A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type

Transcription

1 Interfaces and Free Boundaries (18), 7 6 DOI 1.171/IFB/ A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type MARIE HENRY CMI Université d Aix-Marseille, rue Frédéric Joliot-Curie, 15 Marseille cedex 1, France marie.henry@univ-amu.fr DANIELLE HILHORST Laboratoire de Mathématiques, CNRS and University Paris-Sud Paris-Saclay, 15 Orsay Cedex, France danielle.hilhorst@math.u-psud.fr CYRILL B. MURATOV Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 71, USA muratov@njit.edu [Received January 17 and in revised form 6 November 17] A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in R N, with N >. It is shown that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatio-temporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the considered class of reactiondiffusion equations. 1 Mathematics Subject Classification: Primary 5K57, 5B6, 5B, 5Q. Keywords: Pattern formation, multiscale analysis, singular perturbations, nonlinear dynamics. 1. Introduction It is now well established that nonlinear systems of coupled reaction-diffusion equations may be capable of rich dynamical behaviors that give rise to the emergence of spatio-temporal patterns [, 8, 1, ]. Mathematical studies of patterns are complicated by the fact that even relatively simple systems of reaction-diffusion equations may possess solutions that can be extraordinarily complex [5, 1,, 5, 6]. At the same time, these complex solutions may arise generically in the situations that mimic physically relevant conditions and hence are important to the physical systems these equations model [8, 1, 1, 1]. c European Mathematical Society 18

2 8 M. HENRY, D. HILHORST AND C. B. MURATOV Perhaps the most well-known class of pattern-forming systems exhibiting complex nonlinear behaviors are reaction-diffusion systems of activator-inhibitor type [8, 1]: u t D u C f.u; v/; (1.1) v t D v C g.u; v/: (1.) Here, u D u.y; t/ R is the activator variable, v D v.y; t/ R is the inhibitor variable, f and g are the nonlinearities, and are positive parameters denoting the ratios of the length and time scales of the activator and the inhibitor, respectively, y R N is the spatial coordinate, and t is time. Equations (1.1) and (1.) arise when modeling many applications in physics, chemistry, and biology, from combustion to autocatalytic chemical reactions and biological tissues undergoing morphogenesis [7, 8, 1]. The fact that u is the activator implies that there exists a positive feedback for u in (1.1), which in mathematical terms means that the nonlinearity f obeys the relation > (1.) in some range of values of u and v. Similarly, the fact that v is the inhibitor means that there is no positive feedback for v in (1.), and that there is a negative feedback in the response of v to variations of u. Again, for (1.1) and (1.) this can be expressed as < v/ < ; (1.) for all u and v. In particular, if (1.) holds on an open interval of u for any fixed v, v/=@u < outside the closure of this interval, then f is a cubic-like function. A canonical example is the FitzHugh-Nagumo system, a version of which has the following nonlinearities [1]: f.u; v/ D u u v; g.u; v/ D u v a; (1.5) p ; p where a R C is a fixed parameter. For v. /, f is bistable in the sense that the ordinary differential equation u t D f.u; v/ has two stable solutions h.v/, h C.v/ and one unstable solution h.v/, where h.v/ < h.v/ < h C.v/ are the three solutions of the algebraic solution f.u; v/ D. This is the kind of nonlinearity, which we will consider in this paper. A rich variety of patterns in this class of systems has been observed both numerically and analytically [1, 1, 5]. In view of the great complexity of the observed spatio-temporal dynamics, various types of reductions are usually employed to better understand these nonlinear phenomena. An especially fruitful approach which has been successfully used to study pattern formation, relies on the strong separation of spatial scales between the activator and the inhibitor. These studies are also motivated by the fact that strong time and length scale separation is routinely observed in applications [8]. In the case of (1.1) and (1.) the length scale separation is expressed in the smallness of the parameter in (1.1) [, 5, 1 15, 18, 5, 7]. One can get insights into the pattern formation scenarios by investigating the limit # under various assumptions on the scaling of other parameters with [, ]. Nevertheless, the main difficulty in such an approach lies in the fact that the problem under consideration is intrinsically multiscale. Thus, it is not generally possible to analyze the events leading to the formation of a particular pattern using a single limit procedure. For reaction-diffusion

3 A MULTIPLE SCALE PATTERN FORMATION CASCADE systems of activator-inhibitor type with cubic-like nonlinearity f this point was already recognised in [1 16, 18, 1,, ]. To our knowledge, the first rigorous attempt to analyze the sequence of pattern formation events arising at different time scales in in the class of systems (1.1)-(1.) was made by Sakamoto [8]. More precisely, Sakamoto considered (1.1) and (1.) under the assumptions that the domain is obtained from a fixed bounded domain via rescaling by a factor of 1=, consistent with the expected length scale of stable stationary sharp interface patterns [11, 1, 17, 18,, ], as well as assuming that D O. = /. He was able to prove, under suitable assumptions on the nonlinearities, that the solutions of the initial-value problem for (1.1) and (1.) with the initial data varying on the spatial scale of evolve in several stages on well-separated time scales when 1. These stages can be summarized as follows: 1. The distribution of u approaches sharp interfaces on O. = / time scale;. The interfaces move with normal velocity being a function of the average value of v, while the latter solves an ordinary differential equation, on the O.1/ time scale. Note that in [8] the generation of interface result is proved only under the restrictive assumption that the initial data of v is a constant. It is also conjectured that after the completion of stage above the interface will follow a different motion law on a slower O. = / time scale. One question that naturally arises following the analysis of [8] is whether the formation and evolution of the spatio-temporal pattern can, in fact, be characterized across all time scales for a generic set of initial data, when is sufficiently small. Perhaps even more importantly, one should be interested in what are all possible phenomena that can be observed in the limit # under different assumptions on the scaling of other quantities in the problem, such as or the domain size. It is clear that the case studied in [8] is only one such scenario. What one needs to do is to systematically explore different scaling regimes to search for distinct reduced problems signifying qualitatively different pattern formation scenarios in the class of systems, which we consider. This paper provides a full study (across all time scales) of one family of scalings which leads to the same pattern formation scenario for 1. We are going to consider systems of reaction-diffusion equations of activator-inhibitor type under extra assumptions that D 1=, i.e. that is obtained by rescaling a fixed bounded domain with 1=, and D O. p /; p.; /: (1.6) The first scaling assumption is the same as in [8] and is motivated by the expected scale of stable interfacial patterns [11, 1, 17, 18,, ]. The second scaling assumption is chosen so that it results in the same qualitative limit behavior as # for all systems of activator-inhibitor type. Thus, the considered limit process generates a universality class of pattern-forming systems governed by (1.1) and (1.) []. Let us briefly summarize here the conclusions of our analysis about the sequence of progressively longer evolution stages that will occur starting from the initial data varying on the scale of the domain for 1 under a number of assumptions (for technical details, see the following sections): 1. u is frozen, v reaches its spatial average on the O. = / time scale;. v is frozen, u forms sharp interfaces on the O. p / time scale; since p < = it follows that p = ;

