P_{j}(\delta_{x=y})$
is holomorphic from $K\backslash I$ to $\mathbb{R}$. Furthermore, there exists
positive constants $\kappa_{1}$, $C$ such that for all $\lambda\in K\backslash
I$, we have the following. If $x\geq 0$ or $y\geq 0$,

$\left\lvert\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(x,y)-P_{j}(\delta_{x=y})\right\rvert\leq
C\lvert\lambda\rvert^{j/2}e^{-\operatorname*{\mathrm{Re}}{\sqrt{\lambda}}\lvert
x-y\rvert}\ \omega_{\kappa_{1},0}(x)\ \omega_{\kappa_{1},0}(y).$

If on the contrary, $x\leq 0$ and $y\leq 0$, then

$\left\lvert\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(x,y)-P_{j}(\delta_{x=y})\right\rvert\leq
C\lvert\lambda\rvert^{j/2}e^{-\kappa_{1}\lvert x-y\rvert}.$

###### Proof.

For the case $j=1$ no correction terms appear, and the proof is close to
[FH18]. We compute

$\partial_{x}G_{\lambda}^{\text{kpp}}(\cdot,y)=\frac{1}{\mathcal{W}^{\text{kpp}}_{\lambda}(y)}\begin{cases}\phi^{-}(y)\partial_{x}\phi^{+}&\text{
on }(y,+\infty)\\\ \phi^{+}(y)\partial_{x}\phi^{-}&\text{ on }(-\infty,y)\\\
\end{cases}$

Begin first with case $x>y$. When $x\geq 0$, we use
$\partial_{x}\phi^{+}(x)=-\sqrt{\lambda}e^{-\sqrt{\lambda}}(1+\kappa(x))$
rather than $\phi^{+}(x)=e^{-\sqrt{\lambda}}(1+\kappa(x))$, which gains us the
claimed $\lvert\lambda\rvert^{1/2}$ in comparison with the case $j=0$. When
$x\leq 0$, we project onto the $(\phi^{-},\psi^{-})$ basis:

$\partial_{x}\phi^{+}(x)=\frac{\operatorname*{Det}(\partial_{x}\phi^{+},\psi^{-})}{\operatorname*{Det}(\phi^{-},\psi^{-})}\phi^{-}(x)+\frac{\operatorname*{Det}(\phi^{-},\partial_{x}\phi^{+})}{\operatorname*{Det}(\phi^{-},\psi^{-})}\psi^{-}(x),$

we refer to the above mentioned reference for more details, or to the proof of
proposition 3.8, see appendix B. Since both $\phi^{-}$ and $\psi^{-}$ are
bounded by $e^{\kappa_{1}x}$, and using that
$\operatorname*{Det}(\phi^{-},\psi^{-})>0$ uniformly w.r.t. $\lambda$, we get
to $\lvert\partial_{x}\phi^{+}(x)\rvert\leq
C\lvert\lambda\rvert^{1/2}e^{\kappa_{1}x}$. This conclude the case $j=1$.

When $j=2$, we compute

$\partial_{x}^{2}G_{\lambda}^{\text{kpp}}(\cdot,y)=\delta_{y}+\frac{1}{\mathcal{W}^{\text{kpp}}_{\lambda}(y)}\begin{cases}\phi^{-}(y)\partial_{x}^{2}\phi^{+}&\text{
on }(y,+\infty),\\\ \phi^{+}(y)\partial_{x}^{2}\phi^{-}&\text{ on
}(-\infty,y),\\\ \end{cases}$

and similar computations as above show the claimed estimate for
$\partial_{x}^{2}G_{\lambda}^{\text{kpp}}(x,y)-\delta_{y}$.

For $j\geq 2$, the correction terms can be computed recursively:

$P_{j+1}(\delta_{x=y})=\delta_{x=y}\frac{\phi^{-}\partial_{y}^{j+1}\phi^{+}-\phi^{+}\partial_{y}^{j+1}\phi^{-}}{W_{\lambda}(y)}+\sum_{k=1}^{j-1}a_{k}^{j}(y)\delta_{x=y}^{(k)}$

Then, $\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(x,y)-P_{j}(\delta_{x=y})$ is
handled as above. ∎

We state a similar result for $G_{\lambda}^{\text{sh}}$. Since
$\mathcal{L}^{\text{sh}}$ is a fourth order operator, the proof slightly
differs. Due to its length, we present the proof in the later appendix B. Note
that [HH05] presents similar computations in a viscous shock front context.

######  Proposition 3.8.

Let $K\subset\mathbb{C}$ be a compact set to the right of
$\Sigma_{\text{}}(\mathcal{L}^{\text{sh},-}_{\theta})\cup\Sigma_{\text{}}(\mathcal{L}^{\text{sh},+})$,
for example
$K\subset\left\\{\operatorname*{\mathrm{Re}}(\lambda)\geq-2\eta\right\\}$.
Then $\lambda\mapsto G_{\lambda}^{\text{sh}}(\cdot,y)$ is holomorphic from $K$
to $H^{4}(\mathbb{R}\backslash\\{y\\})$, and there exists $C,\kappa_{2}>0$
such that the following holds. For all $x,y\in\mathbb{R}^{2}$, and $\lambda\in
K$:

$\left\lvert G_{\lambda}^{\text{sh}}(x,y)\right\rvert\leq
Ce^{-\kappa_{2}\lvert x-y\rvert}.$

Furthermore, if $x\geq 0$ or $y\geq 0$, then we have:

(21) $\left\lvert G_{\lambda}^{\text{sh}}(x,y)\right\rvert\leq
Ce^{-\kappa_{2}\lvert x-y\rvert}\omega_{\kappa_{2},0}(y).$

We can now obtain a statement on $G_{\lambda}^{\text{co}}$, similar to the
above proposition 3.6. We use the structure of the equation satisfied by
$G_{\lambda}^{\text{co}}$ to directly transfer the spatial decay from
$G_{\lambda}^{\text{kpp}}$ to $G_{\lambda}^{\text{co}}$.

######  Proposition 3.9.

Let $K\subset\mathbb{C}$ be a compact set to the right of
$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp},-})\cup\Sigma_{\text{}}(\mathcal{L}^{\text{sh},-}_{\theta})\cup\Sigma_{\text{}}(\mathcal{L}^{\text{sh},+})$,
and note $I=(-\infty,0)$. Then $\lambda\mapsto
G_{\lambda}^{\text{co}}(\cdot,y)$ is holomorphic from $K\backslash I$ to
$H^{2}(\mathbb{R}\backslash\\{y\\})$. Furthermore, there exists
$\kappa_{3},C>0$ such that for all $\lambda\in K\backslash I$, we have the
following. If $x\geq 0$ or $y\geq 0$, then:

$\lvert G_{\lambda}^{\text{co}}(x,y)\rvert\leq
Ce^{-\operatorname*{\mathrm{Re}}{\sqrt{\lambda}}\lvert x-y\rvert}\
\omega_{\kappa_{3},0}(x)\ \omega_{\kappa_{3},0}(y).$

If on the contrary $x\leq 0$ and $y\leq 0$,

$\lvert G_{\lambda}^{\text{co}}(x,y)\rvert\leq Ce^{-\kappa_{3}\lvert
x-y\rvert}.$

###### Proof.

Recall that

(22) $(\lambda-\mathcal{L}^{\text{kpp}})G_{\lambda}^{\text{co}}=\beta
G_{\lambda}^{\text{sh}}.$

Since $\lambda$ is located to the right of
$\Sigma_{\text{}}(\mathcal{L}^{\text{kpp}})$, the operator
$\lambda-\mathcal{L}^{\text{kpp}}$ is invertible, and we get

$G_{\lambda}^{\text{co}}(x,y)=\beta\int_{\mathbb{R}}G_{\lambda}^{\text{kpp}}(x,\tau)G_{\lambda}^{\text{sh}}(\tau,y)\mathrm{d}\tau.$

Indeed, we have constructed $G_{\lambda}^{\text{kpp}}(x,y)$ 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 $x\geq 0$
or $y\geq 0$. Then from proposition 3.6, where we neglect the
$\omega_{\kappa_{1},0}(\tau)$ terms, and from the refined estimate (21) in
proposition 3.8, we obtain that

$\lvert G_{\lambda}^{\text{co}}(x,y)\rvert\leq
C\int_{\mathbb{R}}e^{-\operatorname*{\mathrm{Re}}{\sqrt{\lambda}}\lvert
x-\tau\rvert}\omega_{\kappa_{1},0}(x)\
e^{-\kappa_{2}\lvert\tau-y\rvert}\omega_{\kappa_{2},0}(y)\ \mathrm{d}\tau.$

Using triangular inequality $\lvert x-\tau\rvert+\lvert\tau-y\rvert\geq\lvert
x-y\rvert$, we obtain

$\displaystyle\lvert G_{\lambda}^{\text{co}}(x,y)\rvert$ $\displaystyle\leq
Ce^{-\operatorname*{\mathrm{Re}}{\sqrt{\lambda}}\lvert
x-y\rvert}\omega_{\kappa_{1},0}(x)\
\omega_{\kappa_{2},0}(y)\int_{\mathbb{R}}e^{-(\kappa_{2}-\operatorname*{\mathrm{Re}}{\sqrt{\lambda}})\lvert\tau-y\rvert}\mathrm{d}\tau,$
$\displaystyle\leq Ce^{-\operatorname*{\mathrm{Re}}{\sqrt{\lambda}}\lvert
x-y\rvert}\omega_{\kappa_{1},0}(x)\ \omega_{\kappa_{2},0}(y),$

which is the first claimed estimate. Now when $x\leq 0$ and $y\leq 0$, we
first assume that $\kappa_{1}\leq\kappa_{2}$. We introduce
$\kappa_{3}:=\frac{1}{2}\kappa_{1}$, and obtain – again using triangular
inequality – that

$\lvert G_{\lambda}^{\text{co}}(x,y)\rvert\leq
C\int_{\mathbb{R}}e^{-\kappa_{1}\lvert
x-\tau\rvert}e^{-\kappa_{2}\lvert\tau-y\rvert}\mathrm{d}\tau\leq
Ce^{-\kappa_{3}\lvert x-y\rvert}\int_{\mathbb{R}}e^{-\kappa_{3}\lvert
x-\tau\rvert}e^{-(\kappa_{2}-\kappa_{3})\lvert\tau-y\rvert}\leq
Ce^{-\kappa_{3}\lvert x-y\rvert}.$

If on the contrary we have $\kappa_{1}>\kappa_{2}$, we set
$\kappa_{3}:=\frac{1}{2}\min(\kappa_{1},\kappa_{2})$ and obtain the same
inequality. ∎

To finish this section, we show bounds on $G_{\lambda}$ for large $\lambda$.
This is needed to transfer spectral behavior into temporal decay. Indeed,
$G_{\lambda}$ 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 $\lambda$ to the right of
$\Sigma_{\text{}}(\mathcal{L})$, with $\lvert\lambda\rvert\geq R$, and all
$x,y\in\mathbb{R}^{2}$, we have

(23) $\left\lvert
G_{\lambda}^{\text{kpp}}(x,y)\right\rvert\leq\frac{C}{\lvert\lambda\rvert^{1/2}}e^{-\lvert\lambda\rvert^{1/2}\lvert
x-y\rvert},\hskip 40.00006pt\left\lvert
G_{\lambda}^{\text{sh}}(x,y)\right\rvert\leq\frac{C}{\lvert\lambda\rvert^{3/4}}e^{-\lvert\lambda\rvert^{1/4}\lvert
x-y\rvert}.$

The first estimate still holds if $G^{\text{kpp}}$ is replaced by
$G^{\text{co}}$. In particular, for $i\in\\{\text{kpp},\text{sh},\text{co}\\}$

$\left\lVert
G_{\lambda}^{i}\right\rVert_{L^{\infty}(\mathbb{R}_{x},L^{1}(\mathbb{R}_{y}))}\leq\frac{C}{\lvert\lambda\rvert},\hskip
40.00006pt\left\lVert
G_{\lambda}^{i}\right\rVert_{L^{\infty}(\mathbb{R}_{y},L^{1}(\mathbb{R}_{x}))}\leq\frac{C}{\lvert\lambda\rvert}.$

###### Proof.

The last claimed inequalities are easily obtained from the first ones, and
implies that $\lVert G_{\lambda}^{i}\cdot
v\rVert_{L^{p}(\mathbb{R})}\leq\frac{C}{\lvert\lambda\rvert}\lVert
v\rVert_{L^{p}(\mathbb{R})}$ 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
$\int_{\Gamma}e^{\lambda t}G_{\lambda}\mathrm{d}\lambda$, with $\Gamma$ being
a ray that extends at infinity. ∎

###  3.4. Linear dynamic

We now convert spatial localization of spectral Green functions $G_{\lambda}$
into temporal decay in appropriate space using the inverse Laplace transform
and adequate integration contours.

