TRANSITION LAYERS FOR A SPATIALLY INHOMOGENEOUS ALLEN-CAHN EQUATION IN MULTI-DIMENSIONAL DOMAINS

In this paper, we study a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. By upper and lower solution method, we obtain a sufficient condition for a hypersurface S in the domain Ω to support stable transition layers, and a necessary condition for S in Ω to support transition layers, not necessarily stable. In addition, sharp estimates on depths of transition layers have also been derived.

1. Introduction.Some classes of reaction-diffusion equations give rise to sharp transition layers when the diffusion coefficients are very small.Such phenomena have long been known in physics, biology and other areas of science.Since the middle of 1970s, they have become subjects of intensive mathematical studies.
The Allen-Cahn equation where Ω is a bounded smooth domain in R N and ν is the unit outer normal vector to ∂Ω, is a typical example in which sharp transition layers develop and evolve when is sufficiently small.There has been tremendous research dedicated to this problem and its stationary counterpart.See [1], [6], [7], [9], [17] and references therein.From a viewpoint of application, it is natural to consider spatially inhomogeneous problems, since realistically speaking, environments where reaction and diffusion take place are not uniform.In this paper, we focus on the following spatially inhomogeneous Allen-Cahn equation where the coefficient h(x) ∈ C 2 (Ω), a strictly positive and continuous function on Ω, represents spatial inhomogeneity of the media.The nonlinearity f ∈ C 1 satisfies the following conditions: 1392 FANG LI AND KIMIE NAKASHIMA (F1) f has precisely three zeros α − < 0 < α + and satisfies f (α − ) < 0, f (0) > 0, f (α + ) < 0; (F2) , for all u = 0.
Clearly, f (u) = u − u 3 meets with the above three conditions.The aim of this paper is to study the qualitative properties related to stationary transition layered solutions in multi-dimensional domains.In one dimensional case, very precise and complete information concerning steady states with layers near maximum and/or minimum points of h have been obtained in [12], [13].This paper can be viewed as the multi-dimensional counterpart of [13].
Let us formulate our problem more precisely.The stationary problem for (1) is written in the following form: The main approach employed in this paper is the construction of upper and/or lower solutions.Definition 1.1.We say u is a weak upper (lower) solution of (2) if u is bounded and measurable in Ω and Ω [u 2 ∆ϕ + ϕh 2 (x)f (u)]dx ≤ (≥)0 (3) for all ϕ ∈ C 2 (Ω) with ∂ϕ ∂ν = 0 on ∂Ω and ϕ > 0 in Ω.For the problem (2), we first derive conditions on h(x) which guarantee the existence of stable solutions of (2) with sharp transition near a given closed smooth hypersurfaces S in Ω.
Theorem 1.2.Given a C 3 closed hypersurface S in Ω, suppose that on S, h satisfies the following two geometric hypotheses: (H1) ∂h ∂µ − (N − 1)κh = 0, where µ is the unit inner normal vector to S, κ is the mean curvature of S and λ 1 , λ 2 , ..., λ N −1 are the principal curvatures of S, then the equation (2) has at least two stable solutions with transition layers in an -neighborhood of S, for every sufficiently small > 0.
Remark 1.The term "solutions with transition layers in an -neighborhood of S" will be explained after the statement of Theorem 2.1, which implies Theorem 1.2, in Section 2.2.
Notice that, if N = 1, the hypotheses (H1) and (H2) become h = 0 and h > 0 at some point in Ω.These are exactly the hypotheses in [13] which imply the existence of stable steady states with transition layers in one dimensional case.This problem is also considered in [14] and [15].In [15], for the radially symmetric case, the authors obtain a criterion which guarantees the existence of transition layered solutions, not necessarily stable, of (2).
When Ω = B R (B R1 \ B R2 , resp.) and h(x) = h(r) where R > 0, B R = x ∈ R N |x| < R and r = |x|, it is proved in [8, Theorem 1.1] that any nonconstant radially symmetric solution of ( 2) is unstable if However, from Theorem 1.2, when is sufficiently small, the equation ( 2) has stable radially symmetric solutions which have a transition layer in an -neighborhood of any given sphere ) satisfying the hypotheses (H1) and (H2).By direct computation, the hypotheses (H1) and (H2) imply that This is exactly the case in which ( 4) is violated.
