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The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special case. Given rather general initial data, we show that the first component of the solution vector develops a steep transition layer and that all the results mentioned above remain true for this component.


Perturbed Allen-Cahn equation
In some classes of nonlinear diffusion equations, solutions often develop internal transition layers -or "interfaces"-that separate the spatial domain into different phase regions. This happens, in particular, when the diffusion coefficient is very small or the reaction term is very large. The motion of such interfaces is often driven by their curvature. A typical example is the Allen-Cahn equation u t = ∆u + ε −2 f (u), where ε > 0 is a small parameter and f (u) is a bistable nonlinearity, whose meaning will be explained below. A usual strategy for studying such phenomena is to first derive the "sharp interface limit" as ε → 0 by a formal analysis, then to check if this limit gives good approximation of the behavior of actual layers.
In this paper we study a perturbed Allen-Cahn type equation of the form in Ω, and study the behavior of layers near the sharp interface limit as ε → 0.
Here Ω is a smooth bounded domain in R N (N ≥ 2) and ν is the Euclidian unit normal vector exterior to ∂Ω. The nonlinearity is given by f (u) := −W ′ (u), where W (u) is a double-well potential with equal well-depth, taking its global minimum value at u = α ± . More precisely we assume that f is C 2 and has exactly three zeros α − < a < α + such that f ′ (α ± ) < 0, f ′ (a) > 0 (bistable nonlinearity), (1.1) and that The condition (1.1) implies that the potential W (u) attains its local minima at u = α − , α + , and (1.2) implies that W (α − ) = W (α + ). In other words, the two stable zeros of f , namely α − and α + , have "balanced" stability. A typical example is the cubic nonlinearity f (u) = u(1 − u 2 ). The term εg ε represents a small perturbation, where g ε (x, t, u) is a function defined on Ω × [0, +∞) × R. This has the role of breaking the balance of the two stable zeros slightly. In the special case where g ε ≡ 0, problem (P ε ) reduces to the usual Allen-Cahn equation. As we will explain later, our main results are new even for this special case.
For technical reasons we also assume that which, in turn, implies the same boundary condition for g. Apart from these bounds and regularity requirements, we do not make any specific assumptions on the perturbation term g ε . Remark 1.2. The reason why we do not assume more smoothness on g is that we will later apply our results to systems of equations including FitzHugh-Nagumo system, in which g ε loses C 2,1 -smoothness as ε → 0.
It is standard that problem (P ε ) has a unique smooth solution, which we denote by u ε . As ε → 0, a formal asymptotic analysis shows the following: in the very early stage, the diffusion term ∆u is negligible compared with the reaction term ε −2 (f (u) − εg ε (x, t, u)) so that, in the rescaled time scale τ = t/ε 2 , the equation is well approximated by the ordinary differential equation u τ = f (u) + O(ε). Hence, in view of the profile of f , the value of u ε quickly becomes close to either α + or α − in most part of Ω, creating a steep interface (transition layer) between the regions {u ε ≈ α − } and {u ε ≈ α + }. Once such an interface develops, the diffusion term becomes large near the interface, and comes to balance with the reaction term. As a result, the interface ceases rapid development and starts to propagate in a much slower time scale.
To study such interfacial behavior, it is useful to consider a formal asymptotic limit of (P ε ) as ε → 0. Then the limit solutionũ(x, t) will be a step function taking the value α + on one side of the interface, and α − on the other side. This sharp interface, which we will denote by Γ t , obeys a certain law of motion, which is expressed as follows (see Section 2 for details): where V n is the normal velocity of Γ t in the exterior direction, κ the mean curvature at each point of Γ t , It is well known that problem (P 0 ) possesses locally in time a unique smooth solution. Let 0 ≤ t < T max , T max ∈ (0, +∞], be the maximal time interval for the existence of the solution of (P 0 ) and denote this solution by Γ = which represents the formal asymptotic limit of u ε (or the sharp interface limit) as ε → 0. The aim of the present paper is to make a detailed study of the limiting behavior of the solution u ε of problem (P ε ) as ε → 0. Our first main result, Theorem 1.3, describes the profile of the solution after a very short initial period. It asserts that: given a virtually arbitrary initial data u 0 , the solution u ε quickly becomes close to α ± , except in a small neighborhood of the initial interface Γ 0 , creating a steep transition layer around Γ 0 (generation of interface). The time needed to develop such a transition layer, which we will denote by t ε , is of order ε 2 | ln ε|. The theorem then states that the solution u ε remains close to the step functionũ on the time interval [t ε , T ] (motion of interface); in other words, the motion of the transition layer is well approximated by the limit interface equation (P 0 ). Then there exist positive constants ε 0 and C such that, for all ε ∈ (0, ε 0 ) and for all t ε ≤ t ≤ T , where t ε := µ −1 ε 2 | ln ε|, we have (1.14) where N r (Γ t ) := {x ∈ Ω, dist(x, Γ t ) < r} denotes the r-neighborhood of Γ t .
The next theorem is concerned with the relation between the actual interface Γ ε t := {x ∈ Ω, u ε (x, t) = a} and the formal asymptotic limit Γ t , which is given as the solution of (P 0 ). Theorem 1.5 (Error estimate). There exists C > 0 such that Corollary 1.6 (Convergence of interface). There exists C > 0 such that where d H (A, B) := max{sup a∈A d(a, B), sup b∈B d(b, A)} denotes the Hausdorff distance between two compact sets A and B. Consequently, Γ ε t → Γ t as ε → 0 uniformly in 0 ≤ t ≤ T , in the sense of Hausdorff distance.
Note that the estimates (1.15) and (1.16) follow from Theorem 1.3 in the range t ε ≤ t ≤ T , but the range 0 ≤ t ≤ t ε has to be treated by a separate argument since the behavior of the solution in this time range is quite different from that of the later stage.
The estimate (1.14) in our Theorem 1.3 implies that, once a transition layer is formed, its thickness remains within order ε for the rest of time.
Here, by "thickness of interface" we mean the smallest r > 0 satisfying . Naturally this quantity depends on η, but the estimates (1.14) and (1.16) assert that it is bounded by 2Cε (with the constant C depending on η) regardless of the choice of η > 0.
Remark 1.7 (Optimality of the thickness estimate). The above O(ε) estimate is optimal, i.e. the interface cannot be thinner than this order. In fact, rescaling time and space as τ := t/ε 2 , y := x/ε, we get Thus, by the uniform boundedness of u and by standard parabolic estimates, we have |∇ y u| ≤ M for some constant M > 0, which implies From this bound it is clear that the thickness of interface cannot be smaller than M −1 (α + − α − ) ε, hence, by (1.14), it has to be exactly of order ε. Intuitively, this O(ε) estimate follows also from the formal asymptotic expansion (2.3), but the validity of such an expansion is far from obvious for solutions with arbitrary initial data.
Our O(ε) estimate is new, even in the special case where g ε ≡ 0, provided that N ≥ 2. Previously, the best thickness estimate in the literature was of order ε| ln ε| (see [7]), except that X. Chen has recently obtained an O(ε) estimate for the case N = 1 by a different argument (private communication). We also refer to the forthcoming papers [20] and [19], in which the same O(ε) estimate is established for different but related problems. The paper [20] is concerned with a "balanced type" Allen-Cahn equation with large spatial inhomogeneity, namely an equation of the form [19] is concerned with a Lotka-Volterra competition-diffusion system with large spatial inhomogeneity whose nonlinearity is of the balanced bistable type. Remark 1.8 (Optimality of the generation time). The estimate (1.14) also implies that the generation of interface takes place within the time span of t ε . This estimate is optimal. In other words, a well-developed interface cannot appear much earlier; see Proposition 3.10 for details.
The singular limit of the Allen-Cahn equation was first studied in the pioneering work of Allen and Cahn [2] and, slightly later, in Kawasaki and Ohta [23] from the point of view of physicists. They derived the interface equation by formal asymptotic analysis, thereby revealing that the interface moves by the mean curvature. These early observations triggered a flow of mathematical studies aiming at rigorous justification of the above limiting procedure; see, for example, [6], [7,8] and [24,25] for results in the framework of classical solutions, and [15], [4,5] and [21] for the case where Γ t is a viscosity solution of the interface equation.
As for problem (P ε ), whose nonlinearity is slightly unbalanced, the limit interface equation involves a pressure term as well as the curvature term as indicated in (P 0 ). This fact has been long known on a formal level; see e.g. [27]. Ei, Iida and Yanagida [14] proved rigorously that the motion of the layers of (P ε ) is well approximated by the limit interface equation (P 0 ), on the condition that the initial data has already a well developed transition layer. In other words, they studied the motion of interface, but not the generation of interface.