######  Proposition 3.11.

There exists positive constants $C$, $\eta$ such that:

$\lvert\mathcal{G}_{t}^{\text{sh}}(x,y)\rvert\leq Ce^{-\eta
t}e^{-\kappa_{2}\lvert x-y\rvert}.$

In particular, for any $1\leq p\leq+\infty$, we have

(24) $\lVert\mathcal{G}_{t}^{\text{sh}}\cdot w\rVert_{L^{p}(\mathbb{R})}\leq
Ce^{-\eta t}\lVert w\rVert_{L^{p}(\mathbb{R})},$

where we have noted $(\mathcal{G}_{t}^{\text{sh}}\cdot
w)(x):=\int_{\mathbb{R}}\mathcal{G}_{t}^{\text{sh}}(x,y)w(y)\mathrm{d}y$.

###### Proof.

Since $\mathcal{L}^{\text{sh}}$ is sectorial with spectral gap $\eta$ from
propositions 3.1 and 3.3, this proof easily follows from the spatial
localization of $G_{\lambda}^{\text{sh}}$, see propositions 3.8 and 3.10. It
is enough to use inverse Laplace transform with a spectral contour in
$\left\\{\lambda\in\mathbb{C}:\operatorname*{\mathrm{Re}}(\lambda)\leq-\frac{\eta}{2}\right\\}$
for small $\lambda$, while restrict to a contour in
$\left\\{\lambda\in\mathbb{C}:\mathrm{Arg}(\lambda)\leq 3\pi/4\right\\}$ in
the region of large $\lambda$. Up to a change of notation
$\tilde{\eta}=\eta/2$, the proof is complete. ∎

######  Remark 3.12.

We now state a similar result for $\mathcal{G}_{t}^{\text{kpp}}$ and
$\mathcal{G}_{t}^{\text{co}}$. Remark that in [FH18, Proposition 4.1], the
situation is similar except that the weight $\omega_{*}$ was exponentially
decaying both at $+\infty$ and $-\infty$. In our setting, since
$\omega_{*}(x)\to+\infty$ when $x\to-\infty$, 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 $\omega_{\kappa_{1},0}(y)$ obtained for both
$G_{\lambda}^{\text{kpp}}$ and $G_{\lambda}^{\text{co}}$ in the above
propositions 3.6 and 3.9. This allows to bypass the weight at $-\infty$ 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 $\lvert
x-y\rvert$, which transfers into $t^{-3/2}$ decay. To show stronger decay, it
is necessary to keep more powers of $\rho_{*}$, thus weakening the weight. In
our situation, more powers of $\rho_{*}$ would result in the translation
eigenvalue $\lambda=0$ appearing again, due to
$\frac{q_{*}^{\prime}}{\omega_{\text{kpp}}\rho_{*}^{2}}\sim\frac{1}{x}$ at
$x\to+\infty$. To guess what bound may be obtained, one can differentiate
$\frac{1}{t^{1/2}}e^{-\frac{x^{2}}{t}}$ (resp.
$\frac{x}{t^{3/2}}e^{-\frac{x^{2}}{t}}$) in the case where $\lvert
x-y\rvert\geq Kt$ (resp. $\lvert x-y\rvert\leq Kt$)

######  Proposition 3.14.

Recall that $\rho_{*}$ is defined by (10). Fix $0\leq j\leq 1$ and restrict to
$t>1$. Then there exists positive constants $C$, $K$ and $\kappa_{1}$ such
that the following two pointwise estimate hold. If $\lvert x-y\rvert\geq Kt$,
then

$\left\lvert\partial_{x}^{j}\mathcal{G}_{t}^{\text{kpp}}(x,y)\right\rvert\leq\frac{C}{t^{\frac{j+1}{2}}}\left(1+\frac{\lvert
x-y\rvert}{\sqrt{t}}\right)^{j}e^{-\kappa_{1}\frac{\lvert
x-y\rvert^{2}}{t}}h(x,y).$

If on the contrary $\lvert x-y\rvert\leq Kt$, then

$\left\lvert\partial_{x}^{j}\mathcal{G}_{t}^{\text{kpp}}(x,y)\right\rvert\leq\frac{1+\lvert
x-y\rvert^{1+\lfloor j/2\rfloor}}{t^{3/2+\lfloor
j/2\rfloor}}e^{-\kappa_{1}\frac{\lvert x-y\rvert^{2}}{t}}h(x,y).$

In the above expressions, $x,y\mapsto h(x,y)/\rho_{*}(y)^{2}$ is integrable in
$y$ uniformly in $x$:

$0\leq h(x,y)\leq C\begin{cases}\omega_{\kappa_{1},0}(x)\
\omega_{\kappa_{1},0}(y)&\text{if }x\geq 0\text{ or }y\geq 0,\\\
e^{-\kappa_{1}\lvert x-y\rvert}&\text{if }x\leq 0\text{ and }y\leq
0.\end{cases}$

In particular, for $0\leq j\leq 1$, for all $1\leq q\leq p\leq+\infty$, all
$x,y\in\mathbb{R}$ and $t\geq 1$,

(25) $\left\lVert\frac{\partial_{x}^{j}\mathcal{G}_{t}^{\text{kpp}}\cdot
w}{\rho_{*}}\right\rVert_{L^{p}(\mathbb{R})}\leq\frac{C}{t^{\frac{3}{2}-\frac{1}{2p}}}\left\lVert\rho_{*}^{3}w\right\rVert_{L^{q}(\mathbb{R})}.$

All the above still hold when $\mathcal{G}_{t}^{\text{kpp}}$ and $\kappa_{1}$
are replaced by $\mathcal{G}_{t}^{\text{co}}$ and $\kappa_{3}$ respectively.

######  Proposition 3.15.

There exists positive constants $C$, $\kappa_{1}$ and a constant $c\neq 0$
such that for $t>1$, the two point-wise estimate of the above proposition hold
with $j=2$ for

$\left\lvert\partial_{x}^{2}\mathcal{G}_{t}^{\text{kpp}}(x,y)-ce^{-t}\delta_{x=y}\right\rvert.$

Hence, for all $1\leq q\leq p\leq+\infty$, all $x,y\in\mathbb{R}$ and $t\geq
1$,

(26) $\left\lVert\frac{\partial_{x}^{2}\mathcal{G}_{t}^{\text{kpp}}\cdot
w}{\rho_{*}}\right\rVert_{L^{p}(\mathbb{R})}\leq\frac{C}{t^{\frac{3}{2}-\frac{1}{2p}}}\left\lVert\rho_{*}^{3}w\right\rVert_{L^{q}(\mathbb{R})}+Ce^{-t}\lVert
w\rVert_{L^{p}(\mathbb{R})}.$

All the above still hold when $\mathcal{G}_{t}^{\text{kpp}}$ and $\kappa_{1}$
are replaced by $\mathcal{G}_{t}^{\text{co}}$ and $\kappa_{3}$ 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 $\lvert x-y\rvert\geq Kt$, then
$\left\lvert\mathcal{G}_{t}^{\text{kpp}}(x,y)\right\rvert\leq\frac{C}{\sqrt{t}}e^{-\kappa\frac{\lvert
x-y\rvert^{2}}{t}}$,

  * •

else, $\lvert x-y\rvert\leq Kt$ implies
$\left\lvert\mathcal{G}_{t}^{\text{kpp}}(x,y)\right\rvert\leq C\frac{1+\lvert
x-y\rvert}{t^{3/2}}e^{-\kappa\frac{\lvert x-y\rvert^{2}}{t}}$.

Having kept the extra exponential localization –
$\omega_{\kappa_{1},0}(x)\omega_{\kappa_{1},0}(y)$ when $x\geq 0$ or $y\geq 0$
– in our estimates of $G_{\lambda}^{\text{kpp}}(x,y)$, the proof of [FH18]
straightforwardly adapt with this extra decay. The same goes for the other
case $x\leq 0$ and $y\leq 0$.

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

$\partial_{x}^{j}\mathcal{G}_{t}^{\text{kpp}}(\cdot,y)=\frac{1}{2i\pi}\int_{\Lambda}e^{\lambda
t}\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(\cdot,y)\mathrm{d}\lambda,$

We first treat the long time scenario: $\lvert x-y\rvert\leq Kt$. Using
estimate from proposition 3.7, there exists a function $H$ holomorphic w.r.t.
$\lambda$ outside $I:=(-\infty,0)$, such that $\lVert
H(\lambda,\cdot,\cdot)\rVert_{L^{\infty}(\mathbb{R}_{x},L^{1}(\mathbb{R}_{y}))}<+\infty$
and

$\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(x,y)-P_{j}(\delta_{x=y})=\sqrt{\lambda}^{j}e^{-\sqrt{\lambda}\lvert
x-y\rvert}H(\lambda,x,y).$

Setting $\rho=\kappa\frac{\lvert x-y\rvert}{t}$ with $\kappa>0$ to be fixed
later. Close to the origin, we use a parabolic contour:

$\Lambda_{1}(\rho)=\left\\{\lambda(\xi)=(\rho+i\xi)^{2}:\xi\in(-\xi_{0},\xi_{0})\right\\},$

and obtain with $\tilde{H}(\xi,x,y,t):=\frac{1}{\pi}e^{i(2\rho\xi t-\xi\lvert
x-y\rvert)}H_{\lambda(\xi)}(x,y)$ that

$\frac{1}{2i\pi}\int_{\Lambda_{1}}e^{\lambda
t}(\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(\cdot,y)-P_{j}(\delta_{x=y}))\mathrm{d}\lambda=e^{\rho^{2}t-\rho\lvert
x-y\rvert}\int_{-\xi_{0}}^{\xi_{0}}e^{-\xi^{2}t}\tilde{H}(\xi,x,y,t)(\rho+i\xi)^{j+1}\mathrm{d}\xi,$

Due to $\mathcal{L}^{\text{kpp}}$ having real coefficients, we have
$G_{\bar{\lambda}}^{\text{kpp}}=\overline{G_{\lambda}^{\text{kpp}}}$, and the
same goes for $H$. Since $\lambda(-\xi)=\overline{\lambda(\xi)}$ we deduce
that the term below the integral in the right hand side of the above equation
satisfies $z(-\xi)=\overline{z(\xi)}$. As a consequence, we compute

$\displaystyle\frac{1}{2i\pi}\int_{\Lambda_{1}}e^{\lambda
t}(\partial_{x}^{j}G_{\lambda}^{\text{kpp}}(\cdot,y)-P_{j}(\delta_{x=y}))\mathrm{d}\lambda={}$
$\displaystyle e^{(\kappa^{2}-\kappa)\frac{\lvert
x-y\rvert^{2}}{t}}\int_{-\xi_{0}}^{\xi_{0}}e^{-\xi^{2}t}\operatorname*{\mathrm{Re}}\left(\tilde{H}(\xi,x,y,t)(\rho+i\xi)^{j+1}\right)\mathrm{d}\xi,$
$\displaystyle={}$ $\displaystyle e^{(\kappa^{2}-\kappa)\frac{\lvert
x-y\rvert^{2}}{t}}\int_{-\xi_{0}}^{\xi_{0}}e^{-\xi^{2}t}\left(\operatorname*{\mathrm{Re}}{\tilde{H}(\xi,x,y,t)}\operatorname*{\mathrm{Re}}{(\rho+i\xi)^{j+1}}\right.$
$\displaystyle{}\left.-\operatorname*{\mathrm{Im}}{\tilde{H}(\xi,x,y,t)}\operatorname*{\mathrm{Im}}{(\rho+i\xi)^{j+1}}\right)\mathrm{d}\xi.$

