Mathew A Johnson

Pascal Noble

AND L Miguel Rodrigues

Kevin Zumbrun

NONLOCALIZED MODULATION OF PERIODIC REACTION DIFFUSION WAVES: NONLINEAR STABILITY

Extending results of Johnson and Zumbrun showing stability under localized (L 1 ) perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational data ū′ (x)h0(x), where ū is the background profile and h0 is the initial modulation.

Introduction

Stability and behavior of modulated periodic wave trains have received considerable recent attention, both in the reaction diffusion and conservation law contexts; see, for example [S1, S2, DSSS, JZ1, JZ2, JZN, BJNRZ1, BJNRZ2, NR1, NR2] and references therein. The initial mathematical challenge of this problem is that the linearized equations, being periodic-coefficient, have purely essential spectrum when considered as problems on the whole line, so that there is no spectral gap between neutral and other modes, making difficult either the treatment of linearized behavior or the passage from linear to nonlinear estimates.

This issue was overcome in the reaction diffusion context in the late 1990s by Schneider [S1, S2], resolving an at the time 30-year open problem. Using a method of "diffusive stability," Schneider combined diffusive-type linear estimates with renormalization techniques to show that, assuming "diffusive" spectral stability in the sense to be specified later of the background periodic wave, longtime behavior under localized perturbations is essentially described by a scalar linear convectiondiffusion equation in the phase variable, with the amplitude decaying more rapidly.

More recently, Oh-Zumbrun and Johnson-Zumbrun [START_REF] Oh | Stability and asymptotic behavior of traveling-wave solutions of viscous conservation laws in several dimensions[END_REF][START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF] in 2010 and 2011 using rather different tools coming from viscous shock theory have resolved the corresponding problem in the conservation law case, showing that, again assuming diffusive spectral stability of the background periodic wave, behavior under localized perturbations is described, roughly, by a system of viscous conservation laws in the derivative of the phase and other modulation parameters, so that the phase decays at slower, error function rate than in the reaction diffusion case. Further developments include the treatment of nonlinear stability of roll waves of the St. Venant equations [JZN], a set of partially parabolic balance laws of quasilinear type, and the resolution in [START_REF] Barker | Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation[END_REF] of the 35-year open problem of proving nonlinear stability of spectrally stable periodic Kuramoto-Sivashinsky waves.

Applied to the reaction diffusion context [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], the approach of [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF] recovers and slightly extends the results of Schneider, yielding the same heat kernel rate of decay with respect to localized perturbations while removing the assumption of [S1, S2] that nearby periodic waves have constant speed. However, one lesson of the successful analysis in [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], the conservation law case, is that one can close a nonlinear iteration for perturbations decaying at slower, error function rate, yielding hope that one could do better also for the reaction-diffusion case.

As described clearly in [DSSS], formal (WKB) asymptotics suggest the same thought, yielding the description to lowest order of a scalar viscous conservation law (1.1)

k t -ω 0 (k) x = (d(k)k x ) x
for the wave number k = k * (1 + ψ x ), where ω = ψ t is the time frequency and k * ψ is the phase shift. Here ω 0 (k) = -c(k)k and wave speed c(k) are determined through the nonlinear dispersion relation obtained from the manifold of periodic traveling-wave solutions ūk (k(xct)), and d(k) through higher-order asymptotics. See [OZ1, Se, OZ3, NR1, NR2] for corresponding developments in the conservation law case. Note that an integrated version of (1.1) tells that ω 0 (k) is a first approximation of the time frequency ω = ψ t in the low frequency regime. From (1.1), one might hope to treat initial shifts k * ψ| t=0 = k * h 0 for which, not the phase shift k * h 0 , but its derivative k * ∂ x h 0 is localized (L1 ), obtaining the same heat kernel rate of decay for k as found in the conservation law case [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF].

In this note, we show that this is indeed the case, proving by an improved version of the nonlinear iteration scheme of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF][START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF] that diffusively spectrally stable periodic reaction diffusion waves are nonlinearly stable under initial perturbations consisting of a localized perturbation plus a nonlocalized modulation h 0 , ∂ x h 0 ∈ L 1 (R), in phase, with heat kernel rate of decay in the wave number and error function decay in the phase. In a companion paper [START_REF] Johnson | Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation[END_REF], we build on these basic estimates to establish, further, time-asymptotic behavior, validating description (1.1) in the long-time limit. Recall that (1.1) has been validated in [DSSS] in the related, asymptotically-large-bounded-time small-wave number limit and for data in uniformly-local Sobolev spaces 1 , in the somewhat different sense of showing that there exists a δ-family of solutions of the full system δ M -close to a formal expansion2 in δ on intervals [0, T /δ 2 ], where M and T are arbitrarily large and δ is the characteristic size of the wave number of the modulation: that is, they build "prepared" data for bounded intervals rather than work with general ones globally in time.

Consider a periodic traveling-wave solution of a reaction diffusion system

u t = u xx + f (u), u ∈ R n ,
or, equivalently, a standing-wave solution u(x, t) = ū(x) of

(1.2) k * u t = k 2 * u xx + f (u) + k * cu
x , where c is the speed of the original traveling wave, and wave number k * is chosen so that ū(x + 1) = ū(x). For simplicity of notation, we will follow this convention throughout the paper; that is, all periodic functions are assumed to be periodic of period one.

We make the following standard genericity assumptions:

(H1) f ∈ C K (R n ), (K ≥ 3). (H2) Up to translation, the set of 1-periodic solutions of (1.2) (with k replacing k * ) in the vicinity of ū, k = k * , forms a smooth 1-dimensional manifold {ū(k, •)} = {ū k (•)}, c = c(k). Linearizing (1.2) about ū yields the periodic coefficient equation (1.3) k * v t = k * Lv := (k 2 * ∂ 2 x + k * c∂ x + b)v, b(x) := df (ū(x)).
Introducing the one-parameter family of closed Floquet operators

(1.4) k * L ξ := e -iξx k * Le iξx = k 2 * (∂ x + iξ) 2 + k * c(∂ x + iξ) + b
acting on L 2 per ([0, 1]) with densely defined domains H 2 per ([0, 1]), determined by the defining relation (1.5)

L(e iξx g) = e iξx (L ξ g) for g periodic, we define following [S1, S2] the standard diffusive spectral stability conditions:

(D1) σ(L) ⊂ {λ | ℜλ < 0} ∪ {0}. (D2) There exists a constant θ > 0 such that σ(L ξ ) ⊂ {λ | ℜλ ≤ -θ|ξ| 2 } for each ξ ∈ [-π, π).
(D3) λ = 0 is a simple eigenvalue of L 0 .3 Notice in (D1) above, we consider L as a closed operator on L 2 (R) with densely defined domain H 2 (R).

Then, we have the following Main Theorem, extending the results of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF] to nonlocalized perturbations. Here, and throughout the paper, given two real valued functions A and B, we say that A B or that for every x ∈ dom(A) ∩ dom(B), A(x) B(x) if there exists a constant C > 0 such that A(x) ≤ CB(x) for each x ∈ dom(A) ∩ dom(B). Even in a chain of inequalities, we will also feel free to denote by C harmless constants with different values.

Theorem 1.1. Let K ≥ 3. Assuming (H1)-(H2) and (D1)-(D3), let

E 0 := ũ0 (• -h 0 (•)) -ū(•) L 1 (R)∩H K (R) + ∂ x h 0 L 1 (R)∩H K (R)
be sufficiently small, for some choice of modulation h 0 . Then, there exists a global solution ũ(x, t) of (1.2) with initial data ũ0 and a phase function ψ(x, t) with initial data h 0 such that for t > 0 and 2 ≤ p ≤ ∞,

(1.6) ũ(• -ψ(•, t), t) -ū(•) L p (R) , ∇ x,t ψ(•, t) W K+1,p (R) E 0 (1 + t) -1 2 (1-1/p) , ũ(• -ψ(•, t), t) -ū(•) H K (R) E 0 (1 + t) -1 4
, and

(1.7) ũ(•, t) -ū(•) L ∞ (R) , ψ(•, t) L ∞ (R) E 0 .
In particular, ū is nonlinearly (boundedly) stable in L ∞ (R) with respect to initial perturbations v 0 = ũ0ū for which v 0 E := inf ∂xh 0 ∈L 1 (R)∩H K (R) E 0 is sufficiently small.

Remark 1.2. It may seem more natural, and indeed is so, to introduce h 0 and ψ via v(x, 0) = ũ0 (x)ū(x + h 0 (x)), v(x, t) = ũ(x, t)ū(x + ψ(x, t)).

However, in doing so one introduces in the equation for v terms involving only ψ and thus not decaying in time; see Lemma 5.1 below. For this reason we work instead with v(x, t) = ũ(x -

ψ(x, t), t) -ū(x), that is, ũ(x, t) = ū(Y (x, t)) + v(Y (x, t), t) where Y is such that Y (x, t) -ψ(Y (x, t), t) = x, Y (y -ψ(y, t), t) = y .
Notice that we insure the existence of such a map Y by keeping, for any t, ψ(•, t) L ∞ (R) bounded and

ψ x (•, t) L ∞ (R) small. 4 It should be stressed, however, that 5 Y (x, t) = x + ψ(x, t) + O( ψ(•, t) L ∞ ψ x (•, t) L ∞ )
so that we are not so far from the natural (but inappropriate) approach. Furthermore, notice also that introducing the map Y above enables one to go back to the original unknown ũ(x, t) when desired.

Remark 1.3. As noted in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], by Sturm-Liouville considerations, (D1) cannot hold in the scalar case n = 1; hence, our analysis is relevant in the system case only, that is, in the case n > 1.

1.1. Discussion and open problems. The main new observation here is that, by suitable adaptations/improvements, we can incorporate in the nonlinear iteration scheme of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF][START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF] the prescription of an initial phase shift h 0 . Indeed, introducing a supplementary unknown ψ representing phase shift and performing the same nonlinear implicit change of variables (described in Remark 1.2) used in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF] converts the initial nonlinear perturbation ū(• + h 0 )ū(•) coming from such a phase shift, to its "linearization" h 0 ū′ , a term suitable for estimation by our iteration scheme based on linearized estimates. However, there are two important obstacles, both technical and conceptual, that must be overcome in implementing this simple observation to obtain a result.

The first concerns a mismatch between the prescribed initial phase shift h 0 and the phase shift predicted by our linearized estimates based on large-time asymptotics. In [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], this change of variables was used to capture and factor out critical linear contributions in the direction of ū′ . Yet, since only localized contributions were considered in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF][START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], there was no need to prescribe ψ(•, 0), and this issue did not arise. More precisely, there was imposed simply ψ(•, 0) ≡ 0, and the linear response was truncated by a smooth cutoff so as to return zero phase shift at initial time, consistent with the imposed data. Clearly, however, that will not work here, since we require that the initial shift be nonzero.

To this deceptively simple problem, we find a deceptively simple solution, namely to impose ψ(•, 0) = h 0 , as we must, but impose a smooth cutoff not on the linearized solution operator, but at the level of the nonlinear iteration scheme, prescribing ψ for 0 ≤ t ≤ 1 as a convex average of the initial shift h 0 and the time-asymptotic value predicted by our linearized estimates; see (7.1)-(7.2) below. This effectively reproduces in our iteration scheme the same initial layer that was the cause of this technical issue in the first place, while shifting the difficulty from a conceptual to a technical one, namely, to show that the resulting perturbation variable is localized, despite nonlocalized initial data, an issue that we must in any case already face for other reasons.

This leads us to the second and, from a technical point of view, main issue, namely, to estimate the linear contribution of data h 0 ū′ , knowing only that ∂ x h 0 is localized. This essentially reduces to proving that ∂ x S(t)(ū ′ h 0 ) ∼ ū′ s p (t)(∂ x h 0 ), where S(t) is the solution operator of the linear evolution about our wave ū and s p (t) is some averaged solution operator, well-approximated by the solution operator of equation (1.1) linearized about the constant wave number k * . Note that this translates into rigorous estimates the expected modulation-type behavior that dynamics about a periodic wave reduces at first order to dynamics about a constant state in the space of parameters describing the family of nearby periodic waves (here just wave number). In proving so, one has to be able to disregard on one hand the contribution of far-from-coperiodic modes, non-critical close-to-coperiodic modes, and higher-order correction to the critical mode starting from ū′ , and on another hand, the contribution of high frequencies of h 0 to close-to-coperiodic modes of h 0 ū′ . We perform this benefiting from the fact that the Bloch transform is both well-behaved with respect to spectral decompositions and expansions of our operator, and designed to deal with resonance-like contributions. Such details are not easily seen from the Green function description of [JZ1, JZ2, JZN]. Indeed, this seems to be an instance where frequency domain techniques detect cancelation not readily apparent by spatial-domain techniques.

In the process, we obtain an alternative proof of the basic estimates of localized terms in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], carried out entirely within the Bloch transform formulation without passing to a Green function description as in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF]. This elucidates somewhat the relation between the Bloch transform based estimates of [S1, S2] and the Green function based estimates of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF][START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], in particular the role of the Green function (integral kernel) in obtaining the estimates of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF]; see Remark 3.2. We also obtain in passing, somewhat miraculously, the estimates needed to show localization of the terms introduced by our nonlinear truncation, thus overcoming the remaining technical obstacle and completing the circle of estimates needed to close our analysis; see (4.3) and Remark 4.2.3.

Our analysis suggests that one might by an elaboration of the approach used here treat still more general perturbations converging asymptotically to any fixed periodic perturbations at ±∞, not necessarily phase shifts of a single wave, which would then time-exponentially relax to modulations following the dynamics of the corresponding periodic problem; see Remark 4.2.2. This would be an interesting direction for further investigation.

We mention, finally, that the techniques introduced here are not limited to the reaction-diffusion case, but apply equally to the conservative case treated in [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], yielding a comparable result of stability with respect to nonlocalized modulations [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF]. Indeed, this clarifies somewhat the relation between the reaction diffusion and conservation law case, revealing a continuum of models with common behavior lying between these two extremes. These issues will be reported on elsewhere.