Singular limit of reaction-diffusion systems
Our results can be extended to reaction-diffusion systems of the form where D is a positive constant, and f ε , h are C 2 functions such that where f (u) is a bistable nonlinearity satisfying (1.1), (1.2), and f ε 2 , along with its derivatives in u, v, remain bounded as ε → 0; The conditions (F) and (H) imply that the ODE systeṁ has a family of invariant rectangles of the form {|u| ≤ L, |v| ≤ M }, provided that ε is sufficiently small. The maximum principle and standard parabolic estimates then guarantee that every solution of (RD ε ) exists globally for t ≥ 0 and remains bounded as t → ∞ (see Proposition 7.1). Apart from (1.18), we do not make any specific assumptions on the function h. Problem (RD ε ) represents a large class of important reaction-diffusion systems including the FitzHugh-Nagumo system   which is a simplified model for nervous transmission, and the following type of prey-predator system: (1.20) Remark 1.9. In some equations such as the prey-predator system (1.20), only nonnegative solutions are to be considered. In such a case, we replace the condition (1.18) by and assume f ε (0, v) ≥ 0. The rest of the argument remains the same.
Now the same formal analysis as is used to derive (P 0 ) shows that the singular limit of (RD ε ) as ε → 0 is the following moving boundary problem: whereũ is the step function defined in (1.13) and This is a system consisting of an equation of surface motion and a partial differential equation. Sinceũ is determined straightforwardly from Γ t , in what follows, by a solution of (RD 0 ) we mean the pair (Γ,ṽ) := (Γ t ,ṽ(x, t)).
In the case of the FitzHugh-Nagumo system (1.19), while in the prey-predator system (1.20), Note that the positive sign in front of the term c 0ṽ (x, t) in the interface equation implies an inhibitory effect onũ, since the velocity V n is measured in the exterior normal direction, toward whichũ decreases.
Lemma 1.10 (Local existence). Assume that v 0 ∈ C 2 (Ω) and that Γ 0 is a C 2 hypersurface which is the boundary of a domain D 0 ⊂⊂ Ω. Then there exists T max ∈ (0, +∞] such that the limit free boundary problem (RD 0 ) has a unique solution (Γ,ṽ) in the interval [0, T max ).
This existence result was established in [10]. The uniqueness can be obtained by using the estimates in [8].
Hereafter, we fix T such that 0 < T < T max and work on [0, T ]. Our main results for the system (RD ε ) are the following:   Then the same conclusion as in Theorem 1.5 holds for (RD ε ). Moreover, there exists a constant C > 0 such that v ε −ṽ L ∞ (Ω×(0,T )) ≤ Cε. Corollary 1.14 (Convergence of interface). Under the assumptions of Theorem 1.11, the same conclusion as in Corollary 1.6 holds for (RD ε ).
The organization of this paper is as follows. In Section 2, we derive the interface equation (P 0 ) from (P ε ) by formal asymptotic expansions which involve the so-called signed distance function.
In Sections 3 and 4, we present basic estimates concerning the generation of interface for (P ε ). For the clarity of underlying ideas, we first consider the special case where g ε ≡ 0 in Section 3, and deal with the general case in Section 4.
In Section 5 we prove a preliminary result on the motion of interface (Lemma 5.1), which implies that if the initial data has already a welldeveloped transition layer, then the layer remains to exist for 0 ≤ t ≤ T and its motion is well approximated by the interface equation (P 0 ).
Our approach in Sections 3 to 5 is based on the sub-and super-solution method, but we use two completely different sets of sub-and super-solutions. More precisely, the sub-and super-solutions for the motion of interface are constructed by using the first two terms of the formal asymptotic expansion (2.3), while those for the generation of interface are constructed by modifying the solution of the equation in the absence of diffusion: u t = ε −2 f (u).
In Section 6, we prove our main results for (P ε ): Theorems 1.3, 1.5 and their respective corollaries.
In the final section, we study the reaction-diffusion system (RD ε ) and prove Theorems 1.11, 1.13 and their corollaries. These results are obtained by applying a slightly modified version of the results for (P ε ). The strategy is to regard f ε (u, v) as a perturbation of f (u). Indeed, the equation for u However, what makes the analysis difficult is the fact that g ε is no longer a given function but a quantity that depends on the unknown function v ε . In particular, the existence of the limit g ε → g (ε → 0) is not a priori guaranteed, and the estimate (1.6) is far from obvious. As it turns out, the standard L p or Schauder estimates for v ε would not yield (1.6), because of the fact that u ε converges to a discontinuous function as ε → 0. In order to overcome this difficulty, we derive a fine estimate of v ε that is based on estimates of the heat kernel and the fact that u ε remains uniformly smooth outside of an O(ε) neighborhood of the smooth hypersurface Γ t .

