reaction–diffusion systems with mass conservation Asymptotic behavior of equilibrium states of ScienceDirect

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ScienceDirect

J. Differential Equations 264 (2018) 550–574

www.elsevier.com/locate/jde

Asymptotic behavior of equilibrium states of reaction–diffusion systems with mass conservation

Jann-Long Chern

^a,1

, Yoshihisa Morita

^b,2

, Tien-Tsan Shieh

^c,d,^∗

aDepartmentofMathematics,NationalCentralUniversity,Chung-Li32001,Taiwan bDepartmentofAppliedMathematicsandInformatics,RyukokuUniversity,SetaOtsu520-2194,Japan

cNationalCenterforTheoreticalScience,NationalTaiwanUniversity,Taipei106,Taiwan dDepartmentofMathematics,NationalTaiwanUniversity,Taipei106,Taiwan

Received 19December2016;revised 5September2017 Availableonline 22September2017

Abstract

Wedealwithastationaryproblemofareaction–diffusionsystemwithaconservationlawundertheNeu- mannboundarycondition.ItisshownthatthestationaryproblemturnstobetheEuler–Lagrangeequation ofanenergyfunctionalwithamassconstraint.Whenthedomainisthefiniteinterval(0,1),weinvestigate theasymptoticprofileofastrictlymonotoneminimizeroftheenergyasd,theratioofthediffusioncoef- ficientofthesystem,tendstozero.Inviewofalogarithmicfunctionintheleadingtermofthepotential, wegettoascalingparameterκsatisfyingtherelationε:=√

d=√

logκ/κ².Themainresultshowsthata sequenceofminimizersconvergestoaDiracmassmultipliedbythetotalmassandthatbyascalingwith κtheasymptoticprofileexhibitsaparabolainthenonvanishingregion.Wealsoprovetheexistenceofan unstablemonotonesolutionwhenthemassissmall.

MSC:35B35;35B40;35K57

* Correspondingauthor.

E-mailaddresses:[email protected](J.-L. Chern),[email protected](Y. Morita), [email protected],[email protected](T.-T. Shieh).

1 Work partially supportedby Ministryof Science andTechnology of Taiwanunder grantMOST-104-2115-M- 008-010-MY3.

2 WorkpartiallysupportedbyJSPSKAKENHIGrantNumber,26287025,26247013,JSTCRESTGrantNumber JPMJCR14D3,andagrantforoverseasresearch(Kokugai-kenkyu)ofRyukokuUniversity(2016–2017).

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Keywords:Reaction–diffusionsystem;Massconservation;Equilibriumsolution;Asymptoticbehavior;Concentration phenomena;Stability

1. Introduction

In the fields of population biology and cell biology concentration phenomena are often ob- served by aggregation of species and chemical substances respectively. One of the well known models is a Keller–Segel chemotaxis model [21]in which spiky patterns appears by the aggregation of cellular slime mold, though it blows up in a higher dimensional domain (for instance, see [17], [5], [23], [20], [25]and the references therein). In this model the total mass of the slime mold is conserved in a reasonable setting. On the other hand in a study for the cell polarity the authors [19]and [7]proposed simple conceptual models to describe the concentration phenomenon induced by a different mechanism from the chemotaxis model, though the mass conservation property shares in the both models. After their contribution, mathematical studies for the conceptual models are developed in [16], [15], [8], [10]and [9](see also [13], [14], [11]

and [12]). In particular, it is shown in [16], [15]and [8]that the spiky pattern is certainly stable in their model equations.

Motivated by those studies, we are concerned with the following reaction–diffusion system:

u_t =du−g(u+γ v)+v,

v_t=v+g(u+γ v)−v, x∈, (1.1)

with the Neumann boundary condition

∂u

∂ν =∂v

∂ν =0, x∈∂, (1.2)

where is a bounded domain of Rⁿwith smooth boundary ∂and 0 < d <1. We note that the diffusion coefficients of u and v equations are normalized: 1 in the v-equation and d in the u-equation where d stands for the ratio of the two diffusion coefficients. For specific cases g(u) =au/(u²+b) (γ =0)and g(w) =w/(w+1)²(γ =1)are provided by [19], where a, b are positive constants.