4 M. HENRY, D. HILHORST AND C. B. MURATOV FIG. 1. An illustration of the dynamic stages at different time scales via schematic density plots of the activator variable. The interfaces do not move, the spatial average of v evolves by an ordinary differential equation on the O.1/ time scale, where 1 p ;. The interfaces move on the O. p = / time scale with normal velocity depending on the average of v which is slaved to the interface; note that p = 1; 5. We also consider the O. p = / time scale and formally show that interfaces move by nonlocal mean-curvature, recalling that p = p =. This permits to characterize the evolution of patterns from the beginning to the end in the class of systems (1.1) (1.) with 1. We illustrate this progression of stages in Fig. 1. Our paper is organized as follows. In Section we present the main results of this paper. In Section we prove some preliminary estimates on u and v, which in particular imply that u and v are bounded. In Section we deduce from the previous estimates that on the time interval Œ; 1 = j ln j, u is close to its initial condition u and that v is close to the spatial average of v. Following Xinfu Chen [], we obtain in Section 5 that at the time WD p j ln j, the solution u develops an interface and that v stays close to the average of v. In Section 6 we prove that there exists a time of order j ln j such that the interface already formed does not move on the interval Œ ; C. Moreover, in each region, and C, of separated by we deduce that v is approximated by the solution, t 7! Qv.t/, of an ordinary differential equation. In Section 7, we prove that on the time interval Œ C ;, where is of order p =,.u ; v / tends to the solution of a free boundary problem where the motion equation connects the velocity of to the

5 A MULTIPLE SCALE PATTERN FORMATION CASCADE 1 limit Qv of v and that Qv is a solution of an algebraic equation. This leads us to consider in Section 8 a larger time interval, and we formally obtain that the interface moves by a nonlocal mean-curvature flow. Our proofs are based on the comparison principle associated with (1.1) and on a construction of non classical sub- and supersolutions of (1.1). For that purpose we use traveling wave solutions of a related one-dimensional parabolic system and a modified distance function to the interface. Then u is squeezed between two functions which have the profile of the traveling wave and which converge to h C in C and h in. Thus we deduce the convergence of u in each subdomain. Furthermore, the sub- and supersolutions depend on v and hence on h.v / and h.v /. This leads us to smoothly extend the functions h.v/, h.v/ to the whole of R and, as a consequence, the function f.u; v/ D.u h C.v//.u h.v//.u h.v// to the whole of R. We then introduce an extended problem for (1.1) (1.) and prove that its unique solution coincides with.u ; v /.. Statement of results We consider the following system 8 p u t D = u C f.u ; v / in.; T / ˆ< v.p t D = v C g.u ; v / in.; T D T ˆ: u.x; / D u.x/; v.x; / D v.x/ for x (.1) (.) (.) (.) and we suppose that.h 1 / R N.N > / a smooth (C 1 ) bounded domain,.h / < p < =,.H / f.u; v/ D u.1 u / v and g.u; v/ D u v a. As it has been described in the introduction, Problem.P / can be obtained from (1.1) and (1.) by setting D p (cf. (1.6)) and x D 1= y, so that y 7! x maps into. Moreover in what follows we use the notation h.x; t/dx D 1 h.x; t/dx for all functions h. jj.1 First stage: v becomes close to the spatial average of its initial condition in a time of order = j ln j We first prove the following result about the approximate attainment by the v variable of its spatial average in the first stage. Theorem.1 Assume that u and v are in C./ and satisfy the D : Let.u ; v / be the solution of.p / then there exist positive constants 1, M 1 and 1 > such that for all.; 1 jv.x; 1 / v.x/dxj 6 M 1.N C/ ; for all x (.6)

6 M. HENRY, D. HILHORST AND C. B. MURATOV and u.x/ M 1 = p j ln j 6 u.x; 1 / 6 u.x/ C M 1 = p j ln j for all x (.7) where 1 D 1 = j ln j.. Second stage: Generation of interface in a time of order p j ln j We prove in this section a generation of interface result at the time Assuming that D p j ln j where is a positive constant and p.; / (.8) v.x/dx 6 p ; (.) where > is small enough such that Lemma B.1 is valid, then we have the following result Theorem. Let L WD p j ln j C = p j ln j C.N C/ (.1) then there exist positive constants, M and such that for all Œ; / the solution.u ; v / of Problem.P / satisfy that h v M L 6 u.x; / 6 h C v C M L ; 8x ; (.11) u.x; / h C v 6 M L ; 8x ;C ; (.1) u.x; / h v 6 M L ; 8x ; ; (.1) where ;C WD ; x ; u.x/ > h WD x ; u.x/ 6 h v C M L ; v M L : Moreover there exists a positive constant K > such that jv.x; / v j 6 KL : (.1). Third stage: time evolution with a fixed interface The goal of this section is the study of Problem.P / on a time interval Œ ;, where D c p;n j ln j m (.15)

7 with A MULTIPLE SCALE PATTERN FORMATION CASCADE < c p;n min p ;.N C / ; and m > is a constant to be chosen later. Setting x ; u.x/ D h ;C WD we assume that D x ; u.x/ > h v p ; (.16) v ; (.17) and ; WD x ; u.x/ < h is a smooth hypersurface. Moreover we also suppose that u and 8 ˆ< u.x/.h / ˆ: u.x/ h h v > dist.x; / if x ;C v 6 dist.x; / if x ; v ; (.18) v satisfy where dist.x; / denotes the distance function from x to. In this stage we prove that in the time interval Œ ; the interface, already formed in stage, does not move and moreover that as # v tends to a function which only depends on t. Lastly, we define the set B h B WD p where is a small constant introduced in (.). C ; p Theorem. Assuming jc./j. ; C/ (.1) jj where are defined by (B.1) then the initial value problem 8 ˆ<. Qv / t.t/ D Qv.t/ C h C Qv.t/ j;c j C h Qv jj.t/ 1 j;c j a jj.e/ ˆ: Qv./ D v.x/ dx i ; possesses a unique solution t 7! Qv.t/ defined on Œ; C1 such that Qv.t/ B; (.) for all t Œ; C1/. Let Qv ;1 WD lim t!1 Qv.t/ then we have j;c j Qv ;1 C h C. Qv ;1 / C h. Qv ;1 / 1 jj and moreover there exists a positive constant C such that j;c j jj a D ; (.1) j Qv. / Qv ;1j 6 C c p;n m : (.)

8 M. HENRY, D. HILHORST AND C. B. MURATOV Setting u.x; t/ D ( hc Qv.t/ if x ;C [ and t Œ; h Qv.t/ if x ; and t Œ; and denoting by Q d.x; / the signed distance to such that ( Qd.x; / D dist.x; / if x ;C we then obtain the convergence theorem Qd.x; / D dist.x; / if x ; Theorem. Assuming (.1) and.h / there exist positive constants M and > such that for all.; the solution.u ; v / of Problem.P / satisfies that ju.x; t C / u.x; t/j 6 M c p;n ; (.) for all x fx ; j Q d.x; /j > c p;n g and t Œ;. Moreover we also have jv.x; t C / Qv.t/j 6 M c p;n ; (.) for all x and t Œ;.. Fourth stage: Propagation of interface for large time The goal of this stage is to study Problem.P / for t > C. We first consider the limit problem, 8 V n; D p h. Qv / on.s/; s.; QT /; (.5) ˆ<.Q / ˆ: jc h C. Qv / j C h. Qv / 1 jj j jtd D ; jc j jj Qv a D in.; QT /; (.6) where Qv D Qv.s/, C.s/ the interior of.s/,.s/ D n C.s/ and V n; is the normal velocity of.s/ in the direction of C.s/. We note that Qv./ D Qv ;1 and that the velocity V n; only depends on s and we state that.q / is well posed locally in time. Theorem.5 Assume that (.1) holds. There exists QT > such that the free boundary Problem.Q / has a unique smooth solution. Qv.s/;.s// for s Œ; QT. Let u.x; s/ D we then obtain the following convergence result ( hc Qv.s/ if x C.s/ [.s/ and s Œ; QT h Qv.s/ if x.s/ and s Œ; QT ; Theorem.6 Assume that (.1) and.h / hold. There exists a positive constant T such that for all > we have u.x; p = s C C / u.x; s/ 6 ; (.7)

