Fabrice BETHUEL Didier SMETS

(Dedicated to Häım Brezis on the occasion of his 70th birthday)

1 Introduction

1.1 Motivation and setting

This paper is a follow-up of a previous work with Orlandi[3],where we derived an upper bound for the motion of front for gradient systems with potentials having several minimal wells of equal depth.Our approach there is based on the local energy inequality combined with some appropriate parabolic estimates.Our aim in this paper is to extend the analysis in order to derive the precise motion laws for fronts:The approach is however restricted at this stage to scalar equations.We will take advantage in particular of the fact that in the scalar case,stationary solutions can be completely integrated,allowing for refined energy estimates.

It is presumably needless to recall that the study of the motion of fronts for scalar reactiondiffusion equations has already a very long history.In particular,equations of Allen-Cahn type,that is,when the potential possesses only two distinct local minimizers which are nondegenerate,have been extensively studied.Under suitable preparedness assumptions on the initial datum,the precise motion law for the fronts has been derived in the seminal works of Carr and Pego[6](see also[10]).Their approach relies on a careful study of the linearized problem around the stationary front,in particular from the spectral point of view.This type of approach is also sometimes termed the geometric approach(see,e.g.,[8]),since it involves ideas related to central manifold theory.Alternate methods,usually termed energy methods relying on global energy estimates have later been worked out(see[5,11,12]).They are presumably more direct to capture the essence of the slow-motion or metastability of pattern phenomenon,but have been unable at this stage to yield the precise motion law.One of the aims of this paper is therefore to fill the gap between the two methods,and raise the energy methods to the same degree of accuracy as the geometric one.

The motivations of this paper are however manifold.First,as mentioned relying on the results in[3],we wish to recover the precise motion law of Carr and Pego,providing therefore an alternate approach which eludes the use of spectral theory and which also allows for a larger class of initial data.Second,whereas most of the existing literature is devoted to Allen-Cahn type potentials,our method can handle also potentials with several equal-depth wells.Notice that a major difference in the later case is that,whereas only attractive forces between the fronts are present in the case of two wells,repulsive forces may be present when there are more than two wells,inducing important differences in the limiting ordinary differential equations.

Besides this,we are able to handle collisions and splittings,and extend the analysis past these events:Similar issues were addressed and solved in the Allen-Cahn case1Actually,only collisions occur in the Allen-Cahn case,splittings do not.by Chen[8],relying crucially on a comparison principle worked out by Fife and McLeod[9].In our opinion,such an argument cannot be extended for potentials with more than two wells,and when hence repulsive forces are present2As a matter of fact,Proposition 3.1 in[8],which rephrases the Fife-McLeod result,simply does not hold when there are more than two wells..Finally,last but not least,we expect that the approach we develop here can be extended and be used as a model in order to derive the motion law in the case the potential wells are degenerate as well as the case of systems,with possibly additional assumptions on the stationary solutions.

To be more specific,we consider and analyze the behavior of solutions v of one-dimensional reaction-diffusion equations of the following form:

where 0<ε<1 denotes a(small)parameter,v denotes a scalar function of the space variable x∈R and the time variable t≥0,the function V,usually termed the potential,denotes a smooth scalar function on R,and V?denotes its derivative.Notice that equation(PGL)εactually corresponds to the L2gradient-flow of the energy functional Eεwhich is defined for a function u:R?→R by the formula

Our assumptions on the potential V express the fact that it possesses several minimizers which are non-degenerate and are formulated as follows.We assume throughout that V is smooth and satisfies the three conditions:

is a finite set,with at least two distinct elements,that is

(H2)We have that λi≡ V??(σi)>0 is positive for each point σiof Σ.

(H3)We have V(u)→+∞,as|u|→+∞.

A canonical example is given by the function

whose minimizers are σ1=+1 and σ2= −1,with λ1= λ2=2 and which is a potential of Allen-Cahn type.Another example we have in mind and we wish to handle is given by

for which Σ ={(2k+1)π,k ∈ Z}and λi=1.Clearly,the potential given by(1.4)does not satisfy conditions(H1)nor(H3),since it has infinitely many minimizers and does not converge to+∞at infinity.However,the analysis can be carried over for this type of potentials,as Theorem 1.5 below will show.

As in[3],the assumption in this paper on the initial datum(·)=vε(·,0)is that its energy is finite.More precisely,given an arbitrary constant M0>0,we assume throughout the paper that

In particular,in view of the classical energy identity

we have,∀t>0,

so that for every given t≥0,we have V(v(x,t))→0 as|x|→∞.It is then quite straightforward to deduce from assumptions(H0)–(H2)as well as the energy identity(1.5),that v(x,t)→ σ±as x→±∞,where σ±∈Σ does not depend on t.

1.2 Regularized fronts and their evolution

The notion of regularized fronts is presumably central in this paper.It describes a situation where,at some given time t0≥ 0,the solution vεto(PGL)εis close to a chain of stationary solutions which are well separated,and suitably glued together.This occurs,as we will see,when the solution has already undergone a parabolic regularization.The rate of accuracy of the regularization,is described by a parameter δ>0,homogeneous to a length,and which is also related to the distance between two fronts.

We recall that for i∈ {1,···,q − 1},there exists a unique(up to translations)solutionto the stationary equation with ε=1,

with,as conditions at infinity,v(−∞)= σiand v(+∞)= σi+1.We also set,for i∈ {2,···,q},(·)≡ ζi(−·),so thatis the unique,up to translations solution to(1.7)such that v(+∞)=σiand v(−∞)= σi−1.A remarkable fact is that there are no other non-trivial solutions to equation(1.7)than the solutions:In particular there are no solutions connecting minimizers which are not neighbors3The situation might be very different in the case of systems,where anyway the notion of neighbors is perhaps meaningless..Some relevant properties of these solutions ζiwill be collected in Section 3.For i=1,···,q− 1,let zibe a point in the interval(σi,σi+1)where the potential V restricted to[σi,σi+1]achieves its maximum,and set Z={z1,···,zq−1}.Since we consider only the one-dimensional case,any solution ζitakes once and only once the value zi.

Next let t0≥ 0,δ > α1ε,and r≥ δ be given,where α1>0 denotes some constant which will be specified in Subsection 3.2.

Definition 1.1We say that vεsatisfies the preparedness assumption WPε(δ,t)if it satisfies the energy assumption(H0)and if there exists a collection of points{ak(t)}k∈J(t)in R,with J(t)={1,···,?(t)},such that the following conditions are fulfilled:

(WP1)For each k ∈ J(t),there exist a number i(k)∈ {1,···,q},such that

(WP2)For each k∈ J(t),there exists a symbol†k∈ {+,−},such that

where Ik=([ak(t)− δ,ak(t)+δ]for each k ∈ J(t).

In the above definition ρ1>0 denotes a constant which will be defined in Section 3(see(3.24)).Notice that,if we consider more generally,for t≥0,the subset O(t)of R is defined by

If WPε(δ,t)holds,then we have for δ≥ α1ε,

In particular,the points ak(t)are easily shown to be unique(see Section 4),and once their existence has been established,the main focus is then on their evolution in time.We introduce also the quantities

with the convention that the quantity is equal to+∞in case the defining set is empty,and where,for a given index k ∈ J(t),we define the integers j±(k)as j±(k)=i(k)±1,if†i(k)=+,and j±(k)=i(k)∓1,otherwise.We also set

Notice that if WPε(δ,t)holds,then it is a simple exercise to show that,if α1is chosen sufficiently large,then we have

so thatwhere λmin=inf λi.Conversely,given the points{ak},the largest value of δ for which one may expect WPε(δ,t)to hold is precisely of the same order as da(t).

Our first result describes the situation,where the initial datum satisfies the assumption WPε(δ,T).We will show that the motion law for the fronts is governed by a simple first order differential equation,which is of nearest neighbor interaction type.The strength of the interaction of the(k+1)-th fronts on the k-th fronts is governed by the quantitydefined,for a collection of ordered points{a1,···,a?},with a1

with the convention?+1=+∞.The numbersentering in formula(1.15)depend only on the properties of the stationary front ζiand will be explicitly defined in Section 3(see(3.9)).Let us however emphasize that>0.It follows in particular that({ai})>0 if the signs†kand †k+1are the same,and({ai})<0 if they are opposite.Notice also that the quantitydecays exponentially as the distance between two neighboring fronts increases.

We also set

with the convention that

Our first main result shows that the evolution of regularized fronts is related to solutions of a differential equation of the type

where Sidenotes a positive quantity4actually its energyrelated to ζiand(s)stands for some error term which will be shown to be exponentially small.

Theorem 1.1Assume that the potential V satisfies assumptions(H1)–(H3),let ε>0,and let vεbe a solution to(PGL)εsatisfying(H0).Let T ≥ 0 be given.There exists constants α∗>0,c∗>0,0< ν∗<1,ρ∗>0,and S∗>0 depending only on V and M0and a time T=T(T)>T satisfying

such that,if δ ≥ α∗ε and property WPε(δ,T)holds,then we may assert:

(i)For any time t ∈ [T,T]the points{ak(t)}k∈J(T)satisfying(1.8)are unique and welldefined,whereas for any t∈ [T+c∗εδ,T],property WPε(ν∗δ,t)holds.

(ii)Property WPε(ν∗da(T),t)holds for any t in[Ttrans,T],where

(iii)We have|da(T)− da(T)|≥ ρ∗da(T).

(iv)For any time t∈ [T,T],there exists a collection of points{bk(t)}k∈J(T)satisfying the differential equation(1.16)with

such that

A few comments are in order.The first two statements describe how property WPεis propagated by the equation(PGL)ε.Assertion(i)of Theorem 1.1 shows that property WPεremains true,except possibly on an initial boundary layer of order εδ,where the collection of points{ak}k∈Jis however still well-defined,and with some smaller length5Recall that this parameter is supposed to describe the accuracy of the approximation by a chain of stationary solutions glued together.δ?≡ ν∗δ.Assertion(ii)shows,that,after a transition period[T,Ttrans],whose length is small compared to the length of[T,T],the rate of the approximation has improved to δ?≡ ν∗da(T),which as mentioned,is the order of the best rate of approximation possible.

The approximation by the differential equation(1.16)is presented in assertion(iii).Turningfirst to the differential equation(1.16),we notice that two neighboring fronts with the same signs†repel,whereas they attract when these signs are opposite.In particular,we will show in Section 2 that,if there exists some k ∈ {1,···,?}such that†k= −†k+1,then collisions have to occur for the differential equation(1.16).Moreover,if the infimum in(1.13)is achieved at some fronts of opposite signs6As a matter of fact,the purely attractive case,for which †k= −†k+1,for every k,and hence all forces are attractive,occurs for instance for the Allen-Cahn functional.,then the maximal time of existence Tmaxof the differential equation(1.16)satisfies an estimate of the form see inequality(2.5)for a precise statement.On the other hand,if all the signs†kare identical,then the system is purely repulsive,and is then defined for all time,i.e.,Tmax=+∞.Moreover,in that case,the system has actually diffusive properties(see Proposition 2.4 below).

Comparing property(1.21)of the differential equation with(1.17)for the partial differential equation(PGL)ε,we observe that the time T − T is of the same order of magnitude as the one provided in(1.21),and therefore appropriate for comparing the two equations.Moreover,in view of assertion(ii)of Theorem 1.1,we see that a point at least as been moved by at least a distance of order of magnitude da(T),which is indeed the appropriate length scale.On this length scale,it follows from assertion(iv)that the differential equation(7.8)describes,up to some lower order terms,the motion of the front points.

1.3 Collisions of fronts

Whereas collisions in the ordinary differential equation(1.16)represent genuine singularities for the solutions and lead to a maximal time of existence,it is not the case for the partial differential equation(PGL)ε,which in view of its parabolic nature possesses regular solutions for all positive time.The notion of fronts is however only well-defined,in the sense of the previous subsection,if the fronts remain sufficiently well-separated,since their mutual distance should be at least of order α∗ε.Our results below show that collisions in(1.16)induce an intermediate time layer for solutions to(PGL)εor order ε2,where annihilation of fronts takes places.This time layer is actually described by two collisions times:The first one,corresponds to a time where two fronts with opposite signs become α∗ε close,a distance at which the approximation by the differential equation(1.16)no longer remains valid.The existence and properties of the timeare provided in the following result.

Theorem 1.2Let ε >0,T ≥ 0 and δ ≥ β∗ε be given,where β∗≥ 2α∗is some constant depending only on V and M0.Assume that WPε(δ,T)holds and that the signs{†k}k∈Jare not all identical.Then there exists some time,such that the following hold:

(i)For any t∈[T,],property WPε(α∗ε,t)holds and we have

(ii)We have

(iii)We have the upper bound,for some constant C∗>0 depending only on V and M0.