Here, we deal with the case for γ=1 and fix the function g(w)as g(w)= w

(w+1)².

It is known that there exists a unique nonnegative classical solution satisfying the initial condition

(u(x,0), v(x,0))=(u0(x), v0(x)), u0, v0∈C⁰(), u0(x)≥0, v0(x)≥0(x∈) (see [8]and [9]). Under the evolution of the system, the total mass is conserved:

(u(x, t )+v(x, t))dx=

(u₀(x)+v₀(x₀))dx (t≥0).

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Moreover, the system allows a Lyapunov function (see [8]), that is, L(u, v)=

d

2|∇(u+v)|²+(1−d)G(u+v)+d

2(u+v)²+1

2|∇(du+v)|²

dx,

where G(u) :=_w

0 g(u)du. Hence, the asymptotic state as t→ ∞belongs to the set of all the equilibrium solutions (see [4]).

In this paper we study the stationary problem of (1.1)and the asymptotic profile of a stationary solution as d→0, where the problem is given as

du−g(u+v)+v=0,

v+g(u+v)−v=0, x∈, (1.3)

with the functions uand vsatisfying the Neumann boundary condition (1.2)and the total mass constraint

(u+v) dx=M, i.e. u + v =M/|| ≡m. (1.4)

Here, ·is the integral average over : · := 1

||

·dx.

Set w=u +v. By a straightforward computation, the system (1.3)is reduced to the scalar equation for unknown function wand some unknown constant λ:

−dw+(1−d)g(w)+dw=λ (x∈), (1.5) with the Neumann boundary condition ∂w/∂ν =0 on ∂ and the total mass constraint

w dx=M. This scalar equation is the Euler–Lagrange equation of the energy functional E(w)=

d

2|∇w|²+(1−d)G(w)+d 2w²

dx, (1.6)

under the total mass constraint (1.4), where the function Gis given by

G(w)= w

0

g(u) du=log|w+1| + 1

|w+1|−1.

We focus on nonnegative solutions to the system (1.1)minimizing the corresponding energy functional E. Therefore, we shall consider the variational problem

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infAE(w)=inf

A

d

2|∇w|²+(1−d)G(w)+d 2w²

dx, (1.7)

subject to an admissible set

A:= {w∈H¹():

w dx=M, w≥0}.

In particular, we analyze the asymptotic behavior of the solution of (1.7)with least energy as d= ε²→0(ε >0). For simplicity, we assume the domain is a one-dimensional interval =(0, 1).

The corresponding Euler–Lagrange equation is given by

−ε²wxx+(1−ε²)g(w)+ε²w=λ (0< x <1), (1.8) wx(0)=wx(1)=0,

1 0

w dx=M.

Invoking of the argument in [3](see also [24]), we see that the minimizer wε(x)is monotone inx.

We remark that a solution (u^∗, v^∗)of (1.3)is associate with a solution w^∗of (1.5)through the relation

(u^∗(x), v^∗(x))=

M− g(w^∗) +w^∗(x)−M

1−d , g(w^∗) −d(w^∗(x)−M) 1−d

, (1.9) and it is proved in [8]that for the minimizer wε the solution (uε, vε)defined as (1.9)is a stable solution. Moreover, L(u^∗, v^∗) =E(u^∗+v^∗)holds for any solution (u^∗, v^∗)of (1.3)since du^∗+ v^∗is constant, the minimizer of Ethereby provides the minimum of L(u, v).

Let κεbe a positive function defined through the relation ε²=logκ

κ⁴ . (1.10)

It is seen that for >0 small enough the function κ is strictly decreasing and limε→0+κ_ε= +∞. Suppose wεis a global minimizer of the functional

Eε(w)= 1 0

ε²

2|w_x|²+(1−ε²)G(w)+ε² 2w²

dx, (1.11)

among the admissible set A. Without loss generality, we may assume that wε is monotone decreasing.

Theorem 1.1. Let κεbe the solution of (1.10)and the function wεbe a minimizer of the functional (1.11). Then the following results hold:

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(i) The sequence {w_ε}converges to Mδ(x)in the following sense:

1 0

w_ε(x)dx=M, lim

ε→0 sup

η≤x≤1

w_ε(x)=0 (∀η∈(0,1)).

