Journal of Differential Equations. Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities

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1 J. Differential Equations Contents lists aailable at ScienceDirect Journal of Differential Equations Nonlinear conectie stability of traelling fronts near Turing and Hopf instabilities Margaret Beck a,,4, Anna Ghazaryan b,2,5, Björn Sandstede a,,3,4 a Department of Mathematics, Uniersity of Surrey, Guildford, GU2 7XH, UK b Department of Mathematics, Uniersity of North Carolina, Chapel Hill, NC 27599, USA article info abstract Article history: Receied 25 August 28 Reised 4 February 29 Aailableonline2February29 Reaction diffusion equations on the real line that contain a control parameter are inestigated. Of interest are traelling front solutions for which the rest state behind the front undergoes a supercritical Turing or Hopf bifurcation as the parameter is increased. This causes the essential spectrum to cross into the right half plane, leading to a linear conectie instability in which the emerging pattern is pushed away from the front as it propagates. It is shown, howeer, that the wae remains nonlinearly stable in an appropriate sense. More precisely, using the fact that the instability is supercritical, it is shown that the amplitude of any pattern that emerges behind the wae saturates at some small parameterdependent leel and that the pattern is pushed away from the front interface. As a rest, when considered in an appropriate exponentially weighted space, the traelling front remains stable, with an exponential in time rate of conergence. 29 Elseier Inc. All rights resered. * Corresponding author. address: bjorn_sandstede@brown.edu B. Sandstede. Partially supported under NSF grant DMS Supported under NSF grant DMS Partially supported by a Royal Society-Wolfson Research Merit Award. 4 Current address: Diision of Applied Mathematics, Brown Uniersity, Proidence, RI 292, USA. 5 Current address: Department of Mathematics, Uniersity of Kansas, Lawrence, KS 6645, USA /$ see front matter 29 Elseier Inc. All rights resered. doi:.6/j.jde

2 4372 M. Beck et al. / J. Differential Equations Fig.. The left panel illustrates the rightmost spectrum of the linearization about the front in the comoing frame ξ = x ct at μ =. As μ increases, we assume that essential spectrum crosses into the right half plane. The right panel illustrates how perturbations to the front profile are pushed, in the comoing frame, towards ξ = just after bifurcation, whilst their amplitude saturates at O μ.. Introduction Consider the reaction diffusion equation u t = D 2 x u + f u;μ, u Rn, x R,. where D is a diagonal matrix with positie coefficients, μ is a real parameter, and f is smooth with f ;μ = forallμ. We assume that. has, for all μ near zero, a traelling front solution ux, t = u x cμt;μ that moes with speed c = cμ> to the right and satisfies lim u ξ; μ =, ξ lim u ξ; μ = u +. ξ We are interested in such fronts for which the zero rest state behind the front undergoes a supercritical Turing or Hopf instability at μ =. In the coordinate frame x, Turing bifurcations lead to small-amplitude patterns that are spatially-periodic and time-independent, while Hopf bifurcations lead to patterns that are spatially homogeneous and time-periodic. In the moing coordinate ξ = x cμt, these bifurcations are generically caused by two locally parabolic cures of essential spectrum that cross into the right half plane at ±iω, for some critical temporal frequency ω >, as μ is increased through zero see Fig.. Much preious work on traelling fronts and their stability in reaction diffusion systems exists; see, for example, [2] and the references therein. In particar, the arious ways in which a front can lose stability hae been inestigated. The spectrum can destabilize at ±iω due to either a pair of isolated eigenalues or the essential spectrum crossing into the right half plane. The former case is a classical Hopf bifurcation and was first analyzed by Henry [6, 6.4] using center-manifold theory. In the latter case, because the bifurcation is due to continuous spectrum, standard reduction techniques cannot be applied. This type of destabilization, known as an essential instability, was first analyzed in [4]. The authors were interested in determining whether or not such a destabilization led to the creation of modated fronts connecting the remaining stable rest state with the pattern that emerges. In their analysis they distinguished between two distinct cases: when the destabilization is caused by the rest state behind the front and when it is caused by the rest state ahead of the front. The reason for considering these cases separately can be seen by analyzing the Fredholm index of the operator L λ, where L is the linearization about the front. Roughly speaking, the Fredholm index is a measure of the solability of the system L λu = h for a gien h. When the instability is ahead of the front, the Fredholm index of L λ increases from zero to one as λ moesfromrighttoleftthroughthe essential spectrum. This leads to an underdetermined system of equations and the existence of bifurcating modated waes. When the bifurcation is behind the front, the index changes from zero to minus one, leading to an oerdetermined system. Therefore, the only nearby wae that exists after the bifurcation is the original, linearly unstable front. It is the nonlinear stability of this solution that we analyze here.