(iv)For anythen propertyholds.

Notice that the fact that two fronts with opposite signs become close at timeis stated in part(ii).In contrast,fronts with the same signs remain well-separated,as shown by the first assertion.

In order to analyze the annihilation of fronts,we provide first some definitions.Assume therefore that at some time t conditions WPε(δ,t)are satisfied,with δ ≥ α1ε.We say that a point ak0(t)for k0∈J(t)is free,if and only if

where the constant κf>0 depends only on V and M0and will be defined in Section 9,andwith the convention that a0(t)=−∞and a?+1(t)=+∞.We set

Likewise,we say that a point ak0(t)for k0∈ J(t)is purely repulsive if and only if†k0= †k0+1=†k0−1,with the convention that†0= †and †?+1= †?.We set

andNotice that,if a point ak0(t)is purely repulsive,then we haveand hence is free if(t)is sufficiently large,that is,

providedIn view of assertion(i)in Theorem 1.2,this last condition is met in particular for t∈[T,]provided we choose β∗sufficiently large,what we assume from now on.The next results provide the annihilation of at least two fronts with opposite signs,within an additional time of order ε2.

Theorem 1.3Let ε >0,T ≥ 0 and δ ≥ γ∗ε be given,where γ∗is some constant depending only on V and M0.Assume that WPε(δ,T)holds,and that the signs{†k}k∈Jare not all identical.There exists a timesuch that condition WPε(α∗ε,)holds,and such that for some constant Υ depending only on V and M0,

Moreover,the following holds:

(i)We have the inclusionwhere κcis some constant depending only on V and M0.

(ii)We have

(iii)We have for some m≥1,

We notice,combining assertion(ii)and assertion(iii)that the total number of front points has decreased by 2m,that is,

so that the results in Theorem 1.3 do indeed describe the annihilation of at least two fronts,annihilation which occurs on a time interval of order ε2,in view of(1.24).Moreover,in view of assertion(ii),we have a one to one correspondence between free or repulsive points at timeand,each of these points being moved at most at a distance of order ε.The annihilation occurs among the attractive points which are not free,among which m pairs disappear in the process.This annihilation process can then only occur a finite number of times,after which the system becomes purely repulsive,all fronts repelling each other.

1.4 Relaxing the preparedness assumptions

We relax now the preparedness assumptions,and extend our analysis to the case of bounded energy initial data.For that purpose,we make use of the framework and concept developed in[3],and define as there for a scalar function u on R,its front set as the set D(u)defined by

This notion which might be understood as a substitute to the notion of front points defined so far only when assumption WPεholds.The constant μ0>0 which appears in this definition is chosen so that,for i=1,···,q,we have B(σi,μ0)∩ B(σj,μ0)= ∅ for all i?=j in{1,···,q}and≤ V??(y)≤ 2λifor all i∈ {1,···,q}and y ∈ B(σi,μ0).A few elementary arguments yield(see[3,Corollary 1])that,if the map u satisfies the energy bound Eε(u)≤ M0,then there exists?points x1,···,x?in D(u),such that

with the bound?≤?0=on the number of points,where η0>0 is some constant depending only on the potential V.In the context of equation(PGL)ε,we set moreover D(t)=D(vε(·,t)),so that

where the intervalsare disjoint,with a length less than ε?and?(J) ≤ ?.It follows from our definitions of the front set,that in the intervalsthe function vε(·,t)takes values near some of the minimizers,which we denote by σj−(k)= σj+(k−1).The pointsplay a role similar to the front points akin the definition WPε,except that they are now only defined up to a scale of order ε,and that the function is not necessarily close to a stationary front in their neighborhood7in contrast,the results described assuming WPε yield an accuracy of order εlog(),hence extremely sharp when δ is of order 1..As a matter of fact,we notice that,if WPε(δ,t)is satisfied,then,in view of(WP2),we have

for some suitable constant κwdepending only on V and M0.However,the regularizing properties of equation(PGL)εare at work,and drives the function towards a well-prepared case,as our next result shows.

Theorem 1.4Let T ≥ 0 and α > α∗be given and assume that(H0)holds.Then there exists a time t ∈ [T,T+ ω(α)ε2],such that WPε(αε,t)holds with ω(α)=c02M0exp(2ρ1α).Moreover,we have

A general principle might therefore be stated as follows:Up to an error term of order ε2in time and of order ε in space,the system behaves as if it were well-prepared according to assumption WPε.More precisely,after an initial boundary layer in time of size at most ω(α∗)ε2,during which the front set has been moved at distance of size at most κ∗ε,the preparedness assumption WPε(α∗ε,t)is full-filled,so that we are in position to apply Theorems 1.1–1.3,which relate the dynamics to the ODE(1.16).

1.5 Relaxing the assumptions on V

The assumptions on the potential can be modified and in fact actually weakened to handle also other kind of potentials,for instance periodic potentials like(1.4).For that aim,we introduce an alternate set of assumptions on the potential V,which can be stated as follows:

(H)1bisWe have that inf V=0 and that the set of minimizers Σ is a discrete set which contains at least two elements.

We may hence write Σ ={σi}i∈J,where J ⊂ Z,with σi< σj,if i

(H)2bisThe potential V is of class C3with?V??C2(R)<∞.Moreover,we have

(H)3bisThere exists some number ν>0 such that,if i∈ J or i+1∈ J,then we have

Obviously,the potential given in(1.4)satisfies these assumptions,as well as actually any smooth periodic potential having non-degenerate minimizers.We have the following theorem.

Theorem 1.5The results in Theorems 1.1–1.4 hold true if we replace the assumptions(H1),(H2)and(H3)on the potential V by assumptions(H1bis),(H2bis)and(H3bis),respectively.

The argument of the proof of Theorem 1.5 actually relies on an elementary observation.

Proposition 1.1Assume that the potential V satisfies assumptions(H1bis),(H2bis)and(H3bis),and let u be such that Eε(u)≤ M0.Then the limits u(±∞)≡u(x)exist and we have,for some constant A depending only on?V??C2(R)< ∞,ν,and λmin,

Moreover,if vεis a solution to(PGL)εsatisfying(H0),then the limits u(±∞,t)≡do not depend on the time t and hence

In order to prove Theorem 1.5,we then observe that relation(1.29)shows that the solution takes values only on a finite interval of R:We therefore may modify the potential outside of this interval without changing the solution,so that assumptions(H1)–(H3)are fulfilled.We may then rely on our previous results.

1.6 Elements in the proofs

The proofs of our main results contain several distinct ingredients.The starting point is the study of solutions to the perturbed stationary equation,which writes for a scalar function u defined on R as

Our main result concerning equation(1.30),which is completely elementary since it relies essentially on Gronwall’s lemma,is given in Proposition 3.1.It states that,if the function verifies an energy bound of the form Eε(u)≤ M0and ifis sufficiently small,then the function u is close to a chain of stationary solutions,i.e.,heteroclinic solutions,as described in property WPε(δ,t),with a parameter δ proportional toWe use this result with u ≡ vε(·,t)and f(·) ≡ ∂tvε,so that smallness of the dissipationat some time t yields property WPε(δ,t),with a parameter δlarge when dissipation becomes small.Combining this property with the energy identity(1.5),which allows to control dissipation,we show that the flow drives to well-preparedness.A similar result was already established in[3,Theorem 3].However,here we take advantage of an important specificity of the scalar case,which is actually the only one which is used in this paper:Stationary solutions are perfectly known,and can even be integrated thanks to the method of separation of variables.In particular,assumption WPεimplies a kind of quantization of the energy,which,in turn,allows to improve bounds on the dissipation.

The next step is to introduce more dynamics in our arguments.For that purpose,as in[3,Lemma 2],we use extensively the localized version of(1.5),a tool which turns out to be perfectly adapted to track the evolution of fronts,and which writes,for a smooth test function χ with compact support in R,

where the term FSis given by

The first term on the right-hand side of identity(1.31)stands for local dissipation,whereas the second is a flux.The quantity ξ is defined for a scalar function u by

sometimes referred to as the discrepancy term in the literature.It is constant for stationary solutions on some given interval I,i.e.,for solutions to

and it vanishes for finite energy solutions to(1.34)on I=R.Using(1.31)for appropriate choices of test functions,combined with several parabolic estimates,we have shown in[3]the following theorem.

Theorem 1.6Let T>0 be given,and assume that(H0)holds.There exist constants ρ0>0 and α0>0,depending only on the potential V and on M0,such that if r ≥ α0ε,then for every t≥0,

provided

Actually,Theorem 1.6 is established in[3]for general systems,under assumptions on the potential V which are the higher dimensional analogs of(H1)–(H3).In particular,a rather remarkable fact is that the result does not involve any assumption of any kind on the stationary solutions8which is a far more difficult question for systems than in the scalar case.A central idea in the proof is to derive appropriate upper bounds on the discrepancy in region which are far from the front set,as well as a suitable choice of test functions χ for(1.31):They are chosen to be affine near the front sets,so that the second derivative vanishes there,and the flux term needs only to be estimates o ffthe front set.

Theorem 1.6 provides a first estimate of the velocity.This estimate combined with the results of Proposition 3.1,and the energy identity(1.5)is actually already sufficient to prove Theorem 1.4.

In order to establish Theorem 1.1 and derive actually an efficient motion law,we need to derive a far more precise estimate for the discrepancy.In order to sketch the argument,assume that WPε(δ,t)holds for some δ >0,and let ak(t)and ak+1(t)be two front points,with k ∈ J(t).In order to estimate the interaction between these two points,we evaluation ξ(·,t)near the middle pointTo that aim,we use several observations as follows:

(1)The behavior of vεnear the points akis described with high accuracy using the appropriate heteroclinic solutions near the points akand ak+1,say on intervals of the formandwhereis of the same order as δ.We will term this region the inner region.

(2)The heteroclinic solutions are known.

(3)The evolution of the points akis known to be small thanks to Theorem 1.6.

(4)In the outer-region[ak(t)+,ak+1(t) −],the solution is well approximated by the solution to the linearized equation near the minimizer σj(k)+,which turns out to be

The boundary conditions are deduced from the values of the heteroclinc solutions at ak(t)+?δ and ak+1(t)−.

(5)It relaxes very quickly to the solution to the corresponding stationary equation:This time relaxation is described by factors involving terms of the form exp

The expression of the discrepancy ξ for the stationary solution in the outer region then offers,after an appropriate small relaxation time,a good approximation of ξ near the pointWe then use identity(1.31)with functions χ which are affine,except possibly near the points(t),so that the previous expansion can be used.We show that this yields a good approximation of the motion of the front points,leading to the proof of Theorem 1.1.

The proof of Theorem 1.2 is based on the approximation provided by Theorem 1.1 as well as some properties of the system of ordinary differential equations(1.16).The proof of Theorem 1.3 uses extensively,besides the results in Theorem 1.1–1.2,the quantization of the energy.

We describe now the outline of the paper.Since our arguments involve several ordinary differential equations,in particular equations(1.7),(1.16)and(1.30),and since the properties involved are all completely elementary,we wish to present them first.Therefore,we start in Section 2 with some result concerning equation(1.16):These results are only used in the proof of Theorems 1.2–1.3,the reader may therefore skip this part in a first reading of the paper.Section 3 presents some properties of the stationary equations(1.7)and(1.30),in particular properties of the heteroclinic orbits,which are obtained through the method of separation of variables,as well as the statement of proof of Proposition 3.1.In Section 4,we describe several properties related to the well-preparedness assumption WPε,in particular the quantization of the energy,how it relates to dissipation,and its numerous implications for the dynamics.In Section 5,we set up a toolbox,which presents various parabolic linear estimates.These estimates are then extensively used in Section 6,where they provide estimates for(PGL)εon parabolic cylinders,assuming that the map takes values to one of the minimizers σi.A major emphasis is put on the expansion of the quantity ξ,which is estimated sharply near the middle of the cylinder.Section 7 is devoted to the proof of Theorem 1.1,based on formula(1.31)as well as on the expansions of ξ provided in Section 6.Section 8 is devoted to the proof of Theorem 1.2 whereas Section 9 is devoted to the proof of Theorem 1.3.Finally in Section 10,we outline the proof of Theorem 1.5.