9 A MULTIPLE SCALE PATTERN FORMATION CASCADE 5 for all.x; s/ f.x; s/ Œ; T ; jd.x; Q.s//j > g and all small enough. Moreover v.x; p = s C C / Qv.s/ 6 ; (.8) for all.x; s/ Œ; T and small enough. We note that in general the solution of.q / may not exist for all s > because of the possibility of vanishing in finite time. Nevertheless, if the solution of.q / exists globally in time, it must necessarily reach a steady state as s! 1. Indeed, if P.C / is the perimeter of C, which in view of the regularity of coincides with the.n 1/-dimensional Hausdorff measure of, from (.5) and our sign convention on V n; we obtain djc.s/j D ds P C p.s/ h Qv.s/ : (.) At the same time, (.6) is equivalent to jc j D a C Qv h. Qv / jj: (.) h C. Qv / h C. Qv / It is a calculus exercise to show that the right-hand side of (.) is a strictly monotone increasing smooth function of Qv. p =; p =/ for all a. 1; 1/. Therefore, for any given jc j.; jj/ and a. 1; 1/ there is at most one value of Qv that satisfies (.), and this value of Qv is increasing as jc j increases. In particular, introducing jc ;1 j D a C 1 jj; (.1) it follows that Qv > whenever jc j > jc ;1j, and vice versa. At the same time, recalling that the sign of h. Qv / coincides with that of Qv, from (.) we get that jc.s/j! jc ;1 j and Qv.s/! as s! 1, if the solution of.q / exists for all s >. In fact, this convergence is exponential, as can be easily seen from the linearization of (.). Thus, in the limit s! 1 the interface.s/ solving.q / must converge (in Hausdorff sense) to some limiting interface ;1 enclosing a set C ;1 whose measure satisfies (.1), while Qv vanishes asymptotically. We note that the coupling between.s/ and Qv.s/ is such that small deviations of j.s/j from 1.1 C a/jj are restored via the solution of.q / on the time scale of the Fourth stage. This property should naturally be inherited by the solutions of.p / for small enough. Therefore, on longer time scales one should expect that the limiting problem preserves the volume of the zero super-level set of u.; t/ for all t larger than the Fourth stage time scale. Noting that lim! u.x; p = s/ D u.x; s/ D h. Qv.s//! 1 as s! 1 for all x ;1, respectively, with ;1 WD nc ;1, we should then expect lim u.x; t/ dx D a t p = : (.)! In other words, beyond the Fourth stage the evolution governed by.p / should become asymptotically mass-preserving, which is an interesting feature of the considered problem, since a priori problem.p / does not have such a property.

10 6 M. HENRY, D. HILHORST AND C. B. MURATOV.5 Fifth stage: Propagation with non local mean curvature In this stage we assume that there exists a solution of the free boundary problem ( V n D K C C./ Ow 1 R 5 j 5j K C C./ R Ow 5 for all.; T /; 5./ D 5; ; where Ow 5 satisfies with 8 ˆ<.Q 5 / ˆ: Ow 5 D u 5 a in.; T 5 Ow D T 5/; Ow 5 dx D for all.; T 5 /; 1 if x C u 5.x; t/ D 5./; 1 if x 5./; where V n, K, C 5./ and 5./ denote respectively the normal velocity in the direction of C 5./, the sum of principal curvatures (positive if C 5./ is convex), the interior and the exterior of 5./, respectively. We remark that.q 5 / only makes sense if u 5.s; / dx D a. 1; 1/. We formally show that u.; p = s/ solving.p / converges to u 5.; / as tends to zero when.; T 5 / and 5; D ;1, where ;1 is the asymptotic limit of.s/ solving.q / as s! 1, provided it exists. Note that the free boundary problem above preserves j5./j, which in view of (.1) satisfies j5./j D 1 C a jj for all.; T 5 /: (.) This is due, as was already mentioned, to the strong restoring effect on j5./j from the spatial average of v inherited from the Fourth stage. We also note that the resulting volume-preserving nonlocal mean curvature flow is a gradient flow. Hence one expects that the interface 5./ ultimately reaches a steady state as! 1, if the solution is global in time.. Preliminary estimates Lemma.1 Assume that u and v are in C./ and satisfy the homogeneous Neumann boundary conditions (.5); then there exists a unique solution of the system.p / for all < T 6 1. Moreover there exists a positive constant C such that for all > ju.x; t/j C jv.x; t/j 6 C for all x and t > : (.1) Proof. From standard theory for parabolic systems we deduce the existence of a unique solution.u ; v / of.p /. Moreover applying the Corollary 1.8 of [] we obtain the estimate (.1). Next we state some estimates, which will be useful in what follows, namely Lemma. We set v.t/ WD v.x; t/dx; then there exist positive constants QC and 1 such that v.t/ v.x/dx 6 QC t; for all t > ; (.)

11 A MULTIPLE SCALE PATTERN FORMATION CASCADE 7 and Proof. max x jv.x; t/ v.t/j 6 QC.N C/ ; for all t > 1 WD 1 = j ln j: (.) Integrating (.) on and on Œ; t for all t > and using (.1) we obtain that t.v / s.s/ds D jv.t/ v./j 6 C 1 t; which coincides with (.). Next we prove a preliminary estimate, which will be useful to obtain (.), namely krv.:; t/k L./ C kv.:; t/ v.t/k L./ 6 C krv k L./ exp Multiplying (.) by v and integrating the result on we obtain rv rv t D = jv j v g.u ; v /; so that using (.1) This together with the inequality d dt krv k L./ 6 = jv j C = C : kv k L./ > krv k L./ ; gives d dt krv k L./ 6 = krv k L./ C = C ; which by Gronwall Lemma implies krv.:; t/k L./ 6 krv.:; /k L./ exp t C C = = : t C C = = : (.) This together with the Poincaré inequality gives (.). We now prove that there exists a constant K independent of such that Setting s WD jv.x; t/ v.x ; t/j 6 K jx x j; for t > ; x; x (.5) jrv.x; t/ rv.x ; t/j 6 K jx x j = ; for t > ; x; x : (.6) t = we deduce from (.) that v is the solution of v s C Av D v C = g.u; v/ DW G.v; x; s/; where Av D that v C v. As it is done in [8], one can check that there exists a constant C ;p such ka v.s/k L p 6 C ka v./k L p C C ;p ;

