Flux limited Keller-Segel system and kinetic equations

# The flux limited Keller-Segel system; properties and derivation from kinetic equations

Benoit Perthame Sorbonne Université, Université Paris-Diderot, CNRS, INRIA, Laboratoire Jacques-Louis Lions, F-75005 Paris, France Nicolas Vauchelet Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539, Laboratoire Analyse Géométrie et Applications, 93430 Villetaneuse, France  and  Zhian Wang Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
###### Abstract.

The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyperbolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently.

We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system.

###### Key words and phrases:
flux limited Keller-Segel system, chemotaxis, drift-diffusion equation, asymptotic analysis, long time asymptotics
###### 2000 Mathematics Subject Classification:
35A01, 35B40, 35B44, 35K57, 35Q92, 92C17

## 1. Introduction

Chemotaxis, the directed movement of an organism in response to a chemical stimulus, is a fundamental cellular process in many important biological processes such as embryonic development [26], wound healing [39], blood vessel formation [10, 17], pattern formation [6, 34]) and so on. Well-known examples of biological species experiencing chemotaxis include the slime mold amoebae Dictyostelium discoideum, the flagellated bacteria Escherichia coli and Salmonella typhimurium, and the human endothelial cells [29]. Mathematical models of chemotaxis were derived from either microscopic (individual) or macroscopic (population) perspectives, which have been widely studied in the past four decades. The macroscopic chemotaxis model has been first developed by Keller-Segel in [24] to describe the aggregation of cellular slime molds Dictyostelium discoideum and in [25] to describe the wave propagation of bacterial chemotaxis. Because thresholds on the cell number decide when solutions will undergo smooth dispersion or blow-up in finite time, and because of the interest of related functional analysis, this system has attracted an enormous number of studies (cf. [36]).

In this paper, we are interested in the flux-limited Keller-Segel (FLKS) system in the whole space . Some particular form of such system has already been introduced in [20, 11]. It describes the evolution of cell density and chemical signal concentration at and time , and is based on the physical assumption that the chemotactic flux function is bounded, modeling velocity saturation in large gradient environment. It reads

 ⎧⎪⎨⎪⎩∂tρ=DΔρ−div(ρϕ(|∇S|)∇S),x∈Rd, t>0,τ∂tS−ΔS+αS=ρ,ρ(0,x)=ρ0(x)≥0,and S(0,x)=S0(x)if τ=1. (1.1)

We denote the cell total number . This system is conservative, that is

 M=∫Rdρ(t,x)dx,∀t≥0.

Compared to the classical Keller-Segel system, the chemotactic response function depends nonlinearly on the chemical concentration gradient. We assume flux limitation, that means there is a positive constant such that

 maxr∈R+ϕ(r)=ϕ(0),maxr∈R+|rϕ(r)|=A∞. (1.2)

These boundedness assumptions on the flux induce that solutions to (1.1) exist globally in time (see e.g. [19, 11]), unlike the Keller-Segel system for which finite time blow-up may occur.

The motivation to study the FLKS system (1.1) comes from its derivation from mesoscopic kinetic model. The first microscopic/mesoscopic description of chemotaxis model is due to Patlak [35] whereby the kinetic theory was used to express the chemotactic velocity in term of the average of velocities and run times of individual cells. This approach was essentially boosted by Alt [1] and developed by Othmer, Dunber and Alt [32] using a velocity-jump processes which assumes that cells run with some velocity and at random instants of time they changes velocities (directions) according to a Poisson process. The advantage of kinetic models over macroscopic models is that details of the run-and-tumble motion at individual scales can be explicitly incorporated into the tumbling kernel and then passed to macroscopic quantities through bottom-up scaling (cf. [18, 45, 46, 15, 16, 42, 37, 12]), where the rigorous justification of upscaling limits have been studied in many works (see [9, 21, 22, 23, 27] and reference therein). Denoting by the cell number density, at time , position moving with a velocity (compact set of with rotational symmetry), the governing evolution equation of this process is described by a kinetic equation reading as:

 ∂f∂t+v⋅∇xf=∫V(T[S](v,v′)f(t,x,v′)−T[S](v′,v)f(t,x,v))dv′, (1.3)

The tumbling kernel describes the frequency of changing trajectories from velocity (anterior) to (posterior) depending on the chemical concentration or its gradient. Because cells are able to compare present chemical concentration to previous ones and thus to respond to temporal gradients along their pathways, the tumbling kernel may depend on the pathway (directional derivative) and takes the form ([12, 37])

 T[S](v,v′)=λ0+σΨ(DtS),DtS=∂tS+v′∇S, (1.4)

where denotes a basal meaning tumbling frequency, accounts for the variation of tumble frequency modulation and denotes the signal response (sensing) function which is decreasing to express that cells are less likely to tumble when the chemical concentration increases.