Remark 1.4. Without change, we may treat reaction-diffusion systems with more general diagonal (rather than simple Laplacian) diffusion as considered in [DSSS]. More generally, all proofs go through whenever the linearized operator generates a C 0 semigroup and the nonlinear flow satisfies a damping estimate like (6.1); in particular, our results apply to the (sectorial) Swift-Hohenberg equations treated for localized perturbations in [S2] 6 and to the general quasilinear 2r-order parabolic case. It would be interesting to extend to the multi-dimensional case as in [U].

Note: Similar results have been obtained by different means by Sandstede and collaborators [SSSU] using a nonlinear decomposition of phase and amplitude variables as in [DSSS] together with renormalization techniques. We emphasize that, though the results are similar, 7 at a technical level these analyses are quite different. In particular, the methods of [SSSU] do not appear to apply to the conservative case treated in [JZ2, JNRZ2], for which asymptotic behavior is described by a system of viscous conservation laws, whereas our methods, having their origins in shock wave theory, are essentially designed for this; see Remark 5.2 for further discussion.

Preliminaries

Recall the Bloch solution formula for periodic-coefficient operators, (i.e., (e tL g)(ξ, x) = (e tL ξ ǧ(ξ, •))(x), a consequence of (1.5)), where L ξ is as in (1.4),

(2.2) ǧ(ξ, x) := j∈Z ĝ(ξ + 2jπ)e i2πjx (periodic in x)
denotes the Bloch transform of g, ĝ(ξ) := 1 2π R e -iξx g(x)dx the Fourier transform, and

(2.3) g(x) = π -π e iξx ǧ(ξ, x)dξ
the inverse Bloch transform, or Bloch representation of g ∈ L 2 (R).

6 See [BJNRZ1, BJNRZ2, BJNRZ3, JZN] for related analyses in such general settings.

7 For ũ0(•-h0(•))-ū(•) and ∂xh0 sufficiently small in H 2 2 (R), where g H 2 2 (R) := ρg H 2 (R) with ρ(x); = (1+x 2 ) 1/2 , the results of [SSSU] yield the decay rates obtained here in (1.6) assuming smallness in

L 1 (R) ∩ H 3 (R) (reducible to smallness in L 1 (R) ∩ H 1 (R) as described in Remark 4.2, [JZ1]
). That is, they give the same conclusions assuming the stronger localization of initial perturbations of order ∼ |x| -5/2 as compared to the order ∼ |x| -1 assumed here.

Note that, in view of the inverse Bloch transform formula (2.3), the generalized Hausdorff-Young inequality u

L p (R) ≤ ǔ L q ([-π,π],L p ([0,1])) for q ≤ 2 ≤ p and 1 p + 1 q = 1 [JZ1], yields for any 1-periodic functions g(ξ, •), ξ ∈ [-π, π], (2.4) π -π e iξ• g(ξ, •)dξ L p (R) ≤ (2π) 1/p g L q ([-π,π],L p ([0,1])) for q ≤ 2 ≤ p and 1 p + 1 q = 1,
where, here and elsewhere, we are denoting

g L q ([-π,π],L p ([0,1])) := π -π g(ξ, •) q L p ([0,1]) dξ 1/q .
This convenient formulation is the one by which we will obtain all of our linear estimates. Furthermore, notice by (D3) that there exists a simple zero eigenfunction ū′ of L 0 , which by standard perturbation results [K] thus bifurcates to an eigenfunction φ(ξ, •), with associated left eigenfunction φ(ξ, •) and eigenvalue

(2.5) λ(ξ) = aiξ -dξ 2 + O(|ξ| 3 ), |ξ| ≤ ξ 0 ,
where a and d are real with d > 0 by assumption (D2) and the complex symmetry, λ(ξ) = λ(-ξ).

Again by standard perturbation results [K], each of the functions φ, φ, and λ are analytic in ξ and well-defined for |ξ| ≤ ξ 0 sufficiently small.

Basic linear estimates

Loosely following [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF] decompose the solution operator as

(3.1) S(t) = S p (t) + S(t), S p (t) = ū′ s p (t), with (3.2) (s p (t)g)(x) = π -π e iξx α(ξ)e λ(ξ)t φ(ξ, •), ǧ(ξ, •) L 2 ([0,1]) dξ, and
(3.3) ( S(t)g)(x) := π -π e iξx (1 -α(ξ))(e L ξ t ǧ(ξ))(x)dξ + π -π e iξx α(ξ)(e L ξ t Π(ξ)ǧ(ξ))(x)dξ + π -π e iξx α(ξ)e λ(ξ)t (φ(ξ, x) -φ(0, x)) φ(ξ), ǧ(ξ) L 2 ([0,1]) dξ,
where α is a smooth cutoff function such that α(ξ) = 0 for |ξ| ≥ ξ 0 and α(ξ

) = 1 for |ξ| ≤ 1 2 ξ 0 , Π p (ξ) := φ(ξ) φ(ξ), • L 2 ([0,1])
denotes the eigenprojection onto the eigenspace Range{φ(ξ)} bifurcating from Range{ū ′ (x)} at ξ = 0, φ the associated left eigenfunction, and Π := Id-Π p , each well-defined on suppt α ⊂ [-ξ 0 , ξ 0 ].

We begin by reproving the following estimates established (in a slightly different form) in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], describing linear behavior under a localized perturbation g ∈ L 1 (R) ∩ L2 (R).

Proposition 3.1 ([JZ1]). Under assumptions (H1)-(H2) and (D1)-(D3), for all

t > 0, 2 ≤ p ≤ ∞, ∂ l x ∂ m t s p (t)∂ r x g L p (R) min (1 + t) -1 2 (1-1/p)-l+m 2 g L 1 (R) , (1 + t) -1 2 (1/2-1/p)-l+m 2 g L 2 (R) , (3.4)
for 0 ≤ r ≤ K + 1, and for some η > 0 and 0 ≤ l + 2m, r ≤ K + 1,

∂ l x ∂ m t S(t)∂ r x g L p (R) min (1 + t) -1 e -ηt ∂ r x g H l+2m+1 (R) + (1 + t) -1 2 (1/2-1/p)-1 2 g L 2 (R) , (3.5)
Estimates (3.4)-(3.5) were established in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF] by first passing to a Green function formulation. Here, we give an alternative proof within the Bloch formulation that will be useful for what follows.

Proof. (i) (Proof of (3.4)(i)). First, expand

(3.6) (s p (t)g)(x) = π -π α(ξ)e λ(ξ)t e iξx φ, ǧ L 2 ([0,1]) (ξ)dξ = j∈Z π -π α(ξ)e λ(ξ)t e iξx φ(ξ, •), e i2πj • L 2 ([0,1]) ĝ(ξ + 2jπ)dξ = j∈Z π -π α(ξ)e λ(ξ)t e iξx φj (ξ) * ĝ(ξ + 2jπ)dξ,
where φj (ξ) denotes the jth Fourier coefficient in the Fourier expansion of periodic function φ(ξ, •), and z * = z denotes complex conjugate. By Hausdorff-Young's inequality, ĝ L ∞ (R) ≤ g L 1 (R) , and by (D2), |e λ(ξ)t α 1/2 (ξ)| ≤ e -ηξ 2 t , η > 0, while by Cauchy-Schwarz' inequality,

α 1/2 (ξ) j | φj (ξ)| ≤ α 1/2 (ξ) j (1 + |j| 2 )| φj (ξ)| 2 j (1 + |j| -2 ) ≤ Cα 1/2 (ξ) φ(ξ) H 1 ([0,1]) .
Combining these facts, and applying (2.4), we obtain for 1/q + 1/p = 1 (3.7)

s p (t)g L p (R) ξ → e -ηξ 2 t L q ([-π,π]) sup |ξ|≤ξ 0 φ(ξ, •) H 1 ([0,1]) g L 1 (R) (1 + t) -1 2 (1-1/p) g L 1 (R) ,
yielding the result for l = m = r = 0. Estimates for general l, m, r ≥ 0 go similarly, passing ∂ r x derivatives onto φ in the inner product using integration by parts and noting that ∂ l x and ∂ m t derivatives bring down harmless bounded factors (iξ) l and λ(ξ) m enhancing decay.

(ii) (Proof of (3.4)(ii)). The second estimate on s p follows similarly, but substituting for the estimate of term π -π e iξx α(ξ)e λ(ξ)t φ(ξ), ǧ(ξ) L 2 ([0,1]) dξ the slightly simpler estimate

x → π -π e iξx α(ξ)e λ(ξ)t φ(ξ), ǧ(ξ) L 2 ([0,1]) dξ L p (R) (ξ, x) → α(ξ)e -ηξ 2 t | φ(ξ), ǧ(ξ) L 2 ([0,1]) | L q ([-π,π],L p ([0,1])) ξ → e -ηξ 2 t ǧ(ξ) L 2 ([0,1]) L q ([-π,π]) ξ → e -ηξ 2 t L r 1 q ([-π,π]) g L 2 (R) (1 + t) -1 2 (1/2-1/p) g L 2 (R) ,
where 1/r 1 + 1/r 2 = 1 and qr 2 = 2, so that r 1 q = 2q 2-q equals ∞ for p = q = 2, while it equals 2 for q = 1, p = ∞. This verifies the result for l = m = r = 0; estimates for general l, m, r ≥ 0 go similarly, as described in (i) above.

(iii) (Proof of (3.5)(1)). By (D1)-(D2) and Prüss' Theorem [Pr], 8 we have

|e L ξ t (1 -α(ξ))| H l+1 ([0,1])→H l+1 ([0,1]) , |α(ξ)e L ξ t Π(ξ)| H l+1 ([0,1])→H l+1 ([0,1]) e -ηt , η > 0,
8 Here, along with standard parabolic resolvent estimates [He], we are using the fact (by smoothness of coefficients and basic ODE regularity theory) that H l+1 ([0, 1]) and L 2 ([0, 1]) spectra coincide for the operators considered here.

whence, by Sobolev embedding,

|∂ l x e L ξ t (1 -α(ξ))| H l+1 ([0,1])→L p ([0,1]) , |∂ j x α(ξ)e L ξ t Π(ξ)| H l+1 ([0,1])→L p ([0,1]) e -ηt
for 2 ≤ p ≤ ∞. The W l,p (R) norms of the first two terms of (3.3), by (2.4) and Parseval's identity,

ǧ L 2 ([-π,π],H l+1 ([0,1])) ∼ g H l+1 (R) , more precisely 1 2π g 2 H l+1 (R) = ǧ 2 L 2 ([-π,π],L 2 ([0,1])) + ǧ 2 L 2 ([-π,π], Ḣl+1 ξ ([0,1])) := ǧ 2 L 2 ([-π,π],L 2 ([0,1])) + (∂ x + iξ) l+1 ǧ(ξ) 2 L 2 ([-π,π],L 2 ([0,1])) ,
are thus bounded by Ce -ηt g H l+1 (R) . The W l,p (R) norm of the third term may be bounded similarly as in the estimation of s p above, noting that the factor (φ(ξ)φ(0)) ∼ |ξ| introduces an additional factor of (1 + t) -1/2 decay. This establishes the result for m = r = 0; other cases go similarly, noting that ∂ t e L ξ t Π(ξ) = L ξ e L ξ t Π(ξ), with L ξ a second-order operator, so that we may essentially trade one t-derivative for two x-derivatives.

(iv) (Proof of (3.5)(ii)). The second estimate on S follows similarly, but substituting for the estimate of term

π -π e iξx α(ξ)e λ(ξ)t (φ(ξ, x) -φ(0, x)) φ(ξ), ǧ(ξ) L 2 ([0,1]) dξ the estimate x → π -π e iξx α(ξ)e λ(ξ)t (φ(ξ, x) -φ(0, x)) φ(ξ), ǧ(ξ) L 2 ([0,1]) dξ L p (R) (ξ, x) → α(ξ)e -ηξ 2 t |ξ|| φ(ξ), ǧ(ξ) L 2 ([0,1]) | L q (ξ,L p ([0,1])) ξ → |ξ|e -ηξ 2 t ǧ(ξ) L 2 ([0,1]) L q ([-π,π]) ξ → |ξ|e -ηξ 2 t L rq ([-π,π]) g L 2 (R) (1 + t) -1 2 (1/2-1/p)-1 2 g L 2 (R) ,
where 1/p + 1/q = 1 and 1/2 + 1/r q = 1/q, so that r q = 2q 2-q is ∞ for p = q = 2 and 2 for q = 1, p = ∞. Remark 3.2. In [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], by contrast, s p (t) is estimated through its integral kernel e(x, t; y) := (s p (t)δ y )(x) = π -π α(ξ)e λ(ξ)t e iξx φ(ξ, y)dξ, yielding the slightly sharper estimate

s p (t)g L p (R) g L 1 (R) ≤ sup y e(•, t; y) L p (R) ≤ sup |ξ|≤ξ 0 φ(ξ) L ∞ ([0,1]) (1 + t) -1/2(1-1/p) in place of the bound C sup |ξ|≤ξ 0 φ(ξ) H 1 ([0,1]) (1+t) -1/2(1-1/p) computed in (3.7).
For our purposes, this makes no difference, and, as we shall see in Section 4, there can be an advantage in maintaining the separation into distinct frequencies afforded by the Bloch representation.

Linear behavior for modulational data

Next, we consider behavior of (2.1) when applied to modulational data g = h 0 ū′ , the linearized version of a nonlinear modulational perturbation ū(x + h 0 (x))ū(x) ∼ h 0 (x)ū ′ (x). The following estimates, obtained by frequency-domain rather than spatial-domain (Green function) techniques as in [JZ1, JZ2, JZN], together with the associated modified nonlinear iteration scheme of Section 7, below, represent the main new technical contributions of this paper.