First develop $(\rho+i\xi)^{j+1}=\sum_{k=0}^{j+1}{j+1\choose
k}(i\xi)^{k}\rho^{j+1-k}$, and notice that $\rho\leq\kappa K$. Then for $k$
even, we bound

$\rho^{j+1-k}\leq\kappa^{1+j-k}K^{j-\lfloor j/2\rfloor-k/2}\left(\frac{\lvert
x-y\rvert}{t}\right)^{1+\lfloor j/2\rfloor-k/2},$

with $j-\lfloor j/2\rfloor-k/2\geq 0$. Finally, each power of $\xi$ provide a
power of $1/\sqrt{t}$ when integrated:

$0\leq\int_{-\xi^{0}}^{\xi_{0}}\xi^{k}e^{-\xi^{2}t}\mathrm{d}\xi\leq\frac{1}{t^{(k+1)/2}}\int_{\mathbb{R}}\left(\xi\sqrt{t}\right)^{k}e^{-\xi^{2}t}\sqrt{t}\mathrm{d}\xi\leq\frac{C_{k}}{t^{(k+1)/2}}.$

Hence, using that $\operatorname*{\mathrm{Re}}{\tilde{H}}$ is bounded, we
compute that

$\left\lvert\int_{-\xi_{0}}^{\xi_{0}}e^{-\xi^{2}t}\operatorname*{\mathrm{Re}}{\tilde{H}(\xi,x,y,t)}\operatorname*{\mathrm{Re}}{(\rho+i\xi)^{j+1}}\mathrm{d}\xi\right\rvert\leq
C_{j}\frac{1+\lvert x-y\rvert^{1+\lfloor j/2\rfloor}}{t^{3/2+\lfloor
j/2\rfloor}}.$

The terms with $k$ odd, corresponding to the imaginary part of
$(\rho+i\xi)^{j+1}$, are bounded similarly:

$\rho^{j+1-k}\leq\kappa^{j+1-k}K^{j-\lfloor
j/2\rfloor-\frac{k-1}{2}}\left(\frac{\lvert x-y\rvert}{t}\right)^{1+\lfloor
j/2\rfloor-\frac{k+1}{2}}.$

Since $\xi\mapsto\operatorname*{\mathrm{Im}}(\tilde{H}(\xi))$ is odd, it
provide an extra power of $\xi$:
$\lvert\operatorname*{\mathrm{Im}}{\tilde{H}(\xi)}\rvert\leq\xi C$. Hence we
compute that

$\left\lvert\int_{-\xi_{0}}^{\xi_{0}}e^{-\xi^{2}t}\operatorname*{\mathrm{Im}}{\tilde{H}(\xi,x,y,t)}\operatorname*{\mathrm{Im}}{(\rho+i\xi)^{j+1}}\mathrm{d}\xi\right\rvert\leq
C_{j}\frac{1+\lvert x-y\rvert^{1+\lfloor j/2\rfloor}}{t^{3/2+\lfloor
j/2\rfloor}}.$

In the short time scenario: $\lvert x-y\rvert\geq Kt$, we assume $K$ is large
enough so that the parabolic contour $\Lambda_{1}$ is far enough from the
origin, which in turn guarantees that large $\lambda$ estimates from
proposition 3.10 applies. The computations are then similar to above, with
$(\rho+i\xi)^{j+1}$ replaced by

$(\rho+i\xi)^{j}=\sum_{k=0}^{j}{j\choose k}(i\xi)^{k}\rho^{j-k}.$

We now use $\rho=\kappa\frac{\lvert x-y\rvert}{t}$, to obtain that the $k$-th
term in the above some contribute as

${j\choose
k}(i\xi)^{k}\rho^{j-k}=C\sqrt{t}\left(\xi\sqrt{t}\right)^{k}\times\frac{1}{t^{\frac{j+1}{2}}}\left(\frac{\lvert
x-y\rvert}{\sqrt{t}}\right)^{j-k},$

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
$\lambda\in\mathbb{C}$, thus we can move the integration contour to the left
of the imaginary axis, and obtain:

$\frac{1}{2i\pi}\int_{\Lambda}e^{\lambda
t}\delta_{y=x}\mathrm{d}\lambda=ce^{-t}\delta_{x=y}.$

Turning now to the $L^{q}$ \- $L^{p}$ estimate (25), we use the extra
exponential localization we have gained in $h$. First when $\lvert
x-y\rvert\leq Kt$, we use both that $(1+\lvert
x-y\rvert)h(x,y)\leq\rho_{*}(x)\rho_{*}(y)h(x,y)$ and that $(\frac{\lvert
x-y\rvert}{t})^{\lfloor j/2\rfloor}\leq C$ to obtain that

$\int_{\lvert x-y\rvert\leq
Kt}\frac{\partial_{x}^{j}\mathcal{G}_{t}^{\text{kpp}}(x,y)w(y)}{\rho_{*}(x)}\mathrm{d}y\leq\frac{\lVert\rho_{*}^{3}w\rVert_{L^{\infty}(\mathbb{R})}}{t^{3/2}}\int_{\lvert
x-y\rvert\leq Kt}\frac{h(x,y)}{\rho_{*}(y)^{2}}\mathrm{d}y.$

The integral on the right hand side is finite. In the other case, we rather
use that $K\leq\frac{1}{\sqrt{t}}\frac{\lvert x-y\rvert}{\sqrt{t}}$, and
change variables to gain as much decay as needed:

$\displaystyle\int_{\lvert x-y\rvert\geq
Kt}\frac{\partial_{x}^{j}\mathcal{G}_{t}^{\text{kpp}}(x,y)w(y)}{\rho_{*}(x)}\mathrm{d}y$
$\displaystyle\leq\frac{\lVert\rho_{*}w\rVert_{L^{\infty}(\mathbb{R})}}{t^{\frac{j+3}{2}}}\int_{\lvert
x-y\rvert\geq Kt}\left(1+\frac{\lvert
x-y\rvert}{\sqrt{t}}\right)^{j+2}e^{-\kappa_{1}\frac{\lvert
x-y\rvert^{2}}{t}}\mathrm{d}y,$
$\displaystyle\leq\frac{\lVert\rho_{*}^{3}w\rVert_{L^{\infty}(\mathbb{R})}}{t^{3/2}}\int_{\mathbb{R}}(1+z)^{j+2}e^{-\kappa_{1}z^{2}}\mathrm{d}z.$

We have shown $(p,q)=(+\infty,+\infty)$. Then $(p,q)=(1,1)$ reads

$\displaystyle\iint_{\lvert x-y\rvert\leq
Kt}\left\lvert\frac{\mathcal{G}_{t}^{\text{kpp}}(x,y)w(y)}{\rho_{*}(x)}\right\rvert\mathrm{d}y\mathrm{d}x$
$\displaystyle\leq\frac{C}{t}\int_{\mathbb{R}}\left\lVert
x\mapsto\frac{1}{\sqrt{t}}e^{-\kappa\frac{\lvert
x-y\rvert^{2}}{t}}\right\rVert_{L^{1}(\mathbb{R})}\lvert\rho_{*}(y)w(y)\rvert\mathrm{d}y,$
$\displaystyle\leq\frac{C}{t}\lVert\rho_{*}w\rVert_{L^{1}(\mathbb{R})},$

together with
$\lVert\rho_{*}w\rVert_{L^{p}(\mathbb{R})}\leq\lVert\rho_{*}^{3}w\rVert_{L^{p}(\mathbb{R})}$.
The other term $\iint_{\lvert x-y\rvert\geq Kt}$ bound similarly and decay as
fast as needed. The case $(p,q)=(+\infty,1)$ proves similarly. Interpolation
between those three cases leads to the $1\leq q\leq p\leq+\infty$ 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 $\rho_{*}(x)^{2}\geq 1$ is removed. ∎

######  Remark 3.16.

Using standard parabolic regularity, we obtain for $0<t\leq 1$ and $1\leq
p\leq+\infty$ that:

$\left\lVert\mathcal{G}_{t}^{\text{kpp}}\cdot
w\right\rVert_{W^{j,p}(\mathbb{R})}\leq C\left\lVert
w\right\rVert_{W^{j,p}(\mathbb{R})}.$

Then from $\inf(1,\frac{1}{t})\leq\frac{2}{1+t}$, estimates (25)-(26) hold
true for $t\geq 0$, $p=q$, and $\frac{1}{t}$ replaced by $\frac{1}{1+t}$.

######  Remark 3.17.

When $p=+\infty$, inequality (25) is the optimal $t^{-3/2}$ decay. It does not
seem possible to obtain estimates with $p<q$, since $\mathcal{G}_{t}(x,y)$ can
not absorb both integral in $x$ and $y$. Remark that $\rho_{*}^{3}$ is
arbitrary, and could be replaced by $\rho_{*}h$ with
$\int_{0}^{+\infty}\frac{\mathrm{d}y}{h(y)}<+\infty$.

###  3.5. Nonlinear dynamic

Here we finally prove proposition 2.3, _i.e._ that
$U=(u_{1},u_{2})^{\mathrm{T}}$ decays in time provided
$V=(v_{1},v_{2})^{\mathrm{T}}$ is bounded. Throughout this section, we will
assume that there exists $C_{\mu}>0$ such that for all $0\leq t\leq t_{V}$,

(27)
$\left\lVert\frac{\omega_{\text{sh}}}{\rho_{*}}U(t)\right\rVert_{L^{\infty}(\mathbb{R})}=\lVert
V(t)\rVert_{L^{\infty}(\mathbb{R})}\leq C_{\mu}.$

To lighten notations, we may note $C_{\mu}$ instead of $CC_{\mu}$ when $C$ is
a constant not depending on $C_{\mu}$. Remind that $V(t)$ and
$\frac{\omega_{\text{sh}}}{\rho_{*}}U(t)$ differs only by a change of space
variable, so that the first equality is automatically satisfied.

######  Proposition 3.18.

For all $1<p\leq+\infty$, there exists positive constants $C_{\text{stab}}$,
$\eta$, $\delta$, such that if $C_{\mu}\leq\delta$, then the solution $U$ of
(8) emanating from initial condition $U_{0}$ – with $U_{0}\rho_{*}^{3}\in
L^{p}(\mathbb{R})$ – is defined for $0\leq t\leq t_{V}$, and satisfies

$\left\lVert\frac{u_{1}(t,\cdot)}{\rho_{*}}\right\rVert_{L^{p}(\mathbb{R})}\leq
C_{\text{stab}}\frac{\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{p}(\mathbb{R})}}{(1+t)^{\frac{3}{2}-\frac{1}{2p}}},\hskip
40.00006pt\lVert u_{2}(t,\cdot)\rVert_{L^{p}(\mathbb{R})}\leq
C_{\text{stab}}e^{-\frac{\eta}{2}t}\lVert U_{0}\rVert_{L^{p}(\mathbb{R})}.$

Furthermore, $C_{\text{stab}}$ is independent of $C_{\mu}$.

###### Proof.

In this proof, we abbreviate $K_{p}:=\lVert
U_{0}\rho_{*}^{3}\rVert_{L^{p}(\mathbb{R})}$. 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 $\omega_{\text{sh}}$. Then nonlinear terms are seen as linear
ones.