Formal derivation of the interface motion equation
In this section we derive the equation of interface motion corresponding to problem (P ε ) by using a formal asymptotic expansion. The resulting interface equation can be regarded as the singular limit of (P ε ) as ε → 0. Our argument is basically along the same lines with the formal derivation given by Nakamura, Matano, Hilhorst and Schätzle [26], who studied a similar but slightly different type of spatially inhomogeneous equations by formal analysis. Let us also mention some earlier papers [1], [17] and [27] involving the method of matched asymptotic expansions for problems that are related to ours. As in [26], the first two terms of the asymptotic expansion determine the interface equation. Though our analysis in this section is for the most part formal, the observations we make here will help the rigorous analysis in later sections.
Let u ε be the solution of (P ε ). We recall that Γ ε t := {x ∈ Ω, u ε (x, t) = a} is the interface at time t and call Γ ε := t≥0 (Γ ε t × {t}) the interface. Let Γ = 0≤t≤T (Γ t × {t}) be the unique solution of the limit geometric motion problem (P 0 ) and let d be the signed distance function to Γ defined by: where dist(x, Γ t ) is the distance from x to the hypersurface Γ t in Ω. We remark that d = 0 on Γ and that |∇ d| = 1 in a neighborhood of Γ. We then define We also assume that the solution u ε is of the form away from the interface Γ (the outer expansion), and near Γ (the inner expansion), where U j (x, t, z), j = 0, 1, · · · , are defined for x ∈ Ω, t ≥ 0, z ∈ R and ξ := d(x, t)/ε. The stretched space variable ξ gives exactly the right spatial scaling to describe the rapid transition between the regions {u ε ≈ α − } and {u ε ≈ α + }. We normalize U 0 in such a way that U 0 (x, t, 0) = a (normalization conditions). To make the inner and outer expansions consistent, we require that As we will see below this will determine U 0 uniquely, which will then determine U 1 .
In what follows we will substitute the inner expansion (2.3) into the parabolic equation of problem (P ε ) and collect the ε −2 and ε −1 terms. To that purpose we compute the needed terms and get where the functions U i (i = 0, 1), as well as their derivatives, are taken at the point (x, t, d(x, t)/ε). Note also that ∇ and ∆ stand for ∇ x and ∆ x , respectively. Collecting the ε −2 terms yields In view of the normalization and matching conditions, we can now assert that U 0 (x, t, z) = U 0 (z), where U 0 (z) is the unique solution of the stationary problem This solution represents the first approximation of the profile of a transition layer around the interface observed in the stretched coordinates. Note that the integral condition (1.2) guarantees the existence of a solution of (2.5). For example, in the simple case where f (u) = u(1 − u 2 ), we have U 0 (z) = tanh(z/ √ 2). In the general case, the following standard estimates hold: Lemma 2.1. There exist positive constants C and λ such that In addition, U 0 is a strictly increasing function and, for j = 1, 2, Proof. We only give an outline. Rewriting the equation in (2.5) aṡ we see that (U 0 (z), U ′ 0 (z)) is a heteroclinic orbit of the above system connecting the equilibria (α − , 0) and (α + , 0). These equilibria are saddle points, with the linearized eigenvalues {λ − , −λ − } and {λ + , −λ + }, respectively, where Consequently, we have for some constants C 1 , C 2 . The desired estimates now follow by setting λ = min(λ + , λ − ).
Next we collect the ε −1 terms. Recalling that ∇U 0z = 0 and that |∇d| = 1 near Γ t , we get This equation can be seen as a linearized problem for (2.5) with an inhomogeneous term. As is well-known (see, for instance, [26]), the solvability condition for the above equation plays the key role in determining the equation of interface motion. The following lemma is rather standard, but we give an outline of the proof for the convenience of the reader.
Lemma 2.2 (Solvability condition). Let A(z) be a bounded function on −∞ < z < ∞. Then the problem has a solution if and only if Moreover the solution, if it exists, is unique and satisfies for some constant C > 0.
Proof. Multiplying the equation by U 0 ′ and integrating it by parts, we easily see that the condition (2.10) is necessary. Conversely, suppose that this condition is satisfied. Then, since U 0 ′ is a bounded positive solution to the homogeneous equation ψ zz + f ′ (U 0 (z))ψ = 0, one can use the method of variation of constants to find the above solution ψ explicitly. More precisely, where ϕ := U 0 ′ . The estimate (2.11) now follows from the above expression and (2.7).
From the above lemma, the solvability condition for (2.8) is given by which gives Moreover, multiplying equation (2.5) by U 0 ′ and integrating it from −∞ to z, we obtain where we have also used the fact that It then follows, in view of the definition of c 0 in (1.12), that We are now ready to derive the equation of interface motion. Since ∇ d (= ∇ x d(x, t)) coincides with the outward normal unit vector to the hypersurface Γ t , we have d t (x, t) = −V n , where V n is the normal velocity of the interface Γ t . It is also known that the mean curvature κ of the interface is equal to ∆ d/(N − 1). Thus the equation of interface motion is given by: Summarizing, under the assumption that the solution u ε of problem (P ε ) satisfies we have formally proved that the boundary Γ t between Ω − t and Ω + t moves according to the law (2.14).
To conclude this section, we give basic estimates for U 1 (x, t, z), which we will need in Section 5 to study the motion of interface. Substituting (2.13) into (2.8) gives where the variables x, t are considered parameters. The problem (2.15) has a unique solution by virtue of Lemma 2.2. Moreover, since A 0 (x, t, z) remains bounded as ( for some constant M > 0. Similarly, since ∇U 1 is a solution of (2.9) with and since g is assumed to be C 1 in x, we obtain for some constant M > 0.
To obtain estimates as z → ±∞, we first observe that (2.7) implies We then apply the following general estimates: Let the assumptions of Lemma 2.2 hold, and assume further that A(z) − A ± = O(e −δ|z| ) as z → ±∞ for some constants A + , A − and δ > 0. Then there exists a constant λ > 0 such that as z → ±∞.
Proof. We only state the outline. To derive the former estimate, we need a slightly more elaborate version of (2.7).
and (2.22), we see that This implies the former estimate in (2.21), where λ can be any constant satisfying 0 < λ < min(λ − , λ + , δ). Substituting this into equation (2.9) gives the estimate for ψ zz . Finally, the estimate for ψ z follows by integrating ψ zz from ±∞ to z.
From the above lemma and (2.20) we obtain the estimate As a consequence, there is a constant, which we denote again by M , such that Next we consider the boundary condition. Note that (1.7) implies Consequently, from the expression (2.12), or equivalently the expression 3 Generation of interface: the case g ε ≡ 0 This section deals with the generation of interface, namely the rapid formation of internal layers that takes place in a neighborhood of Γ 0 = {x ∈ Ω, u 0 (x) = a} within the time span of order ε 2 | ln ε|. For the time being we focus on the special case where g ε ≡ 0. We will discuss the general case in Section 4. In the sequel, η 0 will stand for the following quantity: Our main result in this section is the following: Theorem 3.1. Let η ∈ (0, η 0 ) be arbitrary and define µ as the derivative of f (u) at the unstable zero u = a, that is Then there exist positive constants ε 0 and M 0 such that, for all ε ∈ (0, ε 0 ), The above theorem will be proved by constructing a suitable pair of suband super-solutions. Note that we do not need condition (1.2) in proving this theorem.