Proposition 4.1. Under assumptions (H1)-(H2) and (D1)-(D3), for all t > 0, 2 ≤ p ≤ ∞,

(4.1) ∂ l x ∂ m t s p (t)(h 0 ū′ ) L p (R) (1 + t) -1 2 (1-1/p)+ 1 2 -l+m 2 ∂ x h 0 L 1 (R) ,
for l + m ≥ 1 or else l = m = 0 and p = ∞, and, for 0

≤ l + 2m ≤ K + 1, (4.2) ∂ l x ∂ m t S(t)(h 0 ū′ ) L p (R) (1 + t) -1 2 (1-1/p) ∂ x h 0 L 1 (R)∩H l+2m+1 (R) ,
and when t ≤ 1

(4.3) ∂ l x ∂ m t (S p (t) -Id)(h 0 ū′ ) L p (R) ∂ x h 0 L 1 (R)∩H l+2m+1 (R) , ∂ l x ∂ m t (s p (t)(h 0 ū′ ) -h 0 ) L p (R) ∂ x h 0 L 1 (R)∩L 2 (R) .
Proof. (i) (Proof of (4.1), l + m ≥ 1). We treat the case l = 1, m = 0; other cases go similarly. First, re-express (4.4)

∂ x (s p (t)(ū ′ h 0 ))(x) = π -π iξα(ξ)e λ(ξ)t e iξx φū ′ , ȟ0 L 2 ([0,1]) (ξ)dξ = j∈Z π -π iξα(ξ)e λ(ξ)t e iξx φ(ξ, y)ū ′ (y), e i2πjy L 2 ([0,1]) ĥ0 (ξ + 2jπ)dξ = j∈Z π -π iξα(ξ)e λ(ξ)t e iξx φ(ξ)ū ′ * j ĥ0 (ξ + 2jπ)dξ = j∈Z π -π ξ ξ + 2πj α(ξ)e λ(ξ)t e iξx φ(ξ)ū ′ * j ∂ x h 0 (ξ + 2jπ)dξ,
where φ(ξ)ū ′ j denotes the jth Fourier coefficient in the Fourier expansion of periodic function φ(ξ)ū ′ . By (2.4) and |e λ(ξ)t α 1/2 (ξ)| ≤ e -ηξ 2 t , η > 0, we thus get

∂ x (s p (t)(ū ′ h 0 )) L p (R) ≤ C(1 + t) -1 2 (1-1/p) ∂ x h 0 L 1 (R) sup |ξ|≤ξ 0 j φ(ξ)ū ′ j 1 + |j| ,
yielding the result together with the Cauchy-Schwarz estimate

j φ(ξ)ū ′ j 1 + |j| ≤ j (1 + |j|) -2 j | φ(ξ)ū ′ j | 2 φ(ξ)ū ′ L 2 ([0,1]) .
(ii) (Proof of (4.1), l = m = 0). 9 This follows by an explicit error function decomposition analogous to that used in [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], p. 18, to treat the conservative case. Computing similarly as in (4.4)

(4.5) (s p (t)(ū ′ h 0 ))(x) = j∈Z π -π 1 ξ + 2πj α(ξ)e λ(ξ)t e iξx φ(ξ)ū ′ * j ∂ x h 0 (ξ + 2jπ)dξ
9 This estimate, which lies apart from the rest of the arguments in the paper, is not needed for the nonlinear iteration. Moreover, the resulting bounds (1.7) may be recovered alternatively by the modulation decomposition of [START_REF] Johnson | Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation[END_REF] without case l = 0. Thus, this case might be skipped by the reader if desired.

we obtain by exactly the same computation as in (i) an estimate on high-frequency terms

x → j =0 π -π e iξx 1 ξ + 2πj α(ξ)e λ(ξ)t φ(ξ)ū ′ * j ∂ x h 0 (ξ + 2jπ)dξ L p (R) ≤ C(1 + t) -1 2 (1-1/p) ∂ x h 0 L 1 (R) sup |ξ|≤ξ 0 j φ(ξ)ū ′ j 1 + |j|
that is stronger than we need, leaving only the estimation of the zero-frequency term

φ(ξ)ū ′ * 0 π -π e iξx α(ξ)e λ(ξ)t ξ ∂ x h 0 (ξ)dξ.
Recalling expansion (2.5) of λ(ξ) and using smoothness of α, we have

(4.6) π -π e iξx α(ξ)e λ(ξ)t ξ ∂ x h 0 (ξ)dξ = +∞ -∞ e iξx e (-iaξ-bξ 2 )t ξ ∂ x h 0 (ξ)dξ - |ξ|≥π e iξx e (-iaξ-bξ 2 )t ξ ∂ x h 0 (ξ)dξ + π -π e iξx O(e -ηξ 2 t | ∂ x h 0 (ξ)|)dξ =: I + II + III.
Terms II and III are readily seen by Hausdorff-Young's inequality to be bounded by a stronger estimate

C ∂ x h 0 L 1 (R) (1 + t) -1 2
(1-1/p) than we need. Summarizing, the only non-negligible term in this computation is term I, which may be explicitly evaluated as the convolution of ∂ x h 0 with e(x, t) := P.V.

+∞ -∞

e iξx e (-aiξ-dξ 2 )t iξ dξ = errfn((xat) 2 / √ t).

Estimating e * ∂ x h 0 L ∞ (R) ≤ e L ∞ (R) ∂ x h 0 L 1 (R) ≤ ∂ x h 0 L 1 (R)
, we are done.

(iii) (Proof of (4.2)). Likewise, this follows by re-expressing (4.7)

S(t)(h 0 ū′ )(x) = π -π e iξx (1 -α(ξ))(e L ξ t ȟ0 (ξ)ū ′ )(x)dξ + π -π e iξx α(ξ)(e L ξ t Π(ξ) ȟ0 (ξ)ū ′ )(x)dξ + π -π e iξx α(ξ)e λ(ξ)t (φ(ξ, x) -φ(0, x)) φ(ξ), ȟ0 (ξ)ū ′ L 2 ([0,1]) dξ = j∈Z π -π e iξx (1 -α(ξ)) (e L ξ t (ū ′ e 2iπj• ))(x) i(ξ + 2πj) ∂ x h 0 (ξ + 2jπ)dξ + j∈Z π -π e iξx α(ξ) (e L ξ t Π(ξ)(ū ′ e 2ijπ• ))(x) i(ξ + 2πj) ∂ x h 0 (ξ + 2jπ)dξ + j∈Z π -π e iξx α(ξ)e λ(ξ)t (φ(ξ, x) -φ(0, x)) φū ′ j (ξ) * i(ξ + 2πj) ∂ x h 0 (ξ + 2jπ)dξ
then estimating as before, where we are using (1α(ξ)) ξ,

Π(ξ)ū ′ = Π(ξ) ( Π(ξ) -Π(0))ū ′ with | Π(ξ) -Π(0)| H l+1 ([0,1])→H l+1 ([0,1]) |ξ|,
and φ(ξ)φ(0) = O(ξ), respectively, to bound the key terms

(1-α(ξ)) i(ξ+2πj) , Π(ξ)(ū ′ e 2ijπ• ) i(ξ+2πj)
, and φ(ξ)-φ(0) i(ξ+2πj)

appearing for j = 0, and are using the Cauchy-Schwarz inequality

j (2πij) l+1 ∂ l+2 x h 0 (ξ + 2jπ) (ξ + 2jπj) l+2 ≤ C j 1 (1 + |j|) 2 j ′ | ∂ l+2 x h 0 (ξ + 2j ′ π)| 2 ≤ C ∂ x h 0 H l+1 (R)
to bound the sum over j = 0 arising for the first and second terms on the righthand side of (4.7) upon estimating as in part (iii) of the proof of Proposition 3.1

|∂ l x e L ξ t (1 -α(ξ))| H l+1 ([0,1])→L p ([0,1]) , |∂ j x α(ξ)e L ξ t Π(ξ)| H l+1 ([0,1])→L p ([0,1]) e -ηt and (ū ′ e 2iπj• ))(x) i(ξ + 2πj) ∂ x h 0 (ξ + 2jπ) H l+1 ([0,1]) (2πij) l+1 ∂ l+2 x h 0 (ξ + 2jπ) (ξ + 2πj) l+2
.

This establishes the result for m = 0; other cases go similarly, noting again that we may trade t-derivatives for x-derivatives using ∂ t e L ξ t Π(ξ) = L ξ e L ξ t Π(ξ),

(iv) (Proof of (4.3)). Expanding S p (t) -Id = (S p (t) -S p (0)) -S(0) = t∂ t S p (s(t)) -S(0) for some 0 < s(t) < t, we obtain the first inequality by combining (4.1) and (4.2). Likewise, the second inequality follows by expanding s p (t)(h 0 ū′ )h 0 = (s p (t)s p (0))(h 0 ū′ ) + (s p (0)(h 0 ū′ )h 0 ) and applying (4.1) together with

s p (0)(h 0 ū′ ) -h 0 L p (R) = π -π e iξx (1 -α(ξ)) j ĥ0 (ξ + 2jπ) φ(ξ)ū ′ * j dξ + π -π e iξx j =0 ĥ0 (ξ + 2jπ) φ(ξ)ū ′ * j -1 dξ L p (R) ≤ C sup ξ≤ξ 0 j |ξ|| φ(ξ)ū ′ j | |ξ+2jπ| ∂ x h 0 L 1 (R) + C ξ → j | ∂xh 0 (ξ+2jπ)| |ξ+2jπ| L 2 (R) ≤ C ∂ x h 0 L 1 (R)∩L 2 (R)
This establishes the case l = m = 0; other cases go similarly.

Remark 4.2. 1. If we split h 0 into high-frequency and low-frequency parts, then the contribution of the high-frequency part decays faster by factor (1 + t) -1/2 in all estimates (4.1)-(4.3). 2. In the estimate of s p , it is easy to see that the same bounds hold if ū′ is replaced by any periodic g ∈ H 1 per ([0, 1]). However, in the estimate of S, replacing ū′ by a periodic g ∈ H 1 per ([0, 1]) that is not in ker Π(0) = Span{ū ′ } introduces a time-exponentially decaying but spatially nonlocalized error π -π e iξx h 0 (ξ) e tL ξ Π(0)(g) dξ, lying at best in L ∞ (R) but unbounded in every L p (R), p < ∞, reflecting the dynamics of the problem on a periodic domain. That is, in this case, S(t)(gh 0 ) ∈ L p (R), p < ∞, violating (4.2) for l = m = 0. To extend our results to the sum of a localized perturbation and a perturbation asymptotic as x → ±∞ to any two fixed periodic waves, not necessarily two shifted copies of the same wave, appears to require the estimation of this latter error term as a time-exponentially decaying function from L ∞ (R) → L ∞ (R), a semigroup/Fourier multiplier problem of concrete technical nature. This would be an interesting direction for further investigation.

3. The estimate (4.3) is the key to handling the "initial layer problem" in our later nonlinear iteration, allowing us to prescribe initial data for ψ as is convenient for the analysis; see Remark 7.2.

Nonlinear perturbation equations

Essentially following [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], for ũ(x, t) satisfying k * ũt = k 2 * ũxx + f (ũ) + k * cũ x and ψ(x, t) to be determined, set (5.1) u(x, t) = ũ(xψ(x, t), t) and v(x, t) = u(x, t)ū(x).

Lemma 5.1 ([JZ1]

). The nonlinear residual v defined in (5.1) satisfies

(5.2) k * (∂ t -L) (v + ψ ūx ) = k * N , k * N = Q + R x + (k * ∂ t + k 2 * ∂ 2 x )S + T , where (5.3) Q := f (v + ū) -f (ū) -df (ū)v, R := -k * vψ t -k 2 * vψ xx + k 2 * (ū x + v x ) ψ 2 x 1 -ψ x , S := vψ x , and
T := -(f (v + ū) -f (ū)) ψ x .
Proof. (Following [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF]. 10 ) From definition (5.1) and the fact that ũ satisfies (1.2), we obtain

(5.4) k * (1 -ψ x )u t + k * (-c + ψ t )u x = k 2 * 1 1 -ψ x u x x + (1 -ψ x )f (u) hence, subtracting the profile equation -k * cū x = k 2 * ūxx + f (ū) for ū, (5.5) k * v t -k * Lv + k * ψ t v x -k * ψ x v t + k * ψ t ūx = Q -ψ x f (u) + k 2 * ψ x 1 -ψ x u x x or (5.6) k * (∂ t -L)v + k * (ψ ūx ) t = Q + R x + (k * ∂ t + k 2 * ∂ 2 x )S -ψ x f (u) + k 2 * (ψ x ūx )
x , and we finish the proof using Lū x = 0 and the profile equation for ū to obtain (5.7)

k * L(ψ ūx ) = k * cψ x ūx + k 2 * ψ x ūxx + k 2 * (ψ x ūx ) x = -ψ x f (ū) + k 2 * (ψ x ūx ) x .
Remark 5.2. The decomposition (5.2) has the effect of grouping linear order source terms ψ t ūx and ψ x ūxx that are individually too large to handle in our later nonlinear iteration into a single term k * (∂ t -L) (ψ ūx ) with explicitly evaluable contribution ψ ūx that may be canceled with nondecaying linear translational modes s p by a judicious choice of ψ (see (7.1) below). For the origins of this approach in the "instantaneous tracking method" of viscous shock theory, see [Z1] (Eq. (2.30)) and especially [HoZ] (in particular, the nearly identical Eq. (5.23) of Cor. 5.4, p. 453); for a general discussion of the method in the context of viscous shock waves, see [Z2]. For related analyses in different contexts featuring "pullback" coordinatizations analogous to (5.1), see [TZ1, Section 3] and [TZ2, Section 2.2.1] (group invariance and uniqueness), [Z3, Section 3] (translationinvariant center-stable manifold), and [TZ3, Theorem 2.2.0] (Nash-Moser uniqueness theorem), and references therein. Similar, "pullback"-type, coordinatizations are used in an important way in [DSSS, SSSU]; however, to avoid any danger of confusion, we emphasize that our approach, here and in [JZ1, JZ2], (i) originates from a different branch of the stability literature [Z1, HoZ], substantially predating [DSSS], 11 and (ii) is fundamentally different from those of [DSSS, SSSU] in the central aspect of the way of detecting cancelation. 12 That is, the common use of coordinatization (5.1), though important, is only a necessary first step eliminating grossly infeasible nondecaying source terms and not the essential point of these various approaches. Our approach here has the advantage that it extends to the more difficult conservation law case [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF][START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF], whose more elaborate asymptotic behavior would appear to greatly complicate an approach via normal forms.