We fix $0<\tilde{\eta}<\eta$ where $\eta$ is given by the exponential decay of
$u_{2}$ at linear level: see (24) in proposition 3.11. For $1<p\leq+\infty$,
we note $\sigma(p):=\frac{3}{2}-\frac{1}{2p}>1$. Then we note

$\Theta_{1}(t):=\sup_{s\in(0,t)}(1+s)^{\sigma(p)}\left\lVert\frac{u_{1}(s)}{\rho_{*}}\right\rVert_{L^{p}(\mathbb{R})}\hskip
40.00006pt\Theta_{2}(t):=\sup_{s\in(0,t)}e^{\tilde{\eta}s}\left\lVert
u_{2}(s)\right\rVert_{L^{p}(\mathbb{R})},$

together with $\Theta(t)=\max(\Theta_{1}(t),\Theta_{2}(t))$, and show that
$\Theta$ is bounded in time. Using (27), we can absorb the unbounded
$\omega_{\text{sh}}$ that comes from the nonlinear terms. We have both

$\lvert\mathcal{N}_{1}(U)\rvert\leq\lvert 3\alpha
q_{*}\rvert\lvert\omega_{\text{sh}}u_{1}\rvert\lvert\omega_{\text{kpp}}u_{1}\rvert+\lvert\alpha\omega_{\text{kpp}}\rvert\lvert\omega_{\text{sh}}u_{1}\rvert^{2}\lvert\omega_{\text{kpp}}u_{1}\rvert\leq
C_{\mu}\lvert\omega_{\text{kpp}}u_{1}\rvert$

and

$\left\lvert\mathcal{N}_{2}(U)\right\rvert\leq
C\lvert\omega_{\text{sh}}u_{1}\rvert\lvert\omega_{\text{kpp}}u_{2}\rvert+C\lvert
u_{2}\omega_{\text{sh}}\rvert^{2}\lvert\omega_{\text{kpp}}u_{2}\rvert\leq
C_{\mu}\lvert\omega_{\text{kpp}}u_{2}\rvert.$

In particular, $U\mapsto\mathcal{N}(U)$ is globally Lipschitz, hence the
solution $U$ is defined in $L^{p}(\mathbb{R})$ for times $0\leq t\leq T$, see
[Hen81]. We can now use the $\omega_{\text{kpp}}$ factor to gain a polynomial
weight – remind that to apply the linear estimates (25), we need one
$\rho_{*}$ to control the Green kernel, and two extra ones to make the
integral w.r.t. $y$ converge: $\omega_{\text{kpp}}\leq\frac{C}{\rho_{*}^{4}}$
– which leads to

$\left\lvert\mathcal{N}_{1}(s,y)\right\rvert\leq\frac{C_{\mu}}{\rho_{*}(y)^{3}(1+s)^{\sigma(p)}}\left\lvert\frac{(1+s)^{\sigma(p)}}{\rho_{*}(y)}u_{1}(s,y)\right\rvert,$

hence

$\left\lVert\rho_{*}^{3}\mathcal{N}_{1}(s)\right\rVert_{L^{p}(\mathbb{R})}\leq\frac{C_{\mu}}{(1+s)^{\sigma(p)}}\Theta_{1}(t).$

Similarly we get to

$\left\lVert\rho_{*}^{3}\mathcal{N}_{2}(U(s))\right\rVert_{L^{p}}\leq
C_{\mu}e^{-\tilde{\eta}s}\left\lVert
e^{\tilde{\eta}s}u_{2}(s,\cdot)\right\rVert_{L^{p}}\leq
C_{\mu}e^{-\tilde{\eta}s}\Theta_{2}(t).$

The Duhamel’s formula decomposes into

$u_{2}(t,x)=\int_{\mathbb{R}}\mathcal{G}_{t}^{\text{sh}}(x,y)u_{2}^{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{R}}\mathcal{G}_{t-s}^{\text{sh}}(x,y)\mathcal{N}_{2}(s,y)\mathrm{d}y\mathrm{d}s.$
$u_{1}(t,x)=\int_{\mathbb{R}}\mathcal{G}_{t}^{\text{kpp}}u_{1}^{0}+\mathcal{G}_{t}^{\text{co}}u_{2}^{0}\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{R}}\mathcal{G}_{t-s}^{\text{kpp}}\mathcal{N}_{1}+\mathcal{G}_{t-s}^{\text{co}}\mathcal{N}_{2}\mathrm{d}y\mathrm{d}s.$

We use Minkowski inequality as well as the injection
$L^{1}((0,t),L^{p}(\mathbb{R}))\subset L^{p}(\mathbb{R},L^{1}(0,t))$ – see
lemma E.3 – onto Duhamel’s formula on $u_{2}$ to obtain that

$\lVert
u_{2}(t)\rVert_{L^{p}(\mathbb{R})}\leq\lVert\mathcal{G}_{t}^{\text{sh}}\cdot
u_{2}^{0}\rVert_{L^{p}}+\int_{0}^{t}\lVert\mathcal{G}_{t-s}^{\text{sh}}\cdot\mathcal{N}(s,\cdot)\rVert_{L^{p}}\mathrm{d}s.$

Now using the linear estimate (24) from proposition 3.11 and the above
nonlinear estimate, we obtain:

$\lVert e^{\tilde{\eta}t}u_{2}(t)\rVert_{L^{p}(\mathbb{R})}\leq C\lVert
u_{2}^{0}\rVert_{L^{p}(\mathbb{R})}+C_{\mu}\Theta_{2}(t)\int_{0}^{t}e^{-(\eta-\tilde{\eta})(t-s)}\mathrm{d}s,$

which after integration and taking the supremum on $t\in(0,\tau)$ reads

$\Theta_{2}(\tau)\leq\frac{C\lVert
u_{2}^{0}\rVert_{L^{p}(\mathbb{R})}}{1-\frac{C_{\mu}}{\eta-\tilde{\eta}}},$

assuming that the denominator is positive. By imposing
$C_{\mu}\leq\delta\leq\frac{\eta-\tilde{\eta}}{2}$, this condition is
fulfilled and we recover $\Theta_{2}(t)\leq 2C\lVert
u_{2}^{0}\rVert_{L^{p}(\mathbb{R})}$ for $0\leq t\leq T$. Remind that $C$ does
not depend on $C_{\mu}$. 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:

$\displaystyle(1+t)^{\sigma(p)}\left\lVert\frac{u_{1}(t,\cdot)}{\rho_{*}}\right\rVert_{L^{p}}\leq{}$
$\displaystyle C\lVert U_{0}\rho_{*}^{3}\rVert_{L^{p}(\mathbb{R})}$
$\displaystyle+\int_{0}^{t}\frac{(1+t)^{\sigma(p)}}{(1+t-s)^{\sigma(p)}}\left(\left\lVert\rho_{*}^{3}\,\mathcal{N}_{1}(s,\cdot)\right\rVert_{L^{p}(\mathbb{R})}+\left\lVert\rho_{*}^{3}\,\mathcal{N}_{2}(s,\cdot)\right\rVert_{L^{p}(\mathbb{R})}\right)\mathrm{d}s,$
$\displaystyle\leq{}$ $\displaystyle
CK_{p}+C_{\mu}\Theta(t)\int_{0}^{t}\left(\frac{(1+t)}{(1+t-s)(1+s)}\right)^{\sigma(p)}\mathrm{d}s.$

Standard computations on integral – see lemma E.4 and remark that
$\sigma(p)>1$ – lead to

$\Theta_{1}(t)\leq CK_{p}+C_{\mu}\Theta(t).$

Summing the above $\Theta_{2}(t)\leq 2CK_{p}$, we get to
$\Theta(t)\leq\frac{CK_{p}}{1-C_{\mu}}$ as soon as $1-C_{\mu}>0$. By assuming
that $C_{\mu}\leq\delta\leq\frac{1}{2}$, we get to $\Theta(t)\leq CK_{p}$,
which implies the claimed temporal decay for $u_{1}$. Remark that
$C_{\text{stab}}:=C$ does not depends on $C_{\mu}$. ∎

##  4\. Perturbations in partially weighted space are bounded in time

###  4.1. Mode filters

Since part of the spectrum of $\mathcal{T}^{-}$ is unstable, the dynamic for
$V$ is unstable at linear level. We count on the nonlinear terms to control
$V$ for large bounded times $t\leq\frac{T}{\mu}$. To do so, we mostly follow
the approach of Guido Schneider, see [Sch94b]. 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 $\mu$.
Let us use a smooth, positive cut-off function $\chi$:

(28) $\chi_{\text{c}}(x):=\begin{cases}1&\text{if }x\in
I_{\text{c}}:=[-\frac{9}{8},-\frac{7}{8}]\cup[\frac{7}{8},\frac{9}{8}],\\\
0&\text{if }x\notin I_{\text{c}}+B(0,\frac{1}{8}),\end{cases}$

with $B(x,r)$ the ball centered at $x$ with radius $r$. We then work in
Fourier space: the matrix $\hat{\mathcal{T}}^{-}(\xi)$ has two eigenvalues

$\lambda_{\text{c}}(\xi)=-(1-\xi^{2})+\mu,\hskip
40.00006pt\lambda_{\text{s}}(\xi)=-d\xi^{2}-2\alpha,$

with associated eigenvectors $\varrho_{i}$, for
$i\in\left\\{\text{s},\text{c}\right\\}$. We note
$\varPi_{i}(\xi)\hat{V}=\langle\hat{V},\varrho_{i}^{*}(\xi)\rangle\varrho_{i}(\xi)$
the respective parallel projections onto each of the eigenspaces, and we
separate critical from stable frequencies:

$\varPi_{\text{c}}V=\mathcal{F}^{-1}\left(\xi\mapsto\chi_{\text{c}}(\xi)\left\langle\hat{V}(\xi),\varrho_{\text{c}}^{*}(\xi)\right\rangle\varrho_{\text{c}}(\xi)\right),\hskip
40.00006pt\varPi_{\text{s}}V=V-\varPi_{\text{c}}V$

Remark that $\chi_{\text{c}}^{2}\neq\chi_{\text{c}}$, so that neither
$\varPi_{\text{c}}$ or $\varPi_{\text{s}}$ are projections. However, using
$I_{\text{c}}^{h}:=[-\frac{5}{4},-\frac{3}{4}]\cup[\frac{3}{4},\frac{5}{4}]$
and
$I_{s}^{h}:=[-\frac{17}{16},-\frac{15}{16}]\cup[\frac{15}{16},\frac{17}{16}]$,
we define $\chi_{\text{c}}^{h}$ and $\chi_{s}^{h}$ with respective support
$I_{\text{c}}^{h}+B(0,\frac{1}{4})$ and $I_{s}^{h}+B(0,\frac{1}{16})$, in a
similar way as (28). Then with

$\varPi_{\text{c}}^{h}V:=\mathcal{F}^{-1}(\chi_{\text{c}}^{h}\varPi_{1}\hat{V}),\hskip
40.00006pt\varPi_{\text{s}}^{h}V:=\mathcal{F}^{-1}((1-\chi_{\text{s}}^{h})\varPi_{1}\hat{V})+\mathcal{F}^{-1}(\varPi_{2}\hat{V}),$

we have $\varPi^{h}_{i}\varPi_{i}=\varPi_{i}$ for
$i\in\\{\text{c},\text{s}\\}$. Ultimately, we will use a scalar “projection”
onto half of the critical Fourier modes, to select only frequencies close to
$\xi=1$:

(29)
$\pi_{1}^{h}V=\mathcal{F}^{-1}\left(\xi\mapsto\chi_{\text{c}}^{h}(\xi)\mathbf{1}_{\xi>0}\frac{1}{\langle\rho_{\text{c}}(1),\rho_{\text{c}}^{*}(\xi)\rangle}\langle\hat{V}(\xi),\varrho_{\text{c}}^{*}(\xi)\rangle\right).$

Such decompositions are well-behaved with Fourier transform. However, we need
to measure objects with $L^{\infty}(\mathbb{R})$ 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 [Sch94c]. We first define a
weight

$\rho_{\mathrm{u,l}}(x):=\frac{1}{1+x^{2}}.$

Then, we say that $u\in L^{2}_{\mathrm{u,l}}(\mathbb{R})$ if

$\lVert
u\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}:=\sup_{y\in\mathbb{R}}\lVert\rho_{\mathrm{u,l}}(\cdot-y)u\rVert_{L^{2}(\mathbb{R})}<+\infty.$

Similarly, we define $H^{s}_{\mathrm{u,l}}(\mathbb{R})$ and its norm by
$\lVert
u\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}:=\sup_{y\in\mathbb{R}}\lVert\rho_{\mathrm{u,l}}(\cdot-y)u\rVert_{H^{s}(\mathbb{R})}$
when $s\geq 0$. Then, we use the following injections to link with Sobolev
spaces, that were used to estimate Green’s kernel.

######  Lemma 4.1.

Let $s\in\mathbb{R}$ with $s\geq 1$. Then the following injections are
continuous: $W^{s,\infty}(\mathbb{R})\subset
H^{s}_{\mathrm{u,l}}(\mathbb{R})\subset L^{\infty}(\mathbb{R})\subset
L^{2}_{\mathrm{u,l}}(\mathbb{R})$.