2 Some Remarks on the Differential Equation(1.16)

2.1 Statement of results

This section,which is independent of our previous analysis,focuses on general properties of the ordinary differential equations(1.16),with an emphasis on estimates for the possible collision time.Therefore,we assume that we are given an integer?∈ N∗,a mapping†from J to{+,−},where J={1,···,?},a solution t?→ b(t)=b1(t),···,b?(t)to the system(1.16),where the constantsare defined according to the definition(1.15),which requires that the value of one of the numbers i(k),for instance i(1)is also given.We consider the solution on its maximal interval of existence,that is,[0,Tmax].We assume moreover throughout this section that the correction terms(s)satisfy the smallness assumption

where qmin=inf{qi}and qmax=sup{qi}.In order to describe the behavior of this system,in particular possible collisions,we are led to introduce the quantity

It turns out that this quantity controls the motion of the points as our next result shows.

Proposition 2.1Let b=(b1,···,b?)be a solution to(1.16)on its maximal interval of existence[0,Tmax]and assume that(2.1)is satisfied.Let 0≤t1≤t2≤Tmaxbe given.For k=1,···,?,we have the bound

where we have set

Notice that we have also the more straightforward inequality

which is a simple consequence of the triangle inequality.

The proof of Proposition 2.1 will be given later.In view of the previous result,it is therefore of importance to derive bounds for db.In this direction,we first have the following result.

Proposition 2.2Let b=(b1,···,b?)be a solution to(1.16)on its maximal interval of existence[0,Tmax]and assume that(2.1)is satisfied.Then,we have,for any t∈[0,Tmax],

where we have set

ProofIt follows from(1.16)and(2.1)that,for any k=1,···,?,we have

and henceIntegrating,we obtain

and the assertion follows as a standard exercise.

In order to derive more refined estimates,we need to take into account the signs of the interactions.For that purpose,we introduce the quantities

with the convention that the quantity is equal to+∞in case the defining set is empty,and we also set Bmax=sup{}and Bmin=inf{}.The main results on this section can be summarized as follows.

Proposition 2.3Let b=(b1,···,b?)be a solution to(1.16)on its maximal interval of existence[0,Tmax]and assume that(2.1)is satisfied.Then,we have,for any time t∈[0,Tmax],

whereandIf all signs{†k}k∈Jhave the same value,then Tmax=+∞.Otherwise,we have the estimate

Given any two times 0≤t1≤t2≤Tmax,we have the estimate

whereis defined in Lemma 2.3 below.Moreover the following inequality holds,in the sense of distributions

Notice that the behavior ofandare very different,the first one measuring the repulsive forces present in the system,whereas the second measures the attractive ones.

Remark 2.1We have stressed so far the behavior of the equation(1.16)for positive times.The properties of the system are actually similar when time flows backwards,i.e.,considering negative times.It suffices to change the attractive forms into repulsive ones and vice-versa to deduce the corresponding results.Notice in particular thatis changed into

Proof of Proposition 2.1(Assuming Proposition 2.3)Integrating inequality(2.3),we obtain

and the conclusion follows invoking(2.6).

The proof of Proposition 2.3,relies on several observations which we present next,the completion of the proof of Proposition 2.3 being presented in a separate subsection.

Our starting point is that,since the system(1.16)involves both attractive and repulsive forces,it is convenient to divide the collection{b1(t),b2(t),···,b?(t)}into repulsive and attractive chains.Consider more generally a positive integer?∈ N∗,set J={1,···,?}and let†be a function from J to{+,−}.We say that a subset A of J is a chain if A consists of consecutive elements.

Definition 2.1Let A={k,k+1,k+2,···,k+m,k+m+1}be an ordered subset of m+2 consecutive elements in J,with m≥0.

(i)The chain A is said to be a repulsive chain,if and only if given two elements i1and i2in J,we have if†i1= †i2.It is said to be a maximal repulsive chain,if there does exists a repulsive chain which contains A strictly.

(ii)The chain A is said to be an attractive chain,if and only if given two elements i1and i2in J,such that|i1−i2|=1,we have †i1= −†i2.It is said to be a maximal attractive chain,if there does exists an attractive chain which contains A strictly.

Notice that,in view of our definition,repulsive or attractive chains contain at least two elements.For a given map†,consider its maximal repulsive chains,ordered according to increasing numbers A1,A2,···,Ap.Consider two consecutive chains Ai={ki,ki+1,ki+2,···,ki+mi,ki+mi+1}and Ai+1={ki+1,ki+1+1,ki+1+2,···,ki+1+mi+1,ki+1+mi+1+1}.It follows from Definition 2.1 that ki+mi+1

is a maximal attractive chain.In particular,we may decompose J,in increasing order,as

where the chains Aiare maximal repulsive chains,the sets Biare maximal attractive chains for i=1,···,p − 1,and the sets B0and Bpare possibly void or maximal attractive chains.Moreover,we have,for i=1,···,p,

2.2 Maximal repulsive chains

In this subsection,we restrict ourselves to the behavior of a maximal repulsive chain A={j,j+1,···,j+m},m ≤ ?− 2 within the general system(1.16).Without loss of generally,we may assume that†i=+for i∈ A.Setting uk=bk+j,we are led to study the function U=(u0,u1,···,um+1).It follows from the fact that b satisfies(1.16),U is moved through a system of m ODE’s,and two differential inequalities as follows:

and

We assume that the solution is defined on I=[0,Tmax],and that at initial time,we have

The behavior of this system is related to the function Fεdefined on Rm+2by

where,for k=1,···,m − 1 and u=(u0,···,um+1),we set

the numbers qk>0,Bk>0 and λj+(k)>0 being computed thanks to(1.15).In the case q0

where=0,for k=0,···,m,we have set

for k=1,···,m+1,we have set

Notice in particular that=for k=0,···,m.We consider

We prove the following proposition in this subsection.

Proposition 2.4Assume that(2.1)is satisfied and that the function U satisfies(2.9)–(2.10)on[0,Tmax]with(2.11).Then,we have,for any t∈[0,Tmax],

The proof relies on several elementary observations,which we present first before completing the proof of Proposition 2.3.We start with some specific properties of the functional F,which are stated in the next lemma.

Lemma 2.1Let U=(u0,···,um+1)be such that u0

and for every k=0,···,m+1,

ProofInequalities(2.15)–(2.16)are direct consequences of the definition(2.12)of F.In view of formula(2.13),if k=0 or k=m+1,there is nothing to prove,provided that we choose μ0≤ 1.Next,let k=1,···,m and consider for instance.we distinguish the following two cases.

Case 1If this case occurs,then,we have,in view of(2.13),and we are done with a choice of γ0≤

Case 2In this case,we repeat the argument with k replaced by k−1.Then either

so thatand we are done,orand we repeat the argument.Since we have to stop at k=0,this leads to the desired inequality.

The next result emphasizes the gradient flow structure of(1.16).

Lemma 2.2Let U be a solution to(2.9)–(2.10)on[0,T],such that(2.11)and(2.1)hold.Then,we have,for every t∈[0,Tmax],

In particular,

ProofCombining equations(2.9)and(2.10)with the chain rule,we are led to

where for the last inequality,we have invoked Lemma 2.1 and inequality(2.1).The second inequality in(2.18)is then a direct consequence of(2.17).Finally,the last inequality of the lemma follows by integration of the differential inequality(2.18).

Proof of Proposition 2.4Combining the last inequality of Lemma 2.2 with inequality(2.15),inequality(2.14)follows.

We complete this section with the following lemma.

Lemma 2.3Let U be a solution to(2.9)–(2.10)on[0,T],such that(2.11)and(2.1)hold.Given any time 0≤ t1≤ t2,we have,with

ProofWe have by the chain rule and inequality(2.15),

The conclusion follows by integration.

2.3 Maximal attractive chains

In this section,we provide a few properties of a maximal attractive chains B={j,j+1,···,j+m},with m ≤ ?− 2 within the general system(1.16):In particular,we show that it generates collisions in finite time,with an upper bound on the collision time.We may assume without loss of generally that †j=+,so that †j+k=sign(−1)k.Defining U as above,the function U still satisfies(2.9),but the inequalities(2.10)are now replaced by

The behavior of the chain B is now still related to the functional Fε(U),where Fεis defined in(2.12),withand hence takes only two values,and λj+(k)= λj+.However,the differential inequality(2.18)is now turned into

which,by integration yields

Proposition 2.5Assume that(2.1)is satisfied and that the function U satisfies the system(2.9)and(2.19)on[0,Tmax]together with(2.11).Then,we have,for any t∈[0,Tmax],

The argument is similar to the proof of Proposition 2.4,we therefore omit it.

Lemma 2.4Assume that(2.1)is satisfied and that the function U satisfies the system(2.9)and(2.19)on[0,Tmax]together with(2.11).Then,we have the estimate

2.4 Proof of Proposition 2.3 completed

Inequalities(2.4)and(2.6)of Proposition 2.3 follow immediately from Proposition 2.4 and Proposition 2.5 applied to each separate maximal chain provided by the decomposition(2.8):We leave the details of the proof to the reader.Inequality(2.5)is then a direct consequence of(2.4).For inequality(2.7),we consider again each maximal attractive chain and notice that,if bkis an element of such a chain which is not at the end points,then we have

and a similar estimate holds for the points which are at the end of the chain.A few elementary arguments then lead to the conclusion.

3 Remarks on Stationary Solutions

In this section,we collect a few elementary results about stationary solutions to(PGL)ε.

3.1 Stationary solutions in R with vanishing discrepancy

Stationary solutions on R may be described by using the method of separation of variable,a tool which cannot be extended to systems.As matter of fact,this simple fact turns out to be crucial,and explains for a large part why the analysis of this paper remains restricted to the scalar case.

Consider more generally an interval I of R and a solution u to(1.34).Multiplying equation(1.34)by u,we are led to the fact that,for any solution u of(1.34),we have

so that ξ is a constant function on I.We restrict ourselves in this section to solutions with vanishing discrepancy,that is which verify

Differentiating(3.2),we verify that any smooth solution to(3.2)is actually a solution to(1.34).We finally solve equation(3.2)by separation of variables.Consider the function ζidefined on the interval(σi,σi+1)by

where ziis defined in the introduction.The map γiis one-to-one from(σi,σi+1)to R,so that we may define its inverse map

from R to(σi,σi+1)as well as the mapWe verify thatas well assolve(3.2)and hence(1.34).The next result,those proof is left to the reader,shows that we have actually obtained all solutions.

Lemma 3.1Let u be a solution to(1.34)on some interval I,such that(3.2)holds,and such that u(x0)∈ (σi,σi+1)for some x0∈ I and some i∈ 1,···,q−1.Then

for some a∈R.

Next,we provide a few simple properties of the functionswhich enter directly in our arguments.In view of the definition(3.4),we have

whereas a change of variable shows that ζihas finite energy given by the formula

It is also straightforward to establish that there exists some constant β1>0,such that,if for some s ∈ R,we havethen

We introduce the constants

so that we obtain the expansions,as u→and as u→

It follows that as x→−∞and as x→+∞,

whereandSimilar asymptotics hold for derivatives.For 0< ε <1 given,and i=1,···,q−1,consider the scaled function.Straighforward computations show that

so there is some constant C>0 which does not depend on r and ε,such that

3.2 Study of the perturbed stationary equation

This section is devoted to some properties of solutions(1.30),that is to the perturbed differential equation uxx= ε−2V?(u)+f on R,where the function f belongs to L2(R).The main result of this section will be to show that,if u has bounded energy and if f is small,then u is close to several translations of the functions ζi,εsuitably glued together.More precisely,we assume throughout this section that

and consider the number

where ρ1and c0are constants depending possibly on M0and which will be determined later(see(3.24)for ρ1and the proof of Lemma 3.4 for c0).Hence,we have

We assume throughout this subsection that

where α1>0 is some constant depending only on V,which will be fixed in the proof of Lemma 3.4 below.This assumption implies in particular

If I is some interval of R and g is a C1function defined on R,it is convenient to introduce the notation

The main result of this section can be stated as follows.

Proposition 3.1Let u be a solution to(1.30)satisfying assumptions(3.12)and(3.15).Then,there exists a collection of points{ak}k∈Jin R,such that the following conditions are fulfilled:

(1),where S0=inf{Si,i={1···,q}.

(2)For each k ∈ J,there exists a number i(k)∈ {1,···,q},such that

(3)The points are well-separated,that is,we have,for

(4)For each k ∈ J,there exists a symbol∈ {+,−},such that we have the estimate,for

(5)Set Ωr(t0)=We have the energy estimate

The proof of Proposition 3.1 will be decomposed into several lemmas.Following the approach of[3],we recast equation(1.30)as a system of two differential equations of first order.For that purpose,we set w=εux,so that(1.30)is equivalent to the system

which we may write in a more condensed form as

where,for x in R,we have set U(x)=(u(x),w(x))and F(x)=(0,f(x)),and where G denotes the vector field on R2given by G(u1,u2)=(u2,V?(u1)).Notice that|∇G(u1,u2)|≤ N(|u1|),where N≥1 is some continuous non-decreasing scalar function.On the other hand,since u is assumed to satisfy the energy bound(3.12),we have

so that we are led to set

We next compare a given global bounded solution u of(1.30)to a possible local solution u0of the unperturbed equation

with comparable initial condition at some point x0∈ R.We denote accordinglyon its maximal interval of existence.As a consequence of Gronwall’s identity,we have the following lemma.