(ii) Set με:=max0≤x≤1w_ε(x) =w_ε(0). There is a constant C1>0and ε1>0such that μ_ε

κ_ε ≤C₁ (0< ε < ε₁).

(iii) Define Wε(x) :=_κ¹_εw_ε _κ^x

ε

. The sequence {W_ε}converges to a function

W0(x)= 1

a−^ax₄² for0< x <

√2 a ,

0 for

√2 a ≤x.

in C_loc^0,αfor 0 ≤α <1, where a=

2√ 2

3M. Furthermore,

εlim→0

μ_ε κ_ε =1

a.

By a straightforward estimate, we will obtain the following Corollary. We leave the proof to readers.

Corollary 1.2. Let w_ε be a minimizer of the functional (1.11)and (uε, v_ε)be the associative solution to the system (1.3)through the relation (1.9). Then the sequence {u_ε}converges to a Delta function Mδ(x) in the same sense describing in Theorem 1.1 (i) and the sequence {v_ε} converges to 0in L^∞(0, 1).

Remark 1.1. Our problem (1.7)is similar to the variational problem arising from the van der Waals/Cahn–Hilliard theory of phase transitions (see [2], [3]),

inf

u∈H¹()

u dx=M

ε²

2|∇u|²+W (u) dx,

where W (u)represents a coarse-grain free energy and usually in the form of double-well potential. Unlike the standard one, our nonlinear potential is indeed dependent on the small parame- terε. As ε→0, the location of two “wells” is getting away. This is the reason that our solution have a “huge” jump near the boundary which is also different from the classical Maxwell solution. Therefore, instead of getting interface of two different phases, we obtain a Dirac Delta function in its asymptotic limit.

Next we consider the existence of another monotone solution. Let 0 < M <1. Then g(M) >

0 and we easily verify that the constant solution w^∗=Mis a nondegenerate local minimizer of the functional Eε and it cannot be a (global) minimizer for sufficiently small ε. Since w_ε is a

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minimizer, we suspect that there is another solution, which might be unstable. This leads us to the next theorem.

Theorem 1.3. Suppose that 0 < M <1. Then equation (1.8)has a strictly monotone decreasing solution w˜_ε(x) whose energy Eε(w˜_ε) is bounded away from 0 as ε→0. The reflected w=

˜

w_ε(1 −x)is a strictly monotone increasing solution. Those solutions are unstable for the gradient flow of Eε(w).

Corollary 1.4. Let w˜_ε be the solution given by Theorem 1.3and let (u^∗, v^∗)be the one defined by (1.9)with w^∗= ˜w_ε. Then (u^∗, v^∗)is unstable in the system (1.1)–(1.2)with =(0, 1)and d=ε².

Remark 1.2. As mentioned before the system (1.1)with (1.2)allows the Turing type instability if g(M) <0, and the instability drives the emergence of the wave patterns in the transient dynamics (see [19]). Thus we were primarily interested in the model equation for this case. On the other hand the above theorem concerns the stable regime for the constant steady state and even this regime the system exhibits a nontrivial structure for the steady state solutions.

Remark 1.3. As seen in §4, our proof of the above theorem is done by a dynamical system argument. Although one might obtain the existence of an unstable solution by utilizing the mountain-pass theorem, we, however, emphasize that the argument here is quite simple.

Remark 1.4. It looks that the variational problem of (1.11)is similar to the stationary problem of the Cahn–Hilliard equation. In fact, the system (1.1)with γ=1 can be formally written as a single equation for w=u +vin what follows. First write (1.1)as

w_t−(1−d)v_t=dw−(1−d)g(w)+(1−d)v, w_t=dw+(1−d)v.

Operating −on the both side of the first equation and differentiating the second one in tyield

−[w_t−(1−d)v_t] = −[dw−(1−d)g(w)+(1−d)v], w_{t t}=dw_t+(1−d)v_t.

By eliminating vterms we obtain

w_{t t}+w_t= −[dw+ ˜g(w)−(1+d)w_t], (1.12) where

˜

g(w):= −(1−d)g(w)−dw.