###### Proof.

Let first recall the definition of
$\lVert.\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}$ that we use. Let
$\rho_{\mathrm{u,l}}(x):=\frac{1}{1+x^{2}}$, then

$\lVert
u\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}:=\sup_{y\in\mathbb{R}}\lVert
u(\cdot-y)\rho_{\mathrm{u,l}}\rVert_{H^{s}(\mathbb{R})}.$

From one hand the injection $H^{s}(\mathbb{R})\subset L^{\infty}(\mathbb{R})$
leads to

$\lVert
u\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}\geq\sup_{y\in\mathbb{R}}\lVert
u(\cdot-y)\rho_{\mathrm{u,l}}\rVert_{L^{\infty}(-1,1)}\geq\frac{1}{2}\sup_{y\in\mathbb{R}}\lVert
u(\cdot-y)\rVert_{L^{\infty}(-1,1)}=\frac{1}{2}\lVert
u\rVert_{L^{\infty}(\mathbb{R})},$

It corresponds to the second injection. From the other hand, $\lVert
u\rho_{\mathrm{u,l}}\rVert_{H^{s}(\mathbb{R})}\leq\lVert
u\rVert_{W^{s,\infty}(\mathbb{R})}\lVert\rho_{\mathrm{u,l}}\rVert_{H^{s}(\mathbb{R})}$
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:
$\hat{M}(\xi)\in\mathfrak{L}(\mathbb{R}^{d},\mathbb{R}^{n})$ with
$Mu:=\mathcal{F}^{-1}(\xi\mapsto\hat{M}(\xi)\hat{u}(\xi))$. Then for $q,s\geq
0$,

$\lVert Mu\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})^{d}}\leq
C(s,q)\lVert\xi\mapsto\langle\xi\rangle^{\frac{s-q}{2}}\hat{M}(\xi)\rVert_{\mathcal{C}^{2}_{b}(\mathbb{R},\mathfrak{L}(\mathbb{R}^{d},\mathbb{R}^{n}))}\lVert
u\rVert_{H^{q}_{\mathrm{u,l}}(\mathbb{R})^{n}},$

with a positive constant $C(s,q)$ independent of $u$ and $M$. Furthermore if
$\alpha$ and $\xi_{0}$ are reals, then

$\lVert x\mapsto e^{i\xi_{0}x}Mu(\alpha
x)\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})^{d}}\leq C\lVert x\mapsto
e^{i\xi_{0}x}u(\alpha x)\rVert_{H^{q}_{\mathrm{u,l}}(\mathbb{R})^{n}},$

where the above constant satisfies

$C\leq
C(s,q)\left\lVert\xi\mapsto\langle\xi\rangle^{\frac{s-q}{2}}\hat{M}\left(\frac{\xi}{\alpha}-\xi_{0}\right)\right\rVert_{\mathcal{C}^{2}_{b}(\mathbb{R},\mathfrak{L}(\mathbb{R}^{d},\mathbb{R}^{n}))}.$

###  4.2. Linear dynamic

######  Lemma 4.3.

Let $T$ and $\mu_{0}$ be positive constants. Then there exists $C>0$ and
$\kappa>0$ such that for all $0<\mu<\mu_{0}$, the following holds. For $s\geq
0$, and $0\leq t\leq\frac{T}{\mu}$,

$\lVert
e^{t\mathcal{T}^{-}}\varPi_{\text{c}}^{h}V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\lVert V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})},$

while for any $t\geq 0$,

$\lVert
e^{t\mathcal{T}^{-}}\varPi_{\text{s}}^{h}V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}\leq
Ce^{-\kappa t}\lVert V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}.$

Both estimates still holds when $\varPi_{i}^{h}$ are replaced by $\varPi_{i}$.

###### Proof.

Since $\mathcal{T}^{-}$ has constant coefficients – see (12) – it acts in
Fourier space through multiplication. Hence we rely on multiplier theory, see
[MS95]. Since at fixed Fourier parameter $\xi\in\mathbb{R}$, the eigenvalues
of matrix $\hat{\mathcal{T}}^{-}(\xi)$ satisfy
$\operatorname*{\mathrm{Re}}{\lambda_{i}(\xi)}\leq\mu$, we obtain using lemma
4.2 that for $t\leq\frac{T}{\mu}$:

$\left\lVert
e^{t\mathcal{T}^{-}}\varPi_{\text{c}}^{h}V\right\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}\leq\left\lVert\exp(t\hat{\mathcal{T}}^{-})\hat{\varPi_{\text{c}}^{h}}\right\rVert_{\mathcal{C}^{2}(\mathbb{R},\mathcal{M}_{2}(\mathbb{R}))}\lVert
V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}\leq Ce^{2t\mu}\lVert
V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})},$

with $\mathcal{M}_{2}(\mathbb{R})$ the set of $2\times 2$ matrices. The case
of $e^{t\mathcal{T}^{-}}\varPi_{\text{s}}^{h}$ adapts easily: for $t\geq 0$
and $\xi$ in the support of $\chi_{\text{s}}$, the eigenvalues of
$\hat{\mathcal{T}}^{-}(\xi)$ satisfy
$\operatorname*{\mathrm{Re}}(\lambda_{j}(\xi))\leq-2\kappa$. To see that
$\varPi_{i}^{h}$ may be replaced by $\varPi_{i}$, simply use that
$e^{t\mathcal{T}^{-}}\varPi_{i}=e^{t\mathcal{T}^{-}}\varPi^{h}_{i}\varPi_{i}$
and that $\lVert\varPi_{i}V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}\leq\lVert
V\rVert_{H^{s}_{\mathrm{u,l}}(\mathbb{R})}$ 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 derivative –
_via_ estimates of the form:

$\lVert V(t)\rVert_{L^{\infty}(\mathbb{R})}\leq C\lVert
V(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq C\vartheta(t)\lVert
V_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq C\vartheta(t)\lVert
V_{0}\rVert_{W^{1,\infty}(\mathbb{R})}.$

###  4.3. Nonlinear dynamic: shadowing the global attractor of the Ginzburg-
Landau equation

Following [GS07], we drive the perturbation using the dynamic at $-\infty$, to
the cost of an extra source term $\mathcal{S}$. We first show that this term
defined by (13) is sufficiently localized in space so that we can extract a
$\lVert\frac{V}{\omega_{\text{sh}}(\cdot-
c_{*}t)}\rVert_{L^{\infty}}=\lVert\frac{U}{\rho_{*}}\rVert_{L^{\infty}}$ from
it. In the rest of this section, we assume that there exists $C_{1}$ and
$t_{U}$ positive constants such that for all $0\leq t\leq t_{U}$,

(30)
$\left\lVert\frac{U(t)}{\rho_{*}}\right\rVert_{L^{\infty}(\mathbb{R})}\leq
C_{1}\frac{\lVert U_{0}\rVert_{X}}{(1+t)^{3/2}}.$

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 $0\leq t\leq t_{U}$.
Furthermore, we recall the notation $\mu=\varepsilon^{2}$, with
$\varepsilon>0$.

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

$\lVert U\rVert_{X}:=\lVert U\rVert_{W^{2,\infty}(\mathbb{R})\times
W^{4,\infty}(\mathbb{R})}+\lVert\rho_{*}^{3}U\rVert_{L^{\infty}(\mathbb{R})}.$

######  Proposition 4.4.

There exist a positive constant $C$ such that for $0\leq t\leq t_{U}$:

$\left\lVert\frac{U(t)}{\rho_{*}}\right\rVert_{W^{2,\infty}(\mathbb{R})\times
W^{4,\infty}(\mathbb{R})}\leq C\frac{\lVert
U_{0}\rVert_{X}}{(1+t)^{\frac{3}{2}}}.$

###### Proof.

First write the Duhamel formula for $u_{1}$, and then differentiate it:

$\partial_{x}^{j}u_{1}(t,x)=\int_{\mathbb{R}}\partial_{x}^{j}\mathcal{G}_{t}^{\text{kpp}}u_{1}^{0}+\partial_{x}^{j}\mathcal{G}_{t}^{\text{co}}u_{2}^{0}\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{j}\mathcal{G}_{t-s}^{\text{kpp}}\mathcal{N}_{1}+\partial_{x}^{j}\mathcal{G}_{t-s}^{\text{co}}\mathcal{N}_{2}\mathrm{d}y\mathrm{d}s.$

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
$\omega_{\text{kpp}}$ in nonlinear terms allows to gain as many powers of
$\rho_{*}(y)$ as needed. We use the same approach for $\partial_{x}^{k}u_{2}$,
the linear estimate
$\lVert\mathcal{G}_{t}^{\text{sh}}u_{2}\rVert_{W^{k,\infty}(\mathbb{R})}\leq
Ce^{-\eta t}\lVert u_{2}\rVert_{W^{k,\infty}(\mathbb{R})}$ comes from standard
parabolic regularity: $\mathcal{L}^{\text{sh}}$ is sectorial with spectral
gap. ∎

######  Proposition 4.5.

There exists positive constants $\delta$ and $C$ such that for all $0\leq
t\leq t_{U}$,

$\lVert\mathcal{S}(\cdot-
c_{*}t,V(t,\cdot))\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\left\lVert\frac{V(t,\cdot)}{\omega_{\text{sh}}(\cdot-
c_{*}t)}\right\rVert_{W^{2,\infty}(\mathbb{R})\times
W^{4,\infty}(\mathbb{R})}\leq C\frac{\lVert U_{0}\rVert_{X}}{(1+t)^{3/2}}.$

###### Proof.

From lemma 4.1, it is enough to bound
$\lVert\mathcal{S}(\tilde{\cdot},V(t))\rVert_{W^{1,\infty}(\mathbb{R})}$. For
$\tilde{x}\leq-1$, we have $\varpi(\tilde{x})=1$, hence

$\mathcal{S}(\tilde{x},V)=(1-q_{*}(\tilde{x}))\begin{pmatrix}3\alpha(1+q_{*}(\tilde{x}))v_{1}\\\
-\gamma v_{2}\end{pmatrix}.$

We show that
$\lVert\omega_{\text{sh}}(1-q_{*})\rVert_{W^{1,\infty}(-\infty,0)}$ is finite.
The equilibrium point $(q,q^{\prime})=(1,0)$ for the front equation is a
saddle, with one positive eigenvalue
$\kappa=(-1+\sqrt{3})\sqrt{\frac{\alpha}{d}}$. Basic ODE dynamic then ensures
that

$\sup_{x\leq 0}\lvert q_{*}(x)-1\rvert e^{-\kappa x}<+\infty,\hskip
40.00006pt\sup_{x\leq 0}\lvert q_{*}^{\prime}(x)\rvert e^{-\kappa x}<+\infty.$

For $\tilde{x}\leq-1$ and $\lvert\theta\rvert$ small enough,
$\omega_{\text{sh}}(\tilde{x})=e^{\theta\tilde{x}}\leq e^{-\kappa\tilde{x}}$,
which leads to

$\left\lVert\mathcal{S}(\cdot-
c_{*}t,V(t,\cdot))\right\rVert_{W^{1,\infty}(-\infty,0)}\leq
C\left\lVert\frac{V(t)}{\omega_{\text{sh}}(\cdot-
c_{*}t)}\right\rVert_{W^{1,\infty}(-\infty,0)}.$

Now for $\tilde{x}\geq 1$, recall the expression of the source term (13). For
linear terms, we use the commutator to gain one derivative: the operator

$\varpi^{-1}(\mathcal{A}-c_{*}\partial_{x})\varpi-(\mathcal{A}-c_{*}\partial_{x})=\varpi^{-1}[\mathcal{A}-c_{*}\partial_{x},\varpi]$

exhibit at most third order derivatives, so that it maps
$W^{2,\infty}(\mathbb{R})\times W^{4,\infty}(\mathbb{R})$ onto
$W^{1,\infty}(\mathbb{R})\times W^{1,\infty}(\mathbb{R})$. Furthermore
$\lVert\varpi^{-1}\partial_{x}\varpi\rVert_{W^{1,\infty}(0,+\infty)}<\infty$.
Altogether, it leads to