The bistable ordinary differential equation
Let us first consider the problem without diffusion: This solution is written in the form where Y (τ, ξ) denotes the solution of the ordinary differential equation Here ξ ranges over the interval (−2C 0 , 2C 0 ), with C 0 being the constant defined in (1.8). We first study basic properties of Y .
which is integrated as follows: We then differentiate equation (3.5) by τ and obtain This last equality, in view of (3.5), completes the proof of Lemma 3.2.
For ξ / ∈ {α − , a, α + }, we define a function A(τ, ξ) by Proof. Differentiating by ξ the equality of Lemma 3.2 leads to These two last results complete the proof of Lemma 3.3.
Next we need some estimates on Y and its derivatives. First, we estimate the speed of the evolution of Y when the initial value ξ lies between α − + η and α + − η.
Corollary 3.5. Let η ∈ (0, η 0 ) be arbitrary. Then there exist positive con- We use this inequality for a < Y (τ, ξ) < α + − η to obtain We also use this inequality for a < ξ < α + − η to obtain Next we use the equality which, in view of (3.9), implies that This proves (3.14). The proof of (3.15) is similar and is omitted.
We now present estimates in the case where the initial value ξ is smaller than α − + η or larger than α + − η.
Lemma 3.6. Let η ∈ (0, η 0 ) and M > 0 be arbitrary. Then there exists a positive constant Proof. Since statement (i) and statement (ii) can be treated in the same way, we will only prove the former. The fact that Y (τ, ξ) remains in the interval [α + − η, α + + M ] directly follows from the bistable properties of f , or, more precisely, from the sign conditions This means that, for any ξ ∈ [α + , α + + M ], we have In view of this, and considering that f ′ (Y ) < 0 for Y ∈ [α + , α + + c], we see from the expression (3.6) that for all τ ≥ τ . It is clear from the same estimate (3.6) that Y ξ ≤C 4 holds also for 0 ≤ τ ≤ τ . We can then use Lemma 3.3 to deduce that The case ξ ∈ [α + − η, α + ] can be treated in the same way. This completes the proof of the lemma.

Construction of sub-and super-solutions
We are now ready to construct the sub-and super-solutions for the study of generation of interface. For simplicity, we first consider the case where In this case, our sub-and super-solutions are given by In the general case where (3.18) does not necessarily hold, we have to slightly modify w ± ε (x, t) near the boundary ∂Ω. This will be discussed later.

Proof of Theorem 3.1
In order to prove Theorem 3.1 we first present basic estimates of the function Y after a time of order τ ∼ | ln ε|.

Optimality of the generation time
To conclude this section we show that the generation time t ε := µ −1 ε 2 | ln ε| that appears in Theorem 3.1 is optimal. In other words, the interface will not be fully developed until t comes close to t ε .
Proof. For simplicity, we deal with the case where (3.18) is valid. In that case, (3.20) holds for all small ε > 0. For each b > 0, we put It follows from this and (3.14) that Now we choose b to be sufficiently large, so that Then the above estimate and (3.20) yield This implies that (1.14) does not hold at t = t ε (b), hence t ε (b) < t ε min . The lemma is proved.

Generation of interface in the general case
In this section we extend Theorem 3.1 to the case where g ε ≡ 0. The proof is more technical than the case g ε ≡ 0, but the underlying ideas are the same. Hence we will basically follow the argument of Section 3, simply pointing out the main differences.