It is worth pointing out that, in Theorem 1.2, for any closed C 3 hypersurface S in Ω, it is not difficult to show the existence of the strictly positive function h ∈ C 2 (Ω) which satisfies the hypotheses (H1) and (H2) on S. Therefore, it is possible for the problem (2) to have stable solutions with layers appearing near any given C 3 closed hypersurface S in Ω as long as h(x) is chosen properly.We include some simple examples of such h(x) at the end of Section 2.1.
We ought to mention that, in [2] and [16], another type of inhomogeneous Allen-Cahn equation, which seems simpler compared to our problem, is considered.
Our next main result deals with a necessary condition for a closed hypersurface S in Ω to support transition layers.
Theorem 1.3.Let S be a closed C 3 hypersurface in Ω. Suppose that there is a sequence of solutions u n of (2) with = n such that S is an isolated limit of the zero level sets of u n , as n decreases to zero, then we have ∂h ∂µ − (N − 1)κh = 0 on S.

Moreover, if
for every point on S, then the layers of the solution u n are in an n | log n | 2neighborhood of S, for n sufficiently large.
Remark 2. We say S in Ω is an isolated limit of the zero level sets of a sequence of solutions u n if there exist a neighborhood Ω S of S in Ω and a sequence of neighborhoods Ω n of S in Ω such that The proof of Theorem 1.3 is based on the sharp estimates of decay rate of transition layers obtained in Proposition 1.
In fact, the condition imposed on S can be weakened, we will be more precise in Section 4. To the best of our knowledge, this is the first time in the literature such a necessary condition is derived.
Comparing Theorems 1.2 and 1.3, we suspect that the following condition is related to the instability of transition layered solutions.Our approach in this paper is both elementary and constructive.The proof itself automatically reveals the nature of those hypotheses required in the multidimensional case.We also remark that our main results in this paper concern sharp transition layers, thus we always assume that the diffusion coefficient is sufficiently small throughout the paper.
This paper is organized as follows.Theorem 1.2 is established in Section 2. Section 3 is devoted to the proof of Proposition 1. Finally, the proof of Theorem 1.3 is included in Section 4.

Existence of stable transition layers.
2.1.Preliminaries.W.l.o.g., let S be a connected C 3 closed hypersurface in Ω.We need to carry out our analysis near S; in particular, we need to compute the Laplacian in the following local coordinate system near S.
There exist It is clear that there exists δ > 0, sufficiently small, such that and the map ] represents the unit inner normal vector to S at the point γ i (s 1 , ..., s N −1 ).First of all, let us rewrite the Laplacian in the local coordinates s 1 ,...,s N −1 , t introduced above.For simplicity, we shall suppress the sub-index i in our computation below (when no confusion arises) and denote the transpose of the vector x = (x 1 , ..., x N ) by x T : (That is, we will be dealing with column vectors.)Setting and we obtain, from the standard computation, that here λ 1 , ..., λ N −1 are the principal curvatures of S. Next if we define, on U ×(−δ, δ), Our upper and lower solutions take the following form , where |t i |, |t 0 | < δ (t 0 to be chosen), v : R → R is a suitable one-dimensional function, and h is the positive function in (2).To verify that the function given by ( 7) is well-defined in S δ , we proceed as follows. If for some i = j, then ) is the unique point on S which is nearest to x, and t i = t j = signed distance between x and S. Thus, for τ ∈ (−δ, δ), Hence the function u given by (7), in terms of v, is well-defined in S δ .
Moreover, we remark that the hypothesis (H1) in the coordinates s 1 ,...,s N −1 , t takes the form and (H2) becomes which, together with (H1), gives that Finally, let us explicitly write down some functions which satisfy the hypotheses (H1) and (H2) on a given closed C 3 hypersurface S. It is routine to verify that where c is any given positive constant and t ∈ (−δ, δ), satisfies the hypotheses (H1) and (H2) on S.And clearly we can always extend the above function h defined in S δ to the whole domain Ω such that it is C 2 differentiable in Ω and positive on Ω.