$\lVert\mathcal{S}(\tilde{\cdot},V)\rVert_{W^{1,\infty}(0,+\infty)}\leq
C\lVert V\rVert_{W^{2,\infty}\times W^{4,\infty}(0,+\infty)}+C\lVert
V\rVert_{W^{1,\infty}(0,+\infty)}^{2}+C\lVert
V\rVert_{W^{1,\infty}(0,+\infty)}^{3}.$

Remark that $\omega_{\text{kpp}}(\tilde{x})=1$, hence the first claimed
estimate is shown. Then
$\frac{V(t,x)}{\omega_{\text{sh}}(\tilde{x})}=\frac{U(t,\tilde{x})}{\rho_{*}(\tilde{x})}$,
and proposition 4.4 ensures the second claimed estimate. The proof is
complete. ∎

######  Remark 4.6.

Since $t\mapsto(1+t)^{-3/2}$ 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 $V\in H^{1}_{\mathrm{u,l}}(\mathbb{R})$,

$\lVert\varPi_{\text{s}}\mathcal{Q}^{-}(\varepsilon\varPi_{\text{c}}V+\varepsilon^{2}\varPi_{\text{s}}V)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\varepsilon^{2}(\lVert\varPi_{\text{c}}V\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+\lVert\varPi_{\text{s}}V\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})})^{2},$

while

$\lVert\varPi_{\text{c}}\mathcal{Q}^{-}(\varepsilon\varPi_{\text{c}}V+\varepsilon^{2}\varPi_{\text{s}}V)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\varepsilon^{3}(\lVert\varPi_{\text{c}}V\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+\lVert\varPi_{\text{s}}V\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})})^{2}.$

###### Proof.

The first estimate is immediate. The second one comes from [Sch94c], and
proves as follows. In Fourier space, multiplication becomes convolution. Since
$\mathrm{Supp}(\chi_{\text{c}})=[-5/4,-3/4]\cup[3/4,5/4]$ and
$\mathrm{Supp}(\chi_{\text{c}}*\chi_{\text{c}})=[-5/2,-3/2]\cup[-1/2,1/2]\cup[3/2,5/2]$
do not intersect, we deduce that

$\varPi_{\text{c}}(\varPi_{\text{c}}V_{1}\times\varPi_{\text{c}}V_{2})=\mathcal{F}^{-1}(\chi_{\text{c}}\times(\chi_{\text{c}}\hat{V}_{1}*\chi_{\text{c}}\hat{V}_{2}))=0.$

Hence lowest order quadratic terms vanish when $\varPi_{\text{c}}$ is applied,
leaving $\varepsilon^{3}$ terms at leading order. ∎

We now begin the proof of proposition 2.4 by decomposing $\mathcal{T}^{-}$ in
Fourier space. This allows to show that after a time
$T_{\text{att}}/\varepsilon^{2}$, the perturbation $V$ is at leading order a
critical oscillating mode.

######  Lemma 4.8 (Attractivity).

Let $T_{\text{att}}>0$ be fixed. There exists a constant $C>0$ – depending on
$C_{1}$ – and a positive constant $\delta$ such that for all $V_{0}$
satisfying

(31) $K_{1}:=\frac{1}{\varepsilon}\left(\lVert
V_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+\lVert
U_{0}\rVert_{X}\right)\leq\delta,$

the solution $V$ to (11) with initial condition $V_{0}$ exists for all time
$0\leq t\leq\frac{T_{\text{att}}}{\varepsilon^{2}}$, and decomposes as
$V=V_{\text{c}}+V_{\text{s}}$, with $\varPi^{h}_{i}V_{i}=V_{i}$. When $0\leq
t\leq\frac{T_{\text{att}}}{\varepsilon}$, it satisfies

$\lVert V_{i}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
CK_{1}\varepsilon,\hskip 40.00006pti\in\left\\{\text{c},\text{s}\right\\},$

while for times $\frac{T_{\text{att}}}{\varepsilon^{2/3}}\leq
t\leq\frac{T_{\text{att}}}{\varepsilon^{2}}$, we have

$\lVert V_{\text{c}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
CK_{1}\varepsilon,\hskip 40.00006pt\lVert
V_{\text{s}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
CK_{1}\varepsilon^{2}.$

Finally, the initial condition $A_{0}$ for (GL) is bounded uniformly in
$\varepsilon$: $\left\lVert
A_{0}\right\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq CK_{1}$, with

(32)
$A_{0}(X):=\frac{1}{\varepsilon}e^{-i\frac{X}{\varepsilon}}\pi_{1}^{h}V_{\text{c}}\left(\frac{T_{\text{att}}}{\varepsilon^{2}},\frac{X}{\varepsilon}\right).$

###### Proof.

Since $\mathcal{T}^{-}$ is sectorial, $V\mapsto\mathcal{Q^{-}}(V)$ is locally
Lipschitz and $t\mapsto\mathcal{S}(\tilde{x},V(t,x))$ 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 $T_{\text{att}}/\varepsilon^{2}$.

To construct $V$, it is enough to solve the following system, with initial
condition $V_{i}(0)=\varPi_{i}V(0)$:

(33)
$\begin{cases}\partial_{t}V_{\text{c}}=\mathcal{T}^{-}V_{\text{c}}+\varPi_{\text{c}}\mathcal{Q}^{-}(V_{\text{c}}+V_{\text{s}})+\varPi_{\text{c}}\mathcal{S}(\tilde{x},V_{\text{c}}+V_{\text{s}}),\\\
\partial_{t}V_{\text{s}}=\mathcal{T}^{-}V_{\text{s}}+\varPi_{\text{s}}\mathcal{Q}^{-}(V_{\text{c}}+V_{\text{s}})+\varPi_{\text{s}}\mathcal{S}(\tilde{x},V_{\text{c}}+V_{\text{s}}),\end{cases}$

and then set $V=V_{\text{c}}+V_{\text{s}}$. A solution
$(V_{\text{c}},V_{\text{s}})$ of the decoupled system (33) is not guaranteed
to write as $V_{i}=\varPi_{i}V$, however the critical-stable separation still
holds, so that lemma 4.3 applies:

$\lVert
e^{t\mathcal{T}^{-}}V_{\text{s}}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}=\lVert
e^{t\mathcal{T}^{-}}\varPi_{\text{s}}^{h}V_{\text{s}}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
e^{-\kappa t}\lVert V_{\text{s}}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}.$

We introduce the local notations $W_{i}:=\frac{V_{i}}{\varepsilon}$ and
$\Theta_{i}(t):=\sup_{0\leq\tau\leq t}\lVert
W_{i}(\tau)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$. Using Duhamel formula
and decay of $\mathcal{S}$, we see that

$\displaystyle\lVert W_{\text{c}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$
$\displaystyle\leq C\lVert
W_{\text{c}}(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+C\varepsilon\int_{0}^{t}({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}\mathrm{d}\tau+C_{1}\frac{\lVert
U_{0}\rVert_{X}}{\varepsilon}\int_{0}^{t}\frac{\mathrm{d}\tau}{(1+\tau)^{3/2}},$
$\displaystyle\leq C_{1}K_{1}+C\varepsilon
t({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}.$

In a similar way, standard integral computations – see lemma E.4 – ensures
that

$\displaystyle\lVert
W_{\text{s}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq{}$ $\displaystyle
Ce^{-\kappa t}\lVert
W_{\text{s}}(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+C\varepsilon({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}\int_{0}^{t}e^{-\kappa(t-\tau)}\mathrm{d}\tau$
$\displaystyle+C_{1}\frac{\lVert
U_{0}\rVert_{X}}{\varepsilon}\int_{0}^{t}\frac{e^{-\kappa(t-\tau)}\mathrm{d}\tau}{(1+\tau)^{3/2}},$
$\displaystyle\leq{}$ $\displaystyle
C_{1}\frac{K_{1}}{(1+t)^{3/2}}+C\varepsilon({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}.$

For $0\leq t\leq\frac{T_{\text{att}}}{\varepsilon}$, we take the sup in the
two above equations to obtain – with
$\Theta:=\max({\Theta_{\text{c}}},{\Theta_{\text{s}}})$ – that $\Theta(t)\leq
C_{1}K_{1}+C\Theta(t)^{2}$. Applying a standard nonlinear argument – see lemma
E.5 – with $K_{1}\leq\delta$ small enough, we recover
$\Theta(T_{\text{att}}/\varepsilon)\leq C_{1}K_{1}$, which is the first
claimed estimate. Now for $T_{\text{att}}/\varepsilon^{2/3}\leq t\leq
T_{\text{att}}/\varepsilon$, we have $\frac{1}{(1+t)^{3/2}}\leq C\varepsilon$,
hence $\lVert W_{\text{s}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C_{1}K_{1}\varepsilon$. This little improvement will allow us to propagate
estimates until time $T_{\text{att}}/\varepsilon^{2}$. For $t\geq 0$, we now
use the new local notation

$V(t+T_{\text{att}}/\varepsilon^{2/3})=:\varepsilon
W_{\text{c}}(t)+\varepsilon^{2}W_{\text{s}}(t),$

together with $\Theta_{i}(t):=\sup_{0\leq\tau\leq t}\lVert
W_{i}(\tau)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$. Remark that from the
above, we know that

$\lVert W_{\text{c}}(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+\lVert
W_{\text{s}}(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq C_{1}K_{1}.$

Then nonlinear estimate from lemma 4.7 ensures that for $t\leq
T_{\text{att}}/\varepsilon^{2}$,

$\displaystyle\lVert
W_{\text{c}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq{}$ $\displaystyle
C\lVert
W_{\text{c}}(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+C\varepsilon^{2}t\,({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}+C_{1}\frac{\lVert
U_{0}\rVert_{X}}{\varepsilon}\int_{0}^{t}\frac{\mathrm{d}\tau}{(1+T_{\text{att}}/\varepsilon^{2/3}+\tau)^{3/2}},$
$\displaystyle\leq{}$ $\displaystyle
C_{1}K_{1}+C({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}.$

Since $(1+T_{\text{att}}/\varepsilon^{2/3}+\tau)^{-3/2}\leq\varepsilon$ when
$\tau\geq 0$, we deduce as above that

$\displaystyle\lVert
W_{\text{s}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq{}$ $\displaystyle
Ce^{-\kappa t}\lVert
W_{\text{s}}(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+C\int_{0}^{t}e^{-\kappa(t-\tau)}\mathrm{d}\tau({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}$
$\displaystyle+C_{1}\frac{\lVert
U_{0}\rVert_{X}}{\varepsilon^{2}}\int_{0}^{t}\frac{e^{-\kappa(t-\tau)}\mathrm{d}\tau}{(1+T_{\text{att}}/\varepsilon^{2/3}+\tau)^{3/2}},$
$\displaystyle\leq{}$ $\displaystyle
C_{1}K_{1}+C({\Theta_{\text{c}}}(t)+{\Theta_{\text{s}}}(t))^{2}.$

Taking the sup for $0\leq t\leq T_{\text{att}}/\varepsilon^{2}$, we obtain as
previously that $\Theta(T_{\text{att}}/\varepsilon^{2})\leq C_{1}K_{1}$ for
$K_{1}$ small enough. This is the second claimed estimate. To bound $A_{0}$,
we use the scaled estimate in lemma 4.2

$\left\lVert X\mapsto
e^{-i\frac{X}{\varepsilon}}Mu\left(\frac{X}{\varepsilon}\right)\right\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq\lVert\xi\mapsto\langle\xi\rangle^{1/2}\hat{M}(\varepsilon\xi+1)\rVert_{\mathcal{C}^{2}_{b}(\mathbb{R},\mathfrak{L}(\mathbb{R}^{2},\mathbb{R}))}\left\lVert
X\mapsto
e^{-i\frac{X}{\varepsilon}}u\left(\frac{X}{\varepsilon}\right)\right\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})},$

with $M=\pi_{1}^{h}$ and
$u=V_{\text{c}}(\frac{T_{\text{apr}}}{\varepsilon^{2}})$. To estimate
$\hat{M}(\varepsilon\xi+1)$, we can restrict to the $\mathcal{C}^{0}_{b}$
norm, since derivative gains us power of $\varepsilon$. Recall that