12 8 M. HENRY, D. HILHORST AND C. B. MURATOV with.; 1/, p > 1. Further since D.A / C 1C./; for N=p > 1 C and.; 1/; we obtain choosing D 15 and p > 8N that 16 v.:; t/ C 1C./ and jv.:; t/j C 1C./ 6 K ; for D =; which gives (.5) and (.6). Applying (.) for all > 1 WD 1 = j ln j where 1 D we obtain that krv.:; t/k L./ C kv.:; t/ v.t/k L./ 6 C krv.:; /k L./ exp j ln j C C = ; so that kv.:; t/ v.t/k L./ D O.= /: (.7) Furthermore in view of (.5)-(.7), we deduce from Lemma. of [8] that for all > 1 WD 1 = j ln j max.x; t/ v.t/ 6 C kv.:; t/ v.t/k Q x N C Q L./ where Q.; 1 and C is a positive constant. Thus (.7) and (.8) with Q D 1 imply (.), which concludes the proof of Lemma... Proof of the first stage: v becomes close to v in a time of order = j ln j Proof of Theorem.1. By (.) with D 1 D 1 = j ln j we obtain that v. 1 / v.x/dx 6 QC 1 = j ln j: This together with (.) gives max x v.x; 1 / v.x/dx 6 C 1.N C/ ; which implies (.6). Next we prove (.7). We set U.x; t/ WD u.x/ C1 p t where C 1 is a constant such that C 1 > C C C.C C 1/ with C defined in lemma. Denoting by L the parabolic operators associated to..1/ one can check that (.8) L.U C ; v / > and L.U ; v / 6 ; (.1) on Œ; 1 and then deduce from the comparison principle that U.x; t/ 6 u.x; t/ 6 U C.x; t/; for all.x; t/ Œ; 1. Applying this with t D 1 we obtain (.7), which completes the proof of Theorem.1.

13 A MULTIPLE SCALE PATTERN FORMATION CASCADE 5. Proof of the second stage: Generation of interface in a time of order p j ln j Proof of Theorem.. Since the details of the computations are given in [] we only give the main p steps of the proof. First we remark that for all v R such that jvj 6 where > is a positive constant, the algebraic equation f.:; v/ D has three solutions h.v/ < h.v/ < h C.v/. As it is done in [], we now introduce an approximation of the function f. To begin with let s 7!.s/ C 1.R/ be a cut-off function satisfying 8.s/ D 1; if jsj 6 1; ˆ<.s/ D ; if jsj > ; <.s/ < 1; if 1 < jsj < ; < s ˆ:.s/ 6 ; if s R; j.s/j 6 ; if s R; u then we set D h.v/ p j ln j u ; C D hc.v/ p j ln j and u D h.v/ p j ln j and f Q u h.v/.u; v/ WD j ln j h C.v/ C C j ln j u C h.v/ j ln j u C.1 C /f.u; v/: (5.1) Thus following the proof of estimate (.8) in [] one can check that jf Q.u; v/ f.u; v/j 6 C f p j ln j; for all u ŒC ; C : (5.) We next show that the solution u can be approximated by the solution of the following ordinary differential equation (! s.; s; w/ D f Q.!; w/; for all s > ;.ODE/!.; ; w/ D ; where Œ C ; C and w B. Replacing by p in the proof of Lemma. in [], one can obtain the following properties of! Lemma 5.1 Assume that Œ C ; C and w B and let!.; s; w/ be the solution of.ode/. Then! C.R R C B/ and!.; s; w/ > : (5.) There exist positive constants and such that for all.; and s >, we have and!.; s; w/ > h C.w/ p j ln j; 8 Œh.w/ C p j ln j; 1/; (5.)!.; s; w/ 6 h.w/ C p j ln j; 8. 1; h.w/ p j ln j ; (5.5) h.w/ p j ln j 6!.; s; w/ 6 hc.w/ C p j ln j; 8 Œ C ; C : (5.6) Moreover, there exists a positive constant C 1 such that for all.; and s Œ; j ln j, we have! j! j 6 C 1 : (5.7) p

14 1 M. HENRY, D. HILHORST AND C. B. MURATOV We are now in a position to prove the generation interface. Let D p j ln j, where is defined in Lemma 5.1. Using (.) we have v.t/ v.x/dx 6 QC p j ln j (5.8) for all t Œ; and then by (.) and the definition of L,(.1), we obtain v.x; t/ v.x/dx 6 v.x; t/ v.t/j C jv.t/ v.x/dx 6 QC L ; (5.) for all.x; t/ Œ 1 ;, which coincides with (.1). Setting and u.x; t/ D u.x; 1 C t/ and v.x; t/ D v.x; 1 C t/; for all.x; t/ Œ; 1 ; (5.1) u.x; t/ D! t u l p where l is a constant to be chosen later. We next prove C = p t j ln j ; p ; v ll ; for all.x; t/ Œ; 1 ; (5.11) u.x; t/ 6 u.x; t/ 6 uc.x; t/; for all.x; t/ Œ; 1 : (5.1) To that purpose, we first compute the derivatives of u, namely p u t D l p! C! s D l p! C f Q!; ; and by (5.7) v C ll ; (5.1) ju j D ju! C jru j! j 6 AQ! ; (5.1) p where AQ is a positive constant. Thus by (5.1), (5.1) and (5.1), (5.), (5.), (5.) we have that L.u ; v / 6 C f p j ln j C QC L ll C! p l C Q A ; for all.x; t/ Œ; 1. Thus L.u ; v / 6 for l > AQ CC f C QC. Similarly, one can check that L.u C ; v / >. Further noting that u.x; / D!.u l = p j ln j; ; v ll / D u.x/ l = p we deduce from (.7) and (5.1) that for l > M 1. By u.x; / 6 u.x; / D u.x; 1 / 6 @n and the comparison principle we obtain (5.1). Let us apply (5.1) at t D from (5.6), (A.11) and (.) that u.x; / D u.x; 1 / > h v C ll p j ln j > h v 1 ; then we deduce.lk C /L :

15 A MULTIPLE SCALE PATTERN FORMATION CASCADE 11 Similarly one can check that u.x; / D u.x; 1 / 6 h C v C.lK C /L ; (5.15) so that (.11) is obtained. Further by (A.11) and (5.) we have u.x; / D u.x; 1 / > h C. v C ll / p j ln j > h. v /.lk C /L provided that u l 1 p l = p j ln j > h. v C ll / C p j ln j: This last condition is satisfied if u > h. v / C CL ; for C >.K l C l C l C /. This together with (5.15) implies (.1). In the same way, one can prove (.1) and this concludes the proof of Theorem.. 6. Proof of the third stage: Time evolution with a fixed interface Proof of Theorem.. We set K. ; v/ WD h C.v/ C.1 /h.v/ v a (6.1) and K.v/ WD K. j;c j ; v/, so that the ODE of the Problem.E/ coincides with jj. Qv / t D K. Qv /: By the Cauchy theorem.e/ admits a unique solution on a maximal time interval I D Œ; QT /. Since v 7! K.v/ is strictly decreasing and since by (.1) K. p / D p j;c j jj 8 a < and K. p / D p j;c j jj 1 a > we deduce that there exists a unique!. p ; p / such that K.!/ D. Moreover we suppose that is small enough to ensure that!. p C ; p /. 1. If! D v, then Qv.t/ D v, for all t Œ; QT / with QT D 1. In this case (.), (.1) and (.) are satisfied.