3 M. Beck et al. / J. Differential Equations Preious stability analyses in the context of this type of bifurcation include the following. For the modated waes that emerge when the bifurcation is ahead of the front, spectral stability was proed in [4]. Nonlinear stability, which does not immediately follow due to the presence of the essential spectrum on the imaginary axis, was proed for Turing bifurcations in [] using weighted spaces and renormalization group techniques. When the bifurcation is behind the front, and so no nearby modated waes exist, one cod expect some form of nonlinear stability for the original, linearly unstable front. This is because, in a comoing frame, the emergent pattern gets pushed towards minus infinity. In addition, because we assume the bifurcation is supercritical, the amplitude of perturbations behind the front shod saturate at some small alue that depends on μ. As a rest, in a function space in which behaior at minus infinity is suppressed, the front shod be stable. In this case, nonlinear stability has been proed for a specific system in which the rest state behind the front undergoes a Turing bifurcation [4]. Howeer, the techniques used there include the maximum principle and energy methods, which are diffict to generalize. Our goal is to show that, for a general class of systems of the form., the front u ξ; μ is nonlinearly stable in an appropriate sense for all μ near zero when the rest state behind it experiences a supercritical Turing or Hopf instability. Our strategy for proing nonlinear stability is as follows. First, using weighted spaces, we show that the front is nonlinearly stable proided the amplitudes of perturbations to the front profile saturate. Afterwards, we utilize mode filters and the Ginzburg Landau equation that goerns the dynamics near the rest state behind the front to establish these a priori estimates. This second step relies heaily on rests by Mielke and Schneider [9,,6,8]. The outline of the remainder of the paper is as follows. In Section 2 we precisely state our assumptions and rests. A priori estimates for perturbations in the weighted space are gien in Section 3. In Section 4, mode filters are used to show that perturbations in the unweighted space remain small. We briefly summarize our rests in Section Set-up, assumptions and rests The function spaces in which we work need to contain functions that are not necessarily localized in space. Hence, we shall work with the so-called uniformly local spaces that are defined as follows. We choose the weight function ρ x = e x and define T y ρ x = ρ x y to be the translation operator T y applied to the weight function. The weighted L 2 norm and the uniformly local L 2 norm are then gien respectiely by u 2 ρ = R ρ x ux 2 dx and u L 2 = sup u T y ρ. y R The uniformly local space L 2 is then defined as L 2 = closure of C bdd R in { u L 2 loc R: u L 2 < }, and the Sobole spaces H s are defined similarly. The translation operator gies uniformity in space for the norm, and taking the closure of smooth functions ensures that the resting space is complete and that the standard Sobole embeddings still hold. Below we work in H so that solutions are defined pointwise. For more information on these spaces, we refer the reader to []. Hypothesis H. For μ =, Eq.. has a traelling-wae solution u x c t for an appropriate wae speed c >, and the wae profile satisfies lim ξ u ξ = and lim ξ u ξ = u + for some u + R n.

4 4374 M. Beck et al. / J. Differential Equations Fig. 2. The rightmost parts of the L 2 spectra of the linearization L and the weighted operator L a with < a areshown for the case where Hypothesis H2 is met. Note that ω = k c > at Turing bifurcations. We now formate the hypotheses on the spectral stability of the front u. Consider the linearized operator L = D ξ 2 + c ξ + f u u ξ; posed in the comoing frame ξ = x c t, and the asymptotic operators L μ = D 2 x + f u;μ, L + = D 2 x + f uu + ; associated with the spatially homogeneous rest states u = and u = u +, formated in the laboratory frame, on the space L 2 with domain H 2. To capture transport, we let ρ a be any smooth, monotone function that satisfies { if ξ, ρ a ξ = e aξ if ξ, 2. with a >. Hypothesis H2. We assume that the following is true: i There is an a > such that the spectrum of the operator L a := ρ al ρa on L 2 lies in the open left half plane for all < a a except for a simple eigenalue at λ =. ii For all μ close to zero, the spectrum of L μ lies in the open left half plane except for two cures gien by λk, μ = λ μ λ 2 μk k 2 + O k k 3, k k, and its complex conjugate, where Re λ 2 >, Re λ > and either Turing: k > and λ =, or Hopf: k =, λ = iω for some ω >. iii The spectrum of L + lies in the open left half plane. As we shall now argue, Hypothesis H2 implies that the spectra of the operators L and L a on L 2 are as shown in Fig. 2. In the laboratory frame x, the critical cures λk, μ in the spectrum of L μ correspond to neutral eigenmodes of the form e ikx u for Turing bifurcations and e iωt u for Hopf bifurcations. The dispersion cure λk, μ in the laboratory frame x becomes λ k, μ; c = λk, μ + ikc in the frame ξ = x ct that moes with speed c; see [5]. Since the spectrum of L contains the dispersion cure λ, ; c see, for instance, [6, Lemma 2 in the appendix to Chapter 5] or [2, 3.4.3], we see that its spectrum indeed touches the imaginary axis at ±iω, where ω = k c at Turing bifurcations. Next, we note that the group elocity dimλ dk k, of the dispersion cure anishes in the laboratory frame x, which implies that the essential spectrum of the weighted linearization L a in the right-moing frame ξ will moe into the left half plane for all sufficiently small

5 M. Beck et al. / J. Differential Equations a > see [3,4], and Hypothesis H2i ensures that no isolated eigenalues of L a are reealed on the imaginary axis when the essential spectrum is moed. Our last assumption is that the bifurcation caused by the critical cure λk, μ is supercritical. To make this precise, consider. near u =, and let ek be an eigenector of the matrix k 2 D + f u ; associated with the eigenalue λ. Upon substituting the ansatz ux, t = ɛe ik x+ω t A ɛx, ɛ 2 t ek + c.c., μ = ρɛ 2, 2.2 into. and expanding in ɛ, we find that the amplitude AX, T satisfies the Ginzburg Landau equation A T = λ 2 2 X A + ρλ A b A 2 A 2.3 for an appropriate alue b C; see [] and references therein. The sign of the real part of the cubic coefficient b determines whether the bifurcation is subcritical or supercritical. We assume the latter: Hypothesis H3. We assume that Re b > so that the bifurcation is supercritical. It is a consequence of Hypothesis H2i that the front u x c t persists for nearby alues of μ. More precisely, there exist a unique profile u ξ; μ and a unique wae speed cμ with u ; = u and c = c so that ux, t = u x cμt;μ satisfies. for all μ near zero. The next theorem shows that these fronts remain nonlinearly stable, in an appropriate sense, for μ near zero and, in particar, the fronts do not feel the linear instability that occurs in their wake. Theorem. Suppose that Hypotheses H H3 are satisfied, then there exist positie constants K, Λ,a, μ, and δ such that the following is true: for all μ μ and any initial condition satisfying, H <δ, the solution of. with initial data ux, = u x;μ + x, exists for all t and can be written as ux, t = u x cμt pt;μ + x cμt, t for an appropriate real-alued function p; furthermore, there is a constant p R such that, t H + pt K, H + μ, ρ a, t H + pt p K e Λ t for all t. In other words, the perturbation ξ, t decays to zero exponentially in time in the weighted norm ρ a H in the comoing frame ξ = x cμt. There are two steps in the proof of this theorem. The first, contained in Section 3, is to show that, if, t H is small for all t, then ρ a, t H K e Λ t for some Λ >. This will follow using a method analogous to that in [4]. The second step, contained in Section 4, is to show that, t H is, in fact, small for all t. This will be proed using the mode filters. 3. A priori estimates in the weighted space To proe that the fronts u x cμt;μ are nonlinearly stable for all μ with μ, we write solutions to. in the form ux, t = u x cμt pt;μ + x cμt, t.