Lemma 3.2Let A=(−b,b)be an interval of R,u be a solution to(1.30)on A and u0be a local solution to(3.25).Assume that for some number a satisfying b≥a>0,we have the inequality

Then u0is well-defined on[−a,+a],and we have

ProofLet I be the largest interval containing 0,such that

On I,since(U−U0)x=G(U)−G(U0)+εF,we obtain the inequality

It follows from Gronwall’s inequality,that,for x ∈ I,

so that by the Cauchy-Schwarz inequality,we are led to the bound,for x∈I,

Hence,if(3.26)is verified,then[−a,a]⊂ I and(3.27)follows.

We will combine the previous lemma with the following lemma.

Lemma 3.3Let u be a solution to(1.30)on R,such that Eε(u)≤ M0<+∞.Then

ProofThis is a direct consequence of the equalityCauchy-Schwarz inequality,and the fact that it is zero at infinity since u has finite energy.

Lemma 3.4Let u be a solution to(1.30)on R satisfying assumptions(3.12)and(3.15)and let x0∈ D(u).There exists some point y0∈ R,some i∈ {1,···,q}and some symbol†∈ {+,−},such that for every 0

Moreover,there exists some constant α1>0 depending only on the potential V,such that,ifthenand

where the constant C>0 depends only on the potential V.

Remark 3.1(1)Sinceit is straightforward to deduce from(3.30)applied withand the properties of the functions(see(3.9))that,if the constant α1is choosing sufficiently large,then there exists some point,such thatand

(2)We also notice that if the constant α1is chosen sufficiently large,

(3)Set

Then,it follows combining(3.11)and(3.31)that

since the function s?→ sexp(−s)is decreasing for large values of s>0 choosing the constant α1sufficiently large,and if we assume df≥ α1ε,we are led to

where S0=inf{Si,i=1,···,q− 1}>0.

ProofGoing back to Lemma 3.2,we consider as solution u0to the unperturbed equation(3.25),the solution obtained choosing as initial conditions u0(x0)=u(x0)and the derivative(x0)in such a way that ξε(u0)(x0)=0.Obviously,it suffices therefore to choose

We impose moreover the sign of(x0)to be the same as the sign of ux(x0),which does not vanish,so that u0is uniquely defined.Since by construction ξε(u0)=0,it follows from Lemma 3.1 that

for some point y0∈ R,some i∈ {1,···,q}and some†∈ {+,−}.Notice that,since x0∈ D(u),there exists a constant c0>0(depending only on the choice of μ0and on the numbers λi,hence on the properties of the potential V),such that

so that by Lemma 3.3,

Since by assumption(3.15),we haveand deduce

We next impose as first condition on c0thatso that we obtain,since 0< ε≤1,

Combining the identitythe bound(3.36)and Lemma 3.3,that are led to

Since u(x0)=u0(x0),we deduce

In view of Lemma 3.2,we estimate

Imposing the additional conditionwe completely determine c0.In view of the definition definition of df,we are hence led to

The inequality(3.30)then follows from Lemma 3.2.For the second assertion,we specify inequality(3.30)for the point x=x0with a=,so that

provided,for the last inequality that α1is chosen sufficiently large.Since x0∈ D(u),we have either|u(x0)− σi|≥ μ0or|u(x0)− σi+1|≥ μ0,and hence

Invoking(3.7),we are led to

Choosing α1possibly even larger so that α1≥ 16β1,we are led tothat is the second assertion follows.We finally turn to the proof of(3.31).For that purpose,we choose a=,so thatand hence,by inequality(3.30),we may decompose u aswhere

Expending accordingly the energy,we derive estimate(3.31).

Remark 3.2The conclusion(3.30)of Lemma 3.4 remains essentially unchanged,if instead of a solution u defined on the whole real line R,we consider a solution on a bounded interval A.In that case,however,we have to replace in our computation the quantity?f?L2(R)by the quantitywhere z is some arbitrary point,which lead,in the conclusion,to changing the constant dfby the constant

Conditions(3.15)–(3.16)also have been changed accordingly.

Proof of Proposition 3.1(Completed)We distinguish two cases.

Case 1D(u)= ∅.In this case,we take J= ∅,and the only thing to be established in this case is estimate(3.21).Since D(u)= ∅,there exists some i∈ {1,···,q},such that|u−σi|≤ μ0.Multiplying equation(1.30)by u−σi,we are led to

and hence,by Cauchy-Schwarz,.In view of(3.14)and the fact that 0<ε<1,this yields to(3.21).

Case 2we construct the points aiby an inductive argument,which stops in afinite numbers of steps.Let first x0be an arbitrary point in D(u).Applying Lemma 3.4 to x0as well as Remark 3.1,we deduce the existence of a point,such that

Applying(3.30)with a=,we are led to

We set a1=,so that(3.18)as well as(3.20)are satisfied for k=1.Moreover,it follows from(3.35)that

We iterative the process considering nextIf this set is empty,then we take J={1},i(1)=i and †1= †,and we stop.Otherwise,we choose some x1∈ Ω1,and argue as we did above,now with x1instead of x0:This yields a point,some number i(2)∈ {1,···,q},some sign †2∈ {+,−},such that

Setting a2=,and invoking(3.35)again,we are led to

Notice also that,by construction,

We construct the set{ak}k∈Jrepeating the previous construction inductively.Since in each iteration the energy in estimates(3.46)decreases by at least an amount of,we stop in at mostnumber of steps,and all assertions,except(3.21)have been verified.In order to establish(3.21),we argue as in case one.We have,integrating by parts in(1.30)for k=1,···,q − 1

so that

By summation,we obtain(3.21),and the proof of Proposition3.1 is complete.

As a by-product of the Proposition 3.1,we may also derive a global estimate.

Lemma 3.5Let u be a solution to(1.30)satisfying assumptions(3.12)and(3.15).then,we have

ProofIt suffices to combine(3.34)and(3.21).

4 Regularized Fronts

4.1 First properties

In this section,we provide some properties of the solution vεto(PGL)εon time slices on which it has already undergone a parabolic regularization,that is when fronts become close to the stationary ones,and are well separated.Such a situation is described in Definition 1.1.

Lemma 4.1If WPε(δ,t0)holds,then WPε(δ?,t0)holds for any α1ε≤ δ?≤ δ.

We leave the proof to the reader.Next,we consider for k∈J(t0)the function

Lemma 4.2Let t0≥ 0,δ > α1ε be given,and assume that vεsatisfies condition WPε(δ,t0).Then,we have,for any 0< δ?≤ δ,

where we have setandwith the numbers>0 having been introduced in(3.9).

ProofWe have,in view of the definition of condition WPε(δ,t0),

whereas by(3.9),we have

In several place,we will assume additionally thatwhere λmax=sup{λi}.Then we obtain under the assumption of Lemma 4.2

For the outer region,we have the following lemma.

Lemma 4.3Let t0≥ 0,δ ≥ α1ε be given,and assume that vεsatisfies condition WPε(δ,t0).Then,we have,for x ∈ [ak+δ,ak+1− δ](resp.x ∈ [ak−1+δ,ak− δ]),

ProofWe writeso that

The conclusion then follows from Lemma(4.1)and assumption WP3.

We complete the subsection with energy estimates.

Lemma 4.4Let t0≥ 0,δ ≥ α1ε be given,and assume that vεsatisfies condition WPε(δ,t0).Then,we have

where ρ2is defined in(3.33)and where the front energy E(t0)is defined by

The proof is similar to the proof of Lemma 3.5,and we omit it.

Notice that the front energy E(t0)may take only a finite number of values,and is hence quantized.We emphasize also that,at this stage,the front energy E(t0)is only defined assuming that condition WPε(δ,t0)holds.However,we leave to the reader to check that the value of E(t0)does not depend on δ,provided of course that δ ≥ α1ε,so that it suffices ultimately,in order to define E(t0),to check that condition WPε(α1ε,t0)is full-filled.

Choosing possibly a larger value for the constant α1,an immediate consequence of Lemma 4.4 as well as the fact that E(t0)may take only a finite number of values is as follows.

Lemma 4.5Let T1≥ T ≥ 0 be given,and assume that conditions WPε(α1ε,T)and WPε(α1ε,T1)hold.Then,we have E(T1) ≤ E(T).Moreover,there exists a positive constant μ1>0,such that,if E(T1)

We next discuss the case of equality E(T1)=E(T),in particular with respect to the L2norm of the dissipation,which is central in several of our arguments.For that purpose consider two times T and T1,such that T1≥T≥0,and set

As a direct consequence of Lemma 4.4 and the global energy identity(1.5),we have the following corollary.

Corollary 4.1Assume δ > α1ε,and let T1≥ T ≥ 0 be such that both WPε(δ,T)and WPε(δ,T1)hold and that E(T)=E(T1).Then,we have

4.2 Finding regularized fronts

Occurrences of well-prepared time slices may be found thanks to Proposition 3.1 and a rough mean-value argument.In many places,we rely on the following observation.

Lemma 4.6Let T ≥ 0,ΔT>0 and δ≥ α1ε be given.If

then,there exists some time t0∈ [T,T+ΔT],such that WPε(δ,t0)holds.

ProofBy the mean-value argument,there exists some time t0∈[T,T+ΔT],such that Consider next the map u=vε(·,t0),so that u is now a solution to(1.30),with source term f=∂t(·,t0).Hence f satisfies(3.14)with df=2δ.The conclusion then follows from Proposition 3.1.

As a matter of fact,one may initiate the search of regularized fronts using(1.5),that is

Therefore,it follows from Lemma 4.6 that it suffices to impose

to deduce the existence of a time t0∈ [T,T+ΔT],such that WPε(δ,t0)holds.In particular,taking δ of the form δ = αε with α ≥ α1,we see that given any T ≥ 0,there exists a time t∈ [T,T+ ω(α)ε2],such that WPε(αε,t)holds with

Since we assume α ≥ α1the front energy E(t)is then well-defined.In other words,on each time interval of size ω(α1)ε2,there exists some time for which the front energy is well-defined.On the other hand,this energy is non-increasing takes only a finite number of values,so that we may expect to find large time intervals,where it remains constant.This is the situation we analyze in the next subsection.

4.3 Propagating regularized fronts

We assume throughout this subsection,that we are given δ > α1ε and two time T1≥ T ≥ 0,such that

Under that assumption,our first result shows that vεremains regularized on almost the whole time interval[T,T1],with a smaller δ though.

Proposition 4.1Assume that assumption(4.10)holds with δ ≥ α2ε,where α2≥ α1is some constant depending only on the potential V and the constant M0.There exists some constant 0< ν1<1,such that given any time t∈ [T+c1εδ,T1],property WPε(ν1δ,t)holds,where

The proof of Proposition 4.1 involves the next result,of possible independent interest.

Lemma 4.7Assume that assumption(4.10)holds with δ ≥ α1ε.We have the estimate,for t∈ [T+ε2,T1],

ProofDifferentiating equation(PGLε)with respect to time,we are led to

It follows from standard parabolic estimates,working on the cylinder Λε=[x−ε,x+ε]×[t−ε2,t]that

where the last inequality follows from Corollary 4.1,and which yields the conclusion.

Proof of Proposition 4.1We divide the proof into several steps.

Step 1Given any time t ∈ [T+c1εδ,T1],we may find some time?t ∈ [t− c1εδ,t],such thatholds,where 0< ν2<1 is some positive constant.

Proof of Step 1In view of Corollary 4.1,we have dissipThe conclusion follows directly from Lemma 4.6,applied with T=t−c1εδ and ΔT=c1εδ,the definition(3.33)of ρ2,and the choiceConcerning the constant ν1,our choice will be

Step 2SetThen,we have,for any si=0,1 and provided that α2is chosen sufficiently large,

Proof of Step 2Since ν1≤ ν2and in view of property WPε(ν2δ,?t)inequality(1.10),we have for i=0,1,

Consider the cylinderand setwhere,for i=0,1,we have defined

It follows from Lemma 4.7 thatThe conclusion that follows from the estimates provided in Proposition 6.1.

Step 3There exists some point z ∈ [ak(t)+δ,ak(t)+2δ],such that

Proof of Step 3It is a direct consequence of the inequality ε−1|ξε|≤ eε(vε),Step 2 for i=1 and a mean-value argument.