10 Note that we have here followed a different convention than in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], reversing the sign of ψ in (5.1) in agreement with formal asymptotics of [Se, DSSS]. The change ψ → -ψ recovers the formulas of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF].

11 The "pullback" coordinatization was introduced in the periodic reaction-diffusion-context in [DSSS], and, separately, in the at-the-time seemingly technically-unrelated periodic conservation law context in [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF].

12 Done here by subtracting out the expected principle linear response k * ψ ūx from the solution and observing that the resulting source terms are small; done in [DSSS, SSSU] by the use of "mode filters," or approximate spectral projectors, and what might be described as the method of normal forms, i.e., successive nonlinear approximations.

Nonlinear damping estimate

Proposition 6.1 ([JZ1]). Assuming (H1)-(H2), let v(•, 0) ∈ H K (R) (for v as in (5.1) and K ≥ 3 as in (H1)) and suppose that for some T > 0, the H K (R) norm of v(t) and ψ t (t) and the H K+1 (R) norm of ψ x (t) remain bounded by a sufficiently small constant for all 0 ≤ t ≤ T . Then there are positive constants θ and C, independent of T , such that, for all 0 ≤ t ≤ T , (6.1)

v(t) 2 H K (R) ≤ C e -θt v(0) 2 H K (R) + C t 0 e -θ(t-s) v(s) 2 L 2 (R) + (ψ t , ψ x )(s) 2 H K (R) ds.
Proof. (Following [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF].) Take for writing simplicity k * = 1. Rewriting (5.6) as

(6.2) (1 -ψ x )v t -v xx -cv x = df (ū)v + Q -(ū x + v x )ψ t + ((ū x + v x )ψ x ) x + (ū x + v x ) ψ 2 x 1 -ψ x x -f (ū + v)ψ x , taking the L 2 (R) inner product against K j=0 (-1) j ∂ 2j x v
1-ψx , integrating by parts, and rearranging, we obtain

d dt v 2 H K (R) (t) ≤ -θ ∂ K+1 x v(t) 2 L 2 (R) + C v(t) 2 H K (R) + (ψ t , ψ x )(t) 2 H K (R) ,
for some θ > 0, so long as (v, ψ t , ψ x , ψ xx )(t) H K (R) remains sufficiently small. Sobolev interpolation

g 2 H K (R) ≤ C-1 ∂ K+1 x g 2 L 2 (R) + C g 2 L 2 (R) gives, then, for C > 0 sufficiently large, d dt v 2 H K (R) (t) ≤ -θ v(t) 2 H K (R) + C v(t) 2 L 2 (R) + (ψ t , ψ x )(t) 2 H K (R)
, from which (6.1) follows by Gronwall's inequality.

Nonlinear iteration scheme

The key idea is, similarly as in the localized case treated in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], starting with ), to choose ψ to cancel s p contributions, as (7.1)

(∂ t -L)(v + ψ ū′ ) = N , v| t=0 = d 0 , ψ| t=0 = h 0 , where d 0 := ũ0 (• -h 0 (•)) -ū ∈ L 1 (R) ∩ H K (R), ∂ x h 0 ∈ L 1 (R) ∩ H K (R
ψ(t) = s p (t)(h 0 ū′ + d 0 ) + t 0 s p (t -s)N (s)ds -(1 -χ(t)) s p (t)(d 0 + h 0 ū′ ) -h 0 + t 0 s p (t -s)N (s)ds ,
where χ(t) is a smooth cutoff that is zero for t ≤ 1/2 and one for t ≥ 1, leaving the system (7.2)

v(t) = S(t)(d 0 + h 0 ū′ ) + t 0 S(t -s)N (s)ds + (1 -χ(t)) S p (t)d 0 + (S p (t) -Id)h 0 ū′ + t 0 S p (t -s)N (s)ds .
We may extract from (7.1)-(7.2) a closed system in (v, ψ x , ψ t ) (and some of their derivatives), and then recover ψ through the slaved equation (7.1).

Remark 7.1. At first sight, we have accomplished nothing by introducing a ψ-dependent change of variable and choosing ψ(0) = h 0 , since we have replaced the nonlocalized perturbation ũ0 (x)ū(x) used in the previous h 0 ≡ 0 setup of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], by a different nonlocalized perturbation d 0 + h 0 ū′ . However, what we really did was replace the asymptotic states ū(x + c ± )ū(x) by their linear approximants c ± ū′ (x), which removes the key difficulty of higher order remainders in the Taylor expansion of nonlinear modulations.

Remark 7.2. Notice that modulational data ū′ h 0 enters in (7.2) only through operators S(t) and (1-χ(t))(S p (t)-Id) for which we have Gaussian decay in L p (R) with respect to ∂ x h 0 L 1 (R)∩H 1 (R) , hence the error incurred by defining ψ by (7.1) instead of the value ψ(t) = s p (t)(h 0 ū′ + d 0 ) + t 0 s p (t-s)N (s)ds exactly canceling s p terms is harmless to our analysis. The choice of (7.1) reflects our need to accommodate the incompatibility between the initial value ψ| t=0 = h 0 prescribed by the spatially-asymptotic behavior of the initial perturbation and the function ψ encoding timeasymptotic behavior of the perturbed solution; that is, it is a device to avoid having to resolve an initial layer near t = 0. 13 Whether this initial layer is an artifact of our analysis or reflects some aspect of short-time behavior is unclear; as the estimates show, this is below our level of resolution.

Nonlinear stability

With these preparations, the proof of stability now goes essentially as in the localized conservative case treated in [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], using the new linear modulation bounds to estimate the new linear term coming from data h 0 ū′ in (7.1) and (7.2). As noted in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], from differential equation (5.2) together with integral equation (7.1)-(7.2), we readily obtain short-time existence and continuity with respect to t of solution (v, ψ t , ψ x ) ∈ H K (R) by a standard contraction-mapping argument treating the linear df (ū)v term of the left hand side along with Q, R, S, T , ψū ′ terms of the righthand side as sources in the heat equation. Associated with this solution define so long as it is finite, (1 + ts) -1 2 (1-1/p)-1/2 (1 + s) -1 2 ds

(
≤ C p (E 0 + ζ(t) 2 )(1 + t) -1 2 (1-1/p) .
Estimate (8.5) yields in particular that (ψ t , ψ x )(t) H K+1 (R) is small, verifying the hypotheses of Proposition 6.1. From (6.1) and (8.4)-(8.5), we thus obtain 13 In the case h0 ≡ 0, this essentially reduces to the simpler device used in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF] to treat the localized case, of substituting χ(t)s p (t) for s p (t). However, the latter is clearly too crude to treat the present case. and achieve the proof of (1.6) for 2 ≤ p ≤ ∞.

Estimate (1.7) then follows through (7.1) using (3.4)(i), by (8.10) ψ(t) L ∞ (R) ≤ CE 0 + CE 0 t 0

(1 + ts) -1 2 (1 + s) -1 2 ds ≤ CE 0 , yielding nonlinear stability by the fact that (8.11) ũ(xψ(x, t), t)ū(xψ(x, t)) = v(x, t) + ū(x)ū(xψ(x, t)), so that

ũ(t) -ū L ∞ (R) ≤ v(t) L ∞ (R) + ū′ L ∞ ([0,1]) ψ(t) L ∞ (R) .
Acknowledgement. Thanks to Björn Sandstede for pointing out the results of [SSSU]. Thanks also to the anonymous referees for several suggestions that substantially improved the exposition. 15 We bound the ∂tS contribution using t 0 s p (ts)∂tS(s)ds = -t 0 ∂t[s p ](ts)S(s)ds + s p (0)S(t)s p (t)S(0).

  (2.1) (S(t)g)(x) := (e tL g)(x) = π -π e iξx (e tL ξ ǧ(ξ, •))(x)dξ

  , ψ x )(t) W K+1,p (R) ≤ C(1 + t) -1 2 E 0 + Cζ(t) 2

	Proof. 14 By (5.3) and corresponding bounds on the derivatives together with definition (8.1), and
	using (6.2) to bound v t ,
	(8.3)	N (t) L 1 (R)∩H 1 (R)	(v, ψ t , ψ x )(t) 2 H 3 (R) ≤ Cζ(t) 2 (1 + t) -1 2 ,
	so long as ζ(t) remains small. Applying the bounds (3.4)(1)-(3.5)(1) and (4.1)-(4.3) of Propositions
	3.1 and 4.1 to system (7.1)-(7.2), we obtain for any 2 ≤ p < ∞ (8.4) v(t) L p (R) ≤ C(1 + t) -1 2 (1-1/p) E 0 + Cζ(t) 2 t 0 (1 + t -s) -1 2 (1-1/p)-1 2 (1 + s) -1 2 ds
	and	2 (1-1/p) ≤ C p (E 0 + ζ(t) 2 )(1 + t) -1
	(8.5)	(ψ t t 0

8.1) ζ(t) := sup 0≤s≤t (v, ψ t , ψ x )(s) H K (R) (1 + s) 1/4 . Lemma 8.1. There exist positive constants C and ε such that if E 0 := (d 0 , ∂ x h 0 ) L 1 (R)∩H K (R) is less than ǫ, then for t ≥ 0 such that ζ(t) is finite and less than ε 1/2 , (8.2) ζ(t) ≤ C(E 0 + ζ(t) 2 ).

  14 Compare to the argument of [JZ2, Lemma 4.2] regarding localized perturbations in the conservative case.H K (R) ≤ C(E 0 + ζ(t) 2 )(1 + t) -1 4 .Combining this with (8.5), p = 2, rearranging, and recalling definition (8.1), we obtain the result.Proof of Theorem 1.1. By short-time H K existence theory, (v, ψ t , ψ x )(t) H K (R) is continuous so long as it remains small, hence ζ remains continuous so long as it remains small. By (8.1), therefore, it follows by continuous induction that ζ(t) ≤ 2CE 0 for t ≥ 0, if E 0 < 1/4C, yielding by (8.1) the result (1.6) for p = 2. Applying (8.4)-(8.6), we obtain (1.6) for 2 ≤ p ≤ p * for any p * < ∞, with uniform constant C. Taking p * > 4 and estimating(8.7) (Q, R, S, T )(t) L 2 (R) (v, ψ t , ψ x , ψ xx )(t) 2 L 4 (R) ≤ CE 0 (1 + t) -3 4in place of the weaker (8.3), then applying (3.4)(ii) in place of (3.4)(i), we obtain (8.8) (ψ t , ψ x )(t) W K+1,p (R) ≤ C(1 + t) -1 2 E 0 + CE 0

	(8.6)	v(t) t 0	(1 + t -s) -1 2 (1/2-1/p)-1/2 (1 + s) -3 4 ds
		≤ CE 0 (1 + t) -1 2 (1-1/p) ,
					-1 2 ds
		+ CE 0	0	t	(1 + t -s) -1 2 (1/2-1/p)-1 2 (1 + s) -3 4 ds
		≤ CE 0 (1 + t) -1 2 (1-1/p)

for 2 ≤ p ≤ ∞.

15

Likewise, using (8.7) together with bound

(Q, T )(t) H 1 (R) + R(t) H 2 (R) + S(t) H 3 (R) ζ(t) 2 (1 + t) -1 2

obtained in the course of proving (8.3), we may use (3.5)(ii) rather than (3.5)(i) to get (8.9)

v(t) L p (R) ≤ C(1 + t) -1 2 (1-1/p) E 0 + CE 0 t 0 e -η(t-s) (1 + s)

Specifically, ψx ∈ H s ul , s ≥ 4, with ψx H s ul sufficiently small, where w H s ul := sup x∈R w H s ([x,x+1]) .

In particular, δ 2 -close to the expansion involving the solution of the second-order equation (1.1).

L0 has always at least the translational zero-eigenfunction ū′ .

This latter comment repairs an omission in[START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF][START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF], where conditions for invertibility were not discussed. This issue is well-discussed in[DSSS], to which we refer the reader for further remarks. See also the more detailed derivation/estimates in[START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF].

This follows from Y (X(y, t), t) -[X(y, t) + ψ(X(y, t), t)] = ψ(y, t)ψ(yψ(y, t), t).

(1-1/p)-1 2 g L 1 (R)∩H l+2m+1 (R) ,

June 8, 2012. Research of M.J. was partially supported by an NSF Postdoctoral Fellowship under NSF grant DMS-0902192. Research of P.N. was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01. Stay of M.R. in Bloomington was supported by French ANR Project no. ANR-09-JCJC-0103-01. Research of K.Z. was partially supported under NSF grant no. DMS-0300487.


Mathew A Johnson

Pascal Noble
email: noble@math.univ-lyon1.fr

AND L Miguel Rodrigues
email: rodrigues@math.univ-lyon1.fr

Kevin Zumbrun
email: kzumbrun@indiana.edu

NONLOCALIZED MODULATION OF PERIODIC REACTION DIFFUSION WAVES: THE WHITHAM EQUATION

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized (L 1 ) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist to leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

Introduction

In this, the second part of a two-part series of papers, we determine the time-asymptotic behavior of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized (L 1 ) perturbation, showing that solutions consist of an (in general nonlocalized) modulation governed by the formal second-order Whitham averaged equations plus a faster-decaying localized residual.

In the first part, [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], we established nonlinear stability under such perturbations, together with detailed rates of decay, using a refined version of the argument used in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF] to show stability under localized perturbations. In each of these cases, the basic approach was to introduce a phase via a phase-dependent change of variables, then separate out from the linearized solution operator slowly-decaying terms corresponding to linear phase shifts, and estimate separately the modulational and nonmodulational parts of the solution. While we reduced at first order the longtime dynamics of the solution to that of phase shifts, no information about the behavior of the phase was given besides decay rates.

Here, we show that, using the same basic linear estimates as in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], but a one-order-higher decomposition of the solution operator, we may describe at a higher precision the asymptotic behavior in terms of a full modulation instead of just a phase shift. This enables us, by a further linear estimate in the spirit of [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], to prove that the principal part of the local wave number appearing in this modulation obeys a convected Burgers equation that is asymptotically equivalent to a solution of an associated Whitham modulation equation of a type derived formally in, e.g., [START_REF] Schneider | Nonlinear diffusive stability of spatially periodic solutions-abstract theorem and higher space dimensions[END_REF][START_REF] Doelman | The dynamics of modulated wave trains[END_REF][START_REF] Whitham | Linear and nonlinear waves[END_REF].