6 4376 M. Beck et al. / J. Differential Equations The real-alued function pt will be defined in more detail below and will allow us to remoe the neutral behaior due to the zero eigenalue. From now on, we fix μ close to zero and consider the dynamics near the front u x cμt;μ. It is conenient to formate all equations in the comoing frame ξ = x cμt. To simplify notation, we define h ξ := u ξ; μ, h pt ξ := h ξ pt = u ξ pt;μ and omit the dependence on μ. Similarly, we write c and f u instead of cμ and f u;μ, respectiely, from now on. In the comoing frame ξ = x ct, Eq.. becomes Substituting the ansatz u t = D 2 ξ + c ξ u + f u. we obtain the system uξ, t = h ξ pt + ξ, t, 3. t = L + ṗh p [ f u h f u h p ] + [ f h p + f h p f u h p ], 3.2 where we use the notation = ξ, recall that p = pt, and set L := D 2 ξ + c ξ + f u h. We shall use the weighted function wξ, t := ρ a ξξ, t, withρ a from 2., which then satisfies the equation where w t = L a w + ṗρ ah p [ f u h f u h p ] w + N w, 3.3 L a = ρ al ρa = D ξ 2 + c 2ρ a 2ρ ξ + a 2 cρ a ρ a ρa 2 ρ a N = [ fu h p + τ f u h p ] dτ = O. ρ a + f u h, ρ a We will consider w H and choose the exponential rate a in the interal, a ], where a is so small that Hypothesis H2i is met and so that there exists a C > for which h ξ ρ a ξ + h ξ ρ a ξ C, ξ R, 3.4 for < a a. We call exponential rates a, a ] admissible. Throughout the remainder of this paper, we will denote by C any generic constant that does not depend on the initial data or on a and μ. Hypothesis H2 implies that the essential spectrum of L a lies strictly to the left of the imaginary axis for all μ near zero and all admissible rates a. The only eigenalue on or to the right of the imaginary axis is the simple eigenalue λ = with eigenfunction ρ a h. Therefore, it is possible to

7 M. Beck et al. / J. Differential Equations define a spectral projection P c a in H onto the one-dimensional center subspace belonging to λ =. The rest of the spectrum will be bounded away from the imaginary axis, and so the complement of P c a is the projection P s a onto the generalized stable eigenspace. In particar, there exists a constant K that is independent of μ so that e Pa slat H K e Λ t 3.5 for t. We combine this information in the following lemma. Lemma 3.. There exist positie constants K, Λ,andμ such that the following holds for any μ < μ.the spectral projection Pa c is well defined and, in fact, gien by P c a w = ψ a, w L 2ρ a h, where ψ a = ρa ψ,andψ spans the kernel of the operator adjoint to L with For P s a = I P c a,wehae3.5. ψa, ρ a h = ψ L 2, h =. L In the decomposition 3., uξ, t = h ξ pt + ξ, t, the shift function pt is not defined uniquely, een if p = is assumed. To aoid ambiguity we require that w, t lies in the range of the projection Pa s for all t for which the decomposition 3. exists. In other words, we require that wξ, t satisfies Pa c w, t = 3.7 for all t, and, applying the projections Pa s and P a c to 3.3, we obtain the system w t = P s a La w + P s aṗρa h p [ f u h f u h p ] w + N w, 3.8 ṗ = ψ a, [ f u h f u h p ]w N w L 2 ψ a, ρ a h p L 2 for w, p. Conersely, solutions w, p of 3.8 automatically satisfy 3.7. Using the initial data p =, the function pt is then defined uniquely. Note that as long as pt remains sufficiently small, the denominator ψ a, ρ a h p L 2 = ψ, h pt L 2 in 3.8 is bounded away from zero due to 3.6. We consider the system consisting of 3.8 coupled to 3.2. Because the nonlinearity f is smooth, there exists a constant K so that the nonlinearity N and the difference in the linearization about h and h p satisfy fu h f u h p H + N H K p + H, 3.9 ṗ K p + H w H for μ close to zero. The second estimate follows from the first and the equation goerning ṗ in 3.8. By [, Lemma 3.3], the linear operators in 3.2 and 3.8 are sectorial operators on H, and the