The perturbed ordinary differential equation
We first consider a slightly perturbed nonlinearity: where δ is any constant. For |δ| small enough, this function is still bistable.
To prove Theorem 3.1, we will construct a pair of sub-and supersolutions for (P ε ) by simply replacing the function Y (τ, ξ) in (3.19) by Y (τ, ξ; δ), with an appropriate choice of δ. For this strategy to work, we have to check that the basic properties of Y (τ, ξ) in Subsection 3.1 carry over to Y (τ, ξ; δ).
First, it is clear that all the differential and integral identities in Subsection 3.1 that follow directly from (3.5) are still valid for (4.4). In particular, Lemmas 3.2 and 3.3 remain to hold if we replace Y (τ, ξ) by Y (τ, ξ; δ), f by f δ and A(τ, ξ) by A(τ, ξ; δ), where .
Next let us show that the basic estimates which we have established in Subsection 3.1 are also valid for Y (τ, ξ; δ). The following lemma, which is an analogue of Lemma 3.4, is fundamental.
Now, it is no trouble to establish an analogue of Lemmas 3.6 and 3.7 with constants independent of δ. We claim, without proof, that:

Construction of sub-and super-solutions
We now construct a pair of sub-and super-solutions by modifying the definition (3.19). We set where the function r(δ, τ ) is given by r(δ, τ ) = C 6 (e µ(δ)τ − 1), and the constant G is chosen such that, for all small ε > 0, which, in view of (1.5), is clearly possible.
Lemma 4.6. There exist positive constants ε 0 and C 6 such that for all ε ∈ (0, ε 0 ), (w − ε , w + ε ) is a pair of sub-and super-solutions for problem (P ε ), in the domain Ω × [0, µ −1 ε 2 | ln ε|], satisfying w − ε (x, 0) = w + (x, 0) = u 0 (x). Proof. First, the same cut-off argument as in Subsection 3.2 enables us to assume (3.18) for simplicity. Hence w ± ε satisfy the Neumann boundary conditions. We define an operator L by and prove below that Lw + ε ≥ 0 by slightly modifying the argument which we have used to prove L 0 w + ε ≥ 0 in Section 3. A straightforward calculation yields If ε 0 is sufficiently small, we note that ±εG ∈ (−δ 0 , δ 0 ) and that, in the range 0 ≤ t ≤ µ −1 ε 2 | ln ε|, . These observations allow us to use the results of the previous subsection with the choices τ := t/ε 2 , ξ := u 0 (x) + ε 2 r(εG, t/ε 2 ) and δ := εG. In particular, the ordinary differential equation (4.4) yields Y τ = f (Y ) + εG, which implies that By the choice of G the first term of the right-hand side member is positive. Using the estimate of A = Y ξξ /Y ξ in Lemma 4.5, we obtain, for a constant C 5 that is independent of ε, In view of (4.3), this inequality implies that, for ε ∈ (0, ε 0 ), with ε 0 small enough, and for C 6 large enough, This completes the proof of the lemma.
Hence, as in Section 3, the comparison principle can be applied to deduce

Proof of Theorem 3.1 for the general case
As in Subsection 3.3, we first present a key estimate of the function Y after a time interval of order τ ∼ | ln ε|. Roughly speaking, a perturbation δ of order ε does not affect the result of Lemma 3.9.
We are now ready to prove Theorem 3.1 in the general case. By setting t = µ −1 ε 2 | ln ε| in (4.11), we get It follows that Hence, as in Section 3, the result (3.2) of Theorem 3.1 is a direct consequence of (4.12) and (4.15). Next we prove (3.3). We take x ∈ Ω such that u 0 (x) ≥ a + M 0 ε; then if we choose M 0 large enough. Using (4.15) and (4.13) we obtain (3.3) which completes the proof of Theorem 3.1.

Motion of interface
In Sections 3 and 4, we have proved that the solution u ε develops a clear transition layer within a very short time. The aim of the present section is to show that, once such a clear transition layer is formed, it persists for the rest of time and that its law of motion is well approximated by the interface equation (P 0 ).
Let us formulate the above assertion more clearly. By taking the first two terms of the formal asymptotic expansion (2.3), we get a formal approximation of a solution up to order ε : Here U 0 , U 1 are as defined in (2.5) and (2.15). The right-hand side has a clear transition layer which lies exactly on Γ t . Our goal is to show that this function is a good approximation of a real solution; more precisely: If u ε becomes close toũ ε at some t = t 0 , then it stays close tõ u ε for the rest of time. Consequently, Γ ε t evolves roughly like Γ t . In order to prove this assertion, we will construct a pair of sub-and super-solutions u − ε and u + ε for problem (P ε ) by slightly modifying the above functionũ ε . It then follows that, if the solution u ε satisfies for t 0 ≤ t ≤ T , which implies that the solution u ε stays close toũ ε . The rest of this section is devoted to the construction of these sub-and super-solutions. We begin with some preparations.

A modified signed distance function
For our later analysis, it is convenient to introduce a "cut-off signed distance function" d, which is defined as follows. First, choose d 0 > 0 small enough so that the signed distance function d defined in (2.1) is smooth in the following tubular neighborhood of Γ: Next let ζ(s) be a smooth increasing function on R such that We then define the cut-off signed distance function d by Note that |∇d| = 1 in the region {(x, t) ∈ Q T , | d(x, t)| < d 0 } and that, in view of (5.2), ∇d = 0 in a neighborhood of ∂Ω. Note also that the equation of motion (P 0 ), which is equivalent to (2.13), is now written as where γ(x, t) is the function defined in (2.16).