This not only gives rigorous validation of the formal Whitham approximation in the strong sense of showing that it describes time-asymptotic behavior, but, through the explicit prescription of initial data coming from our analysis, also gives new information of predictive value not available from the formal asymptotic derivation. Indeed, as discussed in [BJNRZ], the connection between initial perturbation and initial values for the Whitham equations has remained for a long time somewhat mysterious. As a side-consequence, we show that the decay rates of [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF] are sharp.

We recall first the nonlinear stability result of [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF]. Consider a periodic traveling-wave solution u(x, t) = ū(k * (xct)) of reaction diffusion system u t = u xx + f (u), or, equivalently, a standing-wave solution u(x, t) = ū(x) of (1.1) k * u t = k 2 * u xx + f (u)ω 0 u x , where ω 0 (k * ) := -k * c(k * ) is the temporal frequency, c is the speed of the original traveling wave, and wave number k * is chosen so that (1.2) ū(x + 1) = ū(x).

Here and throughout the paper, all periodic functions are assumed to be periodic of period one. We make the following standard genericity assumptions: (H1) f ∈ C K (R n ), (K ≥ 3). (H2) Up to translation, the set of 1-periodic solutions of (1.1) (with k replacing k * ) in the vicinity of ū, k = k * , forms a smooth 1-dimensional manifold {ū(•; k)} = {ū k (•)}, c = c(k). Linearizing (1.1) about ū yields the periodic coefficient equation

(1.3) k * v t = k * Lv := (k 2 * ∂ 2 x -ω 0 (k * )∂ x + b)v, b(x) := df (ū(x)).
Introducing the one-parameter family of closed Floquet operators

(1.4) k * L ξ := e -iξx k * Le iξx = k 2 * (∂ x + iξ) 2 -ω 0 (k * )(∂ x + iξ) + b acting on L 2
per ([0, 1]) with densely defined domains H 2 per ([0, 1]), determined by the defining relation (1.5)

L(e iξx g) = e iξx (L ξ g),

we define following [S1, S2] the standard diffusive spectral stability conditions:

(D1) σ(L) ⊂ {λ | ℜλ < 0} ∪ {0}. (D2) There exists a constant θ > 0 such that σ(L ξ ) ⊂ {λ | ℜλ ≤ -θ|ξ| 2 } for each ξ ∈ [-π, π). (D3) λ = 0 is a simple eigenvalue of L 0 .
1 Notice in (D1) above, we consider L as a closed operator on L 2 (R) with densely defined domain H 2 (R).

The following stability result was established in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], generalizing results of [START_REF] Schneider | Nonlinear diffusive stability of spatially periodic solutions-abstract theorem and higher space dimensions[END_REF][START_REF] Schneider | Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation[END_REF][START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF]. Here, and throughout the paper, given two real valued functions A and B, we say that A B, or that for every x ∈ dom(A) ∩ dom(B), A(x) B(x), if there exists a constant C > 0 such that A(x) ≤ CB(x) for each x ∈ dom(A) ∩ dom(B). Even in a chain of inequalities, we will also feel free to denote by C harmless constants with different values.

Proposition 1.1 ([JNRZ1]

). Let K ≥ 3. Assuming (H1)-( H2) and (D1)-(D3), let

E 0 := ũ0 (• -h 0 (•)) -ū(•) L 1 (R)∩H K (R) + ∂ x h 0 L 1 (R)∩H K (R)
be sufficiently small, for some choice of phase modulation h 0 . Then, there exists a global solution ũ(x, t) of (1.1) with initial data ũ0 and a phase function ψ(x, t) such that, for t > 0 and 2 ≤ p ≤ ∞,

(1.6) ũ(• -ψ(•, t), t) -ū(•) L p (R) , ∇ x,t ψ(•, t) W K+1,p (R) E 0 (1 + t) -1 2 (1-1/p) , ũ(• -ψ(•, t), t) -ū(•) H K (R) E 0 (1 + t) -1 4 ,
and

(1.7) ũ(•, t) -ū(•) L ∞ (R) , ψ(•, t) L ∞ (R) E 0 .
In particular, ū is nonlinearly (boundedly) stable in L ∞ (R) with respect to initial perturbations

v 0 = ũ0 -ū for which v 0 E := inf ∂xh 0 ∈L 1 (R)∩H K (R) E 0 is sufficiently small.
Recall now the formal, Whitham equation, as derived in various contexts and to varying degrees of accuracy in [START_REF] Whitham | Linear and nonlinear waves[END_REF][START_REF] Howard | Slowly varying waves and shock structures in reaction-diffusion equations[END_REF][START_REF] Serre | Spectral stability of periodic solutions of viscous conservation laws: Large wavelength analysis[END_REF][START_REF] Doelman | The dynamics of modulated wave trains[END_REF][START_REF] Noble | Whitham's equations for modulated roll-waves in shallow flows[END_REF][START_REF] Noble | Whitham's modulation equations and stability of periodic wave solutions of the generalized Kuramoto-Sivashinsky equations[END_REF]. By translation-invariance of the underling equations, (1.8) L 0 ū′ = 0, so that by (D3) the zero-eigenspace of L 0 is exactly Range{ū ′ }. Denote by ūad the left, or adjoint, zero eigenfunction of L 0 . Fixing ū, k * , introduce the parametrization

(1.9) ūk (x -β) = ū(k, x -β)
of nearby periodic traveling waves, i.e., 1-periodic solutions of ω 0 (k

)ū k x -k 2 ūk xx -f (ū k ) = 0, ω 0 (k) = -kc(k), for definiteness chosen in such a way that (1.10) ūad (k), ∂ k ū(k) L 2 ([0,1]) = 0.
(This may be achieved by an appropriate translation, by the fact that ūad , ū′

L 2 ([0,1]) = 0.) Then, the formal approximate solution of u t -u xx -f (u) = 0 obtained by a nonlinear WKB expansion is (1.11) u(x, t) ≈ ūκ(x,t) (Ψ(x, t)),
where the wave number κ := Ψ x satisfies the Whitham equation (viscous scalar conservation law)

(1.12)

κ t -(ω 0 (κ)) x = (d(κ)κ x ) x
or equivalently, the phase Ψ satisfies its integral (viscous Hamilton-Jacobi equation)

(1.13) Ψ t -ω 0 (Ψ x ) = d(Ψ x )Ψ xx where, taking (1.10) into account, d(k) = 1 + 2k u ad (k), ∂ k ū′ (k) L 2 ([0,1])
, and ω 0 (k) = -c(k)k is the nonlinear dispersion relation determined by the manifold of periodic traveling-wave solutions ūk (k(xct)) described in (H2). See [START_REF] Noble | Whitham's equations for modulated roll-waves in shallow flows[END_REF][START_REF] Noble | Whitham's modulation equations and stability of periodic wave solutions of the generalized Kuramoto-Sivashinsky equations[END_REF] for a detailed derivation of this kind of nonlinear Whitham's equation in the context of the Saint-Venant and Kuramoto-Sivashinsky equations when the modulation procedure yields a system rather than an equation.

In our context, we look for the evolution of a localized perturbation k = k * h x in our co-moving frame, and so it is enough to retain the quadratic order approximants

(1.14) k * k t + k * q(k) x = k 2 * d(k * )k xx , and
(1.15) k * h t + q(k * h x ) = k 2 * d(k * )h xx with q(k) = -(ω ′ 0 (k * ) + c(k * ))k -1 2 ω ′′ 0 (k * )k 2 .
For, as is well known, (1.12) and (1.14) are "asymptotically equivalent" for such data, in the sense that the difference between solutions of (1.12) and (1.14) decays faster in all L p (R), 1 ≤ p ≤ ∞, than does the solution itself, which, for an initial perturbation with nonzero integral, decays at the rate of a heat kernel. See [DSSS] for a direct derivation of the quadratic Whitham's equation.

The notion of asymptotic equivalence is quantified in the following result.

Lemma 1.2. Let η > 0 be arbitrary. Let κ be a solution of (1.12) with initial datum κ 0 and k be a solution of (1.14) with initial datum

k 0 = κ 0 ( • /k * ) -k * , E 0 := k 0 L 1 (R)∩H 3 (R) sufficiently small. Then, setting κ(x, t) = k * + k(k * (x -c(k * )t), k * t), (κ -κ)(t) L p (R) E 2 0 (1 + t) -1 2 (1-1/p)-1 2 +η , 1 ≤ p ≤ ∞; moreover, if m 0 := R k 0 = 0 and E 1 := E 0 + | • | k 0 L 1 (R) is sufficiently small, then (1.16) k(t) -φ(•, 1 + t) L p (R) , (κ -κ)(t) L p (R) E 2 0 (1 + t) -1
where

(1.17) φ(x, t) = 1 √ t φ x + ω ′ 0 (k * )t √ t
is the unique self-similar2 solution of (1.14) determined by φ = k 0 ; in particular, κ(t) -

k * L p (R) , κ(t) -k * L p (R) |m 0 | (1 + t) -1 2 (1-1/p) .
Proof. See Appendix A.

Our main result is as follows.

Theorem 1.3. Let η > 0 and K ≥ 3. Under the assumptions of Proposition 1.1, let k and h satisfy the quadratic approximants (1.14) and (1.15) of the second-order Whitham modulation equations (1.12) and (1.13) with initial data k| t=0 = k * ∂ x h 0 , h| t=0 = h 0 , and let ψ be the phase prescribed in the proof of Proposition 1.1 in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF] (see (4.4) below). Then, for t > 0, 2 ≤ p ≤ ∞,

(1.18) ũ(• -ψ(•, t), t) -ūk * (1+ψx(•,t)) ( • ) L p (R) E 0 ln(2 + t) (1 + t) -3 4 , k * ∂ x ψ(t) -k(t) L p (R) E 0 (1 + t) -1 2 (1-1/p)-1 2 +η , ψ(t) -h(t) L p (R) E 0 (1 + t) -1 2 (1-1/p)+η .
Remark 1.4. From estimates (1.18)(i) on ũ and (1.6)(ii) and (1.7) on ψ x and ψ, we obtain

ũ(•, t) -ūk * (1+ψx(•,t)) ( Ψ(•, t)) L p (R) E 0 ln(2 + t) (1 + t) -3 4 , 2 ≤ p ≤ ∞,
where Ψ(•, t) is the inverse of y → X(y, t) := yψ(y, t). We insure the existence of such a map by keeping, for any t, ψ(t) L ∞ (R) bounded and

ψ x (t) L ∞ (R) small. Since Ψ(x, t) -(x + ψ(x, t)) = ψ( Ψ(x, t), t) -ψ( Ψ(x, t) -ψ( Ψ(x, t), t), t) one may translate this into a bound (1.19) ũ(•, t) -ūk * Ψx(•,t) ( Ψ(•, t)) L p (R) E 0 ln(2 + t) (1 + t) -3 4 , 2 ≤ p ≤ ∞,
of the form (1.11), or degrade it into3 ũ(•, t)ūk * (1+ψx(•,t)) (• + ψ(•, t)) L p (R) E 0 (1 + t) -1 2 (1-1/p) , obtaining thereby, for η > 0 arbitrary, and 2 ≤ p ≤ ∞, the exact Whitham comparison

(1.20) ũ(•, t) -ūk * +k(•,t) (• + h(•, t)) L p (R) E 0 (1 + t) -1 2 (1-1/p)+η .
Here, we could as well write

(1.21) ũ(•, t) -ū(• + h(•, t)) L p (R) E 0 (1 + t) -1 2 (1-1/p)+η
similarly as in (1.6), since k is negligible at this order of approximation.

From Theorem 1.3 and Lemma 1.2, we see immediately that the decay rates of Proposition 1.1 are sharp. At the same time, we give rigorous validation of the Whitham equation in two ways. The first, more obvious way, is to show through (1.20) that asymptotic behavior consists of modulation by a solution h of the exact Whitham equations. Here, as pointed out in Remark 1.4, modulation in wave number is asymptotically irrelevant, and only phase shift plays a role. The second, less direct, but ultimately sharper (by factor (1 + t) -1/4 to (1 + t) -1/2 in rate of decay) way, is to show through (1.19) that a more accurate description of asymptotic behavior is modulation including both phase shift and variation in wave number by a solution Ψ of an approximate Whitham equation with rapidly decaying error term (term r(t) of (5.4), together with terms of similar order coming from the difference Ψx -Ψ x = O(ψ 2

x ); see Remark A.1). See [DSSS] for validation of the Whitham equation in the altenative sense of building a family of solutions existing on asymptotically-large but bounded intervals and close to a given asymptotic expansion involving a given solution of the Whitham equation.

Remark 1.5. Similarly as in [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF], all of our analysis goes through in the general quasilinear 2rorder parabolic case; in particular, our results extend to the (sectorial) Swift-Hohenberg equations treated for localized perturbations in [S2]. See [BJNRZ], Appendix B, for related analysis.

Remark 1.6. We conjecture that (1.18)(i) and thus also of (1.19) may be improved for p > 2 to

ũ(• -ψ(•, t), t) -ūk * (1+ψx(•,t)) ( • ) L p (R) E 0 ln(2 + t) (1 + t) -1 2 (1-1/p)-1 2
by the use of additional, W k,p → L p estimates outside the scope of this paper; see Remark 4.3.

1.1.

Discussion and open problems. The Whitham modulation equations, and WKB approximations in general, are "magical" prescriptions of great predictive value that may be obtained by formal, "consistency-type," considerations suppressing the sometimes very complicated details of the underlying equations. As a consequence, they often hide the mechanisms leading to the behavior they predict. In particular, rigorous verification of such asymptotic expansions typically comes from techniques apart from the method of derivation of the Whitham equation. Moreover, in the course of verification, these other techniques may often give additional information not found in the formal asymptotics.