8 4378 M. Beck et al. / J. Differential Equations arguments in [6,] therefore imply that solutions exist locally in time, are unique, and depend continuously on the initial conditions. This proes the local existence and uniqueness of the decomposition 3.. Moreoer, the continuous dependence of the solutions on the initial conditions implies that, for any gien < η, there exist a γ > and T >, such that sup t [,T ] pt +, t H η proided, H γ. 3. The maximal time T such that the aboe holds is denoted by T max η. The following lemma states that, as long as the solutions in the unweighted space and p remain small, the solution w in the weighted space will decay exponentially fast in time and control the behaior of p. This is the main rest of this section and is analogous to [4, Lemma 3.2]. Lemma 3.2. Pick Λ such that <Λ<Λ, then there exists an ˆη with < ˆη so that the following is true. If < η < ˆη and w is a solution of 3.3 for which the corresponding solutions and p satisfy 3., then w, t H K e Λt w, H, pt K w, H for all t T max η, for some positie constant K that is independent of μ and η.ift max η =,then there is a p R with pt p K e Λt w, H 3. for all t. Once this lemma has been established, the proof of Theorem will follow if we can proe that T max ˆη = for our particar choice of ˆη. Proof. Consider the equation for w in 3.8 for t [, T max, rewritten here for conenience: w t = P s a La w + P s aṗρa h p [ f u h f u h p ] w + N w. Applying the ariation-of-constants forma to this equation, we obtain w, t = e P s a La t w, + e P s a La t s P s aṗsρa h ps s [ f u h f u h ps ] w, s + N, s w, s ds, where we recall that h pt = h ξ pt depends on t through pt. From 3.5 and 3., it follows that w, t H K e Λ t w, H + K e Λ t s ṗs ρ a h ps H + K ps +, s H w, s H ds.

9 M. Beck et al. / J. Differential Equations Since ρ a and h p ξ = h ξ p are bounded uniformly in p, there exists a constant K 2 > such that Therefore, using 3. we obtain ρ a h p H K 2. w, t H K e Λ t t w, H + K K e Λ t w, H + K K K 2 + η For <Λ<Λ and t T max,define e Λ t s K K 2 + ps +, s H w, s H ds Mt := sup e Λs w, s H. s t e Λ t s w, s H ds. 3.2 After mtiplying 3.2 by e Λt we find that e Λt w, t H K e Λ Λt w, H + K K K 2 + η and therefore If ˆη is chosen so that then we hae t K w, H + K K K 2 + η Mt e Λ Λt s ds K w, H K K K 2 + Λ Λ + K K K 2 + η Mt, Λ Λ Mt K w, H + K K K 2 + η Mt. Λ Λ ˆη 2 or, equialently, ˆη e Λ Λt s e Λs w, s H ds Λ Λ 2K K K 2 +, for < η < ˆη, and therefore w, t H Mt 2K w, H 2K e Λt w, H. 3.3 From 3.3, 3.9, 3. and the assumption that η <, we then obtain

10 438 M. Beck et al. / J. Differential Equations ṗt 2K K η e Λt w, H 2K K e Λt w, H 3.4 for t T max. On the other hand, t pt ṗt ds 2K K w, H e Λs ds 2K K w, H. Λ If T max =, then 3.9, and consequently 3.4 and 3.3, hold for any t. According to 3.4, p = ṗs ds exists, and we hae pt p = ṗs ds for all t. To estimate the conergence rate, we use 3.4 and obtain pt p 2K K w, H e Λs ds 2K K Λ t w, H e Λt. The statements of the lemma then follow with K = max{ K 2, K K 2Λ }; recall that the constants K, K and K 2 are independent of η and T. 4. Estimates in the unweighted space ia mode filters In this section, we proe that, in fact, T max ˆη =. In particar, we will proe the following proposition. Proposition 4.. There exist positie constants K, δ and μ such that, if, H <δ, then the solution to 3.2 exists for each μ with μ μ and satisfies, t H + pt K, H + μ for all t. In particar, T max ˆη = for ˆη = K δ + μ and all μ with μ μ. The proof will consist of two steps. First, we will show that the behaior of solutions to 3.2 is really goerned by the bifurcation at. Then we will show that, because this bifurcation is supercritical, the amplitude of perturbations saturates eentually at O μ in H. The combination of these rests, which leads to the aboe proposition, is contained in Lemma 4.2 below. 4.. Reduction to behaior at minus infinity In order to proe Proposition 4., we first show that the eolution of in the comoing frame is controlled by the dynamics near ξ =. To do this, we will use a method similar to that of [, 5], which we now describe. We write Eq. 3.2 as t = L + N + p,, 4.

11 M. Beck et al. / J. Differential Equations where L = D 2 ξ + c ξ + f u, N = f f u, p, = ṗh p + [ f u h f u ] [ f u h f u h p ] + [ f h p + f h p f u h p ] N. Notice that L and N are respectiely the linearization about the unstable state u = in the comoing frame and the corresponding nonlinearity, where we recall that we assumed that f ; μ = for all μ. The function consists of the drift term ṗh p, the difference between the linearization about the front and about, and the difference between the nonlinearity ealuated at the front and at. On account of Lemma 3.2, we can iew pt as a gien function. Our goal is to use the information in Section 3 about exponential decay in a weighted space and information about p to obtain the following estimates. Lemma 4.. There exists a constant C, depending only on Λ and ˆη,suchthat for all t T max. p,, t H C w, t H Ce Λt w, H This lemma implies that we can think of the eolution of as being goerned by the eolution near the unstable state plus the effects of an inhomogeneity that is exponentially decaying in time. Proof. We will deal with each term inside separately and use that any function in H is defined pointwise by Sobole embedding. Consider the term ṗh p. Since the underlying wae is smooth, we know that h p H K 3 for some constant K 3. Furthermore, Eq. 3.9 and Lemma 3.2 imply that ṗ K p + H w H K 4 e Λt w, H 4.2 for t T max and some constant K 4, which gies the desired estimate for the first term. Next, we write Eq. 3.4 and smoothness of f imply that for < a a and t T max. Similarly, [ f u h f u ] ξ f u h p ξ f u wξ ρ a ξ [ sup fuu u ] hp ξ wξ. u h ρ a ξ [ f u h f u ] H C w H Ce Λt w, H [ f u h f u h p ] ξ [ sup fuu u ] h ξ h ξ pt wξ u h ρ a ξ [ C sup h ξ + ζ ] pt wξ. ζ sup pt ρ a ξ