16 1 M. HENRY, D. HILHORST AND C. B. MURATOV. If v >! then since Qv.t/ and! are two different solutions of.e/ we have Qv.t/ >!; for all t I and then K. Qv.t// < K.!/ D ; for all t I. Thus by the ODE Qv is non increasing and we have! < Qv.t/ < v ; for all t I: (6.) Further by classical argument, one can prove that QT D C1, lim t!1 Qv.t/ D! and that lim t!1. Qv / t D. Thus! D Qv ;1 (6.) and (.), (.1) are satisfied. We now prove (.). Let w.t/ D e t. Qv.t/!/ then using (6.) and the fact that h are non increasing we have w t D e t. Qv!/ C e t K. Qv / K.!/ D e t j;c j jj h C. Qv / h C.!/ C e t 1 so that w.t/ 6 w./ for all t Œ; 1/. This gives 6 Qv.t/! 6 e t v! j;c j h C. Qv / h C.!/ 6 ; jj ; for all t Œ; 1/: Thus for t D D c p;n j ln j m we obtain 6 Qv. C /! 6 C c p;n m, which in view of (6.) coincides with (.).. One can give similar arguments in the case v <! and conclude the proof of Theorem.. In what follows we introduce preliminary notations, which will be useful in the sequel. Let O h and O h be the perturbations of h and h defined in Appendix A. Setting O f.s; v/ WD s O h.v/.s O hc.v//.s O h.v// (6.) we denote by. ; / the solution of the following system 8 p t ˆ< D = C f O. ; / in.; T /. P O / t D = C g. ; / in.; T ˆ: D D T with the initial conditions (6.5) (6.6) (6.7).x; / D u.x/;.x; / D v.x/; for x : (6.8) By standard theory for parabolic systems there exists a unique solution. ; / of. O P / such that j.x; t/j C j.x; t/j 6 OC ; for all x and t > : (6.)

17 A MULTIPLE SCALE PATTERN FORMATION CASCADE 1 We now claim that there exists a positive constant QC such that and.t/./ 6 QC t; for all t > (6.1) max.x; t/.t/ 6 QC.N C/ ; for all t > 1 WD 1 = j ln j: (6.11) x Since.P / and. P O / have the same second parabolic equation the proof of (6.1) and (6.11) are exactly the same as the one of (.) and (.) respectively. Moreover we denote by. ; / the solution of system. P O / with the initial condition.x; / D u.x; / and.x; / D v.x; /; for x : (6.1) We now consider the interface defined by the interface motion equation where D V n; D = p p O h. /; j td D ; (6.1).x; t/dx and V n; denotes the normal velocity of. One can show that Problem (6.1) possesses a unique classical solution on a maximal time interval Œ; Qt. We then deduce from (6.1) and the definition of h O that jv n; j 6 C = p, where C is a constant independent on and Qt. We now set there exists ı such that W R! R N.s; z/ D s C zn.s/i for all < ı 6 ı, is a C 1 diffeomorphism from. ı; ı/ to its range,.ı/. (6.1) This yields setting l p t.t/ WD = p Oh.r/ dr; (6.15) that jl.t/j 6 C = p t, for all t Œ; Qt. Thus the interface is well defined on Œ;, where D c p;n j ln j m. This gives that (6.1) admits a unique classical solution on the time interval Œ;. Note that for t Œ; divides into two subdomains ;.t/, with ;./ D ;. Next we introduce a smooth truncated approximation of the signed distance function to the interface ; more precisely let < r < ı we define d.x; t/ C..; // as 8 ˆ< d.x; t/ D ˆ: r r if x ;C.t/ and dist x;.t/ > r if x ;.t/ and dist x;.t/ > r dist x;.t/ if x ;C.t/ and dist x;.t/ 6 r dist x;.t/ if x ;.t/ and dist x;.t/ 6 r ;

18 1 M. HENRY, D. HILHORST AND C. B. MURATOV and extend it smoothly for x fr = < dist.x;.t// < r g. Moreover we assume that is small enough D /: (6.16) Further let.u.z; V; ı/; C.V; ı// be the solution of the following system 8 ˆ< U zz.z; V; ı/ C C.V; ı/u z C f O.U.z; V; ı/; V / D ı; 8z R;.T W / lim z!c1 U.z; V; ı/ D h ˆ: O C.V; ı/; lim z! 1 U.z; V; ı/ D h O.V; ı/; U.; V / D h O.V; ı/: The basic properties of.u.z; V; ı/; C.V; ı// are recalled in Lemma A.1. We now define for all x and t Œ; U.x; t/ D U d.x; t/ S 1S e mt = ;.t/; S S! (6.17) where S WD c p;n j ln j (6.18) and S 1, S are positive constants to be determined later. Note that by the definitions of L and c p;n (see (.1) and (.16)) we have S > L and lim # We next state four lemmas, which will be useful to prove Theorem.. Lemma 6.1 The functions U.x; t/ satisfy that OL U C ; WD p U C = U C t OL U ; WD p U S D 1: (6.1) = O f U C ; > ; on Œ; (6.) = U O t f U ; 6 ; on Œ; D Œ; : Proof. with It follows from D. Furthermore we have that OL U C ; D I1 C I C I C S S (6.) I 1 WD p U. / t = U z d C U zz.1 jrd j / (6.) I WD U z p =.d / t C p h O ; S S C p = e mt ms 1 S (6.) and I WD O f U; C O f U; ; (6.5)

19 A MULTIPLE SCALE PATTERN FORMATION CASCADE 15 d where the derivatives of U are evaluated at the point.x; t/ C S 1S e mt = ;.t/ S S. Since the estimates of I 1, I and I are standard (see [], [6], [8], [11]) we only give the main steps of the computations. We have by the definition of d and the property (A.6) of U that Uzz.1 jrd j / 6 C sup Uzz. d.x; t/ C S 1S e mt ; =.t/ S S / Moreover since D c p;n j ln j m jd j> r 6 CK 1 sup jd j> r we have for t Œ; e Kj d CS 1 S e mt = j : jd C S 1S e mt j > r S 1 S e m D r S 1 S c p;n for jd j > r ; so that Uzz.1 jrd j / 6 CK 1 e K = Œ r =CS 1 S c p;n 6 C 1 p ; (6.6) where we have used the fact that S c p;n tends to, as #. Noting that d is bounded and also using (A.5), (6.6) and the fact that j./ j 6 C (see(6.6) and (6.)) we deduce from (6.) that I 1 > D 1 p : (6.7) To estimate I, we first note that the motion equation (6.1) together with the mean value theorem and the smoothness of the function d implies.d / t C = p p O h. / 6 Djd j; in Œ; : Substituting this into (6.5) and using (A.1) we obtain I > U z Moreover we have that p = Djd j K S p S C p = me mt S 1 S : (6.8) me mt S 1 S Djd j > Djd C emt S 1 S j C e mt S 1 S.m D/: Substituting this into (6.8) we obtain I > p DU z jd C emt S 1 S j = C U z S h K S p C p = S 1.m D/e mti : Noting that the second term is positive for m > D and using (A.6) combinated with the fact that sup zr jze K z j is bounded we obtain I > D p ; (6.)