Step 4(Proof of Proposition 4.1 Completed)To prove that WPε(ν1δ,t)holds,we have to establish that conditions WP3(ν1δ)and WP4(ν1δ)hold at time t.Condition WP3(ν1δ)is actually an immediate consequence of Step 1.For WP2(ν1δ),we apply Remark 3.2 to the map vε(·,t)on the interval A=[ak(t)− 2δ,ak(t)+2δ]with f= ∂vε(.,t),so that,in view of Lemma 4.7,we deriveComputingaccording to(3.39)wefind,thanks to the definition of ν2,

provided that α2is chosen sufficiently large.This yields

which establishes WPε3(ν1δ)at time t for our choice(4.12)of the constant ν1.

We complete this subsection deriving a simple consequence of Corollary 4.1,which will be used in several places.Consider an arbitrary point a∈R,numbers d>0,r>0 and set

Lemma 4.8Assume that assumption(4.10)holds with δ ≥ α2ε.Let t ∈ [T,T1],a ∈ R,d>0 and r>0 be given.There exists somesuch that

ProofSet t1=inf{t− εr,T}.It follows from Corollary 4.1,the definition of dissip and a standard mean-value argument that,for somewe have the inequality

The results follows by integration and invoking Cauchy-Schwarz inequality.

4.4 First properties of trajectories

In this subsection,we discuss a few elementary properties of the subset O(t)of R,defined in(1.11)in particular in connection with the property WPε(δ,t).As a matter of fact,this is the true for all positive times,as a consequence of a general result of Angenent on parabolic scalar equations9This is the second place where we invoke the fact that the equation is scalar:However,this observation is not crucial in the proof.(see[2]):Moreover,the number of elements in WPε(δ,t)can only decrease.Going back to Proposition 4.1,we see that if(4.10)holds with δ ≥ α2ε,then the number of points in O(t)is constant on the interval[T,T1],and therefore,we may write,for t∈[T,T1],

Concerning the motion of the individual points ak(·),we have the following proposition.

Proposition 4.2Let 0≤T ≤T1and δ be given,and assume that condition(4.10)holds with δ ≥ α3ε,where α3≥ α2is some constant.Then given any times t and t?∈ [T,T1],we have

ProofWe divide the proof into steps.

Step 1Given any times t and t?∈ [T,T1],such that|t− t?|≤ c1εδ and WPε(ν1δ,t?),we have

Proof of Step 1We apply Lemma 4.8 with r=c1δ,a=ak(t)and choose d of the form

where the parameter?is determined bySince we impose that 4?≤ ρ3,there exists,in view of Lemma 4.8,somesuch that

Since the assumption 0< ε <1 holds,the last inequality holds if δ ≥ α3ε,provided that α3≥ α2>0 is chosen sufficiently large.On the other hand,since WPε(ν1δ,t?)holds,we have for any −ν1δ ≤ l≤ ν1δ,

while we have that for any l∈R the estimatewhere K is some constant.We deduce that for any −ν1δ ≤ l≤ ν1δ,we have

Next assume by contradiction that

If(4.21)holds,then we have

Using(4.20)with l=ak(t?)−[ak(t)+d],we are led to

where the last inequality holds since we imposeprovided δ ≥ α3ε and that α3is chosen sufficiently large.Combining this inequality with(4.18),we are led to

provided δ≥α3ε and that α3is chosen sufficiently large.Since|˙vε|≤Kε−1,we deduce,setting

provided δ ≥ α3ε and that α3is chosen sufficiently large.On the other hand,it follows from the definition of ak(t)that|vε(ak(t),t)− zi(k)|=0,so that Minbd=0,a contradiction,and so that(4.21)does not hold,if δ ≥ α3ε.Similarily,one shows

which leads to the conclusion(4.17)withby choosing α4sufficiently large.

Step 2Given any times t and t?∈ [T,T1],such that|t− t?|≤ c1εδ,we have

Proof of Step 2Without loss of generality,we may assume that T≤t≤t?≤T1.This is an immediate consequence of Step 1 and Proposition 4.1.Indeed,if t?>T+c1εδ,then it satisfies,in view of Proposition 4.1,WPε(ν1δ,t?),and the conclusion then follows directly from Step 1.Otherwise,we have t?−T ≤ c1εδ.Since assumptions WPε(δ,T)holds,we deduce from Step 1 that

and the same inequality with t replaced by t?.Combining these two inequalities,the conclusion follows.

Step 3(Proof of Proposition 4.2 Completed)We introduce the intermediate times tn=t+kc1εδ for n ∈ {0,1,···,nf},where nfis the largest integer less thanwith tkf+1=t?.In view of Step 2,we have for n=0,···,nf,

so that adding these inequalities,we are led to

and the conclusion follows for a suitable choice of the constant α3.

If follows from Propositions 4.1–4.2 that if assumption(4.10)is satisfied for some δ ≥ α4ε,then the number of front points ak(t)does not change on the time interval[T,T1],the front points ak(t)are perfectly labelled,continuous in time.Likewise the numbers i(k)and the signs†i(k)do not depend on t.Moreover,an elementary,yet important observation is:

Lemma 4.9Let 0≤T≤T1and δ be given,and assume that condition(4.10)holds with δ ≥ α3ε.Then,given anyand t∈ [T,T1],we have

ProofIf t≥ T+c1εδ,then the conclusion follows immediately from the fact that property WPε(ν1δ,t)holds.If T ≤ t≤ T+c1εδ,then we have for j=1,2,

provided that α3is chosen sufficiently large.On the other hand,it follows from property WPε(δ,T)that|ak1(T)− ak2(T)|≥ δ,and the conclusion follows combining the previous inequalities.

4.5 The stopping time Tsim(δ,T)

Whereas the two previous subsections discussed some consequences of condition(4.10),we provide here a situation where such a condition is met.For that purpose,we will invoke for thefirst time so far the upper bound on the speed of the front set provided in Theorem 1.6.Given a time T ≥ 0 and δ >0,we assume throughout this subsection that WPε(T,δ)holds.We then set

in the case that the set on the right-hand side is not void,and Tsim(T,δ)=+∞ otherwise.We have the following proposition.

Proposition 4.3Assume that WPε(T,δ)holds with δ ≥ α4ε for some constant α4≥ α3.Then condition C(ν4δ,T,Tsim(δ,T))is met,where 0< ν4<1 is some constant.Moreover,we have

ProofWe first establish inequality(4.27).In view of(1.27),we have[−α1ε,α1ε],so that,combining with Theorem 1.6,for r ≥ α0ε,we are led to the inclusion

provided

where the sets Ik,rdenote the intervals Ik,r=[ak(T)−α1ε−r,ak(T)+α1ε+r].Choosing

we deduce that

where we assume that the constant α2is chosen sufficiently large.This proves(4.27).

For the first statement,we notice that,thanks to(4.8)there exists some time T1∈[Tsim−such thatholds,where we set

Next consider the timeUsing a similar argument,we may find some time T2∈such thatholds.It follows thatholds,henceholds.Choosingthe proof is complete.

Combining the previous result with Proposition 4.1,inequality(4.16)of Proposition 4.2 as well as the identity(4.15),we are immediately led to the following statement.

Proposition 4.4Assume that WPε(T,δ)holds with δ ≥ α4ε.Then for any t ∈ [T,Tsim(T,δ)],the points{ak(t)}k∈J(T)satisfying(1.8)are well-defined.Moreover,assumption WPε(ν0δ,t)holds for any t∈ [T+c2εδ,Tsim(T,δ)],where ν0= ν1ν4and c2=c1ν4.Moreover,we have

provided that the constant α4is chosen sufficiently large.

Inequality(4.29)will be used to handle small initial time boundary layers of size ε2,which are related to the parabolic estimates provided in the next section.

5 Linear Parabolic Estimates

In this section,we single out a few linear and mostly elementary parabolic estimates,which will be used directly in the study of the nonlinear equation(PGL)ε.We consider in this section 0<ε<1 a small parameter,the standard space-time cylinder10More general cylinders and solutions may be handled by using translations and scalings.

a smooth function c defined on Λ,such that for all(x,t)∈ Λ ,we have

where λ>0 is a given positive number,and linear parabolic equation

as well as its special case,

Notice that it follows from the maximum principle and(5.2)that we have the inequality

where uεis the solution to(5.4)satisfying uε=|uε|on Π,where

withandIn several places,we will be led to assuming that the function c satisfies the additional condition

The main estimate of this section will be given in Proposition 5.1.It involves also the difference

We start with a few preliminary results.

5.1 Basic estimates

Lemma 5.1Let uεbe a solution to(5.3),such that uε=0 on Π−∪ Π+.We have

for any(x,t)∈ Λ.Moreover,if c satisfies(5.7)then,we have for ε2≤ t≤ 1 and+ε≤x≤−ε,

ProofFor the first statement,we notice that the function h defined by

is a solution to(5.4),and hence,by the maximum principle,we have uε≤ h(x,t).Invoking(5.5),the conclusion(5.9)hence follows.

For the second statement,that is estimate(5.10),we invoke the regularization properties of the heat equation together with a scaling argument.Let(x0,t0)∈Λ be given,such that ε2≤ t0≤ 1 andWe perform the change of variableandso that(x0,t0)corresponds in the new variables(x,t)to the point(0,1).We consider the scaled map

which satisfies the parabolic equation

on the large cylinderand where the function c is defined as

It follows from assumption(5.7)for any given|c(x,t)|≤C.To conclude,we evoke the following standard parabolic estimate.

Lemma 5.2Let u be a smooth real-valued function on Λ and assume

Then,ifdenotes the cylinderthere exists a universal constant C>0,such that

For a proof,we refer to[4,Lemma A.7],where closely related estimates are established.

Proof of Lemma 5.1(Completed)We apply(5.14)to the equation(5.12),restricted to Λ ⊂with d=?c(x,t)uε?L∞([x0−ε,x0+ε]×[t0−ε2,t0])and c=?uε?L∞([x0−ε,x0+ε]×[t0−ε2,t0]).We

are hence led to the inequality,using(5.7),

Invoking(5.9),and going back to the original variables,the conclusion(5.10)follows.

Lemma 5.3Let uεbe a solution to(5.3)such that uε=0 on Π0.There exists a constant C>0 which does not depend on ε nor on λ,such that for all(x,t) ∈ Λ,

Moreover,if the function c satisfies condition(5.7),then,we have

ProofIt follows from the maximum principle that for all(x,t)∈Λ,

where Ψεis the stationary solution to(5.4)given by Ψε(x,t)= Ψ0,ε,where Ψ0,εis the solution to the stationary problem(5.29)with boundary conditions

so that

and in particular,

Combining(5.20)and(5.17),we derive(5.15).The proof of(5.16)follows the same arguments as the proof of inequality(5.10)of Lemma 5.1.Therefore,we omit it.

5.2 Equations with source terms

Next,we let f ∈ C2(Λ),we consider the equation with source term

with boundary condition

Lemma 5.4Let uεbe a solution to(5.21)–(5.22).We have the estimate

Moreover,if c satisfies(5.7)then,we have

ProofBy the maximum principle,it suffices to consider the case f≥0,what we assume throughout the rest of the proof.Invoking the maximum principle once more in that case,we conclude that 0≤ uε≤ uε,where uεis the solution to

with boundary condition uε=0 on Π.We are led therefore to establish the bound(5.23)for the solution uεonly.We extend the function f to the whole of R × [0,1]setting f(x,s)=0,ifGiven t>0,we consider also the functionof the scalar variable x defined by

and the solutionof the differential equation

given by convolution with the corresponding kernel,namely

Next,we consider the function?u defined on R+×[0,1]by(x,s)=(x),so that we immediately derive that

Invoking once more the maximum principle,we deduce that≥ uεon[0,t]×and the conclusion(5.23)follows.

Next,we turn to the proof of(5.24).We argue as in the proof of(5.10),and consider the change of variables and the scaled map given by(5.11),which satisfies in our setting the parabolic equation ∂tu−+c(x,t)u=f,where the function f is defined by f(x,t)= ε2fε(x0+εx,t0+ε2(t−1)).Invoking Lemma 5.2 once more,we deduce that

which yields the conclusion(5.24).