$\hat{M}(\xi)\hat{u}=\chi_{\text{c}}^{h}(\xi)\langle\varrho_{\text{c}}(1),\varrho_{\text{c}}^{*}(\xi)\rangle^{-1}\langle\hat{u},\varrho_{\text{c}}^{*}(\xi)\rangle$

so that Cauchy-Schwartz leads to $\lVert
M(\xi)\rVert_{\mathfrak{L}(\mathbb{R}^{2},\mathbb{R}^{2})}\leq\lvert\chi_{\text{c}}^{h}(\xi)\varrho_{\text{c}}^{*}(\xi)\langle\varrho_{\text{c}}(1),\varrho_{\text{c}}^{*}(\xi)\rangle^{-1}\rvert\leq
C\lvert\chi_{\text{c}}^{h}(\xi)\rvert$. Here we assume that the support of
$\chi_{\text{c}}$ is so small that
$\langle\varrho_{\text{c}}(1),\varrho_{\text{c}}^{*}(\xi)\rangle$ does not
vanish by continuity. We emphasize that this support is still independent of
$\mu$, so that the separation of frequencies comes with spectral gap. Using
the support of $\chi_{\text{c}}^{h}$, we obtain

$\lVert\xi\mapsto\langle\xi\rangle^{1/2}\hat{M}(\varepsilon\xi+1)\rVert_{\mathcal{C}^{2}_{b}(\mathbb{R},\mathfrak{L}(\mathbb{R}^{2},\mathbb{R}))}\leq\frac{C}{\sqrt{\varepsilon}}.$

Now rescaling the $L^{2}_{\mathrm{u,l}}(\mathbb{R})$ norm, we get

$\lVert X\mapsto
e^{-i\frac{X}{\varepsilon}}u(X/\varepsilon)\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}\leq\sqrt{\varepsilon}\lVert
e^{-ix}u\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}\leq\sqrt{\varepsilon}\lVert
u\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}.$

Hence we have shown that $\lVert
A_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq C\varepsilon^{-1}\lVert
V_{\text{c}}(\varepsilon^{-2}T_{\text{apr}})\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}\leq
C_{1}K_{1}$.

∎

Now that $V$ is at leading order a critical oscillating mode, we can
approximate it by
$\psi(\varepsilon,A):=\varepsilon\psi_{\text{c}}(\varepsilon,A)+\varepsilon^{2}\psi_{\text{s}}(\varepsilon,A)$,
with $A$ the solution to the GL equation with initial condition $A_{0}$, see
(32), with the critical part of $\psi$ being:

(34)
$\psi_{\text{c}}(\varepsilon,A)(t,x)=(e^{ix}A(\varepsilon^{2}t,\varepsilon
x)+c.c)\varrho_{\text{c}},$

and the stable part:

$\psi_{\text{s}}(\varepsilon,A)(t,x)=\lvert A(\varepsilon^{2}t,\varepsilon
x)\rvert^{2}\varrho_{0}+e^{ix}\partial_{X}A(\varepsilon^{2}t,\varepsilon
x)\varrho_{1}+e^{2ix}A(\varepsilon^{2}t,\varepsilon x)^{2}\varrho_{2}+c.c.$

All $\varrho_{i}$ and $\varrho_{\text{c}}$ are two-dimensional vectors with
explicit expressions, see appendix A. The choice of $\psi_{\text{s}}$ is made
clearer therein, it ensures that the error of approximation
$R=V-\psi(\varepsilon,A)$ stays small when time evolves.

######  Lemma 4.9 (Global attractor for Ginzburg-Landau).

Let $T,X\mapsto A(T,X)$ be a solution of the Ginzburg-Landau equation (GL).
Then there exists a constant $C_{GL}>0$ such that

$\lVert A(T)\rVert_{W^{1,\infty}(\mathbb{R})}\leq C_{GL}+e^{-T/2}\lVert
A(0)\rVert_{W^{1,\infty}(\mathbb{R})}.$

###### 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

$\lVert A(T)h\rVert_{L^{2}(\mathbb{R})}\leq C_{GL}+e^{-2T} \lVert
A(0)h\rVert_{L^{2}(\mathbb{R})}$

where $h_{a}(X)=\frac{1}{1+(X-a)^{2}}$. The same estimate holds for
$\partial_{x}A$. It leads to

$\lvert A(T,X)\rvert\leq\frac{1}{h(X)^{2}}\left(C_{GL}+e^{-T} \lVert
A(0)h\rVert_{H^{1}(\mathbb{R})}\right),$

which after taking the supremum over $a\in\mathbb{R}$ writes $\lVert
A(T)\rVert_{L^{\infty}(\mathbb{R})}\leq C_{GL}+e^{-T} \lVert
A(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$. The bound on
$\partial_{X}A(T)$ 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 $t\leq\varepsilon^{-7/4}T_{\text{apr}}$, 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
$\varepsilon^{-2}T_{\text{apr}}$, which is optimal due to the linear growth of
critical modes $\lVert
e^{t\mathcal{T}^{-}}\varPi_{\text{c}}V_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\geq
Ce^{\varepsilon^{2}t}\lVert\varPi_{\text{c}}V_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$.
In our case, the slowly decaying inhomogeneous term $\mathcal{S}$ prevents us
to go beyond $\varepsilon^{-7/4}T_{\text{apr}}$.

######  Lemma 4.10 (Approximation).

For $T_{\text{apr}}>0$ fixed, there exists a positive constant
$C_{\text{apr}}$ such that the following holds. Let
$t_{0}\geq\frac{T_{\text{att}}}{\varepsilon^{2}}$, $A_{0}\in
H^{1}_{\mathrm{u,l}}(\mathbb{R})$ and a solution $V$ of (11) such that

$\lVert R(t_{0})\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}=\lVert
V(t_{0})-\psi(\varepsilon,A_{0})\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq\varepsilon^{5/4}.$

Note $A$ the solution of (GL) with initial condition $A_{0}$ at $t=t_{0}$, and
$R(t)=V(t)-\psi(\varepsilon,A(t))$. Then for $t\geq t_{0}$, the error term
decomposes as $R(t)=R_{\text{c}}(t)+R_{\text{s}}(t)$, with
$\varPi^{h}_{i}R_{i}(t)=R_{i}(t)$. Furthermore, for $0\leq
t-t_{0}\leq\frac{T_{\text{apr}}}{\varepsilon^{1/4}}$,

$\lVert R_{i}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C_{\text{apr}}\varepsilon^{5/4},\hskip
40.00006pti\in\left\\{\text{c},\text{s}\right\\},$

while for $\frac{T_{\text{apr}}}{\varepsilon^{1/4}}\leq
t-t_{0}\leq\frac{T_{\text{apr}}}{\varepsilon^{7/4}}$,

$\lVert R_{\text{c}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C_{\text{apr}}\varepsilon^{5/4},\hskip 40.00006pt\lVert
R_{\text{s}}(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C_{\text{apr}}\varepsilon^{9/4}.$

###### Proof.

Insert $R(t)=V(t)-\psi(\varepsilon,A(t))$ in (11) to obtain that for $t\geq
t_{0}$, $R$ satisfies

$\partial_{t}R=\mathcal{T}^{-}R+2B(\psi,R)+\mathcal{Q}^{-}(R)+\mathcal{S}(t)+\mathrm{Res}(\psi),$

where $B$ is a symmetric bilinear function that satisfies
$\mathcal{Q}^{-}(V)=B(V,V)$, and the residual term is

$\mathrm{Res}(\psi)=-\partial_{t}\psi+\mathcal{T}^{-}\psi+\mathcal{Q}^{-}(\psi).$

We decompose Duhamel formula into a critical and stable part:

$\varPi_{i}R(t)=e^{t\mathcal{T}^{-}}\varPi_{i}R(t_{0})+\int_{t_{0}}^{t}e^{(t-s)\mathcal{T}^{-}}\varPi_{i}\left(2B(\psi,R)+\mathcal{Q}^{-}(R)+\mathcal{S}+\mathrm{Res}(\psi)\right)\mathrm{d}s,$

with $i\in\left\\{\text{c},\text{s}\right\\}$. To close a nonlinear argument,
we follow [Sch94b]. Since $A$ satisfies (GL), the leading order terms in the
residual $\mathrm{Res}(\psi(\varepsilon,A))$ vanish, see appendix A. Hence we
can control $\mathrm{Res}(\psi(\varepsilon,A))$ for times smaller than
$\varepsilon^{-2}T_{\text{apr}}$, 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
$\lVert\mathcal{S}(t,\cdot)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\varepsilon^{3}$ when $t\geq
t_{0}\geq\frac{T_{\text{att}}}{\varepsilon^{2}}$. This implies that

$\left\lVert\int_{t_{0}}^{t}e^{(t-\tau)\mathcal{T}^{-}}\varPi_{\text{c}}\mathcal{S}(\tau)\mathrm{d}\tau\right\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C(t-t_{0})\varepsilon^{3},$

while

$\left\lVert\int_{t_{0}}^{t}e^{(t-\tau)\mathcal{T}^{-}}\varPi_{\text{s}}\mathcal{S}(\tau)\mathrm{d}\tau\right\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\varepsilon^{3}\int_{t_{0}}^{t}e^{-\kappa(t-\tau)}\mathrm{d}\tau\leq\varepsilon^{3}.$

Hence when $t-t_{0}\leq\varepsilon^{-1/4}T_{\text{apr}}$, the estimates on
$\mathcal{S}$ can be sub-summed into the one on
$\mathrm{Res}(\psi(\varepsilon,A))$. Now for
$\varepsilon^{-1/4}T_{\text{apr}}\leq
t-t_{0}\leq\varepsilon^{-7/4}T_{\text{apr}}\leq$, we can once again sub-summed
$\mathcal{S}$ into $\mathrm{Res}(\psi(\varepsilon,A))$. 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 $\lVert
V(0)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq C\lVert
V_{0}\rVert_{W^{1,\infty}}\leq\varepsilon K_{1}\leq\varepsilon\delta$, we wait
a time $t_{0}:=\varepsilon^{-2}T_{\text{att}}$, see lemma 4.8. Now $V$
decomposes as $V=V_{\text{c}}+V_{\text{s}}$, with the leading order
$V_{\text{c}}$ writing as:

$\hat{V_{\text{c}}}(t_{0},\xi)=\mathcal{F}(x\mapsto\varepsilon
e^{ix}A_{0}(\varepsilon x))(\xi)\varrho_{\text{c}}(\xi)+c.c,$

_i.e._ $V_{\text{c}}(t_{0})$ is obtained as the modulation of a large profile
$x\mapsto A_{0}(\varepsilon x)$, with $\lVert
A_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq CK_{1}$ from lemma 4.8. We
propagate this ansatz by decomposing $V(t,x)=\psi(\varepsilon,A)(t,x)+R(t,x)$,
see (34). Applying lemma 4.9 and $\lVert
A_{0}\rVert_{L^{\infty}(\mathbb{R})}\leq C\lVert
A_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$ from lemma E.3, we conclude
that $\lVert A(\varepsilon^{2}t)\rVert_{W^{1,\infty}(\mathbb{R})}\leq
C_{GL}+CK_{1}$ for all times $t\geq 0$. However, $A$ and $\psi(\varepsilon,A)$
does not depend on the same space variable, so that rescaling
$H^{1}_{\mathrm{u,l}}(\mathbb{R})$ norms loses us a $\varepsilon^{-1/2}$.
Instead, we use lemma E.3 and its injections to rescale in
$W^{1,\infty}(\mathbb{R})$ space:

$\displaystyle\lVert\psi_{\text{c}}(\varepsilon,A)(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$
$\displaystyle\leq C\lVert x\mapsto A(\varepsilon^{2}t,\varepsilon
x)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})},$ $\displaystyle\leq C\lVert
x\mapsto A(\varepsilon^{2}t,\varepsilon x)\rVert_{W^{1,\infty}(\mathbb{R})},$
$\displaystyle\leq C\lVert
A(\varepsilon^{2}t)\rVert_{W^{1,\infty}(\mathbb{R})},$ $\displaystyle\leq
C_{GL}+CK_{1}e^{-t/\varepsilon^{2}}$

Recall the expression of $K_{1}$ (31). Similar estimates lead to
$\lVert\psi_{\text{s}}(\varepsilon,A)(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C_{GL}+CK_{1}e^{-\frac{t}{\varepsilon^{2}}}$. Now for $t\geq t_{0}$ we can
estimate