20 16 M. HENRY, D. HILHORST AND C. B. MURATOV for small enough. Next we estimate I. Using (A.1) and (6.11) with t > we obtain that I > c O.x; t/.t/ > D.N C/ : (6.) Substituting (6.7),(6.) and (6.) into (6.) we deduce that f OL U C ; > S S p.d 1 C D / D.N C/ : Thus for S > D 1 C D C D we obtain OL.U C ; / >, which coincides with (6.). One can prove the inequality (6.1) in a similar way. Next we prove the following inequalities on the initial functions, namely Lemma 6. U.x; / 6.x; / 6 U C.x; /; for all x : (6.1) Proof. In view of (6.1) and (.) we first note that./ v D v. / v 6 QC p j ln j: (6.) Further if d.x; / 6 M L then x ; and thus by.h / u.x/ h v 6 dist.x; / 6 M L ; so that x ;. Thus we deduce from (6.1), (6.1) and (.1) that.x; / D u.x; / 6 h v C M L 6 h Moreover since U is strictly increasing we have v C M S : (6.) U C.x; / D U d.x; / C S 1 S = ;./; S S > O h./; S S Further since by (6.1)./ S S D v. / S S we deduce from (6.) and (.) that./ S S. p C ; p / for small enough, so that by (A.) Oh./; S S D h./ S S : Then by (A.1), (A.1) and (6.) we obtain U C.x; / > h./ C K S S > h v K QC p j ln j C K S S : Thus for S > M C K QC K and small enough we obtain by (6.) that.x; / 6 U C.x; / for all x such that d.x; / 6 M L : (6.)

21 A MULTIPLE SCALE PATTERN FORMATION CASCADE 17 If d.x; / > M L then choosing S 1 > M we have by (A.5) and (A.7) successively that U C.x; / > U M L = ;./; S S > O h C../; S S / K 1 e By (A.1), (A.1) and (6.) we obtain as previously that U C.x; / > h C v C S K S K QC p j ln j K 1 e K M L = : K M L = ; and then by (.11) one has for S > M C K QC C K 1 K and small enough that.x; / 6 U C.x; / for all x such that d.x; / > M L : This together with (6.) implies that.x; / 6 U C.x; /, for all x. Similarly one can show that.x; / > U.x; /, for all x. This completes the proof of Lemma 6.. Next we prove the following result Lemma 6. There exists a positive constant L 1 such that.x; t/ hc O.t/ 6 L1 c p;n j ln j if d.x; t/ > S 1 c p;n j ln j (6.5) and.x; t/ O h.t/ 6 L1 c p;n j ln j if d.x; t/ 6 S 1 c p;n j ln j; (6.6) for all.x; t/ Œ;. Proof. Applying the comparison principle to the equation (.1) with the functions U we deduce from (6.16), the lemmas 6.1 and 6. that U.x; t/ 6.x; t/ 6 U C.x; t/ for all.x; t/ Œ; : (6.7) For d.x; t/ > S 1 c p;n j ln j we have by (6.18) that d.x; t/ S 1e mt S > S 1 c p;n j ln j S 1 e m S D S 1 c p;n j ln j: This in view of (A.5), (A.7), (A.1) and the fact that lim # c p;n = D C1 implies.x; t/ > U.x; t/ > U.S 1 c p;n j ln j = ;.t/; S S / > h O C..t/; S S / K 1 e K S 1 c p;n j ln j = > O h C..t// K S S K 1 S > O h C..t// L 1S ; (6.8) for L 1 > K S C K 1 and small enough. Moreover we have by (A.1) that.x; t/ 6 U C.x; t/ 6 O h C..t/; S S / 6 O h C.t/ C K S S 6 O h C.t/ C L 1 S : This together with (6.8) implies (6.5). Similarly one can check (6.6).

22 18 M. HENRY, D. HILHORST AND C. B. MURATOV Lemma 6. There exists L > such that l.t/ 6 L = p j ln j (6.) and for all t Œ;..t/ Qv.t/ 6 L c p;n j ln j ; (6.) Proof. By (6.15) we have jl.t/j 6 = p C c p;n j ln j m for all t Œ;, which implies (6.). We now show (6.). Indeed, integrating (6.6) on we first note that. / t.t/ D.t/ C 1 jj.x; t/dx C.x; t/dx a where ;C ; D.t/ C h O C..t//j;C.t/j C h O..t//j;.t/j jj D C 1 jj ;C.x; t/ O hc..t// dx C 1 jj h.t/ C O C..t//j;C.t/j C h O..t//j;.t/j jj C 1 jj J ;C 1 WD J ;C WD J ; WD J ; WD By (6.5) we have that J ;C 1 C J ;C C J ; C J ; fx; d.x;t/>s 1 c p;n j ln jg fx; 6d.x;t/<S 1 c p;n j ln jg fx; d.x;t/6 fx; >d.x;t/> ;C J 1 S 1 c p;n j ln jg S 1 c p;n j ln jg ; a h.x; t/ O hc.t/i dx h.x; t/ O hc.t/i dx h.x; t/ O h.t/i dx a h.x; t/ O h.t/i dx:.x; t/ O h..t// dx (6.1) 6 L1 c p;n j ln jjj: (6.) Since h O C. / and are bounded, we obtain ;C J 6 C S1 c p;n j ln j: (6.) Similarly, one can prove that jj ; j C jj ; j 6 QC S 1 c p;n j ln j, which we substitute into (6.1) to deduce also in view of (6.), (6.) that satisfies O. / t.t/ D.t/ C hc.t/ j;c.t/j C h O jj.t/ j;.t/j a C! 1.t/

23 A MULTIPLE SCALE PATTERN FORMATION CASCADE 1 where j! 1.t/j 6 C c p;n j ln j, for all t Œ;. Thus setting OK. ; v/ WD Oh C.v/ C.1 / O h.v/ v a (6.) we obtain Since Qv. p C ; p j;c. / t.t/ D OK.t/j jj / we have in view of Theorem. ;.t/ C! 1.t/: (6.5) j;c j. Qv / t D OK ; Qv : jj Thus also using (6.5) we deduce for all t Œ; that. Qv /.t/ D 6 t t j;c j OK ; Qv.s/ jj j Qv.s/.s/j C I.s/ C II.s/ j;c OK.s/j jj ;.s/ ds C Qv././ ds C C c p;n j ln j C j Qv././j; (6.6) where I.s/ D Oh C. Qv /j;c j C h O. Qv /j; j jj II.s/ D Oh C. /j;c j C h O. /j; j jj Oh C. /j;c j C h O. /j; j jj ; Oh C. /j;c j C h O. /j; j jj : By (A.) we have that I.s/ 6 K Qv.s/.s/: (6.7) Moreover since j; j j; j D j;c j j;c j and since O h are uniformly bounded one gets II.s/ 6 C./j;C j j;c j: (6.8) Further by (6.) one gets j;c j j;c j 6 C. / = p j ln j, which in view of (6.8) implies This with (6.), (6.6) and (6.7) gives that. Qv /.t/ 6 t II.s/ 6 C.; / = p j ln j:.1 C K /j Qv.s/.s/jds C C.; / = p j ln j C C c p;n j ln j t 6.1 C K / j Qv.s/.s/jds C C c p;n j ln j ; C QC p j ln j