Construction of sub-and super-solutions
As we stated earlier, we now construct sub-and super-solutions by modifying the functionũ ε in (5.1). Concerning the second term U 1 , which is defined in (2.15), the terms ∆U 1 and U 1t do not make sense as we only assume that g(·, ·, u) ∈ C 1+ϑ, 1+ϑ 2 . In order to cope with this lack of smoothness, we replace U 1 by a smooth function U ε 1 , which is defined by where the variables x, t, ε are considered parameters. Using (1.5) and the same arguments as in the end of Section 2, we obtain estimates analogous to (2.18) and (2.19), with a constant M independent of ε: Moreover, g ε being C 2 in x and C 1 in t, ∆ x U ε 1 and U ε 1t are solutions of (2.9) with A = ∆ x A ε 0 and A = A ε 0t , respectively. Thus, in view of (1.3), we obtain with some constant C independent of ε. Similarly, (1.5) and Lemma 2.3 yield estimates analogous to (2.23) and (2.24) for U ε 1 , with C and M independent of ε: In the rest of this section, C and M will stand for the constants that appear in inequalities (5.8)-(5.11). Note also that (1.7) implies the Neumann boundary conditions (2.26) for U ε 1 . We look for a pair of sub-and super-solutions u ± ε for (P ε ) of the form where p(t) = −e −βt/ε 2 + e Lt + K, The main result of this section is the following: Lemma 5.1. Choose β, σ > 0 appropriately. Then for any K > 1, there exist constants ε 0 , L > 0 such that, for any ε ∈ (0, ε 0 ), the functions (u − ε , u + ε ) are a pair of sub-and super-solutions for (P ε ) in the domain Ω × [0, T ].

Proof of lemma 5.1
By virtue of (2.26) and the fact that ∇d = 0 near ∂Ω, we have What we have to show is and that Lu − ε ≤ 0. We will prove only the former inequality for u + ε , since the latter follows by the same argument.

The term E 1
Direct computation gives In virtue of (5.18) and (5.20), we have Combining this, (5.19) and the inequality σ ≤ σ 2 , we obtain I ≥ 2σβ. Consequently, we have

The term E 2
First, in the region where |d| ≤ d 0 , we have |∇d| = 1, hence E 2 = 0. Next we consider the region where |d| ≥ d 0 . We deduce from Lemma 2.1 and from (5.10) that : We remark that 0 < K − 1 ≤ p ≤ e LT + K. Consequently, if we assume

The term E 3
By (5.4) and (2.16), we have Since γ is of class C 1+ϑ, 1+ϑ 2 by virtue of (1.5), we see that the interface Γ t is of class C 3+ϑ, 3+ϑ 2 . Therefore both ∆d and d t are Lipschitz continuous near Γ t . It follows that there exists a constant N > 0 such that: Applying Lemma 2.1 and the estimate (5.10) we deduce that Thus, recalling that |p| ≤ e Lt + K, we obtain where C 3 := 2N C and C 3 ′ := 2N C/λ.

The term E 4
In view of (1.4) and (5.10), both g u and |U ε 1z | are bounded by some constant C. Hence, substituting the expression for p t and q, we obtain where C 4 := C + σ(C + M F 2 ).

The term E 5
In view of (2.16), the term |γ| is bounded by Using (1.4) and (5.11), we easily obtain |E 5 | ≤ C 5 , where C 5 depends only on C, M , F 2 .

Completion of the proof
Collecting all these estimates gives where C 8 := C 2 + KC 3 + C 3 ′ + C 5 + C 6 + C 7 . Now we set which, for ε 0 small enough, validates assumptions (5.19) and (5.21). For ε 0 small enough, the first term of the right-hand side of (5.22) is positive, hence The proof of Lemma 5.1 is now complete, with the choice of the constants β, σ as in (5.16), (5.17).
6 Proof of the main results Here d 0 (x) :=d(x, 0) denotes the signed distance function associated with the hypersurface Γ 0 . Now we define functions H + (x), H − (x) by One then easily checks, using (6.7) and (6.1), that, for ε 0 small enough, for which completes the proof of Theorem 1.3.

Proof of Theorem 1.5
In the case where µ −1 ε 2 | ln ε| ≤ t ≤ T , the assertion of the theorem is a direct consequence of Theorem 1.3. Thus, all we have to consider is the case where 0 ≤ t ≤ µ −1 ε 2 | ln ε|. We first need the following lemma concerning Y , the solution of the perturbed ordinary differential equation (4.4).
Now we turn to the proof of Theorem 1.5. We first claim that there exists a positive constant M 2 such that for all t ∈ [0, µ −1 ε 2 | ln ε|], To see this, we choose M 0 large enough, so that M 0 ≥ C 8 + 2C 6 holds in addition to (3.2), (3.3) and (3.4). We then choose M 2 > M 1 , where M 1 is as defined in (6.2). In view of this last condition, we see that if ε 0 is small enough and if d 0 (x) ≥ M 2 ε, then for 0 ≤ t ≤ µ −1 ε 2 | ln ε|, ← thanks to (4.16) ≥ a + C 8 ε.
This inequality and Lemma 6.1 imply w − ε (x, t) > a, where w − ε is the subsolution defined in (3.19). Consequently, by (3.20), In the case where d 0 (x) ≤ −M 2 ε, similar arguments lead to u ε (x, t) < a. This completes the proof of (6.11). Note that we have proved that, for all Since Γ t depends on t smoothly, there is a constantC > 0 such that, for all and (6.14) As a consequence of (6.11) and (6.13) we get which completes the proof of Theorem 1.5.
Proof of Corollary 1.6. In view of Theorem 1.5 and the definition of the Hausdorff distance, to prove this corollary we only need to show that for some constant C ′ > 0. To that purpose let C ′ be a constant satisfying C ′ > max(C, C), where C is as in Theorem 1.3 andC as in (6.14). Choose t ∈ [0, T ] and x 0 ∈ Γ t arbitrarily and, n being the Euclidian normal vector exterior to Γ t at point x 0 , define a pair of points: x + := x 0 + C ′ εn and x − := x 0 − C ′ εn.

Application to reaction-diffusion systems
In this section we discuss the singular limit of the reaction-diffusion system (RD ε ) and prove Theorems 1.11, 1.13 and their corollaries. Our strategy is to regard the first equation of (RD ε ) as a perturbed Allen-Cahn equation and apply what we have already proved for this equation.