If wt tis absent from (1.12)and g˜is a cubic function, it looks close to the viscous Cahn–Hilliard equation

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(1−ν)u_t= −(u+f (u)−νu_t), (ν∈(0,1)) proposed by [18].

Remark 1.5. As stated in the first part of this section our model equation was presented by a conceptual model for the cell polarity. As a similar model system there is the following one as proposed in [22]

u_t= −μ[(u+v)u−(1−(u+v))v],

v_t=Dv+μ[(u+v)u−(1−(u+v))v], (1.13) where uand v stand for the density of a proliferating population and a migrating population in tumour cells respectively. (u +v)is the probability that an immotile cells becomes motile, μis the exchange rate of phenotype of the cells, and Dis the diffusion coefficient for v. This model is called “go-or-rest” model in [22]and numerical simulations are done for the function given by

(ρ):=1

2(1+tanh(α[ρ^∗−ρ])),

where αand ρ^∗is positive parameter. By a biological reason the diffusion of uequation is absent.

We notice

(u+v)u−(1−(u+v))v=(u+v)(u+v)−v.

The system (1.1)with γ=1 could be regarded as a singularly perturbed system of (1.13)with g(w) =(w)wby the normalization as D=1 and μ =1. Since our nonlinearity is different from this case, the above results cannot apply to g(w) =(w)w. Our study, however, would provide a direction of the study to this case.

Remark 1.6. As for the case when γ=0 and g(u) =au/(u²+b) (a, b >0), which appears in [7], [19], [15], [16]and [10], similar results to Theorems 1.1 and 1.3can be obtained by simple modification of the argument in the present paper. We don’t state this case in detail to avoid repeating similar arguments.

Remark 1.7. We believe that in a domain with higher dimension the qualitative behaviors of stationary solutions to the equations (1.1)–(1.2)and their structural stability can be understood through our analytical techniques in the paper. Thus, in this case, we conjecture the minimizers of energy functional (1.6)concentrate at points with maximal curvature on the boundary as d →0. In other words, the limiting function is still a Dirac mass at those points. The detailed discussion will be carried out in the forthcoming paper.

The rest of the paper is organized as follows: in the next section we give the proof for (i) and (ii) of Theorem 1.1and in §3we complete the proof of Theorem 1.1by proving (iii). The proof of Theorem 1.3is given in §4.

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2. Proof of Theorem 1.1(i) and (ii)

In this paper, we concern with the equilibrium states of the system (1.1)which is corresponding to minimizers of the variational problem (1.7). The existence and regularity results of minimizers are obtained by the standard direct method in the Calculus of Variation and the standard elliptic theory. For fixed ε, the minimizer wε satisfies the system (1.8)where λis the corresponding Lagrange multiplier of the variational problem (1.7).

We define

φ_ε(x):=

2Mκ(1−κx) (0≤x≤1/κ),

0 (1/κ≤x≤1).

Lemma 2.1. There are constants C0>0and ε0>0such that Eε(φ_ε)≤C₀logκε

κε

(0< ε≤ε₀). (2.1) Proof. We simply write κ instead of κεbelow. Plugging the test function in the functional, we compute the energy term-by-term:

1 0

{(φ_ε)_x}²dx= 1/κ 0

2Mκ² 2

dx=4M²κ³, 1

0

(φ_ε)²dx= 1/κ 0

4M²κ²(1−κx)²dx=4 3M²κ, 1

0

log(φε+1)dx= 1

κ + 1 2Mκ²

log(2Mκ+1)−1 κ, and

1 0

1 φε+1 −1

dx= 1

2Mκ²log(2Mκ+1)−1 κ.

Therefore, we have

Eε(φε)=2M²ε²κ³+(1−ε²) 1

κ + 1 2Mκ²

log(2Mκ+1)−1 κ

+(1−ε²) 1

2Mκ²log(2Mκ+1)−1 κ

+2 3Mε²κ.

Since the definition of κ=κ_ε, we see

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ε²κ³=logκ κ , which implies the desired inequality (2.1). 2

From the above lemma we see that the minimizer converges to zero almost everywhere as ε→0. In addition we have the next lemma.

Lemma 2.2. Let με=max0≤x≤Lw_ε(x), where wε is the minimizer of (1.11). We have

ε→+lim0με= +∞.