The first goal of the present paper is to derive the FLKS system (1.1) as the parabolic limit of the kinetic equation (1.3)-(1.4) and relate the flux limiting function to . In particular, we introduce a new rescaling, related to the stiffness of signal response, which has been shown to be important to describe the traveling pulses of bacterial chemotaxis observed in the experiment [41, 40, 13] and is related to instabilities both of the FLKS system and the kinetic equation [38, 8]. In particular, we wish to go further than the case proposed in [40], when the response function is bi-valuated step (stiff) function: , where the parabolic limit equation of (1.3) is

 ∂tρ=Δρ−div(ρu[S]),u[S]=J(St,|∇S|)∇S|∇S|.

with denoting a macroscopic quantity depending on and/or (cf. [12, 40]). Our method of proof is based on the method of moments and on compactness estimates to treat the nonlinearity.

Our second goal is to study the long time behaviour of solutions to the FLKS system (1.1) and the existence of stationary radial solutions. Contrary to the Keller-Segel system for which finite time blow-up of weak solutions is observed, solutions to (1.1) under assumption (1.2) exist globally in time. For a study of the long time convergence towards radially symmetric solutions for the Keller-Segel system, which may be computed explicitely, we refer to [7]. Yet, we do not have an explicit expression of radially symmetric solutions for system (1.1). Then, we prove that when the degradation coefficient is positive (), diffusion takes the advantage over attraction. On the contrary, when the degradation coefficient is disregarded (), the total mass of the system, denoted , appears to be an important parameter. Indeed, when , we observe a threshold phenomenon, with a critical mass in dimension , for the existence of radial stationary solution.

More precisely, our main results may be summarized as follows :

• Radial stationary solutions when . (Theorem 3.1)
For , there are no positive radially symmetric steady state solutions to (1.1) with finite mass .
For , system (1.1) has positive radially symmetric steady state if and only if .

• Long time behaviour in one dimension when . (Corollary 4.2)
For and , for any , there exists a unique stationary solution . Moreover, denoted by is the solution of the dynamical system (1.1) with and . We have

 limt→+∞W2(ρ(t),¯ρ)=0,

where denotes the Wasserstein distance of order .

• Long time behaviour. (Theorem 5.1)
In dimension or . Let be a solution of (1.1) on . If and , or if , and is small enough, then we have for any ,

 ∥ρ(t)∥Lp(Rd)≤Ct−d2(1−1p),

where is a nonnegative constant. Notice that this estimate on the time decay is the same as the one for the heat equation.

The situation in bounded domain is quite different. Indeed, existence of steady state solutions for on bounded domain with Neumann boundary conditions has been investigated in [11] in one dimension. Based on a bifurcation analysis, they observe spiky solutions when the chemotactic sensibility is large. See also [28, 31, 43, 44] for spiky steady states in chemotaxis models.

The outline of the paper is as follows. In the next section, we derive the flux-limited Keller-Segel model (1.1) from the kinetic system with the appropriate scaling. Section 3 deals with the existence of radially symmetric stationary states in dimension greater than . The one dimensional case is investigated in section 4. The study of the long time behaviour is performed in section 5 where Theorem 5.1 is proved. Then we summarize briefly our results in a conclusion and provide open questions related to this work. Finally, an appendix is devoted to some technical lemma useful throughout the paper.

## 2. Derivation of FLKS from kinetic model

Our approach uses the stiffness parameter and a smoothed stiff response function . In other words, we consider the following smooth stiff tumbling kernel

 T[S](v,v′)=λ0+σΨ(DtS/ε).

A possible example, as suggested in [41], is . Other examples include, for instance, . The case corresponds to a stepwise stiff response function mentioned in the introduction. In [41], it has been measured that . For convenience, we write for some scaling constant and rewrite above tumbling kernel as

 T[S](v,v′)=λ0+χεΨ(DtS/ε). (2.1)

In this paper, we shall take as a scaling parameter and derive the parabolic limit of kinetic models of chemotaxis which turns out to be the FLKS model (1.1) as long as the response function is bounded.