The present case is no exception. The Whitham modulation equations, based on slowly varying perturbations, can be related rather directly at the linearized, spectral level, to perturbation expansions of small Floquet-number/eigenvalue modes; see, in various contexts, [START_REF] Doelman | The dynamics of modulated wave trains[END_REF][START_REF] Noble | Whitham's equations for modulated roll-waves in shallow flows[END_REF][START_REF] Noble | Whitham's modulation equations and stability of periodic wave solutions of the generalized Kuramoto-Sivashinsky equations[END_REF]. Moreover, the same type of critical modes expansion, followed carefully, leads to the linearized estimates of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF]. However, to carry this analysis to the nonlinear level involves significant technical challenges, and appears to require more indirect methods, motivated by, but at a technical level quite different from, the formal Whitham modulation.

In [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF] and the present paper, we have broken the nonlinear analysis into two distinct pieces, focused respectively on decay and asymptotic behavior. In [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], by a judiciously chosen nonlinear transformation involving an implicitly prescribed shift in phase, we showed how to convert rigorously the picture of behavior afforded by formal asymptotics to a system of integral equations exhibiting the expected nonlinear decay. At a technical level, this could be understood as identifying the main part of the linearized solution operator as a linearized phase shift, and separating off this part of the behavior by a counterbalancing nonlinear change in phase. Put most naively, the analysis is driven by the observation that the critical mode given by the kernel of the linearized Bloch operator L 0 about the wave at Bloch frequency ξ = 0 is

(1.22) φ(0, •) = ū′ ,
where ū′ represents instantaneous translation.

The above analysis gives a simple and self-contained argument yielding sharp rates of decay to a phase modulation of the wave. In the present analysis, we show how to extract from the system of integral equations derived in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF] an approximate differential equation governing the phase, and to connect this equation to the formally predicted Whitham equation. The first step is to go one step further in the decomposition of the solution, Taylor expanding the critical mode φ(ξ, x) of L ξ about ξ = 0, and observing that, under an appropriate normalization (see (1.10) and (2.6)), the first order corrector is

(1.23) ∂ ξ φ(0, •) = ik * ∂ k u(•; k * )
corresponding at linear level to modulation in the wave number k; see Lemma 2.1 below. As often happens for higher-order correctors, the corresponding nonlinear correction can be made in simple, linear fashion (see (4.6)), since this term is fast enough decaying that nonlinear effects are negligible.

This yields a new residual z decaying at rate (4.8) faster by factor (1 + t) -1/4 to (1 + t) -1/2 than that of the residual v of [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF]. Once this is done, we may, isolating explicit terms k in the ∂ k ū direction, and discarding as asymptotically negligible all terms in the integral equations that are not linear or quadratic in k, obtain, evaluating the resulting quadratic interaction coefficients, an integral equation that may be recognized as the Duhamel (variation of constants) representation of a forced Burgers equation corresponding to (1.14). From this description and our previous bounds on the residual, we then readily obtain the convergence result of our main Theorem 1.3.

Taken together, these two analyses give a blueprint for connecting formal asymptotics to the integral equations natural for analysis. This should carry over to other interesting settings; indeed, we have already carried out in [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] the extension to the much more complicated conservation law case. We note that, despite its difficulty, the analysis in the end, both for decay and for convergence to the Whithams approximant, is quite short and easily verified. In particular, there is little advance preparation by formal asymptotics, different from the approach set out in [DSSS].

We note also that our approach gives "direct" access to bounds, i.e., we "solve" rather than "impose." As noted earlier, we obtain as a result somewhat sharper bounds (1.19) than what is available from comparison to the exact Whitham equation, a result that would at least not be easily derived by starting from the exact Whitham solution to begin with. That is, the system of integral equations we derive contains more information than the formal Whitham approximation.

One interesting problem for future investigation is the rigorous verification of the spectral stability conditions in interesting situations either by analysis or numerical proof. Another, very interesting direction, is the derivation of pointwise estimates on nonlocalized perturbations similarly as has been done for localized solutions in [J].

A simplifying aspect of the present analysis is that we were able to separate the analyses of decay and behavior. However, there could be an advantage in combining these, in that Burgers equation is known to decay even for large perturbations, and, thanks to the maximum principle, decays in L ∞ . Thus, one might hope by such a simultaneous argument structure to treat the case that ψ x is large but bounded in L 1 but only small in L ∞ . This would be a very interesting extension to carry out. Finally, as described in [DSSS], nonlinear stability of shock-type solutions, for which not only ψ but k approaches different endstates, is a challenging and very interesting open problem, which appears to require substantial new ideas beyond those introduced here; in particular, perturbation around such shock-type solutions and not a background periodic wave.

Note: Similar results have been obtained by different means by Sandstede and collaborators [SSSU] using a nonlinear decomposition of phase and amplitude variables combined with a renormalization iteration process as in [S1, S2]. Specifically, they obtain the bound (1.21) assuming the somewhat stronger localization of initial perturbations of order ∼ |x| -5/2 as compared to the order ∼ |x| -1 assumed here; see footnote 7, [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF]. Our bound (1.19), though not of the same form, gives more precise information by factor (1 + t) -1/4 to (1 + t) -1/2 in the rates of decay; note also that this takes into account modulation in wave number, whereas the approximation of [SSSU] does not. The results of [SSSU] on the other hand include also convergence (at the same rate) to a "nonlinear diffusion wave" φ as described in (1.17). Combining (1.20) with (1.16), we recover this bound as well, but with localization ∼ |x| -2 closer to the |x| -5/2 assumption of [SSSU].

Preliminaries

Recall the Bloch solution formula for periodic-coefficient operators,

(2.1) (S(t)g)(x) := (e tL g)(x) = π -π e iξx (e tL ξ ǧ(ξ, •))(x)dξ, 4
where L ξ is as in (1.4),

(2.2) ǧ(ξ, x) := j∈Z ĝ(ξ + 2jπ)e i2πjx (periodic in x)

denotes the Bloch transform of g, ĝ(ξ) := 1 2π R e -iξx g(x)dx the Fourier transform, and

(2.3) g(x) = π -π
e iξx ǧ(ξ, x)dξ the inverse Bloch transform, or Bloch representation of g ∈ L 2 (R). The generalized Hausdorff-Young inequality u L p (R) ≤ (2π) 1/p ǔ L q ([-π,π],L p ([0,1])) for q ≤ 2 ≤ p and 1 p + 1 q = 1 [JZ1], together with (2.3), yields for any 1-periodic function g(ξ, •)

(2.4) π -π e iξ• g(ξ, •)dξ L p (R) ≤ (2π) 1/p g L q ([-π,π],L p ([0,1])) for q ≤ 2 ≤ p and 1 p + 1 q = 1
where, here and elsewhere, we are denoting

g L q ([-π,π],L p ([0,1])) := π -π g(ξ, •) q L p ([0,1]) dξ 1/q
. By (D3), the zero eigenfunction ū′ of L 0 is simple, whence by standard perturbation results bifurcates to an eigenfunction φ(ξ, •), with associated left eigenfunction φ(ξ, •) and eigenvalue

(2.5) λ(ξ) = aiξ -dξ 2 + O(|ξ| 3 ),
where a and d are real and d > 0 by assumption (D2) and complex symmetry λ(ξ) = λ(-ξ), each of φ, φ, λ analytic in ξ and defined for |ξ| ≤ ξ 0 , ξ 0 being positive and sufficiently small. Before refining linear estimates of [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], we need some extra spectral preparation. For this purpose, we set φ(0) = ūad , assume parametrization is normalized according to (1.10) and normalize eigenfunctions in such a way that

(2.6) ūad , φ(ξ) L 2 ([0,1]) = φ(0), φ(ξ) L 2 ([0,1]) = 1
for all ξ ∈ [-ξ 0 , ξ 0 ]. A similar preparation was already needed in [DSSS], see [START_REF] Doelman | The dynamics of modulated wave trains[END_REF]Section 4.2].

Lemma 2.1 ( [DSSS]). Assume (H1)-( H2) and (D3), normalize according to (1.10) and (2.6). Then

(2.7)

∂ ξ φ(0, •) = ik * ∂ k u(•; k * )
and the constants a and d in (2.5) are

(2.8) a = -k * c ′ (k * ), d = k * (1 + 2k * u ad , ∂ k ū′ (k * ) L 2 ([0,1]) ).
Proof. To expand the equation

L ξ φ(ξ) = λ(ξ)φ(ξ), we split L ξ k * L ξ = L (0) + ik * ξL (1) + (ik * ξ) 2 L (2) .
Then we find ik 1) ū′ . Yet differentiating the profile equation with respect 1) ū′ . Thus, taking scalar product with ūad leads to

* aū ′ = L (0) ∂ ξ φ(0) + ik * L (
to k yields -k * c ′ (k * )ū ′ = L (0) ∂ k ū(k * ) + L (
a = u ad , L (1) ū′ L 2 ([0,1]) = -c ′ (k * ) hence L (0) ∂ ξ φ(0) = L (0) ∂ k ū(k * ) and therefore ∂ ξ φ(0) -∂ k ū(k * ) ∈ ker L 0 = Cū ′ .
Normalizations (1.10) and (2.6) reduce it to (2.7). Afterwards, expanding further, we find

k 2 * c ′ (k * )∂ k ū(k * ) -k * d ū′ = 1 2 L (0) ∂ 2 ξ φ(0) + (ik * ) 2 L (1) ∂ k ū(k * ) + (ik * ) 2 L (2) ū′ .
Taking scalar product with ūad again gives

d = k * u ad , L (1) ∂ k ū(k * ) + L (2) ū′ L 2 ([0,1]
) . which, thanks to the explicit form

L (1) ∂ k ū′ + L (2) ū′ = 2k * ∂ k ū(k * ) ′ + c(k * )∂ k ū(k * ) + ū′ ,
completes the proof.

From now on, although we do not repeat them, we always assume the above normalizations.

Linear estimates

Now, refining slightly the decomposition of [START_REF] Johnson | Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations[END_REF][START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], decompose the solution operator as

(3.1) S(t) = R p (t) + R(t), R p (t) = (ū ′ + k * ∂ k ū∂ x )s p (t), with (3.2) (s p (t)g)(x) = π -π e iξx α(ξ)e λ(ξ)t φ(ξ, •), ǧ(ξ, •) L 2 ([0,1]) dξ, and
(3.3) ( R(t)g)(x) := π -π e iξx (1 -α(ξ))(e L ξ t ǧ(ξ))(x)dξ + π -π e iξx α(ξ)(e L ξ t Π(ξ)ǧ(ξ))(x)dξ + π -π e iξx α(ξ)e λ(ξ)t (φ(ξ, x) -φ(0, x) -ξ∂ ξ φ(0, x)) φ(ξ), ǧ(ξ) L 2 ([0,1]) dξ,
where α is a smooth cutoff function such that α(ξ) = 0 for |ξ| ≥ ξ 0 and α(ξ

) = 1 for |ξ| ≤ 1 2 ξ 0 , (3.4) Π p (ξ) := φ(ξ) φ(ξ), • L 2 ([0,1])
denotes the eigenprojection onto the eigenspace Range{φ(ξ)} bifurcating from Range{ū ′ } at ξ = 0, φ the associated left eigenfunction, and Π := Id -Π p , each well-defined on suppt(α) ⊂ [-ξ 0 , ξ 0 ].

Proposition 3.1 ([JZ1, JNRZ1]). Under assumptions (H1)-( H2) and (D1)-(D3), for all t > 0, 2 ≤ p ≤ ∞,

∂ l x ∂ m t s p (t)∂ r x g L p (R) min (1 + t) -1 2 (1-1/p)-l+m 2 g L 1 (R) , (1 + t) -1 2 (1/2-1/p)-l+m 2 g L 2 (R) , (3.5)
for 0 ≤ r ≤ K + 1, and for some η > 0 and 0 ≤ l + 2m, r ≤ K + 1,

∂ l x ∂ m t R(t)∂ r x g L p (R) min (1 + t) -1 2 (1-1/p)-1 2 g L 1 (R)∩H l+2m+1 (R) , e -ηt ∂ r x g H l+2m+1 (R) + (1 + t) -1 2 (1/2-1/p)-1 g L 2 (R) , (3.6)
Proof. The estimates on s p were proved in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF]. Using (2.4) together with spectral projection/direct computation for low Floquet numbers and standard semigroup estimates for high Floquet numbers, estimates on R are proved exactly as were the estimates on S in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], with the observation that the substitution of factor (φ(ξ)φ(0) -ξ∂ ξ φ(0)) in R for factor (φ(ξ)φ(0)) in S introduces an additional factor of |ξ| in the estimates, hence an additional (1 + t) -1/2 factor of decay. As we will build on the main arguments of these proofs to establish further linear estimates, we briefly recall them here.