From now on, we denote s = (s 1 , ..., s N −1 ) when there is no confusion.

Construction of Stable Transition Layers.
Our first main result here establishes the existence of a pair of stable transition layers of the problem (2), which includes Theorem 1.2, namely, where h > 0 on Ω and f ∈ C 1 satisfies (F1), (F2) and (F3): (in the Introduction) (F1) f has precisely three zeros α − < 0 < α + and satisfies Theorem 2.1.Suppose that, on a C 3 closed hypersurface S ⊂ Ω, the following hypotheses (H1) and (H2) hold: where µ is the unit inner normal vector to S, κ is the mean curvature of S and λ 1 , λ 2 , ..., λ N −1 are the principal curvatures of S. Then (2) has two stable solutions with transition layers in an -neighborhood of S, for every sufficiently small > 0.
Remark 3. We say a nontrivial solution u of (2) has a transition layer in an -neighborhood of S in Ω if for any σ > 0, there exists a positive constant C, independent of , such that when d(x, S) > C , either , where S δ is specified in (5).
In order to define and verify our upper and lower solutions, the following three lemmas are necessary.

Lemma 2.2. There exists a unique solution to the following problem
This lemma is rather standard.We omit the proof.
Let R and R be two positive numbers such that and where φ is the unique solution to the problem (10) and The constant β here is to be specified later.Now denote The following two lemmas give estimates for φ and {τ i } 4 i=1 respectively.Lemma 2.3.Let φ(z) be a unique solution of (10).There exist some positive constants 0 < A 1 < A 2 such that the following inequalities hold This lemma can be proved by following similar arguments in [4, Propostion 4.1].We omit the proof here.3 when is sufficiently small.
Proof.By the definition of τ 2 , it satisfies Using ( 14) and Lemma 2.3, we get Consequently, holds when is sufficiently small.The other inequality for τ 2 holds in the same way.By the same argument, we obtain the other inequalities.
Next, we set Finally, we are ready to define our two pairs of weak upper and lower solutions.For simplicity, we first denote the subdomain enclosed by S, by D; the subdomain of Ω complement to the region enclosed by the surface ∂S δ \ D, by Ω out ; and the subdomain enclosed by ∂S δ D, by Ω in , where S δ is specified in (5).See Fig. 2.1.
We define where the constant γ is to be specified later.Clearly, U + L , U − L , U + R and U − R are well-defined for every sufficiently small in view of our arguments in Section 2.1 following (7).As for the continuity of those functions, let us explain it by taking U + L for example.Notice that, because of Lemma 2.4 and the fact that L is continuous in Ω due to the definitions of functions in ( 15) and ( 16).
It follows immediately from the definitions of these functions.
Let us specify β in (12) and γ in (16) as follows.According to the definition of θ in (14), straightforward calculation shows that where Obviously, there exists a constant N 1 > 0 independent of > 0 such that Next recalling the definition of R in (11), it is obvious that there exists sufficiently large Setting choose β > 0 sufficiently small and γ > 0 sufficiently large such that and Our main lemma is as follows.
Obviously, due to standard results (cf.e.g., [18,Theorem 3.6] and [11, Theorem 4]), Lemmas 2.5 and 2.6 immediately yield that (2) has two stable solutions u 1 and u 2 , for sufficiently small > 0.Moreover, because of the definitions of the functions U + L , U − L , U + R and U − R , it is routine to check that the transition layers of these two stable solutions u 1 and u 2 are in an -neighborhood of S, for sufficiently small > 0. Therefore, as long as Lemma 2.6 is verified, the proof of Theorem 2.1 is complete.
Proof of Lemma 2.6.We only show that U + L is a weak upper solution of (2) since the other cases can be handled similarly.
For convenience we first denote the subdomain of Ω complement to the region enclosed by the surface by Ω 1 ; the subdomain enclosed by the surface by Ω 2 ; and the subdomain by Ω 0 .Fig. 2.2 shows the graph of U + L along t-axis.