12 4382 M. Beck et al. / J. Differential Equations Appealing again to 3.4, this gies It remains to estimate the expression [ f u h f u h p ] H Ce Λt w, H. [ f hp + f u h p f h p ] N = [ f h p + f u h p f h p ] [ f f u ]. Rearranging the terms and using that f ;μ = forallμ, weget [ f h p + f ] [ f h p f ] + [ f u f u h p ] = [ fu + sh p f u sh p ] dsh p C h p = h p ρ a w C w. f uu sh p [h p, ] ds Therefore, [ f h p + f u h p f h p ] N H Ce Λt w, H, which proes the lemma Boundedness of solutions ia mode filters Since the estimate for in Lemma 4. is in H and does therefore not depend on the underlying referenceframe,wecanwriteeq.4.fortheeolutionofthesmallperturbation in the original frame x and obtain t = D 2 x + f ;μ + p, = D 2 x + f ;μ + O e Λt. 4.3 In particar, for sufficiently large times, the dynamics of x, t ought to be goerned by the dynamics of the reaction diffusion system t = D x 2 + f ;μ. 4.4 Since is small, this suggests that we use the Ginzburg Landau formalism for Turing or Hopf bifurcations, which describes the dynamics of the enelopes AX, T of modated waes of the form x, t = Ṽ δ A := δe ik x+iω t A δx,δ 2 t ek + c.c., 4.5 for μ δ 2 and with ek gien ia λk, ek = ˆL ik, ek, by the Ginzburg Landau equation A T = λ 2 2 X A + μλ δ 2 A b A 2 A. 4.6 For Re b >, which we assumed in Hypothesis H3, Eq. 4.6 has a bounded attractor, and solutions are therefore bounded uniformly in time; see [9]. Furthermore, it was shown in [8,] that these

13 M. Beck et al. / J. Differential Equations properties of the Ginzburg Landau equation imply that solutions of 4.4 belonging to sufficiently small initial data stay small for all times. The proof of the latter assertion is based upon the use of mode filters, which separate the neutral part of the continuous spectrum near the imaginary axis from the rest of the spectrum and allow one to decompose the solution of 4.4 into a center-unstable component, goerned by the Ginzburg Landau equation, and a stable component, goerned by an exponentially decaying semigroup. The additional Oe Λt term that appears in 4.3 preents us from applying the rests of [8,] directly to Eq Howeer, the ideas and techniques deeloped in those papers are still applicable, and by establishing the so-called approximation and attractiity properties introduced there, we will show that solutions to 4.3 with sufficiently small initial data remain small in H for all t. These two properties will be stated precisely below but, essentially, the approximation property says that, gien any solution A of 4.6, there is a solution of 4.3 that is close to Ṽ δ A in an appropriate sense for large but finite times. In other words, any small solution that looks like a modated wae at a gien initial time will continue to look like a modated wae for large finite times. The attractiity property states that, gien any solution of 4.3, there is a solution A of 4.6 such that Ṽ δ A is close to, again in an appropriate sense, for large but finite times. In other words, all small solutions will eentually look like a modated wae for a large finite time. Proing this is more diffict and requires the use of the mode filters deeloped in [8]. These two properties together will then gie the proof of Proposition 4.. We state the details for Turing bifurcations only and remark that the modifications for the Hopf case require the use of the rests of [9]. First, we define precisely the spaces in which we will work. Let = H R, Rn and S t be the semiflow associated with Eq. 4.3 in. In addition, let G T be the semiflow associated with 4.6 in Y A = H R, C. Local existence for both these semiflows follows from standard arguments. As mentioned aboe, it is known that the flow for 4.6 is globally bounded [9]. We will also need to use the fact that solutions to 4.6 remain in a ball of radius O μ /δ in H for all t. This rest, for the space H rather than L2 or L, follows from the energy estimates of [9, 3]. To define the mode filters, we introduce a cutoff function ˆχ C R, [, ] that satisfies { if k [ γ, γ ], ˆχk = if k / 2γ, 2γ, where γ is some small constant that is independent of μ and δ. Let L μ = D 2 x + f u;μ be the linearized operator at u = in the original frame x and denote its Fourier transform and associated adjoint by ˆL = ˆL k, μ and ˆL = ˆL k, μ, respectiely. For k close to k, these operators hae the eigenalue λk, μ from Hypothesis H2ii, plus its complex conjugate, and we denote by êk, μ and ê k, μ the respectie eigenectors with ê, ê = for allk, μ. We now define the operators P ± in Fourier space as the mtiplication operators P ±û k = ˆχk k ê k, μ, ûk êk, μ and the associated complex-alued ersions ˆp ± ia ˆp ± û = ˆχk k ê k, μ, ûk. These mtipliers, and the ones to follow, depend on μ, but we shall suppress this dependence in our notation. Note also that the functions êk, μ and ê k, μ need only be defined in balls of radius 2γ around ±k for the aboe definitions to be meaningf. We also define P ± mfû k = ˆχ 2k k ê k, μ, ûk êk, μ,