24 M. HENRY, D. HILHORST AND C. B. MURATOV for all t Œ;. Thus by Gronwall s Lemma we deduce. Qv /.t/ 6 C c p;n j ln j e.1ck /t ; for all t Œ;. Now recalling that D c p;n m j ln j and choosing m > 1 C K we obtain (6.), which ends the proof of Lemma 6.. We are now in a position to prove Theorem.. Proof of Theorem.. Let M WD max. QC C L ; L 1 C L K /, we have by (6.11) and (6.) that.x; t/ Qv.t/ 6.x; t/.t/ C.t/ Qv.t/ 6 M c p;n j ln j ; Further since Qv. p C ; p p / we deduce that C ; p for.x; t/ Œ; : (6.) ; for all.x; t/ Œ; (6.5) and small enough. Thus by (A.) h. O / D h. / and h O. / D h. /, so that both problems. P O / and.p / coincides. This gives in view of (6.8) that.x; t/ D.x; tc / D u.x; tc / and.x; t/ D.x; tc / D v.x; tc /; (6.51) for all.x; t/ Œ;. Then (.) follows directly from (6.) and we also deduce from (6.) and (6.51) that v.x; C / Qv. / 6 L c p;n j ln j : (6.5) Let x such that j Q d.x; /j > c p;n and let s Œ; T. We first assume that Q d.x; / > c p;n then we obtain from (6.) that d.x; t/ > r = > S 1 c p;n j ln j; or d.x; t/ > dist.x; / L = p j ln j > S 1 c p;n j ln j; for small enough. Thus by Lemma 6., (A.11) and (6.) we deduce.x; t/ h C Qv.t/ 6.x; t/ h C.t/ C hc..t// h C Qv.t/ 6.x; t/ h C.t/ C K.t/ Qv.t/ 6 L 1 c p;n j ln j C L K c p;n j ln j 6 M c p;n : This together with (6.51) gives (.) in the case Q d.x; / > c p;n. Similarly one can prove (.) in the case Q d.x; / 6 c p;n, which completes the proof of Theorem..

25 A MULTIPLE SCALE PATTERN FORMATION CASCADE 1 7. Proof of the fourth stage: Propagation of interface for large time Proof of Theorem.5. Let ı and be defined by (6.1), we set for r R fs C zn.s/; 6 z < rg; if r > ;.r/ WD fs C zn.s/; r < z 6 g; if r < ; then by [1] we have, since V n only depend on s, that vol.r/ D P.r/; (7.1) where P is a polynomial function with coefficients which only depend on. We now assume that. Qv.s/;.s// has a unique smooth solution of.q / on a time interval Œ; QT and we set l.s/ WD s V n;./d D p s h Qv./ d; (7.) so that.s/ D f C l.s/n./; g where n./ is the normal of at. Then we have jl.s/j 6 ı on a time interval, which we denote again by Œ; QT and thus by (7.1) This yields in view of the assumption (.1) that jc.s/j D jc./j C P l.s/ : (7.) jc.s/j. ; C/; jj for all s in a time interval, still denoted by Œ; QT. Thus by (.6) and Lemma B.1 below, we obtain jc Qv.s/ D W.s/j ; for all s Œ; QT ; jj so that in view of (7.) C./ C P l.s/ Qv.s/ D W jj and l s D p C h W./ C P l.s/ jj ; (7.) for all t Œ; QT. We now consider the following ODE Y.S / s D H.Y /; for s Œ; QT Y./ D ; with H.:/ D p jc h ı W./j C P.:/. By the Cauchy Lipschitz theorem we have that.s jj / admits a unique solution on a maximal time interval.s 1 ; s / with s 1 < < s. Thus choosing QT jc.; s / such that jl.ts/j 6 ı and./j C P.l.s//. ; C/ and setting jj C./ C P l.s/ Qv.s/ WD W and.s/ WD f C l.s/n./; g jj we conclude that. Qv ; Theorem.5. / is the unique solution of.q / on Œ; QT, which ends the proof of

26 M. HENRY, D. HILHORST AND C. B. MURATOV The goal of this stage is to study Problem.P / on the O. p introduce the corresponding change of variable, namely = / time scale. This leads us to s WD = p.t /: (7.5) and to deduce from Problem. O P / the following system 8. ˆ< / s D = C 1 f O. ; / in.; T /; =. P O /. / s D p = C p = g. ; / in.; T /; ˆ: with the @n D D T (7.6) (7.7) (7.8).x; / D.x; C / D u.x; C / for x ; (7.).x; / D.x; C / D v.x; C / for x : (7.1) By (A.1) we have for. p C ; p / that.x; s/ D.x; p = s C C / D u.x; p = s C C / (7.11).x; s/ D.x; p = s C C / D v.x; p = s C C /: (7.1) Moreover let be the interface defined by the motion equation V n; D p O h.s/ ; j sd D ; (7.1) where V n; is the velocity of. One can prove that Problem (7.1) admits a unique classical solution on a time interval Œ; Qs, for some positive constant Qs. We then deduce from (7.1) and the construction of h O that jv n; j 6 C, where C is a constant independent on and Qs. This yields setting l.s/ WD p s Oh.z/ dz; (7.1) that jl.s/j 6 C Qs, for all s Œ; Qs. Thus the interface is well defined on Œ; ı C /, where ı is defined by (6.1). This gives that (7.1) admits a unique classical solution on the time interval Œ; Qs, with < Qs < ı C. Further for s Œ; Qs, divides into two subdomains, ;.s/. Let < r < ı we introduce as in the stage 6 a smooth truncated approximation of the signed distance function to the interface, namely 8 ˆ< d.x; s/ D ˆ: r r if x ;C.s/ and dist x;.s/ > r if x ;.s/ and dist x;.s/ > r dist x;.s/ if x ;C.s/ and dist x;.s/ 6 r dist x;.s/ if x ;.s/ and dist x;.s/ 6 r ;

27 A MULTIPLE SCALE PATTERN FORMATION CASCADE and extended smoothly for x fr = < dist.x;.s// < r g. Moreover we also assume that We set and we define for s D Qs /: (7.15) T WD min. QT ; Qs /: (7.16) where U.x; s/ D U d.x; s/ R 1 R e Qms = ;.s/; R R ; (7.17) R D c p;n j ln j (7.18) and R 1, R and Qm are positive constants to be determined later. As in Section. we now prove that U are sub and super-solution of (7.6). Lemma 7.1 The functions U.x; s/ satisfy that OL.U C ; / WD.U C / s = U C 1 f O.U C = ; / > ; on Œ; T (7.1) OL.U ; / WD.U / s = 1 U f O.U = ; / 6 ; on Œ; T D Œ; T : Proof. It follows from (7.15) that D. Moreover we have L.U C ; / D I 1 C I C I C R R = (7.1) I 1 WD U. / s U z d C 1 = U zz.1 jrd j / (7.) I WD 1 = U z..d / s C C p O h. ; R R / C e Qms QmR 1 R / (7.) I WD 1 = O f.u C ; / O f.u C ; / ; d.x;s/cr 1R e Qms ; = where the derivatives of U are evaluated at the point.s/; R R. Since the computations are similar to those done in Lemma 6.1 we only give the main estimates. We have by the definition of d and (A.6) that Uzz.1 jrd j / 6 CK1 sup e Kj d CR 1 R e Qms = j jd j> r