We claim that under conditions (8), ( 9), ( 18) and (19), for all > 0 sufficiently small.Since the proof of this claim is lengthy, we prefer to finish the rest of the proof first based on the inequality (20).Obviously, U + L is bounded and measurable in Ω.
Therefore for any ϕ ∈ C 2 (Ω) with ∂ϕ ∂ν = 0 on ∂Ω and ϕ > 0 in Ω, the divergence theorem implies that Applying the divergence theorem again, we have where ν 1 , ν 2 are the unit outer normal on S 1 , S 2 to Ω 1 , Ω 2 , respectively.It follows from the definition of the function U + L in ( 16) that Therefore, combined with (20), we have and thus U + L is a weak upper solution of (2).Now let us verify (20).All the positive constants mentioned in the following are independent of > 0. Set Clearly, −τ 1 < z < τ 2 because of the definition of Ω 0 .We only show our proof when 0 ≤ z < τ 2 , that is 0 It can proved similarly when −τ 1 < z ≤ 0. From (6), we deduce

FANG LI AND KIMIE NAKASHIMA
Here , where m(s, t) ≡ 1 By Lemma 2.2 and the definition of v + in (12), direct calculation yields that for some positive constant c 1 .Hence, combined with Lemma 2.4, we have for some positive constant c 1 , where We claim that there exists ρ(γ) where c 2 , c 2 (c 2 < c 2 ) are positive constants independent of γ and .By ( 8) and ( 9), there exists α > 0 such that Then it is easy to check that for − γ < t < 0 where c 3 is a positive constant satisfying Here M 1 < ∞ by Lemma 2.2 and obviously M i , 2 ≤ i ≤ 5, are all bounded.From (24), if − γ ≤ t ≤ − γ + l √ γ , where l > 0 is small enough and independent of γ and , we have m(s, t) ≤ 0. This means that m(s, t) ≤ 0 holds if Setting The claim is proved.Now let us go back to (21) and continue to show that We will handle the following three cases separately.Case 1: 0 ≤ z ≤ N * , where N * is from (17).On this interval, there exists some constant c 4 > 0 such that Using ( 21), (25), (26) and Lemma 2.3, we have Then (18) yields that the right hand side of the above inequality is negative if is sufficiently small.Case 2: N * ≤ z ≤ ρ(γ).On this interval m(s, t) ≤ 0 from (23) and φ (z) ≥ 0.
Moreover, it follows from ( 17) and ( 22) that Hence we conclude that 1 if is sufficiently small.Case 3: ρ(γ) ≤ z ≤ τ 2 .On this interval, it holds that where c 5 is a positive constant.Notice that (27) also holds on this interval.Therefore, together with (19), we have 1 if is sufficiently small.The proof is complete.
3. Depth of transition layers.In this section, we will provide the estimates concerning the depth of the transition layers.Let u be a nontrivial solution of the problem (2) as follows on ∂Ω and Z be the set of zeros of u .Set d(x) = dist(x, Z ).Our estimates are stated as follows.
Proposition 1.There exist η 1 > 0, C 1 > 0, independent of , such that either of the following inequalities holds for sufficiently small.
Remark 4. We want to point out that these estimates hold for any non-trivial solutions of the problem (2).Moreover, in this paper, these estimates play an important role in the proof of Theorem 1.3.
The proof of Proposition 1 is lengthy.To better explain our idea, we present the main steps of the proof in a series of lemmas.Lemma 3.1.There exists M 0 > 0 such that for M ≥ M 0 , the following problem has the unique positive solution w M : Moreover, w M is radially symmetric and 0 ≤ α + − w M (0) ≤ cM −2 , where M ≥ M 0 and c is a positive constant independent of M .