14 4384 M. Beck et al. / J. Differential Equations and similarly ˆp ±, which hae smaller support in Fourier space. Using these operators, we can define mf the mtipliers P c := P + + P, P s := P c, P c mf := P + mf + P mf, P s mf := P c mf, which filter either the stable or center modes, and the associated complex-alued operators ˆp c, ˆp s, ˆp c mf, and ˆps mf. It was shown in [7, Lemma 5] that the operators j P mf and j P defined aboe in Fourier space correspond to bounded linear operators P j mf and P j from H s into itself for any s. Furthermore, P j mf and P j commute with L and map their ranges into itself. Though the operators P j mf and P j are not projections, we hae P c P c mf = P c, which we shall use below. mf Next, define the scaling operator R δ by R δ ux = uδx and the mtiplication operator Θ by Θux = e ikx ux, which is just a translation operator in Fourier space. To relate the reaction diffusion system and the Ginzburg Landau equation, we use the modified ansatz x = V δ A := δθf [ ˆχkêk + k, μfr δ A ] + c.c. : Y A, where F denotes the Fourier transform. The map that sends a solution to its approximation by extracting its critical modes is gien by A δ : Y A, A δ u = δ R /δθ p + u. 4.7 We may now state, in terms of V δ and A δ, the attractiity property that will be used to connect the flows for Eqs. 4.3 and 4.6. Let B Z R denote the ball of radius R in Z centered at. Proposition 4.2 Attractiity. For each r >, there exist positie constants r, r,r,t and δ such that sup inf S T B Y δr A B Y /δ 2 V δa Y r δ 5/4, 4.8 A R t [,T /δ 2 ] S t B δr B δ r 4.9 for all δ and μ with <δ<δ and μ δ 2. Remark 4.. Proposition 4.2 asserts that we can extend the time interal on which, t is defined and remains small from t T max to δ 2 t T, for some T. Technically, we cannot extend this time interal for without also doing so for p. Howeer, one can see that, on any time interal where, t H remains small, w, t H decays exponentially, and pt must therefore also remain small due to Eq We will use this fact below. Proof of Proposition 4.2. Since the emphasis in this proof is on the temporal behaior of solutions, we write throughout t and wt instead of, t and w, t. Allnormsaretakenwithrespectto the spatial ariable for fixed time t. The method of proof is similar to that of [8, Lemma ], except that we need to take the term p, into account. We will show the following: there exist a T with < T and a constant C, independent of μ and δ, such that, for any satisfying δr, the corresponding solution to 4.3 satisfies t Cδ for all δ 2 t T. Furthermore, we can write T /δ 2 = δu c + δ 2 u s, where u j C and P j u j = u j for j = c, s. To see why this leads to the statement of the proposition, define AT = A δ T /δ 2.

15 M. Beck et al. / J. Differential Equations We claim that this defines the element of V δ B Y A R to which S t is attracted, in the sense of the proposition. Indeed, it is straightforward to show that P c u = V δ A δ u for each u with P c u = u, and we therefore hae S T /δ 2 V δ AT = δu c + δ 2 u s V δ [ Aδ δuc + δ 2 u s ] Y δ P c u c V δ A δ u c Y + Cδ 2 Cδ 2. Hence, it suffices to show how solutions of 4.3 can be controlled using the mode filters. Using Lemmas 4. and 3.2, we can write p, x, t = Hx, twx, t, where Hx, t is smooth and bounded uniformly for t T max, and wx, t denotes, with a slight abuse of notation, the function wξ, t written in the frame x, t. We recall that, as we shall use only H estimates in the remainder of this section, the frame does not matter. Next, we set c, = δ P c mf and s, = δ P s mf, and substitute = δ c + δ s into Eq. 4.3 to get δ t c + t s = δl c + L s + N δ c + δ s + Htwt. We now define c and s to be solutions to the following integral equations: c t = e L t c + δ P c mf e L t τ [ N δc τ + δ s τ + Hτ wτ ] τ dτ, 4. s t = e L t s + δ P s mf e L t τ [ N δc τ + δ s τ + Hτ wτ ] τ dτ. 4. Using that the mode filters and the semigroup commute, we can proe as in [8, Lemma 4] that, for each fixed T, there are constants K, K 2 and κ > such that P c mf el t u H K u H, P s mf el t u H K 2 e κt u H for all t with δ 2 t T. The constant K in the aboe estimate will, in general, depend on T due to the growth of order e μt e μt /δ 2 in the center directions. Howeer, as long as μ δ 2,thiswill not affect our rest, because below we choose < T. We proceed now as follows: First, the system 4.3 has a unique solution p, on the interal t T max, which we use to define the inhomogeneity Htwt. Using this information and the estimate from Lemma 4., we see that the system of integral equations defines a contraction in the ball of radius 2R centered at c, s in C bdd [,δ 4 ], H H, proided δ 4 T max.wenowestimate c and s for t δ 4 to show that the components c,s indeed remain bounded on this time interal from which we can then infer, ia the relation = δ c + δ s and by Lemma 3.2 and Remark 4., that T max must be at least as large as δ 4. Using and the fact that w δ, wefindfor t δ 4