5.3 Stationary solution to the linearized problem

In this subsection,we consider the intervaland the solution Uεto the stationary problem for a given parameter λ>0

whereandare given.The solution to(5.29)is easily integrated as

where the constants Aεand Bεare deduced from the values ofandby

We introduce a quadratic form related to the discrepancy defined for a scalar function u by

Lemma 5.5The function Q(Uε)is constant onwith value

The proof is a straightforward computation,which is left to the reader.Similarly,we also have,concerning the energy,for any

In view of our subsequence analysis of the nonlinear problem(PGL)ε,we are led to introduce various additional assumptions onand the function c.First,we consider the case,whereandare of the same order of magnitude,that satisfies an inequality of the type

where K0is some given positive constant.In that case,we have the expansion

where the error term R0,εsatisfies,for every 0<ε<1,the bound

5.4 Comparison with the stationary solution to(5.4)

Next,we set

and consider the solution Uεto(5.29)with corresponding boundary conditions.Our next results describes the possible relaxation of a given solution uεto(5.3)to the stationary solution Uε.In order to state our result,we introduce appropriate notions of oscillations for the various parts of the boundary Π,namely first

and

Proposition 5.1With the notation above,we have the estimate,for(x,t)∈Λ,

Moreover,if c satisfies(5.7),then,we have,for t≥ ε2,

ProofWe may decompose uεas

whereis the solution to(5.3)defined on Λ,such that

where the function u1,εis the solution to(5.3)defined on Λ,such that

and finally u2,εis the solution to(5.21)with boundary condition u2,ε=0 on Πand source term f given by

The function u0,εis estimated thanks to Lemma 5.1,which yields

whereas the function u1,εis estimated thanks to Lemma 5.3,which yields

In order to estimate the function u2,ε,we will invoke Lemma 5.4,and for that purpose,we needfirst to bound the source term f given by(5.41).We have

In view of Lemma 5.4,we are therefore led to estimate the integral

We leave it as an exercise to the reader to verify that

so that

Combining(5.42)–(5.43)and(5.47),we derive(5.38).For(5.39),we use the corresponding estimates,observing that

5.5 Estimates for the quadratic part of the discrepancy

In this subsection,we wish to derive some estimates for Qλ(uε),there Qλis defined in(5.31).Furthermore,we restrict ourselves to the case that there exists some given constant?>0,such that

and we finally also assume that

Lemma 5.6Assume that uεis a solution to(5.3),and that assumptions(5.33)and(5.48)–(5.49)are full-filled.Then,we have,for(x,t)∈Λ,

where the error term satisfies the estimate,for every 0<ε<1 and t≥0,

and

For the proof of Lemma 5.6,we expand uε=Uε+rε,where rε=(uε− Uε),so that by Cauchy-Schwarz inequality,

We then estimate the right-hand side of this inequality thanks to the estimates for rεprovided in Proposition 5.1,inequality(5.32)for Dλ(Uε)as well as the expansions(5.34)and(5.35).We omit the details.

In the asymptotic limit ε → 0,the bound(5.51)shows the term R0,ε(x,t)is indeed an error term only in the case x is small.In particular,ifand 0< ε<1,then,we have

6 Relaxation to the Stationary Equation o ffthe Front Set

The purpose of this section is to obtain a precise expansion of the discrepancy function ξ(,t),when computed far from the front set.For that purpose,we are led to analyze in details a typical situation we present next.Let(x0,t0)be a given arbitrary point in R×R+.For r>0,we consider the space-time cylinder

whereand a solution vεto(PGL)ε.We assume throughout this section that the front set of vεdoes not intersect Λr(x0,t0),that is,we assume that there exists some i∈ {1,···,q},such that,∀(x,t)∈ Λr(x0,t0),

Following the notation introduced in Section 5,we set−σi,and define Uε,ras the solution to

Set

whereWe assume in this section that θbdsatisfies the following smallness assumption:For some fixed constant?>0,

We assume similarly that

Notice in particular that,if(6.3)–(6.4)are satisfied,then we have

The main result of this section is as follows.

Proposition 6.1Let(x0,t0)be in R×R+,let r≥ε be given,and let vεbe a solution to(PGL)ε,such that(6.3)–(6.4),(5.33)and(H0)hold.Then,we have for(x,t) ∈ Λr(x0,t0),

and,if t≥ ε2,

Moreover,we have for every 0<ε<1,

where the error terms satisfies the estimate

and

ProofIn view of the scale and translation invariance of the equation(PGL)ε,we are led to introduce the new parameter,which satisfies assumption 0

so that v?is now a solution to(PGL)?,and the original domain of interest Λr(x0,t0)is changed into the standard cylinder Λ.On the other hand,since σiis a minimizer for the potential V which is assumed to be smooth,we may expend the potential V as

and its derivative V?near σias

where Φ and ϕ are some smooth functions.Setting w?=v?− σion Λ we are led to rewrite the equation(PGL)εas

where the function ciis defined on the cylinder Λ as c(x,t)= λi− ς?(x,t)with

It satisfies therefore,in view of assumption(6.4),the estimate

We are hence in position to apply Proposition 5.1 to the equation(6.13):Estimates(5.38)–(5.39)combined with the inequalities(6.3)and(6.14),then lead directly to(6.6).Turning to(6.9),we write

so that(6.9)is a direct consequence of Lemma 5.6 together with the smallness assumptions(6.3)–(6.4),which lead to(6.5)and allow to bound suitably the term

Finally,we end the section with a crude estimate,which will also be used in some places.

Lemma 6.1Let(x0,t0)be in R×R+,let r≥ ε be given,and let vεbe a solution to(PGL)ε,such that(6.3)–(6.4),(5.33)and(H0)hold.Then,we have for(x,t)∈ Λr(x0,t0),

The proof is a direct consequence of Proposition 6.1 and inequality(5.32).

6.1 Expansions and bounds for ξassuming WPε(δ,T)

Appropriate expansion for ξ are the central tool in order to derivate the motion law of the fronts.Throughout this section,given δ >0,we assume that WPε(δ,T)holds for some time T ≥ 0 and given some δ >0.For times t ∈ [T,Tsim(δ,T)],we consider the intervals[ak(t),ak+1(t)].Our purpose is to provide some accurate estimates for the discrepancy ξ on these intervals.It turns out actually that our estimates are essentially relevant only for points near the center

of the interval,provided that the length dk(t)=|ak+1(t)−ak(t)|of the interval is not too large compared to δ.For that purpose,we will use in some places the condition

with the last inequality being a consequence of the inequality ρ4≤ ρ3,where we set

Our next result is central in the derivation of the motion law.

Proposition 6.2Let T ≥ 0 be given,and assume that WPε(δ,T)holds for some δ ≥ α5ε,for some constant α5≥ α4.Given any time T+ε2≤ t≤ Tsim(δ,T)and k ∈ J(T),such that WSk(t)holds,we have,for x∈[ak(t),ak+1(t)],

where({ai(t)}is defined in(1.15)and the error terms satisfy the estimate,for positive constants K1>0 and?6,

and

Notice that the previous result yields a precise expansion of the discrepancy provided that the following conditions are met:

(i)The distance between the points ak(t)and ak+1(t)can be compared to the length δ,i.e.,condition WSk(t)is met.

(ii)The point x is close to the center

(iii)The time t is not too close to the initial time T.

More precisely,a direct consequence of Proposition 6.2 is as follows.If WSk(t)is satisfied and

then,we have

where

Proof of Propostion 6.2We first describe the general outline of the proof.We will work on a cylinder of the form

where the parameter r>0 homogeneous to a length is defined by

We then divide this cylinder(t)into a region close to the front sets near ak(t)and ak+1(t),termed here the “inner region”,and the rest of the cylinder,termed the “outer”region.In the inner region,we will show that the solution remains close to an optimal profile,whereas in the outer region,we are in position to apply Proposition 6.1,with the main point in the proof being somehow to glue together the inner and the out region.

In order to define the outer region,we argue as in Lemma 4.8,so that we may find some numberwhere ν5is defined in(6.16),such that

We then define the outer region as

Adapting the notation of Section 6 to the present framework,we are led to set wε=vε−σj+(k),

and θ0accordingly(see(6.2)).Finally,we notice that there exist some time T ≤ t?

In order to apply Proposition 6.1 to Λout(t),we need to deduce several estimates forθbd···,which are mainly derived from estimates on the inner region.We divide the remainder of the proof into several steps.

Step 1Estimates for

Setandis sufficiently large,we have

and

ProofIt follows from condition WPε(ν1δ,t?)that for the time t?defined above and for anywe have

Indeed,in view of the definition(6.16)of ν5and since ρ3≤ ρ1,we have

so that Lemma 4.2 and the previous remark apply.We turn first to the estimate for.We choosewhereas

Hence,inequality(6.24)yields

Combining(6.27)with(6.28),we deduce that provided δ ≥ ε,

On the other hand,it follows from the definition ofthat

Combining(6.27),(6.29)–(6.30),(6.23)and the fact that ρ4≤ ρ3,we derive(6.25)for γε.We derive the corresponding estimate forusing the same argument.For the proof of(6.26),we observe that,as a consequence of our construction of δ and the definition(6.16)of ν5,we have

so that,ifis sufficiently large,thenfrom which we deduce the conclusion.

Step 2Estimates for θbdand θ0.

We have

Proof of Step 2The estimate for θbdis a direct consequence of(6.23)together with the inequality ρ4≤ ρ3,whereas the estimate for θ0follows directly from Lemma 4.3.

Step 3Proof of Proposition 6.2 completed.

We are now in position to apply Proposition 6.1 on the cylinder Λout(t)with

so that assumptions(6.3)–(6.4)and(5.33)are satisfied,with a constant K0depending only on the numbers B±,and provided that the ratiois sufficiently large.We have therefore,for every 0<ε<1 and every s∈[t−εr,t],

where the error term satisfies the estimates

and

We next use formula(6.31)at time s=t,and distinguish two cases.If,thenso that

and then,we absorb this term in the second term on the right-hand side of(6.32)at the cost of a larger constant C.Otherwise,that is,ifthen C2,ε(x,t)has exactly the form provided in(6.19),and we are done for this part of the error terms.It remain to check that the other error terms have the announced behavior.For that purpose,we first notice that it follows from Step 1 that

where,and that

Combining(6.31)and(6.34)–(6.35),we obtain the desired result by choosing

We also need to handle the case,where the assumption WSk(t)is not met.In that direction,we are not able to provide an expansion,but only an upper bound,which turns out to be sufficient for our further analysis.

Proposition 6.3Let T ≥ 0 be given,and assume that WPε(δ,T)holds for some δ ≥ α6ε.Given any time t ∈ [T,Tsim(δ,T)],such that inequality WSk(t)does not hold,we have for

with

The proof is essentially the same as the proof of Proposition 6.2,with

the main point being to replace the definition of r given in(6.22)by the new choiceWe leave the details to the reader.Finally,in some place,we will need another estimate somewhat in the same spirit as Proposition 6.3 provided by the following lemma.

Lemma 6.2Let T ≥ 0 be given,and assume that WPε(δ,T)holds for some δ ≥ α6ε.Given any time t∈ [T,Tsim(δ,T)],such that t≥ ε2,given k ∈ 1,···,?(t)−1 and x ∈ [ak(t),ak+1(t)],we have

where γ(x,t)=inf{|x−ak(t)|,|x−ak+1(t)}.

The proof is a rather direct consequence of inequality(5.32),therefore we omit it.

7 Motion Law for Fronts

The purpose of this section is to provide the proofs of Theorems 1.1–1.2.To that aim,we will combine the result of the previous section with the motion law for the local energy(1.31)by making use of test function of a special type,which we describe in this section.

7.1 Approximating the points ak(t)using the energy density

Let t≥ 0 and let k ∈ J(t)be given.Since the expansion for the discrepancy ξ is only accurate near the points,we are led to introduce intervals Ik(t)of the following form:

where,for the construction of the pointswe distinguish several cases.

Case 1None of the conditions WSk(t)and WSk−1(t)holds.In that case,we set

Case 2At least,one of the conditions WSk(t)or WSk−1(t)holds.Without loss of generally,we may assume that WSkholds,with the other case being handled in a similar way.Then,we distinguish once more two subcases.If

then we set

Otherwise,we set

Notice that,if condition(7.2)is met,thenWe then construct a test function χ ≡ χk,thaving the following properties:

One may check that the set of functions verifying these three conditions is not void.Notice that=0 on the intervaland hencehas support on Vk(t)=Ik(t)In view of Proposition 6.2,we introduce that the stopping time Tk(δ,t)is defined by

We consider the energy densityas well as the related quantity

where qk=Si(k),

We claim that bk,t(s)offers a good approximation of ak(s).

Lemma 7.1Let T ≥ 0 be given,and assume that WPε(δ,T)holds for some δ ≥ α6ε.We have,for any k ∈ J(T)and s ∈ [T+c2εδ,Tk(δ,T)],

where we have set Dk(T)=inf{L(δ),(T),(T)}and for the last inequality at the cost of a possible larger choice of the constant α6.