Proof.Obviously, there exists M 0 > 0 such that the constant solution w = 0 of (30) is unstable if M ≥ M 0 .In this case it is well known (see [10, Theorem 3.1], for example) that (30) has a positive solution.Because f satisfies (F3) in the Introduction, we can prove the uniqueness of the positive solution to (30).We denote this solution by w M .The solution w M is radially symmetric by the Gidas-Ni-Nirenberg symmetry result [3].Now, suppose that the set {x ∈ B M | w M > α + } is not empty.By applying maximum principle in any connected component of this set, it is easy to derive a contradiction.Hence, α + − w M (0) ≥ 0.
Next, let µ 1 denote the first eigenvalue of Then the first eigenvalue of Denote the first eigenfunction by ϕ 1 .The fact that ϕ 1 is radially symmetric follows from the simplicity of the first eigenvalue.Moreover, because of the conditions imposed on f , there exists σ > 0, such that In fact we will prove that for M ≥ M 0 Suppose that there exists some M > M 0 such that Then from (30) and (31) we have Obviously, when l is sufficiently small, lϕ 1 < w M in B M .As l increases, there are only two possibilities because w M is bounded.The first one is that there exists some l such that lϕ 1 touches w M at some points in B M and lϕ 1 ≤ w M in B M .This is a contradiction to strong maximum principle [5,Page 35].The second one is that there exists some l such that lϕ 1 (M) = w M (M).This contradicts to Hopf boundary lemma [5,Page 34].The proof is complete.
Let us prepare some notations for convenience and then continue the proof of Proposition 1. Set By the properties of f , we can choose the constants H close to −f (α + ) and A > 0 large enough such that and where c and M 0 are the constants appearing in Lemma 3.1.For any x 0 ∈ Ω satisfying d(x 0 ) ≤ 3 A, (28) or (29) holds if C 1 > 0 is large enough.Now it remains to establish the proposition for x 0 ∈ Ω satisfying d(x 0 ) > 3 A.
It is clear from the definition of d(x) that either u > 0 or u < 0 holds in B(x 0 ; d(x 0 )).W.l.o.g., we assume that u > 0 in B(x 0 ; d(x 0 )).With the help of the previous lemma, we have the following preliminary estimate.Lemma 3.2 (See Fig. 3.1).For any fixed Proof.It is obvious that u > 0 in B(a; A).To estimate u , we first consider Then By Lemma 3.1 and (33), (36) has a solution w 1 satisfying which directly imples that (35) has a solution v 1 and Notice that v 1 satisfies Then it is easy to check that, for any 0 ≤ l ≤ 1, lv 1 also satisfies (38) because of the property (F3) of f (in the Introduction).Suppose that there exists x ∈ B(a; A) such that v 1 (x) > u (x).Observe that when l is close to 0, lv 1 (x) < u (x) for x ∈ B(a; A), since u > 0 in B(x 0 ; d(x 0 )) ⊃ B(a; A).Increasing l, a contradiction can be derived by the same argument at the end of the proof of Lemma 3.1.Hence v 1 (x) ≤ u (x) for x ∈ B(a; A).Combined with (37), we obtain the estimate (34).
The proof is complete.
Next, for any fixed Define where The key ingredient in our proof is the following lemma.
Remark 5.This lemma is proved by constructing a pair of upper and lower solutions.We want to point out that the lower solution constructed in the proof below is crucial.Actually, because of the relation between the problems (39) and (40), our lower solution provides the lower bound to the solution of the problem (39).While, at the same time, thanks to Lemma 3.2, the solution of the problem (39) is comparable to u , the solution of the problem (2).And Proposition 1 follows.We will strictly demonstrate these arguments after the proof of Lemma 3.3.
Proof of Lemma 3.3.Set where , η and γ are sufficiently small such that where H comes from (32), and About the function w(r) defined in (41), direct calculations yield that, for M 2 < r ≤ M , w (r) = −bη exp(−η(M − r)) < 0, and for 0 Moreover, it is routine to check that w ( M 4 ) > 0 and w ( M 2 ) < 0 due to the condition M > 2η γ in (42).Therefore, there exists l * ∈ ( M 4 , M 2 ) such that w (l * ) = 0.In fact, from previous calculations, it is obvious that r = l * is the only critical number of w(r) in (0, M ).See Fig. 3 and then show that w − (z) is a lower solution to (40).