(i) (Proof of (3.5)). In the case l = m = r = 0 estimates on s p follows from, for either s = 1 or s = 2, introducing s ′ such that 1/s + 1/s ′ = 1,

x → π -π e iξx α(ξ)e λ(ξ)t φ(ξ), ǧ(ξ) L 2 ([0,1]) dξ L p (R) (ξ, x) → α(ξ)e λ(ξ)t | φ(ξ), ǧ(ξ) L 2 ([0,1]) | L q ([-π,π],L p ([0,1])) ξ → e -ηξ 2 t L rs,p ([-π,π]) ξ → α(ξ) 1/2 | φ(ξ), ǧ(ξ) L 2 ([0,1]) | L s ′ ([-π,π]) (1 + t) -1 2 (1/s-1/p) ξ → α(ξ) 1/2 | φ(ξ), ǧ(ξ) L 2 ([0,1]) | L s ′ ([-π,π]) ,
where 1/p + 1/q = 1 and 1/s ′ + 1/r s,p = 1/q, so that 1/r s,p = 1/s -1/p. Here we have used the fact that by (D2) for some η > 0, |e λ(ξ)t α 1/2 (ξ)| ≤ e -ηξ 2 t . Now for s = 2, we combine this with Cauchy-Schwarz inequality and Parseval identity that yield

ξ → α(ξ) 1/2 | φ(ξ), ǧ(ξ) L 2 ([0,1]) L 2 ([-π,π]) ≤ sup |ξ|≤ξ 0 φ(ξ, •) L 2 ([0,1]) ǧ L 2 ([-π,π],L 2 ([0,1]) sup |ξ|≤ξ 0 φ(ξ, •) L 2 ([0,1]) g L 2 (R)
while for s = 1 we write

φ(ξ), ǧ(ξ) L 2 ([0,1]) = j∈Z φj (ξ) * ĝ(ξ + 2jπ)
where φj (ξ) denotes the jth Fourier coefficient in the Fourier expansion of periodic function φ(ξ, •), and z * = z denotes complex conjugate, and apply Hausdorff-Young's inequality, ĝ

L ∞ (R) ≤ g L 1 (R) with Cauchy-Schwarz' inequality, α 1/2 (ξ) j | φj (ξ)| ≤ α 1/2 (ξ) j (1 + |j| 2 )| φj (ξ)| 2 j (1 + |j| -2 ) ≤ Cα 1/2 (ξ) φ(ξ) H 1 ([0,1]) .
to get

ξ → α(ξ) 1/2 | φ(ξ), ǧ(ξ) L 2 ([0,1]) | L ∞ ([-π,π]) sup |ξ|≤ξ 0 φ(ξ, •) H 1 ([0,1]) g L 1 (R) .
This yields the result for l = m = r = 0. Estimates for general l, m, r ≥ 0 go similarly, passing ∂ r

x derivatives onto φ(ξ) in the inner product using integration by parts and noting that ∂ l x and ∂ m t derivatives bring down bounded factors (iξ) l and λ(ξ) m enhancing decay through

ξ → |ξ| l+m e -ηξ 2 t L rs,p ([-π,π]) (1 + t) -1 2 (1/s-1/p)-l+m 2 (ii) (Proof of (3.6))
. By (D1)-(D2), benefiting from standard parabolic resolvent estimates [He] together with the fact that (by basic ODE regularity theory) H l+1 ([0, 1]) and L 2 ([0, 1]) spectra coincide, we may apply Prüss' Theorem [Pr] and get

|e L ξ t (1 -α(ξ))| H l+1 ([0,1])→H l+1 ([0,1]) , |α(ξ)e L ξ t Π(ξ)| H l+1 ([0,1])→H l+1 ([0,1]) e -ηt , η > 0,
whence, by Sobolev embedding,

|∂ l x e L ξ t (1 -α(ξ))| H l+1 ([0,1])→L p ([0,1]) , |∂ j x α(ξ)e L ξ t Π(ξ)| H l+1 ([0,1])→L p ([0,1]) e -ηt for 2 ≤ p ≤ ∞. The W l,p (
R) norms of the first two terms of (3.3), by (2.4) and Parseval's identity,

1 2π g 2 H l+1 (R) = ǧ 2 L 2 ([-π,π],L 2 ([0,1])) + (∂ x + iξ) l+1 ǧ(ξ) 2 L 2 ([-π,π],L 2 ([0,1])) ,
are thus bounded by Ce -ηt g H l+1 (R) . The W l,p (R) norm of the third term may be bounded similarly as in the estimation of s p above, noting that the factor (φ(ξ)φ(0) -ξ∂ ξ φ(0)) = O(ξ 2 ) introduces an additional factor of (1 + t) -1 decay. This establishes the result for m = r = 0; other cases go similarly, noting that ∂ t e L ξ t Π(ξ) = L ξ e L ξ t Π(ξ), with L ξ a second-order operator, so that we may essentially trade one t-derivative for two x-derivatives.

Proposition 3.2 ([JNRZ1]

). Under assumptions (H1)-( H2) and ( D1)-( D3), for all t > 0 and 2 ≤ p ≤ ∞,

(3.7) ∂ l x ∂ m t s p (t)(h 0 ū′ ) L p (R) (1 + t) -1 2 (1-1/p)+ 1 2 -l+m 2 ∂ x h 0 L 1 (R) ,
when l + m ≥ 1 or else l = m = 0 and p = ∞, and, for 0

≤ l + 2m ≤ K + 1, (3.8) ∂ l x ∂ m t R(t)(h 0 ū′ ) L p (R) (1 + t) -1 2 (1-1/p)-1 ∂ x h 0 L 1 (R)∩H l+2m+1 (R) ,
and when t ≤ 1

(3.9) ∂ l x ∂ m t (R p (t) -Id)(h 0 ū′ ) L p (R) ∂ x h 0 L 1 (R)∩H l+2m+1 (R) , ∂ l x ∂ m t (s p (t)(h 0 ū′ ) -h 0 ) L p (R) ∂ x h 0 L 1 (R)∩L 2 (R) .
Proof. Again, estimates on s p were proved in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF] whereas the proof of those on R p and R goes exactly as the proof of the corresponding estimates for S p and S in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], noting that the substitution of factor (φ(ξ)φ(0) -ξ∂ ξ φ(0)) in R for factor (φ(ξ)φ(0)) in S introduces an additional factor of (1 + t) -1/2 in the decay rate for R. We briefly outline the new points of the arguments.

(i) (Proof of (3.7)). The inequality (3.7) was established in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], Proposition 4.1.

(ii) (Proof of (3.8)). Contribution of the last term in (3.3) is bounded as in the proof of (3.7). For the remaining terms, we first split

(h 0 ū′ )ˇ(ξ, x) = ū′ (x) ȟ0 (ξ, x) = j∈Z ū′ (x)e 2πjx h 0 (2πj + ξ)
then apply to each term of the sum the semigroup bounds as above and, after noting that (1α(ξ)) ξ and that

Π(ξ)ū ′ = Π(ξ) ( Π(ξ) -Π(0))ū ′ with | Π(ξ) -Π(0)| H l+2m+1 ([0,1])→H l+2m+1 ([0,1]) |ξ|,
achieve the proof with the Cauchy-Schwarz inequality

x → ū′ (x) ∂ x h 0 (ξ) H l+2m+1 ([0,1]) + j =0 x → ū′ (x)e 2iπjx i(ξ + 2πj) ∂ x h 0 (ξ + 2πj) H l+2m+1 ([0,1]) | ∂ x h 0 (ξ)| + j =0 (2πj) l+2m+1 (∂ l+2m+2 x h 0 )(ξ + 2πj) (ξ + 2πj) l+2m+2 | ∂ x h 0 (ξ)| + j 1 (1 + |j|) 2 j ′ | (∂ l+2m+2 x h 0 )(ξ + 2πj ′ )| 2 | ∂ x h 0 (ξ)| + j∈Z | (∂ l+2m+2 x h 0 )(ξ + 2πj)| 2 , the L 2 ([-π, π]) norm of the last quantity being bounded by ∂ x h 0 H l+2m+1 (R) . (iii) (Proof of (3.9)). Expanding R p (t) -Id = (R p (t) -R p (0)) -R(0) = t∂ t R p (s(t)) -R(0)
for some 0 < s(t) < t, we obtain the first inequality by combining (3.7) and (3.8). The second inequality was established in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], Proposition 4.1.

We require also the following key new estimates, proved by similar techniques. The first statement reflects the fact that modulations approximately travel at reference group speed a. The second one quantifies the fact that the (parabolic) second-order linearized modulation equation is given by (3.13).

Lemma 3.3. Under assumptions (H1)-( H2) and ( D1)-( D3), for all t > 0, 2 ≤ p ≤ ∞,

(∂ t -a∂ x )(s p (t)g) L p (R) min (1 + t) -1 2 (1-1/p)-1 g L 1 (R) , (1 + t) -1 2 (1/2-1/p)-1 g L 2 (R) , (3.10)
(3.11) (∂ t -a∂ x )(s p (t)(h 0 ū′ )) L p (R) (1 + t) -1 2 (1-1/p)-1 2 ∂ x h 0 L 1 (R) .
Proof. Differentiating (3.2), we have

(3.12) (∂ t -a∂ x )(s p (t)g)(x) = π -π (λ(ξ) -aiξ)e iξx α(ξ)e λ(ξ)t φ(ξ), ǧ(ξ) L 2 ([0,1]) (ξ)dξ with λ(ξ) -iaξ = O(|ξ| 2
), whence the result follows, again by (2.4).

Proposition 3.4. Assuming (H1)-( H2) and ( D1)-(D3), let σ(t) be the solution operator of the convected heat equation

(3.13) u t = au x + du xx ,
where a, d are as in (2.5), and let g be a periodic function on [0, 1], g ∈ L 2 ([0, 1]). Then, for all t > 0, 2 ≤ p ≤ ∞, and l ∈ N,

(3.14) ∂ l x s p (t)(h 0 g) -φ(0), g σ(t)(∂ l x h 0 ) L p (R) (1 + t) -1 2 (1-1/p)-1 2 t -l-1 2 ∂ x h 0 L 1 (R) . Proof. Expressing φ(0), g L 2 ([0,1]) (σ(t)∂ l x h 0 )(x) = R (iξ) l e iξx e (iaξ-dξ 2 )t φ(0), g L 2 ([0,1]) h 0 (ξ)dξ ∂ l x s p (t)(gh 0 )(x) = π -π (iξ) l e iξx α(ξ)e λ(ξ)t j φ(ξ), ge i2πj• L 2 ([0,1]) h 0 (ξ + 2jπ)dξ
and subtracting, we obtain, for l ∈ N,

φ(0), g σ(t)(∂ l x h 0 ) -∂ l x (s p (t)(gh 0 )) (x) = R\[-π,π] e iξx e (iaξ-dξ 2 )t (iξ) l-1 φ(0), g ∂ x h 0 (ξ)dξ + π -π e iξx (1 -α(ξ))e (iaξ-dξ 2 )t (iξ) l-1 φ(0), g ∂ x h 0 (ξ)dξ + π -π e iξx α(ξ)(iξ) l-1 (e (iaξ-dξ 2 )t -e λ(ξ)t ) φ(0), g ∂ x h 0 (ξ)dξ + π -π e iξx α(ξ)e λ(ξ)t (iξ) l-1 φ(0) -φ(ξ), g ∂ x h 0 (ξ)dξ - π -π e iξx α(ξ)e λ(ξ)t (iξ) l j =0 ( φ(ξ)g) * j 1 i(ξ + 2jπ) ∂ x h 0 (ξ + 2jπ)dξ.
where ( φ(ξ)g) j denotes the jth coefficient in the Fourier expansion of periodic function φ(ξ)g. Using Haussdorff-Young estimates twice, the first term is bounded in L p (R) by

ξ → |ξ| l-1 e -dξ 2 t L q (R\[-π,π]) g L 2 ([0,1]) φ(0) L 2 ([0,1]) ∂ x h 0 L 1 (R) e -ηt t -(l-1) ∂ x h 0 L 1 (R) ,
where q is such that 1/p + 1/q = 1 and η > 0. Using (1α(ξ)) |ξ|,

φ(ξ) -φ(0) L 2 ([0,1]) ≤ |ξ| sup |ξ ′ |≤ξ 0 ∂ ξ φ(ξ ′ ) L 2 ([0,1]) , λ(ξ) -(iaξ -dξ 2 ) = O(|ξ| 3 ) to get |e (iaξ-dξ 2 )t -e λ(ξ)t | = e -dξ 2 t |1 -e (λ(ξ)-(iaξ-dξ 2 ))t | |ξ| 3 e -dξ 2 t
and by Cauchy-Schwarz estimate sup

|ξ|≤ξ 0 j =0 ( φ(ξ)g) * j 1 i(ξ + 2jπ) ∂ x h 0 (ξ + 2jπ) ∂ x h 0 L 1 (R) sup |ξ|≤ξ 0 j =0 | ( φ(ξ)g) j |
(1 + |j|)

∂ x h 0 L 1 (R) sup |ξ|≤ξ 0 φ(ξ)g L 2 ([0,1]) g L 2 ([0,1]) sup |ξ|≤ξ 0 φ(ξ) L ∞ ([0,1]) ∂ x h 0 L 1 (R) ,
we may apply (2.4) to the other terms and bound them in L p (R) by

ξ → |ξ| l e -ηξ 2 t L q ([-π,π]) ∂ x h 0 L 1 (R) (1 + t) -1 2 (1-1/p)-l 2 ∂ x h 0 L 1 (R) ,
with q is such that 1/p + 1/q = 1 and η > 0.

4. Nonlinear stability estimates 4.1. Perturbation equations and nonlinear decomposition. We now refine the nonlinear perturbation equations of [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF]. First, recall the iteration scheme introduced in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF]. For ũ satisfying k * u t = k 2 * u xx + f (u) + k * cu x , we introduced the nonlinear perturbation (4.1) v(x, t) = ũ(xψ(x, t), t)ū(x)

and showed that it satisfies

(4.2) k * (∂ t -L) (v + ψ ūx ) = k * N := Q + R x + (k * ∂ t + k 2 * ∂ 2 x )S + T, where (4.3) Q := f (v + ū) -f (ū) -df (ū)v, R := -k * vψ t -k 2 * vψ xx + k 2 * (ū x + v x ) ψ 2 x 1 -ψ x , S := vψ x , and
T := -(f (v + ū) -f (ū)) ψ x .
Defining the phase ψ implicitly by (4.4)

ψ(t) = s p (t)(d 0 + h 0 ū′ ) + t 0 s p (t -s)N (s)ds -(1 -χ(t)) s p (t)(d 0 + h 0 ū′ ) -h 0 + t 0 s p (t -s)N (s)ds ,
where d 0 := ũ0 (•h 0 (•))ū and χ is a smooth cutoff such that χ(t) is zero for t ≤ 1/2 and one for t ≥ 1, applying Duhamel's principle (variation of constants), and rearranging terms, we then obtained the integral representation (4.5)

v(t) = S(t)(d 0 + h 0 ū′ ) + t 0 S(t -s)N (s)ds + (1 -χ(t)) ū′ s p (t)(d 0 + h 0 ū′ ) -h 0 ū′ + t 0 ū′ s p (t -s)N (s)ds ,
closing the system for (v, ψ), where S(t) := S(t) -ū′ s p (t). As described in [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF], the choice (4.4) is designed to cancel, modulo an "initial layer" term (1χ)(. . . ) vanishing for t ≥ 1, any ū′ s p terms that would otherwise occur in the description of v, that is, to extract as much as possible the expected phase shift from the solution ũ. Indeed, it is essentially uniquely determined by this requirement together with the requirement that ψ(•, 0) = h 0 . The crucial fact making possible the estimates of Proposition 1.1 is that the nonlinear term N is of quadratic order in terms of v and derivatives of ψ decaying at the rate of a heat kernel, whereas propagators S and ∇ x,t s p decay at the rate of a differentiated heat kernel, leading to a closable iteration in variables (∇ x,t ψ, v) and derivatives. This is analogous to the situation of the standard iteration scheme used to show decay with respect to localized (L 1 ) data of a Burgers equation

k t -k xx = ( 1 2 k 2 )
x treated as a perturbation of the heat equation, which is perhaps natural in view of the expected asymptotics (1.14). We refer the reader to [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF] for further details; see also [START_REF] Zumbrun | Refined Wave-tracking and Nonlinear Stability of Viscous Lax Shocks[END_REF][START_REF]Asymptotic behavior of multi-dimensional scalar viscous shock fronts[END_REF] for related arguments in the context of viscous shock stability.