Preliminaries: global existence
Before studying the singular limit of (RD ε ), we first show that the solution of this system exists globally for t ≥ 0, provided that ε is sufficiently small. Recall that the system (RD ε ) is written in the form   where h(u, v) satisfies the hypothesis (H). The standard parabolic theory guarantees the existence of local solutions for (RD ε ). In order to prove that the solution exists globally for t ≥ 0, it suffices to show that the solution remains uniformly bounded. This will be done by using the well-known method of invariant rectangles.
Given arbitrary u 0 , v 0 ∈ C(Ω), we choose a constant L > 0 such that Such a constant L exists since f (u) > 0 for u < α − and f (u) < 0 for u > α + . By hypothesis (H), we can choose a constant M 1 satisfying along with the condition (1.18), namely Now we consider the rectangle It follows from (7.1) that, for all sufficiently small ε > 0, 3) The inequalities (7.2) and (7.3) imply that the rectangle R is a positively invariant region for the system of ordinary differential equations since the vector field (ε −2 f ε (u, v), h(u, v)) points inwards everywhere on the boundary of R. The maximum principle then implies that R is also positively invariant for the system (RD ε ). Consequently, since (u 0 (x), v 0 (x)) ∈ R for x ∈ Ω, we have so long as the solution is defined. This uniform bound then implies that the solution exists globally for t ≥ 0. In the case of equations for which only nonnegative solutions are to be considered (see Remark 1.9), we can argue just similarly, by replacing R by the rectangle R + := {(u, v) | 0 ≤ u ≤ L, 0 ≤ v ≤ M 1 }. Summarizing, we have proved the following proposition: In the case where the conditions of Remark 1.9 apply, assume further that u 0 , v 0 ≥ 0. Then there exists ε 0 > 0 such that for any ε ∈ (0, ε 0 ), the solution of (RD ε ) exists globally for t ≥ 0 and is uniformly bounded.
Remark 7.2. For the details of the method of invariant rectangles, we refer the reader to the book [28,Chapter 14,Corollary 14.8]. See also [12] and [11]. It should be noted that [22] makes a much earlier study of invariant rectangles for a finite-difference scheme for reaction diffusion systems.

Proof of the main results
Now we turn to the reaction-diffusion system (RD ε ) and explain our strategy for proving Theorems 1.11, 1.13 and their corollaries.
In what follows, we fix the initial data (u 0 , v 0 ) and denote by (u ε , v ε ) the solution of the system (RD ε ). The solution of the associated moving boundary problem (RD 0 ) will be denoted by (Γ,ṽ).
Given a function v(x, t) on Ω × [0, +∞), we set where f 1 , f ε 2 are as in (1.17). The first equation of (RD ε ) is then written in the form so that u ε (x, t) is the solution of (P ε ) with the choice of the perturbation term g ε (x, t, u) = g ε [v ε ](x, t, u). On the other hand, the equation of surface motion in the limit problem (RD 0 ) is written in the form so that Γ is the solution of (P 0 ) with the choice g(x, t, u) = g[ṽ](x, t, u). Thus Theorems 1.11, 1.13 and their corollaries will follow from what we have shown for the single equation (P ε ). In order for Theorems 1.3 and 1.5 for (P ε ) to be applicable to the present reaction-diffusion system (RD ε ), all we have to do is to verify the conditions (1.3) to (1.6). More precisely, we have to show that, for all small ε > 0, are defined by (7.4) and since f 1 , f ε 2 are smooth, it suffices to prove the following estimates for some C > 0 and for all small ε > 0: The estimates (7.7) and (7.8) are elementary, but (7.9) requires far more elaborate analysis. In this subsection we prove (7.7), (7.8) and give an outline of the proof of (7.9). A full proof of (7.9) will be given later.
In the mean while, the same rescaling argument as in Remark 1.7 yields Combining (7.12) and (7.8), we see that h(u ε , v ε ) hence, by the Schauder estimate, Here the constant C may depend on the choice of ϑ ∈ (0, 1). On the other hand, I 1 is bounded in C 2,1 (Ω × [0, T ]) since v 0 ∈ C 2 (Ω). Combining these, . Substituting this into (7.10) yields the second inequality in (7.7).
Outline of the proof of (7.9). We decouple the system (RD ε ) as follows.
As mentioned earlier, u ε (x, t) is the solution of (P ε ) with the choice of the perturbation term g ε (x, t, u) = g ε [v ε ](x, t, u), that is, Once the solution u ε is determined, v ε is the solution of the problem with the choice u = u ε . This means that v ε (x, t) is a fixed point of the following map Φ ε := Φ 2 • Φ ε 1 : where Φ ε 1 maps a function v(x, t) to the solutionū ε (x, t) of (⋆) for the choice g ε (x, t, u) = g ε [v](x, t, u), and Φ 2 maps a functionū(x, t) to the solution v(x, t) of (⋆⋆) for the choice u =ū.
On the other hand, as for the limit problem (RD 0 ), the solutionṽ(x, t) can be regarded as a fixed point of the map Φ 0 := Φ 2 • Φ 0 1 : where Φ 0 1 maps a function v(x, t) to the step function is the solution of the equation of surface motion and Ω − (Γ) denotes the region enclosed by the hypersurface Γ and Ω + (Γ) the region between ∂Ω and Γ.
Given δ 0 > 0, we define t max = t max (δ 0 ) > 0 by The key estimates for proving (7.9) are the following: There exist constants δ 0 > 0 and C > 0 such that, for any t ∈ (0, t max ], we have Claim 7.4. There exists a constant A > 0 such that, for any v satisfying the estimates (7.7), (7.8) and the Neumann boundary conditions, and for any t ∈ (0, T ], we have The proof of these claims will be given later. For the moment, let us simply mention that Claim 7.4 can be shown by the following two-step argument: first, our results on the single equation (P ε ) yields in the sense that the transition layer of Φ ε 1 (v) and that of Φ 0 1 (v) are within an O(ε) distance; this observation and an estimate of the heat kernel yield (7.15). To prove Claim 7.3, we also use a similar estimate of the heat kernel, see Lemma 7.6 and Subsection 7.7 for details.
The rest of this section gives a detailed account of the proof of (7.9). We begin with some notations to clarify the statements of the above claims.