For |z| ≤ l * , it follows from the definition of w and (32) that For l * < r ≤ M , r = M /2, where r = |z|.Here, the point r = M /2 is excluded since g(r) is discontinuous at r = M /2.First, direct calculations give that where w(r) ≤ ξ(r) ≤ α + .Next, by the definition of g(r), l * ∈ ( M 4 , M 2 ) and (32), we derive that Then we handle the following three cases separately.
Case 1: By (42), ( 43) and (45), we have Case 3: M 2 < r ≤ M .Since in this case g(r) = 0, by (42) and (45), it is easy to see that ∆w − + f (w − ) ≥ 0. Therefore it is routine to check that w − (z) is a weak lower solution to (40).On the other hand side, w + (z) ≡ α + is an upper solution to (40) and obviously w + (z) > w − (z) in B(0; M ).Then due to standard results (cf.e.g., [18,Theorem 3.6] and [11,Theorem 4]), there exists a solution, denoted by w 2 , to (40) and w − (z) ≤ w 2 (z) ≤ w + (z) in B(0; M ).Now it is time to put together Lemmas 3.2 and 3.3 and finish the last piece of the proof of Proposition 1. Recalling (39) and (40), from Lemma 3.3, we automatically get a solution, denoted by v 2 , to (39) by setting Moreover, because of Lemma 3.2, we may derive the following estimate by applying the similar arguments as we have used to prove Lemma 3.2.Notice that where w − is the lower solution defined in (44).The inequality u < α + is obtained as follows.Suppose that the set {x ∈ Ω | u > α + } is not empty.A contradiction may be derived by applying maximum principle in one connected component of this set.Moreover, the possibility that u (x) = α + at some point in Ω contradicts to strong maximum principle.Now the inequality (28) is established when we assume that u > 0 in B(x 0 ; d(x 0 )).Obviously, the inequality (29) can be proved similarly when assuming u < 0 in B(x 0 ; d(x 0 )).The proof of Proposition 1 is complete.
4. Characterize the location of transition layers.In this section, we will prove a stronger result, that is Theorems 4.1 and 4.2, which includes Theorem 1.3.(2) For every = n , (2) has a solution u n with the property that at least one connected component of where ) < η and µ is the unit normal vector to S ∩ Λ and κ is the mean curvature of S ∩ Λ.
Remark 6. Theorem 4.1 reveals that the equation (2) only allows layers near a curve satisfying the hypothesis (H1).Here the term "layer" includes clustering layers such as in [11].Note that the hypersurface S in Theorem 4.1 can intersect the boundary of Ω.
Theorem 4.2.Suppose that S ∩ Λ mentioned in Theorem 4.1 is a closed hypersurface and the conditions (1), ( 2), (3) in Theorem 4.1 still hold.Denote Υ ≡ S ∩ Λ and choose η small enough such that for every point on Υ, then we have when n is sufficiently large, where the constant C > 0 is independent of n, µ is the unit inner normal vector to Υ, κ is the mean curvature of Υ and λ 1 , λ 2 , ..., λ N −1 are the principal curvatures of S.
The rest of the section is devoted to the proofs of Theorems 4.1 and 4.2.Since the proof is a long story, we divide it into three subsections.
A contradiction will be derived later in Section 4.2.
W.l.o.g., suppose that S ∩ Λ divide the domain Λ into two subdomains.Let D denote the one which makes the unit normal vector µ at P to be the unit inner normal vector of ∂D, and We may choose n larger to guarantee that δ n is small enough when necessary.There exist a C 3 hypersurface S in D and a small constant δ > 0 such that the following properties hold: where h(s, t) = h(σ(s, t)) = h(x 1 , ..., x N ).