Proof. We will prove by contradiction. Assume that there exists some sequence {ε_j}converging to 0 and some constant C >0 such that μεj ≤C for all j. Choose a point w0< Msufficiently small in an open interval (0, 1)such that the intersection of the tangent line

y=G(w₀)(w−w₀)+G(w₀),

and the graph y=G(w)is achieved at the point w=w₀and w1> C. Note that G(w₀)=G(w1)−G(w0)

w₁−w₀ .

Noticing that y=G(w)is convex up to w=1 and concave for w >1, and invoking of the L^∞-boundedness of the sequence {wε_j}, we see

G(w_ε_j)≥G(w₀)(w_ε_j−w₀)+G(w₀).

Therefore,

Eεj(w_ε_j)= 1 0

ε² 2

dw_ε_j dx

²+(1−ε²_j)G(w_ε_j)+ε² 2 w²_ε

j

dx

≥(1−ε_ε²

j) 1 0

(G(w₀)(w_ε_j−w₀)+G(w₀)) dx

>1 2

G(w₀)(M−w₀)+G(w₀)

>0

for εj small enough. The constant lower bound of the energy Eε_j(w_ε_j)contradict to the result of the previous Lemma 2.1:

Eε_j(wε_j)≤C₀logκ(ε_j) κ(ε_j) →0.

This implies

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ε→+lim0μ_ε= +∞. 2 By the above lemma we can assert (i) of Theorem 1.1.

Lemma 2.3. For the same μεin Lemma 2.2 1 2√

2εμε(logμε)^1/2≤Eε(wε) (2.2) holds.

Proof. Without loss of generality, we may assume wε(x)is strictly monotone decreasing. In- deed, any stable solution is monotone by [3]but the constant solution w=Mof (1.8)cannot be the minimizer because of Lemma 2.1. Thus, we have με=w_ε(0). Let ξ =ξ_ε be the point such that

μ_ε

2 =w_ε(ξ ).

Put ϕε(x) :=w_ε(x)/μ_ε. We have ϕ(0) =1, ϕ(ξε) =1/2 and

σ_ε:=Eε(w_ε)= 1 0

ε²μ²_ε

2 {(ϕ_ε)_x}²+(1−ε²)G(μ_εϕ_ε)+ε²μ²_ε 2 ϕ_ε²

dx

where

G(μεϕε)=log(μεϕε+1)+ 1

μ_εϕ_ε+1 −1=logμ_ε

2 +log(√

μεϕε+ 1

√μ_ε)+ 1

μ_εϕ_ε+1−1.

Because limε→0μ_ε= +∞and the definition ξε, we have log(√

μεϕε+ 1

√μ_ε)+ 1

μ_εϕ_ε+1 −1≥log(

√με

2 + 1

√μ_ε)+ 1

μ_ε+1−1>0, on the interval [0, ξ_ε]. Thus,

G(μ_εϕ_ε)≥1

2logμ_ε (0≤x≤ξ_ε), for small ε. Utilizing this, we estimate

σε≥

ξε

0

ε²μ²_ε{(ϕ_ε)_x}²

2 +(1−ε²)G(μεϕε) dx

≥

ξ_ε

0

1

2{εμ_ε(ϕ_ε)_x}²+1

4 logμ_ε 2

dx

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≥ 1

√2

ξε

0

εμε

logμε|(ϕε)x|dx (putz=ϕx(x))

= 1

√2εμ_ε logμ_ε

1/2 1

(−1)dz= 1 2√

2εμ_ε

logμ_ε. 2

We complete the proof of Theorem 1.1(ii). Combining Lemmas 2.1 and 2.3, we obtain εμ_ε

2√ 2

logμε≤Eε(uε)≤C₀logκ_ε

κ_ε =C0εκε

log(κε), (2.3)

where we used (1.10). Furthermore, 2√

2C0κ_ε

log(κε)=2√ 2C0κ_ε

log(2√

2C0κ_ε)−log(2√ 2C0)

≤2√ 2C0κ_ε

log(2√

2C0κ_ε),

for εsmall enough. Combining the fact that the function s(logs)^1/2is monotone increasing for large s >0, we conclude that μ_ε≤2√

2C0κ_ε. This proves (ii) of Theorem 1.1. 2 3. Proof of Theorem 1.1(iii)

3.1. Some estimates

In order to prove (iii) of Theorem 1.1we need elaborate estimates. We rewrite the equation (1.8)as

ε²

2 w²_x=(1−ε²)G(w)+ε²

2 w²−(λεw+Aε), (3.1) w_x(0)=w_x(1)=0,

1 0

w dx=M.