16 4386 M. Beck et al. / J. Differential Equations c t C c + Cδ CR + Cδ 3 4 R 2 + C w δ C c τ Y + s τ 2 C t dτ + e Λτ w Y δ dτ since t δ 4 and, again for t δ 4, s t Y Ce κt s Y + Cδ + C e κt τ Λτ w Y δ dτ e κt τ c τ + s τ 2 dτ CR e κt + CδR 2 + C w e Λ/δ 4 e κ/δ 4 δ C 2 e κt + δ. 2 Choosing t = δ 4, we can conclude from the last estimate that s δ 4 Y C 2 e κ/δ 4 + δ C 2 δ. 2 We now exploit this better estimate to bound the solution oer the longer time interal [, T /δ 2 ]. We define u c = c /δ /4 and u s = δ s /δ /4 and substitute = δu c + δ 2 u s into Eq. 4.3 to arrie at the integral equation u c t = e L t u c + δ P c mf e L t τ [ N δuc τ + δ 2 u s τ + H τ + δ ] 4 w τ + δ 4 dτ, 4.2 u s t = e L t u s + δ 2 P s mf e L t τ [ N δuc τ + δ 2 u s τ + H τ + δ ] 4 w τ + δ 4 dτ. Local existence of solutions is clear, and we therefore need to show that solutions u c, u s remain bounded on the interal [, T /δ 2 ], uniformly inδ. The key obseration that allows us to obtain the necessary estimates of the right-hand side of 4.2 is due to Schneider [6] who proed that P c mf B[P c mf u, P c mf u]= for any bilinear form B of the form B[u, ] =ut B, where B C n n. We therefore write N = B[, ]+Ñ, Ñ = O 3, and note that cubic nonlinearities do not pose any problems for estimates of the below type [8]. Thus, to keep the analysis simple, we focus, from now on, on the quadratic terms and remark that the analysis to follow can be extended easily to account for the nonlinearity Ñ using Ñ + 2 C From 4.2, we obtain

17 uc t Y e L t u c Y + Cδ 2 M. Beck et al. / J. Differential Equations t uc τ us τ + δ us τ 2 dτ + Ce Λ δ /4 δ w Y, us t Y e L t u s Y + C uc t 2 + δ uc t us Y t Y + δ 2 us t 2 Ce Λ δ /4 Y + w Y. δ For j = c, s, we introduce the ariables U j t := sup u j τ Y, t T /δ 2, τ t and get U c t C uc + Cδ 2 [ Λ Uc τ U s τ + δu s τ 2] dτ + Ce δ /4 w Y, 4.3 δ U s t C us Y + [ C U c t 2 + δu c tu s t + δ 2 U s t 2] + Ce δ Λ δ /4 w Y. The constant C that appears in 4.3 does not depend on the initial data or on δ or T with T. We now choose constants K c and K s so that K c C uc + Ce Λ δ /4, K s C us + Ce Λ δ /4 for all releant initial data and alues of δ, and define K s := 4 K s + 6CK 2 c. Next, we pick T so that < T < min{, log 2/[4CK s ]}. AslongasU c t 4K c and U s t Ks for t T /δ 2,wehae U s t K s + 6CK 2 c + C[ 4δK c U s t + δ 2 U s t 2]. Therefore, if δ is chosen sufficiently small so that then, in fact, 4Cδ K c + Cδ 2 4 K s + 6CKc 2 2, U s t K s. 2 Furthermore, substituting this bound into the equation for U c,wehae 2 U c t K c + CδT K s + 4Cδ 2 Ks U c τ dτ

18 4388 M. Beck et al. / J. Differential Equations for all t with δ 2 t T. Gronwall s inequality then implies that U c t K c + CδT K 2 s e 4CK s T 3 4 K c due to our choice of T, proided δ is so small that CδT K 2 s K c/4. This means that the preceding estimates hold true for all t in the interal [, T /δ 2 ] as claimed. Proposition 4.3 Approximation. For all positie R,T,andr, there exist positie constants r 2 and δ such that the following is true for all δ and μ with μ δ δ.ifa B Y A R and S T /δ 2 with S T /δ 2 V δ A r δ 5/4,then sup t T /δ 2 S t ST /δ 2 V δ Gδ 2 t A r 2 δ 5/ Proof. This statement is a generalization, in two ways, of the standard Ginzburg Landau approximation theorems found, for example, in [5,8] or the reiew []. First, the parameter δ that measures the size of the solutions is typically taken to be O μ. Howeer, we need to allow for solutions that do not necessarily shrink to zero as μ does. This type of extension has been discussed in [], and a similar technique can be used here. Second, we need to account for the term p,, which we are iewing as an inhomogeneity. To deal with this, the statement of the proposition has been slightly modified so that the approximation does not occur until a time T /δ 2. To see why this works, suppose we allow the solution with initial data to eole up to the time T /δ 2. The proof of the preceding proposition shows the flow is well defined for this long time and that the solution remains bounded. From the time T /δ 2 onwards, the inhomogeneity p, will therefore be exponentially small in δ, een when we replace its argument x by x/δ. As a rest, one can follow the arguments in [8, 3.2] to proe approximation by subsuming the inhomogeneity into the residual terms. By combining these rests with the fact that solutions of 4.6 with initial data of size O μ /δ remain small, we can now proe that stays small in for all time. Lemma 4.2. Under the assumptions of Propositions 4.2 and 4.3, there exist positie constants T,T and δ such that, for all δ, μ and r with μ δ<δ and r sufficiently large, we hae S T +T /δ 2B δr B δr. In particar, solutions S t with initial conditions in B δr stay bounded and exist for all time. Proof. This is essentially the same proof as in [8, ], but we restate it here for conenience. Since the bifurcation is supercritical, the rests of [9] imply that all solutions of 4.6 satisfy lim sup T AT Y A C μ /δ. Furthermore, they imply that there exists an R such that, for each R >, there exist positie constants R and T so that G T B Y A R B Y A R and T [,T ] Choose r independent of δ and sufficiently large so that G T B Y A R B Y A R. 4.5 V δ B Y A R B δr /2, 4.6 which can be done simply by the definition of V δ. By attractiity, we know that there are positie constants R, T and r so that S T /δ 2 B δr Ur δ 5/4 Vδ B Y A R