28 M. HENRY, D. HILHORST AND C. B. MURATOV and for all s Œ; T d C R 1 e Qms R > r R 1 R e QmT ; for jd j > r ; so that, since lim # R D, ju zz.1 jrd j /j 6 CK 1 : Substituting this into (7.) and also using the fact that U, d and U z are bounded and that by (7.7), j. / tj 6 p = C we deduce that To estimate I we remark that d satisfies.d / s C I 1 > F 1 p = : (7.) p O h. / 6 QDjd j; in Œ; T ; which we substitute into (7.) to obtain in view of (A.), (A.5), (A.1) that Moreover since I > 1 = U z QDjd j K R R C Qme Qms R 1 R : Qme Qms R 1 R QDjd j > QDjd C e Qms R 1 R j C e Qms R 1 R. Qm QD/; we deduce as in the proof of (6.) that I > F ; (7.5) for small enough. As it is done in the proof of (6.) we obtain from (A.1) and (6.11) that I > c O f =.x; p = s C C /. p = t C C / > F =.N C/ : (7.6) Substituting (7.), (7.5) and (7.6) into (7.1) we deduce OL.U C ; / > 1 = R R F 1 p = F F 1 =.N C/ ; for all.x; s/ Œ; T. Thus for R > F 1 C F C F we obtain OL.U C ; / >, which coincides with (7.1). One can prove the inequality (7.) in a similar way. Next we state the following estimates on the initial condition, namely Lemma 7. Proof. We first recall that by (6.5) and (6.51) U.x; / 6.x; / 6 U C.x; /; for all x : v. C / p C ; p : (7.7) First case: d.x; / 6 c p;n. Then by definition of d we have Q d.x; / D dist.x; / 6 c p;n and thus using (7.) and (.).x; / D u.x; C / 6 h C. Qv. // C M c p;n = : (7.8)

29 A MULTIPLE SCALE PATTERN FORMATION CASCADE 5 Further by (A.5), (7.1), (7.7) (A.) and (A.1) we obtain U C.x; / D U d.x; / C R 1 R = ;./; R R > O h../; R R / D O h.v. C /; R R / D h.v. C / R R / > h.v. C // C K R R : This in view of (6.5) and (A.11) gives U C.x; / > h Qv. / K L c p;n j ln j C K R R ; which together with (7.8) implies.x; / 6 U C.x; / for R > M C K L K. Second case: d.x; / > c p;n. We note that d.x; / C R 1R > R 1R and then as previously using (A.7), (A.1), (7.7), (7.1), (.) and (A.11) we obtain U C.x; / > U R1 R ; =./; R R > h O C./; R R R 1R K 1 e K = > h C Qv. / K L c p;n j ln j C K R R K 1 R ; for small enough. Thus for R > M CK L CK 1 K we deduce in view of (.) and (7.) that.x; / D u.x; C / 6 U C.x; / for all x such that d.x; / > c p;n. Finally we have obtained that.x; / 6 U C.x; / for all x. Similarly, one can check that.x; / > U.x; / for all x, which ends the proof of Lemma 7.. Next we prove the following result Lemma 7. There exist two positive constants, R and R, such that.x; s/ hc O equation..s/ 6 R R if d.x; s/ > R R (7.) and for all s Œ; T. Proof..x; s/ O h.s/ 6 R R if d.x; s/ 6 R R ; (7.) Using the comparison principle we deduce from lemmas 7.1 and 7. that U.x; s/ 6 u.x; s/ 6 U C.x; s/: (7.1) For d.x; s/ > R R, where R > R 1 C R 1 e QmT we obtain that d.x; s/ R 1e Qms R > R R R 1 QmT R > R 1 R : This in view of (7.1), (A.7) and (A.1) implies that.x; s/ > U.x; s/ > U R1 R = ; > O h C.s/; R R K 1 e.s/; R R K R 1 R = > O h C.s/ K R R K 1 R > h C.s/ R R ; (7.)

30 6 M. HENRY, D. HILHORST AND C. B. MURATOV for R > K R C K 1 and small enough. Moreover we have by (A.1) that.x; s/ 6 U C.x; s/ 6 O h C.s/; R R 6 O h C.s/ C K R R 6 O h C.s/ C R R : This combined with (7.) implies (7.). Similarly one can check (7.). Lemma 7. There exists a function L C.Œ; T / such that l tends to L uniformly on Œ; T, as tends to. Moreover L.s/ D s lim Oh.z/ dz for all s Œ; T : (7.) # Further L is differentiable almost everywhere on Œ; T and there exists a positive constant L such that j.l / s.s/j 6 L almost everywhere on Œ; T : (7.) Proof. We deduce from (7.1) that l and.l / s are bounded uniformly with respect to on a time interval Œ; T. Thus there exist a function L and a subsequence of, which we denote again by such that l tends to L uniformly on Œ; T, as tends to. This together with (7.1) gives (7.). Further since h O is smooth and is bounded on Œ; T we have that L is a Lipschitz function. Thus L is differentiable almost everywhere on Œ; T and.l / s.s/ D lim Oh.s/ ; for almost s Œ; T : # This with the fact that O h is smooth and is bounded implies that.l / s is bounded for almost s Œ; T, which coincides with (7.). Lemma 7.5 tends uniformly to Qv on Œ; T : (7.5) L.s/ D Proof. In what follows we check that satisfies = p. / s D C O h C. p s h Qv.z/ dz; for all s Œ; T : (7.6) /j;c j jj C O h. / 1 j;c j jj a C!.s/; (7.7)./ D Qv ;1 C!.s/; (7.8) where j!.s/j 6 CR and!.s/ 6 C c p;n m by (7.1) that, for all s Œ; T. We first prove (7.8). We have./ D v C D Qv 1 C Qv 1 C Qv. C / C Qv. C / C v. C / :

31 A MULTIPLE SCALE PATTERN FORMATION CASCADE 7 Thus (7.8) follows directly from (.) and (.). Since the proof of (7.7) is very similar to the proof of (6.5), we omit the details of the computation. Integrating (7.7) on we obtain = p. / s.s/ D.s/ C 1 jj.x; s/dx C.x; s/dx a where J Q ;C 1 WD J Q ;C WD J Q ; WD J Q ; WD ;C ; D.s/ C h O C..s//j;C.s/j C h O..s//j;.s/j jj J Q ;C 1 C J Q ;C C J Q ; C J Q ; C 1 jj fx; d.x;s/>r R g fx; 6d.x;s/<R R g h.x; s/ O hc.s/i dx h.x; s/ O hc.s/i dx fx; d.x;s/6 RR h.x; s/ O h fx; >d.x;s/> R R g a ; (7.).s/i dx h.x; s/ O h.s/i dx: As it is done in the proof of Lemma 6., we deduce from Lemma 7. and the fact that h. / and are bounded that j J Q ;C i j 6 CR for i D 1; and jj Q ; i j 6 CR for i D ; which we substitute into (7.) to deduce that satisfies (7.7). Moreover by the motion equation (7.1) we obtain as it is done in the proof of Theorem.5 that j;c.s/j D j;c j C P l.s/ ; where P is the polynomial function introduced in (7.1). This together with Lemma 7. yields that which satisfies We set and By (.1),.t/ jj./ jj j;c j tends to a function uniformly on Œ; T, (7.).s/ D j;c j C P L.s/ ; for all s Œ; T : (7.1). ; v/ WD K. ; v/ C v D h C.v/ C h.v/.1 / a (7.) O. ; v/ WD O h C.v/ C O h.v/.1 / a: (7.). ; C/ thus there exists a time, which we denote again by T such that. ; C/, for all t Œ; T. Thus setting.s/ D W..s/ / we deduce from Lemma B.1 p p jj that. C ; / and jj ;.s/.s/ D : (7.)