$\displaystyle\lVert V(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$
$\displaystyle\leq\lVert\psi(\varepsilon,A)(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+\lVert
V(t)-\psi(\varepsilon,A)(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})},$
$\displaystyle\leq C\varepsilon(1+K_{1}e^{-t/\varepsilon^{2}})+\lVert
R(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})},$

and it only remains to bound the error $R(t)$. We initialy decompose it as
$R_{\text{c}}(t_{0})=V_{\text{c}}(t_{0})-\varepsilon\psi_{\text{c}}(A_{0})$
and
$R_{\text{s}}(t_{0})=V_{\text{s}}(t_{0})-\varepsilon^{2}\psi_{\text{s}}(A_{0})$.
From the definition of $A_{0}$,

$\pi_{1}^{h}V_{\text{c}}(t_{0},x)=\varepsilon e^{ix}A_{0}(\varepsilon x),$

while definitions of $\psi_{\text{c}}$ (34) and $\pi_{1}^{h}$ (29) ensure that

$\pi_{1}^{h}(\varepsilon\psi_{\text{c}}(A_{0}))(x)=\mathcal{F}^{-1}\left(\chi_{\text{c}}^{h}\mathbf{1}_{\xi>0}\mathcal{F}(\varepsilon
e^{iy}A_{0}(\varepsilon y)\right)=\varepsilon
e^{ix}\mathcal{F}^{-1}(\chi_{0}\mathcal{F}(A_{0}(\varepsilon y))).$

where $\chi_{0}(\xi)=\chi_{\text{c}}^{h}(\xi-1)\mathbf{1}_{\xi>1}$ is
supported near $\xi=0$. Hence

$\pi_{1}^{h}(V_{\text{c}}(t_{0})-\varepsilon\psi_{\text{c}}(A_{0}))(x)=\varepsilon
e^{ix}\mathcal{F}^{-1}\left((1-\chi_{0})\mathcal{F}(A_{0}(\varepsilon\cdot))\right).$

Since $R_{\text{c}}(t_{0})$ has compact support in Fourier space, we can gain
one derivative, and treat separately the behavior close to $1$ and $-1$:

$\lVert R_{\text{c}}(t_{0})\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\lVert\pi_{1}^{h}R_{\text{c}}(t_{0})\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}+\lVert\pi_{-1}^{h}R_{\text{c}}(t_{0})\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}.$

We first rescale space, then apply the scaled estimate from lemma 4.2, and
finally use the localization of $\chi_{0}$ together with the bound on $A_{0}$
– see lemma 4.8 – to obtain that

$\displaystyle\lVert\pi_{1}^{h}R_{\text{c}}(t_{0})\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})}$
$\displaystyle\leq\frac{1}{\sqrt{\varepsilon}}\left\lVert
x\mapsto\pi_{1}^{h}R_{\text{c}}\left(t_{0},\frac{x}{\varepsilon}\right)\right\rVert_{L^{2}_{\mathrm{u,l}}(\mathbb{R})},$
$\displaystyle\leq\sqrt{\varepsilon}\left\lVert(1-\chi_{0}(\varepsilon\xi-1))\langle\xi\rangle^{-1/2}\right\rVert_{\mathcal{C}^{2}_{b}(\mathbb{R})}\lVert
X\mapsto e^{iX}A_{0}(X)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})},$
$\displaystyle\leq\varepsilon^{3/2}CK_{1}.$

The conjugated part $\pi_{-1}^{h}R_{\text{c}}(t_{0})$ bound identically.
Furthermore:

$\lVert R_{\text{s}}(t_{0})\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq\lVert
V_{\text{s}}(t_{0})\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}+\varepsilon^{2}\lVert\psi_{\text{s}}(A_{0})\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\varepsilon^{2}+C\varepsilon^{2}\lVert x\mapsto A_{0}(\varepsilon
x)\rVert_{H^{2}_{\mathrm{u,l}}(\mathbb{R})}.$

Using that $x\mapsto A_{0}(\varepsilon x)$ has compact support in Fourier
space, we can gain two derivatives, inject into $L^{\infty}(\mathbb{R})$,
scale, and finally inject back into $H^{1}_{\mathrm{u,l}}(\mathbb{R})$. It
reads: $\lVert x\mapsto A_{0}(\varepsilon
x)\rVert_{H^{2}_{\mathrm{u,l}}(\mathbb{R})}\leq C\lVert
A_{0}\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}$. Hence $\lVert
R(t_{0})\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq\varepsilon^{5/4}$, and
lemma 4.10 ensures that

$\lVert R(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq C\varepsilon^{5/4}$

for times $t_{0}\leq t\leq t_{1}=t_{0}+\varepsilon^{-7/4}T_{\text{apr}}$.
Applying the same Lemma as many time as needed, we conclude that the above
bound on $R(t)$ holds for times $t_{0}\leq t\leq
t_{n}=t_{0}+n\varepsilon^{-7/4}T_{\text{apr}}$, without deterioration of the
constant. Hence

$\lVert V(t)\rVert_{H^{1}_{\mathrm{u,l}}(\mathbb{R})}\leq
C\varepsilon(1+K_{1}e^{-t/\varepsilon^{2}})+C\varepsilon^{5/4}\leq
C\sqrt{\mu}(1+K_{1}e^{-t/\mu}),$

and the proof is complete. ∎

##  Appendix A Ginzburg-Landau equation

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

(35)
$\partial_{t}V=\mathcal{T}_{0}V+\mathcal{T}_{\mu}V+\mathcal{N}_{2}(V)+\mathcal{N}_{3}(V),$

where the linear terms are given by:

$\mathcal{T}_{0}=\begin{pmatrix}d\partial_{x}^{2}-2\alpha&\beta\\\
0&-(1+\partial_{x}^{2})^{2}\end{pmatrix},\hskip
40.00006pt\mathcal{T}_{\mu}=\begin{pmatrix}0&0\\\ 0&\mu\end{pmatrix},$

while the quadratic and cubic terms write as:

$\mathcal{N}_{2}(V)=\begin{pmatrix}-3\alpha{v_{1}}^{2}\\\
\gamma{v_{1}}{v_{2}}\end{pmatrix}\hskip
40.00006pt\mathcal{N}_{3}(V)=-\begin{pmatrix}\alpha{v_{1}}^{3}\\\
\sigma{v_{2}}^{3}\end{pmatrix}.$

In the following, we note $M(X)$ the symbol of the constant coefficient
operator $\mathcal{T}_{0}$, such that $\mathcal{T}_{0}=M(\partial_{x})$. We
also note $\varepsilon=\sqrt{\mu}$. Assume that the solution to such system
writes according to the following ansatz:

$V(t,x)=\varepsilon(e^{ix}A(\varepsilon^{2}t,\varepsilon
x)+c.c)\varrho_{\text{c}}+\varepsilon^{2}(A_{0}\varrho_{0}+e^{ix}A_{1}\varrho_{1}+e^{2ix}A_{2}\varrho_{2}+c.c).$

Here and in the following $c.c$ is the complex conjugate: $z+c.c$ stands for
$z+\bar{z}=2\operatorname*{\mathrm{Re}}{z}$. The amplitude $A\in\mathbb{C}$ is
the main unknown, while each $A_{i}(\varepsilon^{2}t,\varepsilon x)$ is an
amplitude we will fix later. The vector $\varrho_{\text{c}}$ is chosen to be
an eigenvector for matrix $M(i)$, associated to eigenvalue $0$, while the
vectors $\varrho_{i}$ will be fixed later. We also introduce
$\varrho_{\text{c}}^{*}$, in the kernel of $M(i)^{*}$, normalized so that
$\langle\varrho_{\text{c}},\varrho_{\text{c}}^{*}\rangle=1$. We propose

$\varrho_{\text{c}}=\begin{pmatrix}\beta\\\ d+2\alpha\end{pmatrix},\hskip
40.00006pt\varrho_{\text{c}}^{*}:=\begin{pmatrix}0\\\
\frac{1}{d+2\alpha}\end{pmatrix}.$

We inject the ansatz in system (35), and then identify terms of order
$\varepsilon^{l}e^{ikx}$ for $l\in\mathbb{N}^{*}$ and $k\in\mathbb{Z}$. As
usual for such computation, the $\varepsilon$ order will be trivially
satisfied. Then $\varepsilon^{2}e^{ikx}$ terms will determine amplitudes
$A_{k}$ and vectors $\varrho_{k}$. Finally, the $\varepsilon^{3}e^{ix}$
equation will lead to Ginzburg-Landau equation. Hence, we may neglect all
terms of order $\varepsilon^{3}e^{2ix}$ and $\varepsilon^{3}e^{0}$. We easily
obtain:

$\partial_{t}V=\varepsilon^{3}(e^{ix}\partial_{T}A\varrho_{\text{c}}+c.c)$

with slow time variable $T=\varepsilon^{2}t$. Similarly, we note
$X=\varepsilon x$ the large space variable. To compute $\mathcal{T}_{0}u$, we
use a formal Taylor expansion, see lemma E.1:

$\displaystyle\mathcal{T}_{0}(\varepsilon e^{ix}A\varrho_{\text{c}})$
$\displaystyle{}=\varepsilon e^{ix}M(i+\partial_{x})A\varrho_{\text{c}},$
$\displaystyle{}=\varepsilon
e^{ix}\left(A\,M(i)\varrho_{\text{c}}+\varepsilon\partial_{X}A\,M^{\prime}(i)\varrho_{\text{c}}+\frac{1}{2}\varepsilon^{2}\partial_{XX}A\,M^{\prime\prime}(i)\varrho_{\text{c}}\right)+\mathcal{O}(\varepsilon^{4}).$

The computation can be easily adapt for
$\mathcal{T}_{0}(\varepsilon^{2}e^{2ikx}A_{k}\varrho_{k})$, for
$k\in\\{0,1,2\\}$. The $\mathcal{T}_{\mu}(V)$ term is easily developed, and we
are left with the nonlinear terms. We note
$B:\mathbb{R}^{2}\times\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ a
symmetric bilinear application that satisfies $B(V,V)=\mathcal{N}_{2}(V)$. It
is given by:

$B(V,W)=\begin{pmatrix}-3\alpha v_{1}w_{1}\\\
\frac{\gamma}{2}(v_{1}w_{2}+v_{2}w_{1})\end{pmatrix}.$

Hence, we obtain:

$\displaystyle\mathcal{N}_{2}(V)={}$
$\displaystyle\varepsilon^{2}\left(e^{2ix}A^{2}\mathcal{N}_{2}(\varrho_{\text{c}})+c.c+2\varepsilon^{2}\lvert
A\rvert^{2}\mathcal{N}_{2}(\varrho_{\text{c}})\right)+\varepsilon^{3}(e^{ix}2AA_{0}B(\varrho_{\text{c}},\varrho_{0})+c.c)$
$\displaystyle{}+\varepsilon^{3}(e^{ix}2\bar{A}A_{2}B(\varrho_{\text{c}},\varrho_{2})+c.c)+\mathcal{O}_{\lvert
k\rvert\neq 1}(\varepsilon^{3}e^{ikx}+\varepsilon^{4}).$

Similarly, the cubic term develop as:

$\mathcal{N}_{3}(V)=\varepsilon^{3}(e^{i}x3A\lvert
A\rvert^{2}\mathcal{N}_{3}(\varrho_{\text{c}})+c.c)+\mathcal{O}_{\lvert
k\rvert\neq 1}(\varepsilon^{3}e^{ikx}+\varepsilon^{4}).$

We now collect previous calculus, and identify terms of same order
$\varepsilon^{l}e^{ikx}$. The $\varepsilon^{2}e^{ikx}$ equations write

$\partial_{X}A\,M^{\prime}(i)\varrho_{\text{c}}+A_{1}\,M(i)\varrho_{\text{c}}=0,$
$A_{0}M(0)\varrho_{0}+2\lvert
A\rvert^{2}\mathcal{N}_{2}(\varrho_{\text{c}})=0,\hskip
40.00006ptA_{2}M(2i)\varrho_{2}+A^{2}\mathcal{N}_{2}(\varrho_{\text{c}})=0.$