ProofSince in view of Proposition 4.4,property WPε(ν0δ,s)holds for s ∈ [T+c2εδTk(δ,T)],we deduce from(1.9)–(1.10)that

where K>0 is some constant.On the other hand,we have,going back to(7.5),

Finally,by Corollary 4.1,we also haveso that the conclusion follows combining the previous inequalities,and imposing additionally that ρ3≤ρ1ν0.

7.2 A first motion law for the points ak(s)

Our previous estimates lead to us directly to the following result,which is the building block in the proof of Theorem 1.1.

Proposition 7.1Let T ≥ 0 be given,and assume that WPε(δ,T)holds for some δ ≥ α6ε.Let t∈ [T,Tsim(δ,T)]and consider k ∈ J(T).If t≥ ε2,it holds for s ∈ [t,Tsim(δ,T)],that

wherewith L(δ)defined in Proposition 6.3.Moreover,if one of the conditions WSk(t)or WSk−1(t)holds,and

then,we have for s ∈ [t,Tk(t,δ)],where Tkis defined in(7.4),the differential equation

where the error term satisfies the estimate

ProofInequality(7.6)is a rather direct consequence of Proposition 6.3,the definition of our test function,and the motion law for the local energy(1.31),which yields

wherewith the inclusion being a consequence of the definition(7.4)of the stopping time Tk.

Turning to(7.8)we may assume that,for instance WSk(t)(δ)holds,that is,we are either in Case 1 or Case 2 of the definition(7.1)of Ik.We have for s ∈ [t,Tk(ρ,t)],

Next we remark that,in view of the properties of the χ,we have

Moreover,in view of the construction of χ,we haveSince we assume that WSk(t)holds,we may invoke on the interval(t),the bound(6.21),so that finally our computation yields

with the estimateIn view of Lemma 7.1,we may write

with the estimate.Similar estimates hold for the integralif WSk−1holds,which lead to equation(7.8)in that case.Otherwise,we are in Case 2 of the definition(7.1)of Ik,and thenturns out to be lower order compared toso that(7.8)holds likewise.

We complete this section with a lower bound for T(t,δ).

Lemma 7.2Under the assumptions of Proposition 7.1,we have the lower bound

where the constant S2is defined in Proposition 2.2.

ProofWe first observe,combining the definition(7.4)of T(t,δ)and the result of Lemma 7.1,that for some k0∈J(t),

We then invoke several properties of the differential equation(1.16)presented in Section 211which are actually independent of the present discussion..First,in view of Proposition 2.1,we have,for any k∈J(t),

so that,if α6is chosen sufficiently large,we obtain,combining the two previous inequalities,

We finally invoke Proposition 2.2 to deduce that

and the conclusion follows,choosing possibly α6sufficiently large.

We deduce from the previous result.

Corollary 7.1Let T ≥ 0 be given,and assume that WPε(δ,T)holds for some δ ≥ α7ε,where α7≥ α6is some constant.Then,we have for any times inf{ε2,t} ≤ s1≤ s2≤ T(t,δ),

If moreover we havethen we have that for a constant 0< ρ7< ρ1,the following holds:

If Dk(t)=L(δ),then(7.11)holds for all inf{ε2,t} ≤ s1≤ s2≤ T(t,δ).

ProofInequality(7.11)is derived integrating inequality(7.6)in time.The first inequality in(7.12)is then derived immediately,noticing thatFinally,the second inequality of(7.12)follows combining estimate(4.29)of Proposition 4.4,Lemma 7.1 and(7.12)with possibly a judicious tuning of the constants α7and ρ7.The last statement is proven similarly.

7.3 Proof of Theorem 1.1

Step 1Defining the time T(T),proofs of(1.17)and assertions(i)–(ii).

We first consider the stopping time T(T,δ)defined in Lemma 7.2.We then choose the constant S∗,such that S∗=S2,so that we are immediately led to the inequality

where Trefis defined in(1.17).On the other hand,in view of definitions(4.26)and(7.4),we also have the inequality T(T,δ)≤ Tsim(T,δ).Imposing furthermore that the constant α∗in the statement of Theorem 1.1 satisfies the condition α∗≥ α4,we are in position to apply Proposition 4.4,which yields that WPε(ν0δ,t)holds for any t ∈ [T+c2εδ,T(T,δ)]and that the points{ak(t)}k∈J(T)are well-defined for t ∈ [T,T+c2εδ],where the constants c2and ν0are defined in Proposition 4.4.

We next introduce a new length scaledefined by

In order to define T,we distinguish two cases.

Case 1= δ.

In this case,we set

With this choice,(1.17)follows from(7.13),whereas as already mentioned,Proposition 4.4 shows that WPε(ν0δ,t)holds for any t∈ [T+c2εδ,T].Since in the case considered here,we have da(T)≤ 44ρ1δ,it follows that property WPε(ν∗da(T),t)holds for any t∈ [T+c2εδ,T],provided that the constant ν∗satisfies the conditionsand α∗≥ 44ρ1α4.We impose also c∗=c2,and we verify that(1.17)as well as assertions(i)–(ii)have been established in the case considered here.

Case 2

We introduce first ΔT anddefined by

It follows from Lemma 4.6 and(4.8)that there exists some time Treg∈[T,T+ΔT],such that WPε(Treg,?δ)holds.We define T as

First notice that,provided δ ≥ α8ε,where α8is some positive constant.Hence,imposing the additional condition α∗≥ α7,we are led to T+ΔT+c2ε?δ∈[T,],and hence Treg∈[T,].We claim that

Indeed,in the case considered here,we have≥ 2δ,so thatOn the other hand,by Corollary 7.1 and(7.12),we have,for any k∈J(T),

Hence,in view of the definition of T,we deduce T(Treg,2δ) ≥ T(T,δ),provided that α8is chosen sufficiently large,and the claim(7.16)follows.Then this establishes inequality(1.17)in Case 2.

In order to establish assertion(ii),we invoke again Proposition 4.4.It yields that for anyand hence anypropertyi.e.,propertyholds.Hence,propertyholds for anyprovided that ν∗is chosen so thatThis establishes assertion(ii)whereas assertion(i)follows from the fact that Ttrans≤ T(T,δ).In view of the previous discussion,we are now in position to fix the value of the constant ν∗as

Notice that a number of our other constants have been determined so far,namely,besides ν∗also S∗and c∗.We however still have left open the choice for α∗and ρ∗.

Step 2Defining the points bk(t),proof of assertion(iv).

In order to define the points{bk(t)}k∈J(T),we distinguish two cases.

Case 1Dk(T)

In this case,one,at least of the conditions WSk(t)or WSk−1(t)holds.We then set

and define the family{bk}k∈J(T)on[T,Ttrans]as the unique solution to the system of ordinary differential equations(1.16),with initial datum at time Ttransgiven by

with the coefficientstaken as=Ck±(Ttrans)for any t∈[T,Ttrans].Since the two definitions and the desired estimates are somewhat different,we handle the intervals[T,Ttrans]and[Ttrans,T]separately.

For the interval[Ttrans,T],in view of the choice(7.18)of the function bk,the statement of assertion(iv)on the time interval[Ttrans,T]is essentially a consequence of Proposition 7.1 and Lemma 7.1.Indeed,condition(7.7)is clearly satisfied for t≥Ttrans,since,in view of our constructions,we have for any k∈J(T),provided δ ≥ α∗ε,and that the constant α∗is chosen sufficiently large.On the other hand,we know,thanks to assertion(ii)that WPε(2ν∗da(T),)is satisfied.It follows that the functions bkare solutions to a system of the form(1.16)on the interval[Ttrans,T],and that estimate(1.19)holds for the whole interval[t,T],provided that we choose

Turning to(1.20)for the interval[Ttrans,T],we invoke Lemma 7.1 at timesince,as mentioned WPε(ν∗da(T),)holds,where ν∗is fixed in(7.17).It yields for s ∈ [+c2εda(T),T]that

where the last inequality holds,if we impose additionally

and provided that α∗is chosen sufficiently large.Since+c2εda(T)≤ Ttrans,provided δ ≥ α8ε,inequality(7.21)holds in particular for s ∈ [Ttrans,T].Hence inequality(7.21)combined with the choice(7.18)of the functions bkleads directly to(1.20)for the interval[Ttrans,T],and establishes assertion(iv)on the interval[Ttrans,T].

Turning to the interval[T,Ttrans],we notice that it follows directly from our definition(7.19)that the function bkis a solution to a differential equation of the desired form with the desired estimate(1.19)for the coefficients.It remains to establish(1.20)for the interval[T,Ttrans].For that purpose,we relay on several distinct observations.First,it follows from Corollary 7.1,inequality(7.12)that

so thatprovided that α∗is choose sufficiently large.

Next,in view of the equation(1.16)for{bk}k∈J(T),we may apply Lemma 2.2 and Remark 2.1 to assert that

so that

provided that α∗is chosen sufficiently large.Hence we deduce integrating inequality(2.3)between T and Ttrans,we are led to

Combining(7.23)–(7.24)and(7.21)for t=Ttrans,we derive(1.20)on the interval[T,Ttrans],if we impose,besides(7.20)and(7.22)the conditions ρ∗≤ ρ7and ρ∗≤

Case 2Dk(T)≥ L(δ)

In this case,define the family{bk}k∈J(T)on[T,T]as the unique solution to the system of ordinary differential equations(1.16),with initial datum at time Ttransgiven

and with the coefficientstaken asfor any t∈[T,T].

One verifies with the same argument as above that with this choice of function bkinequalities(1.19)are automatically satisfied.For inequality(1.20),we apply the last statement in Corollary 7.1 and inequality(4.29)in Proposition 4.4:Since properties WPε(δ,T)and WPε(ν∗da(T))hold,

Using the same arguments as for(7.24)but now on the whole interval[t,T],we obtain

Combining(7.26),(7.27)and the definition(7.25),we derive the desired conclusion(1.20)in the case considered.

Step 3Proof of assertion(iii).

The stopping time T is defined in Step 1,where we distinguish two cases.We provide here a proof in the second case,the proof in the first case being readily the same(and even possibly a little simpler).In view of(7.23),we first observe that,we have,for any k∈J(T),and if δ ≥ α∗,

On the other hand,in view of the definition(7.15)of T,there exists some k0∈J(T),such that

Combining(7.28)with(7.29),we are hence led to

provide that the constant α∗is chosen sufficiently large.Combining this inequality with(1.20),we deduce that

provided once more that the constant α∗is chosen sufficiently large.Since{bk}k∈J(T)is a solution to the differential equation(1.16),we may invoke Proposition 2.1 to assert that

provided again that the constant α∗is chosen sufficiently large.Invoking once more(1.20),wefinally deduce

which yields the desired result,provided that ρ∗is chosen sufficiently small.

8 Collisions

The proof of Theorem 1.2 relies the results in Theorem 1.1,combined with various properties of solutions to the differential equation(1.16).We present next the other main observations which lead to the proof of Theorem 1.2 as separate subsections.

8.1 Comparing with the differential equation(1.16)

We first notice that,under the assumptions of Theorem 1.1,a rather direct consequence of inequality(1.20)is that,for any T≤t≤T(T)(resp.Ttrans≤T≤T(T)),

where the subscript b refers to the solution{bk}k∈J(T)to(1.16)described in Theorem 1.1.Hence,we may use the properties of the differential equation(1.16)presented in Section 2 to derive related results for the partial differential equation.For instance,applying Proposition 2.3 to{bk}k∈J(T),we are led,for t∈ [T,T)]and δ ≥ α∗ε,to the estimates

where Λ0>0 and Λ1>0 are two constant depending only on the constants in Proposition 2.3.As an immediate consequence of(8.2),we obtain for t∈[T,T)],

where γ>0 is some constant.We next present a few observations and constructions which enter into the proof.

8.2 The stopping time T1

We introduce a new stopping time T1defined by

so that T1(T)

with the convention T0(T)=T.Upper bounds for T1,expressed in terms of da(T)will be needed for our proofs.A first one is derived from(8.2),which yields

Lemma 8.1Assume that property WPε(δ,T)holds,and that δ ≥ β2ε,where β2≥ β1is some constant.Assume thatThen we have

whereIf furthermore,

then,we have the identities

ProofBy the assumption,the inequalityholds,so that we have,on one hand,whereas on the other hand,we have by the definition of T1,the inequalityGoing back to(8.2),we check that,ifthenand the first assertion follows.We leave the second assertion to the reader.

Our previous discussion leads us to distinguish whether condition(8.8)is met or not.