Since we assume wεis a monotone decreasing function, the first equation can be written as

w_x= −

2 ε²

(1−ε²)G(w)+ε²

2 w²−(λ_εw+A_ε)

,

thus

− dw

2

ε² (1−ε²)G(w)+^ε₂²w²−(λεw+Aε) =dx.

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Set w₊=w(0)and w₋=w(1). The values w₊, w₋, λ_εand Aεsatisfy

(1−ε²)G(w₊)+ε²

2 w²₊=λεw₊+Aε, (3.2)

(1−ε²)G(w₋)+ε²

2 w²₋=λεw₋+Aε, (3.3)

w₊

w₋

dw

2

ε² (1−ε²)G(w)+^ε₂²w²−(λ_εw+A_ε)

=1, (3.4)

w₊

w₋

w dw

2

ε² (1−ε²)G(w)+^ε₂²w²−(λ_εw+A_ε)

=M. (3.5)

We investigate the behavior of w₊, w₋, λ_ε and Aε as ε→0 in the following discussion. First, invoking of (3.1), we have

Eε(w)= 1 0

ε²

2 |wx|²+(1−ε²)G(w)+ε² 2w²

dx

= 1 0

2(1−ε²)G(w)+ε²w²−(λ_εw+A_ε)

dx

= 1 0

ε²|w_x|²+(λ_εw+A_ε)

dx.

We claim the following lemma:

Lemma 3.1. Let w(x)be the monotone decreasing minimizer of (1.11). Then there is an εc>0 such that for 0 < ε < εc

w₋=w(1)=O

logκε

κε

, (3.6)

C₁logκε

κε ≤λ_ε≤C₂logκε

κε

, (3.7)

Aε=O logκ_ε

κ_ε

(3.8) hold, where C1and C2are positive constants.

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Proof. Because

(1−ε²)G(w₋)≤Eε(w)≤Clog(κε) κ_ε , and

G(w₋)=w₋ 1 0

G(tw₋) dt=w²₋ 1 0

(1−t )G(tw₋) dt,

we have

w₋=O

logκε

κε

,

which implies (3.6).

Solving the first two equations for λε, we find

λ_ε=(1−ε²)(G(w₊)−G(w₋))+^ε₂²w²₊−^ε₂²w₋²

w₊−w₋ .

Because w₋→0 and ε²w²₊→0, we obtain

εlim→0

λε−logμ_ε μ_ε

=0.

Moreover, by με≤2√ 2C0κε,

logμε

με = κε

με

1 κε

(logκ_ε+log(με/κ_ε))

≥C1

logκ_ε κ_ε , therefore,

C₁logκε

κε ≤λ_ε. Since the energy Eε(w_ε)can be written as

Eε(w_ε)= 1 0

ε²|(w_ε)_x|²+(λ_εw_ε+A_ε)

dx,

this lead us to the inequalities

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0≤ 1 0

(λ_εw+A_ε) dx≤C₀logκε

κε

,

thus

0≤λ_εM+A_ε≤C₀logκ_ε κε

. (3.9)

From (3.3)

λ_εw₋=(1−ε²)G(w₋)+ε²

2w₋² −A_ε>−A_ε follows. Applying this to (3.9)yields

λ_εM−λ_εw₋=λ_ε(M−w₋) < λ_εM+A_ε≤C₀logκ_ε κ_ε , thus

λ_ε≤C₂logκε

κε

. We have (3.7).

Finally, by

−λεM≤Aε≤C0

logκ_ε κ_ε , we can easily obtain (3.8)and conclude the proof. 2 3.2. Rescaling the equation

We complete the proof for (iii) of Theorem 1.1by applying the result of this subsection. Here, we are going to find the limiting profile of the boundary layer at x=0.