Here we may need make U and V smaller if necessary.(P4) There exists an open set Q ⊂ U such that P * ∈ γ(Q) and According to the assumptions of Theorem 4.1, either u n (x) < 0 or u n (x) > 0 in (S η \ S δn ) ∩ D. W.l.o.g.we assume that u n (x) < 0 in (S η \ S δn ) ∩ D. In this (2) to derive a contradiction, we have to worry about the boundary of Σ.This is why Σ is chosen in this special way.4.2.Proof of Theorem 4.1.To avoid the distraction of complicated calculations and make our idea transparent, we present our main steps in the form of lemmas and derive a contradiction to finish the proof of Theorem 4.1 first.Then we provide proofs of these lemmas.
Remark 8.By the definition of u ].For > 0 sufficiently small, we have u where Lemma 4.5.For > 0 sufficiently small, First of all, Lemma 4.4 guarantees that at the boundary of Σ, u (t0) (x) > u (x) for t 0 ∈ [−2δ n , δ 2 ].Then, setting t 0 = δ 2 , we have u ( δ 2 ) (x) > u (x) for x ∈ Σ, because of Lemma 4.5.Therefore, when we let t 0 decreases from δ 2 to −2δ n , thanks to the strong maximum principle and Hopf boundary lemma, Lemma 4.3 and Remark 8 following it imply that for any t which obviously contradicts Lemma 4.6.Theorem 4.1 is proved.Now we only need prove these lemmas.The proof of Lemma 4.3 is similar to that of Lemma 2.6.Proposition 1 is used frequently in the proofs of Lemmas 4.4 and 4.5.
Proof of Lemma 4.3.According to the definition of u (t0) (x) in (50), it is easy to check that 2 ∆u (t0 Then we only need show that The proof is similar to that of Lemma 2.6.Therefore, we mainly explain how to modify the proof of Lemma 2.6.
By setting h(s, τ )dτ, z = y and repeating similar calculation we have carried on to obtain (21), we get for some positive constant C 3 , where h sj s k dτ .

)
By the assumptions of Theorem 4.2, either u n (x) < 0 or u n (x) > 0 on (Υ η \ Υ δn )∩ D, where D denotes a domain bounded by Υ. W.l.o.g.we assume that u n (x) < 0. Set = n for simplicity.For each t 0 ∈ [M n | log | 2 , η/4], define u (2)Proof of Theorem 4.2.For the rest of this section, we focus on proving Theorem 4.2.The idea in the proof of Theorem 4.2 is quite similar to that in the proof of Theorem 4.1.We will also assume that our claim is false, construct a series of proper functions based on this assumption, and then derive a contradiction by comparing these functions with the solution of the problem(2).The best part of the proof is the construction of proper functions.To avoid unnecessary repetition and keep the proof concise, we only emphasize the different parts and skip some calculations when they are similar to previous proofs.Now let us begin our proof.Suppose that the statement of Theorem 4.2 is not true.This means that there exists{M n > 0} | log n | 2 , and Γ n ∩ Υ δn \ Υ Mn n | log n | 2 = ∅,for n large.W.l.o.g., assume thatΓ n ∩ Υ δn \ Υ Mn n | log n| 2 ∩ D = ∅,(57)where D denotes a domain bounded by Υ.Since Υ = S ∩ Λ is a C 3 closed hypersurface, there exists the local coordinate system in Υ η introduced in subsection 2.1.Choose η smaller if necessary.By Theorem 4.1, we already have * + O( 3 ) < 0 for > 0 sufficiently small.4 )dτ < − τ 2 .Then similar to Lemma 4.3, we claim that for any t 0 ∈ [M n | log | 2 , η/4] For x ∈ Σ 0 , by the same arguments and calculations in Lemma 4.3, we get 1 (x)) ≤ 0 in Σ 0 ∪ Σ 1 ∪ Σ 2 , (61)provided > 0 is sufficiently small.Obviously, in Σ 1 ∪ Σ 2 , 2 ∆u (t0) 1 (x) + h 2 (x)f (u (t0) 1 (x)) ≤ 0.Moreover, |t−t 0 | ≤ C 5 | log |.Then, combined with Lemma 2.4 and the assumption (60), we obtain thatm 2 (s, t) ≤ −A | log | 2 + C 5 |t − t 0 | 2 + C 5 |t − t 0 | ≤ − A 2 | log | 2