8.3 The iterative construction(8.5)when condition(8.8)holds

When condition(8.8)holds,we introduce the integer nfdefined as

where the time Tjis defined by Tj=Tj(T)for j=0,···,nf.We denote bythe solution to the ordinary differential equation(1.16)defined on Ij=[Tj,Tj+1]obtained invoking Theorem 1.1.We describe first some elementary properties.

Lemma 8.2Assume that(8.8)and WPε(δ,T)hold with δ ≥ β2ε.For k=0,···,nf,we have

and

For any j=1,···,nfand every s ∈ [Tj,Tj+1],propertyholds,and we have

ProofThe identities(8.9)and(8.10)follow directly from Lemma 8.1,whereas the last assertions are a direct consequence of Theorem 1.1 assertion(ii).

Remark 8.1As rather direct consequence of inequality(8.10)we deduce that for some constant Λ3>0,and provided that β2is chosen sufficiently large,

The main result in this subsection are summarized in the following proposition.

Proposition 8.1Assume that(8.8)and WPε(δ,T)hold with δ ≥ β2ε.Then,we have the inequalityfor every s ∈ [T,Tnf]and

Proofwe introduce the stopping time s1defined by

if the set on the right-hand side is not empty,and s1=Tnfotherwise.Let j1∈N be such thatSincewe deduce,combining with inequality(8.3)applied for the time Tj1,thatIt then follows from the definition of nfthat

Hence,if we choose the constant β2sufficiently large,then we haveIt follows,invoking inequality(8.12)that

In view of inequality(2.7)of Proposition 2.3,we have

where the last inequality is a consequence of(8.11),provided that we choose the constant β2sufficiently large.We then deduce that for s∈[s0,s1],we have

Integrating this inequality and using again(8.11),we are led to

where Λ4>0 is some constant.If β2is chosen sufficiently large,then we deduce from(8.14)thatHence,s1=Tnfwhich establishes the first assertion in Proposition 8.1.Inequality(8.13)follows going back to(8.14).

8.4 The iterative construction(8.5)when condition(8.8)fails

We begin this subsection with a preliminary results,which is somewhat a counterpart to Lemma 8.1.

Lemma 8.3Assume that property WPε(δ,T)holds with δ ≥ β2ε,and assume that(8.8)does not hold. Then we have

ProofIn view of our assumption and since 0< ρ∗≤ 1,we haveIt follows on the other hand from(8.3)thatIn view of the definition of T1,we haveso that eitherorThe first equality is excluded:Indeed,if it were true,then we would have,combining with our previous inequality,

a contradiction with the fact that(8.8)does not hold,if we choose the constant β2sufficiently large.Hence we conclude thatwhich,combined once more with(8.3),leads to the conclusion.

When condition(8.8)fails,a new stopping time is introduced,related to the integer mfdefined by

Proposition 8.2Assume that property WPε(δ,T)holds with δ ≥ β2ε,and assume that(8.8)does not hold.We have for j=1,···,nf,

For every s ∈ [Tj,Tj+1],propertyholds andMoreover,

ProofThe proof is very similar to the proof of Proposition 8.1 and relies both on Theorem 1.1 and Lemma 8.3.We therefore omit the details.

8.5 Proof of Theorem 1.2 completed

We choose throughout β∗= β2.We distinguish two cases.

Case AInequality(8.8)does not hold.

In this case,we are in position to apply the results of Proposition 8.2 in Subsection 8.4.As a matter of fact,we choose

where the value of mfis provided by(8.16).For this choice,imposing C∗≥ λ3and c∗≥ λ,all the statements provided in Theorem 1.2 are provided by the results of Proposition 8.2.

Case BInequality(8.8)holds.

Here,we are in position to apply first the results of Subsection 8.3.We introduce the time T0=Tnf,where Tnfis provided by Proposition 8.1.Hence,we obtain the boundsand

It follows that inequality(8.8)holds for the time T0,and we may therefore then argue as in thefirst step,setting

It follows from inequality(8.18)of Proposition 8.2 that

where the last inequality follows from the boundMoreover,we haveThe conclusion then follows combining(8.19)–(8.20)and an appropriate choice of the constants.

9 Annihilations

The purpose of this section is to provide the proof of Theorem 1.3.

9.1 Dissipation of energy

Proposition 9.1Let 0 ≤ T1

providedwhere the constant μ1is introduced in Lemma 4.5.

ProofAssume by contradiction that inequality(9.1)does not hold.In view of Lemma 4.5,we then necessarily have E(T1)=E(T2),and we are in position to apply Lemma 4.9,which yields in particular,

In view of our assumption on T2−T1,we deduce thatdefined in Theorem 1.2 belongs to[T1,T2].From the very definition ofwe obtainwhich contradicts(9.2)and hence completes the proof.

We notice that,under the assumptions of Theorem 1.2 that E(s)=E(T)for any s∈[T,T−col].We next define the timefor which a dissipation has undergone.

Proposition 9.2Let T ≥0 and δ>0 be given,suppose that the assumptions of Theorem 1.2 are full-filled and that moreover δ ≥ β4ε,where β4≥ β3is some constant depending only on V and M0.Then there exists some timesuch thatholds,and moreover(1.24)holds,for some constant Υ depending only on V and M0.

ProofWe impose first thatTurning to Theorem 1.2,by continuity of the function δ−(·),we may assert that there exists some timesuch that

and,in view of assertion(iii),that property WPε(β3ε,T1)holds.On the other hand,assuming that the constant β4is given,it follows from Lemma 4.6 and relation(4.9)that there exists some timesuch that condition WPε(β4ε,T2)holds.Moreover,by definition,we have

We choose next the constant β4sufficiently large,such that it satisfies the additional condition

so thatIt follows hence from Proposition 9.1 thatSetting=T2the conclusion follows with Υ =2ω(β4).

9.2 The fate of fronts betweenand

Proposition 9.3Let T ≥0 and δ>0 be given,suppose that the assumptions of Theorem 1.2 are full-filled and that moreover δ ≥ β4ε.There exists a constant κc>0,such that

If for someis well separated from the other fronts in the sense of(1.22),then we have

ProofRecall that in view of Theorem 1.3,we haveso that,in view of Theorem 1.6,we have,for some constant κ1>0,such that,for any

On the other hand,by(1.27),we havewhich yields(9.4),choosing κc≥ 2κw+2κ1,the inclusion(9.4).

Assume next that(1.22)holds for somewith a constant κf>0 yet to be determined.We set

andIf we impose the condition κf>10κc,then the second assertion of Proposition 9.3 essentially reduces to prove that these sets are singletons.Imposing also that κf>4κ1,we see in view of(9.5)thaton the intervalwhich is not empty in view of our constraint on κf,and similarily thaton the intervalHence,needs to connect between the pointsandthe values σj(k)−to σj(k)+,and hence we deduce that

To complete the proof,it remains finally to show that Ok0()reduces to a single point.For that purpose,we invoke once more the localized energy inequality(1.31),with a test function χ such that 0≤χ≤1 and

Writing(1.31)for this choice of test function,we are led to

On the other hand,in view of the energy quantization property expressed in Lemma 4.4 which can easily be localized,we deduce that

Since vε(·,)needs to connect between the points ak0()− κ1ε and ak0()+κmfε the values σj(k)−to σj(k)+,there exits some m0∈ Jk0(),such that Si(m)=Si(k0),and the previous inequality becomes

The right-hand side of this inequality can be made arbitrarily small,choosing κfand possibly also β4sufficiently large,whereas the right-hand side is either zero or bounded below by a positive constant.Hence for a suitable choice of the constants,we are led toyielding hence the desired conclusion.

9.3 Proof of Theorem 1.3 completed

First notice that the timehas been defined in Proposition 9.2,and the estimate(1.24)as well as the conditionhave been established there.It remains hence to prove assertions(i)–(iii).Assertion(i)rephrases the inclusion(9.4)of Proposition 9.3.Assertion(ii)follows from the second assertion in Proposition 9.3,as well,for the part concerning repulsive points,inclusion(1.23)and the discussion thereafter.

The proof of Assertion(iii)requires some additional discussion.Arguing as for(9.7),we deduce that

so that,in view of the inequalitywe are led to the inequality

We next invoke several observations.The first one is that the setcan be decomposed aswhere all of the setsare maximal attractive chains.We notice that a maximal attractive chain involves only one heteroclinic orbit denoted here ξm,so that

In view of the continuity of the front sets properties described in Subsection 9.2,each maximal attractive chaingives rise to a corresponding maximal attractive chainforso thatthe number of elements inis odd if the number of elements inis odd,even but possibly zero if the number of elements inis even.Moreover,invoking the localized energy identity(1.31)with appropriate test functions as for the proof of(7.16),we obtain

On the other hand,inequality(9.8)is turned into

Hence,combining the two last inequalities,we deduce that there exists some m0,such thatThe conclusion follows,taking into account that the numbers involved are positive integers with the same parity.

10 Relaxing the Assumptions on the Potential

In this section,we outline the main points of the arguments of the proofs of Proposition 1.1 and Theorem 1.5.An important step is to check that the result stated in[3,Lemma 1]remains valid under assumptions on the potential we consider here.More precisely,we have the following lemma.

Lemma 10.1Assume that the potential V satisfies assumptions(H1)–(H3).Let u be such that Eε(u)<+∞.There exist constants η0>0 and N>0 depending only on?V??C2(R),ν and λmin,such that,if,for a∈R,we have

then there exists some σi∈ Σ,such that

The proof is parallel to the proof of Lemma 1 in[3]and is left to the reader.We notice also that(1.25)remains also valid with?≤?0,where the constant?0depends only on?V??C2(R),ν and λmin.

10.1 Sketch of the proof of Proposition 1.1

Since?[a,a+1]eε(u(x))dx→0 as a→ ±∞,so that,in view of(10.1),we deduce that there exists some σ+∈Σ(resp.σ−∈Σ),such that|u(x)−σ+|→0 as x→+∞(resp.as x→−∞).In view of(1.25),we have

On the other hand,by embedding,we have

so that|u(R)|≤?0M0,and(1.28)follows.Finally,we leave the last assertion to the reader.

10.2 Sketch of the proof of Theorem 1.5

In view of Proposition 1.1 and relation(1.29),we know that vεtakes values in some interval of the form[σ+−A,σ++A],with A depending only on?V??C2(R),ν and λmin.We may then construct a potential,such that=V on the set[σ+−2A,σ++2A]which full-fills conditions(H1)–(H3).It follows that the function vεis also a solution to(PGL)εfor the potential,and we are hence in position to apply the results in Theorems 1.1–1.4,which leads to the desired conclusion.One may also check that the extensionmight be constructed in such a way that all constants involved in the theorems depend only on?V??C2(R),νand λmin.

[1]Alikakos,N.D.,Bronsard,L.and Fusco,G.,Slow motion in the gradient theory of phase transitions via energy and spectrum,Calc.Var.Partial Differential Equations,6,1998,39–66.

[2]Angenent,S.,The zero set of a solution of a parabolic equation,J.Reine Angew.Math.,390,1988,79–96.

[3]Bethuel,F.,Orlandi,G.and Smets,D.,Slow motion for gradient systems with equal depth multiple-well potentials,J.Di ff.Equations,250,2011,53–94.

[4]Bethuel,F.and Smets,D.,Slow motion for equal depth multiple-well gradient systems:The degenerate case,Discrete Contin.Dyn.Syst.,33(1),2013,67–87.

[5]Bronsard,L.and Kohn,R.V.,On the slowness of phase boundary motion in one space dimension,Comm.Pure Appl.Math.,43,1990,983–997.

[6]Carr,J.and Pego,R.L.,Metastable patterns in solutions of ut=?2uxx−f(u),Comm.Pure Appl.Math.,42,1989,523–576.

[7]Carr,J.and Pego,R.L.,Invariant manifolds for metastable patterns in ut=?2uxx−f(u),Proc.Roy.Soc.Edinburgh Sect.A,116,1990,133–160.

[8]Chen,X.,Generation,propagation,and annihilation of metastable patterns,J.Differential Equations,206(2),2004,399–437.

[9]Fife,P.and McLeod,J.B.,The approach of solutions of nonlinear diffusion equations to travelling front solutions,Arch.Ration.Mech.Anal.,65(4),1977,335–361.

[10]Fusco,G.and Hale,J.K.,Slow-motion manifolds,dormant instability,and singular perturbations,J.Dynam.Differential Equations,1,1989,75–94.

[11]Grant,C.P.,Slow motion in one-dimensional Cahn-Morral systems,SIAM J.Math.Anal.,26,1995,21–34.

[12]Otto,F.and Rezniko ff,M.G.,Slow motion of gradient flows,J.Differential Equations,237,2007,372–420.