Define rescaled functions Wε:ε:=(0, κε) →Rby W_ε(y)= 1

κ_εw_ε y

κ_ε

. The equation (1.8)becomes

−W_ε+(1−ε²) logκ_ε

κ_ε²W_ε

(κ_εW_ε+1)²+ 1

κ_ε²W_ε= κ_ελ_ε

logκ_ε on_ε, W_ε(0)=W_ε(κ_ε)=0,

κε

0

W (y) dy=M. (3.10)

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The first of the equation (3.1)is re-written as (W_ε(y))²= 2

logκ_ε

(1−ε²)

log(κεWε+1)+ 1

κ_εW_ε+1−1

+ε²κ_ε²

2 W_ε²−(λ_εκ_εW_ε+A_ε)

(3.11) in the interval ε.

Because the sequence {με/κε}is bounded, this implies {Wε}is bounded in L^∞-norm. From (3.11), we also know {W_ε}is bounded in W^1,^∞. Thus, there exists a subsequence {εj}such that {W_ε_j}converges to a function W0in C^0,α()on any compact subset ⊂(0, ∞)for 0 ≤α <1 and the subsequence {μ_ε_j/κ_ε_j}converges to some constant.

Lemma 3.2. Suppose that wεis a minimizer for the variational problem

inf

w∈AEε(w)= inf

w∈A

1 0

ε²

2|w_x|²+(1−ε²)G(w)+ε² 2w²

dx.

Then there exists a subsequence {Wε_j}and a function W0∈C_loc^0,αsuch that the subsequence Wε_j

converges to W₀in C_loc^0,αwhere α∈ [0, 1).

Lemma 3.3. Suppose W_ε_j→W₀in C_loc^0,α. For any y∈supp(W0), we have

jlim→∞

log(κε_jW_ε_j(y)+1) logκ_ε_j =1.

Proof. For y∈supp(W0), we have

log(κεjW_ε_j(y)+1)

logκε_j −log(κεjW₀(y)+1) logκε_j =

log

1+^W^εj^(y)⁻^W⁰^(y)

W0(y)+_κεj¹

logκε_j →0 and

jlim→∞

log(κεjW₀(y)+1) logκε_j = lim

j→∞

logκ_ε_j +log(W0(y)+_κ¹

εj)

logκε_j =1. 2

By (3.11)we have W_ε(y)

= − 2

logκ_ε

(1−ε²)

log(κεW_ε+1)+ 1

κ_εW_ε+1−1

+ε²κ_ε²

2 W_ε²−(λ_εκ_εW_ε+A_ε) 1/2

.

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Using {W_ε}converges to W₀ in W_loc^1,p(0, ∞)and the above estimates together with (3.7)and (3.8), we find that the limiting function W0should satisfies the equation

W= −

2χsupp(W )−2a W1/2

(3.12) on the half real line (0, ∞). The solution of this equation is

W0(y)= 1

a−^ay₂² for 0< y <

√2 a ,

0 for

√2

a ≤y. (3.13)

Here, we set ato be a constant such that a=limj→∞

κ_εj

μ_εj. The exact value of the constant awill be determined later.

3.3. L¹convergence

Although the mass constraint (3.10) holds for any ε >0, it is not guaranteed that M= _∞

0 W₀(x) dx holds. The reason is that the convergence Wεj(x)to W0(x)is only locally uni- form in C^0,α for 0 ≤α <1. We compute the limit ^κ_εj

0 W_ε_j(x)dx below. Apply the change of the independent valuable xby z=w_ε(x), we have

M= 1

0

w_ε(x) dx=

με

γε

√ε 2

√ z

_ε(z)dz,

where we put

_ε(z):=(1−ε²)G(z)+ε²

2z²−λ_εz−A_ε, γ_ε:=w_ε(1) (recall με=wε(0)). We note

_ε(γ_ε)=(μ_ε)=0.

Moreover, G(x)is a monotone increasing and

G(z)=g(z) >0 (0≤z <1), G(z) <0 (1< z).

Hence,

G(z)−G(γε) z−γε

(z∈(γ_ε,1]), G(με)−G(z)

με−z (z∈ [1, με)) are monotone increasing and decreasing respectively.