Recall from the introduction that ψ x approximates perturbation in wave number according to the formal Whitham approximation. Accordingly, refining the decomposition (4.1), we recall Lemma 2.1 and define the new nonlinear perturbation variable

(4.6) z(x, t) := ũ(x -ψ(x, t), t) -ū(x) -∂ k ū(x)k * ∂ x ψ(x, t) = v(x, t) -∂ k ū(x)k * ∂ x ψ(x, t),
approximately separating out expected modulation in the wave-number along with the phase shift.

Lemma 4.1. For ψ defined as in (4.4), the nonlinear residual z defined in (4.6) satisfies (4.7)

z(t) = R(t)(d 0 + h 0 ū′ ) + t 0 R(t -s)N (s)ds + (1 -χ(t)) R p (t)(d 0 + h 0 ū′ ) -h 0 ū′ -∂ k ū k * ∂ x h 0 + t 0 R p (t -s)N (s)ds . Proof. Using z(t) = v(t)-∂ k ū k * ψ x (t), R(t) = S(t)-∂ k ū k * ∂ x s p (t) and R p (t) = (ū ′ +∂ k ū k * ∂ x )s p (t), equation (
4.7) follows immediately from (4.4) and (4.5).

4.2. Refined stability estimate. Comparing (4.5) and (4.7), we see that we have by these manipulations effectively exchanged for S and s p the faster-decaying propagators R and R p in the representations of z vs. v. With these improvements, we may refine Proposition 1.1 as follows.

Proposition 4.2. Under the assumptions of Proposition 1.1, for t > 0, 2 ≤ p ≤ ∞, we have the sharpened estimate

(4.8) z(t) L p (R) E 0 ln(2 + t) (1 + t) -3 4 .
Proof. From the basic bounds (1.6) of Proposition 1.1, we have N (t) H 1 (R) ≤ CE 0 (1 + t) -3 4 . Applying the bounds of Propositions 3.1 and 3.2 to system (4.7), we obtain for any 2 ≤ p ≤ ∞ (4.9)

z(t) L p (R) ≤ CE 0 (1 + t) -1 2 (1-1/p)-1 2 + CE 0 t 0 (1 + t -s) -1 2 (1/2-1/p)-1 (1 + s) -3 4 ds ≤ CE 0 ln(2 + t) (1 + t) -3 4 .
Remark 4.3. We expect that it should be possible to improve (4.8) to

z(t) L p (R) E 0 ln(2 + t) (1 + t) -1 2 (1-1/p)-1 2 , 2 ≤ p ≤ ∞,
by substituting in (4.9) W k,p bounds on N and W k,p → L p bounds on the solution operators R p and R. However, to obtain such W k,p → L p bounds for p > 2 would appear to require techniques outside the Hausdorff-Young-type estimates used in this paper, perhaps pointwise bounds as in [J], or (operator-valued) Fourier multiplier techniques as suggested in Remark 4.2.2, [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF].

The Whitham equation

We complete our investigation by connecting to the Whitham modulation equation.

Lemma 5.1. Assuming (H1)-( H2), ( D1)

-(D3), k * N (t) = f p k 2 * ψ x (t) 2 + r(t)
, where

(5.1)

f p = 1 2 d 2 f (ū)(∂ k ū, ∂ k ū) + k * ∂ k ū′′ - 1 k * df (ū)∂ k ū + ū′′ -a∂ k ū′ is periodic and r(t) L 1 (R) E 2 0 (1 + t) -1 .
Proof. Immediate from Lemma 4.1, (1.6), (4.8), and ψ t (t)aψ x (t) L 2 (R) = O((1 + t) -3/4 ), a consequence of Lemma 3.3.

Lemma 5.2. Setting e := φ(0), f p L 2 ([0,1]) , we have e = 1 2 ω ′′ 0 (k * ).

Proof. We may rewrite

f p = -L∂ k ū + 2k * ∂ k ū′′ -ω ′ 0 (k * )∂ k ū′ + ū′′ + 1 2 d 2 f (ū)(∂ k ū, ∂ k ū) thus e = φ(0), 2k * ∂ k ū′′ -ω ′ 0 (k * )∂ k ū′ + ū′′ + 1 2 d 2 f (ū)(∂ k ū, ∂ k ū) L 2 ([0,1]) = 1 2 ω ′′ 0 (k * ) .
The last equality comes differentiating twice the profile equation with respect to k (see [DSSS]).

Proof of Theorem 1.3. With estimates on k * ∂ x ψ(t) from Proposition 1.1, the estimate on ũ follows (recall Remark 1.4) from the bound

(5.2) ũ(• -ψ(•, t), t) -ūk * (•) -∂ k ūk * (•)k * ∂ x ψ(•, t) L p (R) E 0 ln(2 + t) (1 + t) -3 4 established in Proposition 4.2 for 2 ≤ p ≤ ∞.
Using Duhamel's principle we may write

(5.3) k(t) = σ(t)k * ∂ x h 0 + t 0 σ(t -s)∂ x (ek 2 (s)))ds ,
where σ is the constant-coefficient solution operator defined in Proposition 3.4. On the other hand, by Propositions 3.1, 3.2, and 3.4, (5.4)

k * ψ x (t) = k * χ(t) ∂ x s p (t)(d 0 + h 0 ū′ ) + t 0 ∂ x s p (t -s)N (s)ds + (1 -χ(t))k * ∂ x h 0 , = σ(t)k * ∂ x h 0 + t 0 σ(t -s)∂ x (e(k * ψ x ) 2 (s))ds + r(t), where r(t) L p (R) E 0 ln(2 + t) (1 + t) -1 2 (1-1/p)-1 2 , 2 ≤ p ≤ ∞ .
Here, we have used (3.14) and

∂ x (ψ 2 x (s)) L 1 (R)∩L 2 (R) ψ x (s) L 2 (R)∩L ∞ (R) ψ xx (s) L 2 (R) (1 + s) -1
to bound (1 + ts) 1 2 (1-1/p) (ts) -12 (1 + s) -1 ds.

Thus, subtracting (5.3) from (5.4), and defining δ := k * ψ xk, we have Defining, for some η > 0,

ν(t) := sup p∈[2,∞] sup 0≤s≤t δ(s) L p (R) (1 + s) 1 2 (1-1/p)+ 1 4 -η ,
we thus obtain by the standard heat bounds σ(t)∂ x f L p (R) ≤ Ct -1 2 (1/q-1/p)-1 2 f L q (R) when 1 ≤ q ≤ p ≤ ∞, that, for 2 ≤ p ≤ ∞, δ(t) L p (R) t/2 0 (ts) -1 2 (1-1/p)-1 2 eδ(s)(k(s) + k * ψ x (s)) L 1 (R) ds + t t/2 (ts) -1 2 (1/2-1/p)-1 2 eδ(s)(k(s) + k * ψ x (s)) L 2 (R) ds + r(t) L p (R) t/2 0 (ts) -1 2 (1-1/p)-1 2 ν(t)E 0 (1 + s) -1+η ds

+ t t/2 (t -s) -1 2 (1/2-1/p)-1 2 ν(t)E 0 (1 + s) -5 4 +η ds + E 0 ln(2 + t) (1 + t) -1 2 (1-1/p)-1 2 E 0 (ν(t) + 1)(1 + t) -1 2 (1-1/p)-1
2 +η , giving ν(t) ≤ C η E 0 (1 + ν(t)) , whence, if E 0 < 1/(2C η ), we have (noting that ν by standard theory remains bounded) that ν(t) ≤ 2C η E 0 . This gives k(t)k * ∂ x ψ(t) L p (R) = δ(t) L p (R) ≤ 2C η (1 + t) -1

Expressing by Duhamel's formula δ(t) = σ(t) δ(0) + t 0 σ(ts)∂ x F(s)ds, where F(t) = O((|k(t)| + |φ(t + 1)|) δ(t)), and, for any η ′ > 0, estimating as before, we obtain by a contraction argument similar to the above that δ(t) L p (R)

E 1 (1 + t) -1 2 (1-1/p)-1 2 +η ′ , 1 ≤ p ≤ ∞, provided E 1 is small enough (depending on η ′ ). Indeed one may prove for ν(t) := sup With a slight bit of additional effort, one can check that k * Ψx satisfies the Whitham equation or its quadratic approximant only up to a truncation of the order ∂ 2 x (ψ 2 x ) ∼ ∂ x (kk x ). Once integrated, this means that the formal Whitham equation (1.12) derived for k * (1 + ψ x ) can be expected to hold for k * Ψx only to the (asymptotically equivalent) quadratic order considered here. In particular, if we were to derive higher order expansions for the equation satisfied by Ψx , as we could in principle do, these would not in general agree with the Whitham equation (1.12). In any case, while equation (1.14), most easily obtained by switching to a comoving reference frame x → xat, is designed to match, about a given wave, exactly the order of description we attain here, equation (1.12) is just a convenient way to piece together the two first orders of a nonlinear WKB expansion; thus (1.12) could in principle be meaningful even when dealing with a modulated background wave (instead of a true wave) but should not improve in accuracy its quadratic approximant about a given wave. For further discussion of this and related issues, see [START_REF] Doelman | The dynamics of modulated wave trains[END_REF][START_REF] Noble | Whitham's equations for modulated roll-waves in shallow flows[END_REF][START_REF] Noble | Whitham's modulation equations and stability of periodic wave solutions of the generalized Kuramoto-Sivashinsky equations[END_REF].

  s)∂ x (eδ(s)(k(s) + k * ψ x (s)))ds + r(t).

  ∂ x s p (ts)f p k 2 * ψ x (t) 2 ds-s)∂ x (e(k * ψ x ) 2 (s))ds L p (R) (ts) -1 2 ∂ x (ψ 2 x (s)) L 1 (R)∩L 2 (R) ds

		t
	0 0	σ(t t (1 + t -s)	1 2 (1-1/p)

t 0 t 0

  ≤ C ′ η ′ E 1 (1 + ν(t)) and conclude that ν(t) ≤ 2C ′ η ′ E 1 if C ′ η ′ E 1 < 1/2. Remark A.1. The order O(∂ x (kδ)) ∼ O(∂ x (kk x )) ∼ O(∂ x (k 3 )) ofneglected terms in the argument above is consistent with the order neglected throughout the paper. Indeed, as emphasized by Remark 1.4, the true local wave number is k * Ψx (where Ψ(•, t) is the inverse of y → X(y, t) := yψ(y, t)) which differs from k * (1 + ψ x ) by an O( ψ x 2 ) term since Ψx (x, t) -[1 + ψ x (x, t)] = ψ x ( Ψ(x, t), t)ψ x ( Ψ(x, t)ψ( Ψ(x, t), t), t) Ψx (x, t)(1ψ x ( Ψ(x, t), t)).

	p∈[1,∞]	sup 0≤s≤t	δ(s) L p (R) (1 + s)	1 2 (1-1/p)+ 1 2 -η ′
	that ν(t)

L0 has always at least the translational zero-eigenfunction ū′ .

(1-1/p)-1 2 +η , 1 ≤ p ≤ ∞;

More exactly self-similar in a frame moving with linear group velocity.

We use here Ψ(•, t) -( • + ψ(•, t)) L p (R) ψ(•, t) L ∞ (R) ψx(•, t) L p (R) ; see [JNRZ2].

In other words, (e tL g)(ξ, x) = (e tL ξ ǧ(ξ, •))(x) a consequence of (1.5).

(1-1/p)-1 2 +η , completing the result for kk * ψ x . A similar computation yields the result for hψ.

Acknowledgement. Thanks to Björn Sandstede for pointing out the results of [SSSU]. Thanks also to the anonymous referees for several helpful suggestions that greatly improved the exposition.

June 8, 2012. Research of M.J. was partially supported by an NSF Postdoctoral Fellowship under NSF grant DMS-0902192. Research of P.N. was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01. Stay of M.R. in Bloomington was supported by French ANR project no. ANR-09-JCJC-0103-01. Research of K.Z. was partially supported under NSF grant no. DMS-0300487.

Appendix A. Asymptotic equivalence of scalar equations Lemma 1.2 follows by a simplified (but also somewhat modified; this does not immediately follow from the results stated in [Ka, LZ]) version of the arguments used in [Ka, LZ] to prove corresponding but weaker versions in the system case. We include a proof here, both for completeness, and to motivate the more complicated comparison arguments appearing in the main body of the paper.

Proof of Lemma 1.2. By the general, system, results of [Ka], 5 provided E 0 := k 0 L 1 (R)∩H 3 (R) is sufficiently small, we have for k := κk * and 1 ≤ p ≤ ∞, the "heat-type" bounds

Setting δ := κκ, we have, subtracting and rearranging,

with δ| t=0 = 0, where a = -dq(k * ), d = d(k * ) are constant. By Duhamel's formula,

where σ is the solution operator of the convected heat equation u tau xdu xx = 0. Applying the standard heat bounds σ(t

5 See [START_REF] Kawashima | Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications[END_REF]Remark 4.2] improving the result of [START_REF] Kawashima | Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications[END_REF]Theorem 4.2] in the strictly parabolic case. Bounds (A.1) are proved by estimates similar to those of this paper and [START_REF] Johnson | Nonlocal modulation of periodic reaction diffusion waves: nonlinear stability[END_REF]; see for example (A.2) just below.