19 M. Beck et al. / J. Differential Equations uniformly in δ and μ, where U r denotes the neighborhood of size r. ForthisalueofR,thereisthen a T so that 4.5 holds. Furthermore, by the approximation property 4.4, there is an r 2 so that S T /δ 2 [ ST /δ 2 B ] 4.4 δr U r2 δ 5/4 Vδ GT B Y A R 4.5 Y U r2 δ 5/4 Vδ B A 4.6 R Furthermore, it follows from 4.9, 4.4 and 4.5 that these solutions are bounded by max{ r δ,2r δ} on the interal [,T + T /δ 2 ]. We can now repeat and iterate the preceding argument with r = 2R, which proes the rest. Proposition 4., and therefore Theorem, follows now from the aboe rests upon taking δ := μ when, H μ, while taking δ :=, H when, H > μ. 5. Discussion We proed nonlinear stability of fronts near a supercritical Hopf or Turing bifurcation of the rest state left behind by the front. Specifically, we proed that small bounded perturbations stay bounded for all times and are pushed away from the front interface towards the wake of the front. Similar rests to the ones obtained here for general reaction diffusion systems were preiously obtained in [4] for a specific model problem in which the front undergoes a Turing bifurcation. To proe our stability rest, we combined the approach from [4] with techniques from [9,6,8], where a priori bounds of small-amplitude solutions were established using the Ginzburg Landau formalism. Following the arguments in [], it shod be possible to show that the dynamics in the wake of the front is goerned by the associated Ginzburg Landau equation. For Hopf bifurcations, the dynamics of the Ginzburg Landau equation depends strongly on the coefficients λ 2 and b discussed in Section 2. Depending on these coefficients, the prealent dynamics may consist of stable oscillatory waes or of spatio-temporally complex patterns, which will appear with small but finite amplitude in the wake of the front. Our rests show that the front will timately outrun these structures in its wake. We mention that this rest was preiously deried by Sherratt [2] through a formal analysis for fronts near supercritical Hopf bifurcations in the case when these can be described by λ-ω systems. We refer the reader to [7,2,2] for numerical simations and applications to predator prey systems. We did not consider Turing Hopf bifurcations, where both k and ω are nonzero in the original frame x. In this case, the dynamics near the destabilizing rest state can be captured formally by a system of coupled Ginzburg Landau equations that describe small left- and right-traelling waes of the form e ik x±ω t. No rigorous approximation or alidity rests are known in this case. Finally, we mention that the ideas from [4] hae recently been used in [2], see also [3], to proe nonlinear stability of combustion fronts. The key difficties in the situation discussed in [2] are that there are mtiple fronts that decay algebraically to the same rest state as x and that both rest states hae essential spectrum up to the imaginary axis. The approach discussed in [4] allowed the author to obtain a priori estimates that guarantee nonlinear stability. References B δr. [] T. Gallay, G. Schneider, H. Uecker, Stable transport of information near essentially unstable localized structures, Discrete Contin. Dyn. Syst. Ser. B [2] A. Ghazaryan, Nonlinear stability of high Lewis number combustion fronts, Indiana Uni. Math. J. 29, in press. [3] A. Ghazaryan, Y. Latushkin, S. Schecter, A.J. de Souza, Stability of gasless combustion fronts in one-dimensional solids, preprint, 28. [4] A. Ghazaryan, B. Sandstede, Nonlinear conectie instability of Turing-unstable fronts near onset: A case study, SIAM J. Appl. Dyn. Syst [5] A. an Harten, On the alidity of the Ginzburg Landau equation, J. Nonlinear Sci [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 98.

20 439 M. Beck et al. / J. Differential Equations [7] A.L. Kay, J.A. Sherratt, On the persistence of spatiotemporal oscillations generated by inasion, IMA J. Appl. Math [8] P. Kirrmann, G. Schneider, A. Mielke, The alidity of modation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A [9] A. Mielke, Bounds for the solutions of the complex Ginzburg Landau equation in terms of the dispersion parameters, Phys. D [] A. Mielke, The Ginzburg Landau equation in its role as a modation equation, in: Handbook of Dynamical Systems II, North-Holland, Amsterdam, 22, pp [] A. Mielke, G. Schneider, Attractors for modation equations on unbounded domains Existence and comparison, Nonlinearity [2] B. Sandstede, Stability of traelling waes, in: Handbook of Dynamical Systems II, North-Holland, Amsterdam, 22, pp [3] B. Sandstede, A. Scheel, Absolute and conectie instabilities of waes on unbounded and large bounded domains, Phys. D [4] B. Sandstede, A. Scheel, Essential instabilities of fronts: Bifurcation, and bifurcation failure, Dyn. Syst [5] B. Sandstede, A. Scheel, Defects in oscillatory media: Toward a classification, SIAM J. Appl. Dyn. Syst [6] G. Schneider, A new estimate for the Ginzburg Landau approximation on the real axis, J. Nonlinear Sci [7] G. Schneider, Error estimates for the Ginzburg Landau approximation, Z. Angew. Math. Phys [8] G. Schneider, Global existence ia Ginzburg Landau formalism and pseudo-orbits of Ginzburg Landau approximations, Comm. Math. Phys [9] G. Schneider, Hopf bifurcation in spatially extended reaction diffusion systems, J. Nonlinear Sci [2] J.A. Sherratt, B.T. Eagan, M.A. Lewis, Oscillations and chaos behind predator prey inasion: Mathematical artifact or ecological reality?, Philos. Trans. R. Soc. Lond. Ser. B [2] J.A. Sherratt, Inading wae fronts and their oscillatory wakes are linked by a modated traelling phase resetting wae, Phys. D