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We first show

εlim→0

1 γε

√ε 2

√ z

_ε(z)dz=0. (3.14)

By ε(γ_ε) =0, we write

_ε(z)=(1−ε²)(G(z)−G(γ_ε))+ε²

2 (z²−γ_ε²)−λ_ε(z−γ_ε)

=(z−γ_ε)h_ε(z), h_ε(z):=(1−ε²)G(z)−G(γ_ε)

z−γ_ε +ε²

2(z+γ_ε)−λ_ε,

where hε(z)can be continuously extended up to z=γεas hε(γε) =(1 −ε²)g(γε) +ε²γε−λε. Since hε(z)is monotone increasing in [γε, 1], we have

1 γ_ε

√ε 2

√ z

_ε(z)dz≤ ε

√2

√ 1 h_ε(γ_ε)

1 γ_ε

√ z z−γε

dz.

Invoking of γ_ε=O(√

logκ_ε/κ_ε), we can conclude (3.14).

We next deal with the integral over the range [1, με]. Take θεsuch that

εlim→0θ_ε=0, lim

ε→0κ_ε^θ^ε= ∞, and define

β_ε:=κ_ε¹⁻^θ^ε. (3.15)

For instance take θε=1/√

logκ_ε. We separate the integral as

με

1

√ε 2

√ z

_ε(z)dz=Iε+Jε,

I_ε:=

β_ε

1

√ε 2

√ z

ε(z)dz, J_ε:=

μ_ε

βε

√ε 2

√ z

ε(z)dz.

We prove limε→0Iε=0. By the change of the valuable z=κεζ in the integral we obtain

I_ε=

βε/κε

1/κε

√1 2

εκ_ε²ζ

√_ε(κ_εζ )dζ=

1/κ _ε^θε

1/κε

√1 2

ζ

_ε(κ_εζ )/logκ_εdζ,

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where we used εκ_ε²=√

logκ_ε. Recall ε(μ_ε) =0 and put αε:=μ_ε/κ_ε. Then _ε(κ_εζ )=(α_ε−ζ )q_ε(ζ ),

q_ε(ζ ):= −(1−ε²)G(κεαε)−G(κεζ ) α_ε−ζ −ε²κ_ε²

2 (ζ+α_ε)+λ_εκ_ε For ζ ∈ [1/κε, β_ε/κ_ε]= [1/κε, 1/κ^θ^ε],

q_ε(ζ )≥ −(1−ε²)G(κεαε)−G(1) αε−1/κε −ε²κ_ε²

2 (1/κ^θ^ε+α_ε)+λ_εκ_ε. (3.16) As seen in the proof of Lemma 3.1, we have

λε=(1−ε²)(G(μ_ε)−G(γ_ε))+^ε₂²μ²_ε−^ε₂²γ_ε²

μ_ε−γ_ε .

Applying this equality to the right hand side of (3.16), we assert that there is c1>0 such that q_ε(ζ )/logκ_ε≥c₁/logκ_ε

holds for every small ε >0. Indeed, we can compute the leading terms of the right hand side of (3.16)as

κε

G(μ_ε)−G(γ_ε)

μ_ε−γ_ε −G(κ_εα_ε)−G(1) α_ε−1/κε =κε

G(μ_ε)−G(γ_ε)

μ_ε−γ_ε −G(μ_ε)−G(1) μ_ε−1

= κ_ε

(μ_ε−γ_ε)(μ_ε−1){−G(μ_ε)(1−γ_ε)+(G(1)−G(γ_ε))μ_ε+G(γ_ε)−G(1)γε}. This term is bounded away from zero as ε→ 0 because of κ_ε/μ_ε =O(1), G(μ_ε)/μ_ε = O(logκ_ε/κ_ε).

We compute

1/κ _ε^θε

1/κε

√ ζ

α_ε−ζdζ

=2

3{(α_ε−1/κ_ε^θ^ε)^3/2−(α_ε−1/κε)^3/2} −2αε(

α_ε−1/κε^θ^ε−

α_ε−1/κε)

≤ 2αε(1/κε^θ^ε−1/κε)

α_ε−1/κε^θ^ε+√

α_ε−1/κε

≤ α_ε/κ_ε^θ^ε

α_ε−1/κε^θ^ε

.

Thus,