Some notations
Given any functionḡ(x, t, u) satisfying the conditions (1.3) and (1.5), we can define a classical solution of the interface equation (P 0 ) on some time interval 0 ≤ t < T max (ḡ). We denote this solution by Γ t [ḡ] in order to clarify its dependence onḡ. More precisely, Γ t [ḡ] is a solution of the problem Also, we denote by u ε [ḡ](x, t) the solution of the problem Once the interface Γ t [ḡ] is given, we denote by Ω − t [ḡ], Ω + t [ḡ] the region enclosed by Γ t [ḡ] and the one enclosed between ∂Ω and Γ t [ḡ], respectively. As in (1.13), we define the step functionũ[ḡ](x, t) bỹ for t ∈ [0, T max (ḡ)). V (x, 0) = v 0 (x) in Ω. (7.19) In view of (7.4) and the above notations, the solution (u ε , v ε ) of (RD ε ) is expressed as On the other hand the solution (Γ,ṽ) of (RD 0 ) is expressed as the step functionũ in (RD 0 ) being given bỹ Finally, the maps Φ ε 1 , Φ 0 1 and Φ 2 are now written as Φ ε

Interface motion under perturbation
In this subsection we show that the interface Γ t [ḡ] depends continuously on the pressure term induced byḡ. To this end, we first fix constants C * , T * > 0, ϑ ∈ (0, 1), and denote by Y the set of functionsḡ(x, t, u) on Ω × [0, Proposition 7.5. Letḡ ∈ Y. Let T ∈ (0, T max (ḡ)). Then there exist positive constants δ, K, M such that, for anyg ∈ Y satisfying g −ḡ L ∞ (Ω×(0,T )×R) ≤ δ, (7.21) it holds that T max (g) > T , where we recall that T max (g) is the maximum time of existence of a classical solution of Problem (P 0 g ). Furthermore, for each t ∈ [0, T ], Proof. First, the assertion that T max (g) > T , forg sufficiently close toḡ, follows from the standard local existence theory for quasi-linear parabolic equations. In fact, by using appropriate parametrization, one can express Γ t [ḡ] and Γ t [g], as graphes over M, where M is a N −1 dimensional manifold without boundary, and transfer the motion equations (P 0 g ) and (P 0 g ), into quasi-linear parabolic equations on the manifold M, at least locally in time. For more details we refer to [9]. Sinceg andḡ satisfy (7.20), and since the embedding is compact if 0 < ϑ ′ < ϑ, the assumption g −ḡ L ∞ ≤ δ implies g(·, ·, u) −ḡ(·, ·, u) where c(δ) is a constant satisfying c(δ) → 0, as δ → 0. Consequently, the coefficients appearing in (P 0 g ) and (P 0 g ) satisfy Hence, the two solutions Γ t [ḡ] and Γ t [g] stay close to each other, at least locally in time, and, by repeating this argument, one can prove that T max (g) > T , for δ sufficiently small. Next we prove the estimate (7.22). This will be done by using the maximum principle. Let us introduce some notation. For eachḡ ∈ Y, we denote by d(x, t;ḡ) the signed distance function associated with the interface Γ t [ḡ]. ByΓ t Γ t we mean thatΓ t lies inside ofΓ t . Clearly we have (7.23) Now we choose t 0 ∈ [0, T ] arbitrarily and put The comparison principle then yields Thus, in order to prove (7.22), it suffices to show that there exist constants K, M > 0 such that, for all small η 0 > 0, for 0 ≤ t ≤ t 0 . We will only show the latter inequality for Γ t [ḡ + η 0 ] since the former can be shown in the same manner.
Recall that d(x, t;ḡ) satisfies the equation ( Then it follows from the above inequality that This inequality and the fact that d new (x, 0) = d(x, 0 ;ḡ) imply thatΓ t satisfies On the other hand, Γ t [ḡ + η 0 ] satisfies    By the comparison principle, we obtain Consequently, for 0 ≤ t ≤ t 0 . The lemma is proved.
where u ± It suffices to prove this estimate on a small interval [0, t 0 ], since the estimate for the remaining interval [t 0 , T ] will follow by simply choosing a large constant C (since G is bounded for t large). Hereafter, we choose t 0 sufficiently small. Then, for 0 < t ≤ t 0 , G(x, y, t) is well approximated by the fundamental solution on the entire space R N : G 0 (x, y, t) := 1 (4πDt) N/2 exp − |x − y| 2 4Dt .
In particular, there exists a constant C > 0 such that 0 < G(x, y, t) ≤ C G 0 (x, y, t) for x, y ∈ Ω, 0 < t ≤ t 0 , (see, for example, [16, Section IV.2]). Thus it suffices to prove (7.33) for G 0 instead of G. Given x ∈ Ω, let x 0 be the point on Γ that is closest to x, and let n(x 0 ) be the outward normal to Γ at x 0 . Then x − x 0 = d(x)n(x 0 ). Define where · denotes the Euclidean inner product in R N and span w the line spanned by the vector w. This gives an orthogonal decomposition R N = Y ⊕ Y 0 , and x 0 + Y is the tangent hyperplane of Γ at x 0 . Since Γ is smooth, it is expressed locally as the graph of a map defined on a subset of Y . More precisely, there exist a smooth map h : Y → Y 0 and a constant δ > 0 such that h(0) = 0, ∇h(0) = 0, and that S := { x 0 +ỹ + h(ỹ) ,ỹ ∈ Y , |ỹ| < δ } ⊂ Γ, dist(x 0 , Γ \ S) ≥ δ. Since |x − y| ≥ |d(x)| for every y ∈ Γ and since This and (7.34) yield |x − y| ≥ δ 2 for y ∈ Γ \ S.
Then the curvatures of Γ(r) are uniformly bounded as r varies, which implies that there exists some constant C > 0 such that Γ(r) G(x, y, t) dS y ≤ C √ t for 0 < t ≤ T, r ∈ [−r 0 , r 0 ].
The estimate (7.31) now follows by integrating in r.