### 2.1. Rescaling of the kinetic equation

We summarize the condition on the response function as follows:

 Ψ∈C1∩L∞(R),Ψ′(z)<0∀z∈R. (2.2)

Applying the parabolic scaling into (1.3) with the tumbling kernel (2.1), and recovering by for convenience, we get

 ε2∂∂tfε(t,x,v)+εv⋅∇xfε(t,x,v)=Lε[Sε](fε) (2.3)

with

 Lε[S](f)=∫V(Tε[S](v,v′)f′−Tε[S](v′,v)f)dv′,f′:=f(t,x,v′),f:=f(t,x,v),

and

 Tε[S](v,v′)=λ0+χεΨε[S](v,v′),  Ψε[S](v′,v)=Ψ(ε∂tS+v⋅∇xS). (2.4)

Since the chemical production and degradation are much slower than the movement (cf. [41, 13]), we assume prior to the microscopic scaling that the equation for is given by

 τ∂tS=ΔS+ε2(ρ−αS),

where , and is a constant denoting chemical decay rate.

After rescaling, we may state the complete problem we are interested in. On one hand, the equation for the chemical concentration with the parabolic scaling reads as

 {τ∂tSε−ΔSε+αSε=ρε,ρε(t,x)=∫Vfε(t,x,v)dv,Sε(0,x)=S0(x)∈L1+∩L∞(Rd)for τ=1. (2.5)

On the other hand, substituting (2.4) into (2.3), we get the final form of the kinetic equation

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩ε2∂∂tfε(t,x,v)+εv⋅∇xfε(t,x,v)=λ0∫V(f′ε−fε)dv′+χε∫V(Ψ∗ε[Sε]f′ε−Ψε[Sε]fε)dv′,fε(0,x,v)=f0(x,v)∈L1+∩L∞(Rd×V),Ψε[Sε]=Ψ(ε∂tSε+v⋅∇xSε), Ψ∗ε[Sε]=Ψ(ε∂tSε+v′⋅∇xSε). (2.6)

### 2.2. Well-posedness, a priori estimates and compactness

The well-posedness and macroscopic limits of kinetic models of chemotaxis where the tumbling kernel depends on the chemical concentration or its spatial derivative have been extensively studied e.g. in [32, 33, 9, 21, 5] either formally or rigorously, based on the advanced functional analytical tools available for kinetic equations. When the tumbling kernel depends on the pathway derivative , the formal limits have been studied in [33, 12] and rigorous justification was given in [14] for the two species case. The well-posedness of equations (2.3)-(2.5) and the limit as are the direct consequence of the results of [14]. For completeness, we present, without proof, the following result

###### Theorem 2.1 (Existence, a priori estimates).

Let and assume (2.2). There exists a unique global solution of (2.5)-(2.6), , . Moreover, there is a constant , independent of , such that

 e−Ct∫Rd×Vfε(t)2dxdv+λ04ε2∫t0∫Rd×V×V|f′ε−fε|2dvdv′dxds≤∥f0∥22, (2.7)
 ∥ρε∥L2([0,T]×Rd)+∥Jε∥L2([0,T]×Rd)≤C(T)eCT,Jε:=1ε∫Vvfεdv. (2.8)

The flux in (2.8) arises because integration of (2.6) with respect to gives

 ∂ρε∂t+divxJε=0. (2.9)
###### Lemma 2.1 (Strong local compactness on ∇Sε(t,x)).

The signal function is uniformly bounded and is strongly locally compact in for all .

###### Proof.

We use that is bounded in from Theorem 2.1.

From usual elliptic or parabolic regularizing effects (see Lemma A.2 in Appendix), and using only the above bounds for , we conclude that is bounded in , with ( in dimensions , in dimension 4).

 τ∂t(∂tSε)−Δ(∂tSε)+α(∂tSε)=∂tρε=−divJε,Jε∈bddL2([0,T]×Rd)

and gives , for any any thanks to the direct estimates on the heat kernel (see (A.1) in Appendix with , , ).

Next, we notice by a similar argument, that is bounded in for . Compactness in for also follows from the convolution formula. Finally, we write

 τ∂t(∂t∇Sε)−Δ(∂t∇Sε)+α(∂t∇Sε)=−∇divJε,

and thus we conclude that , for all , thus providing time compactness. ∎

As a conclusion, we may extract a subsequence such that locally in all spaces , , . Notice that the bounds above tell us that, as ,

 Ψε(n)[Sε(n)]=Ψ(ε(n)∂tSε(n)+v⋅∇xSε(n))→Ψ(v⋅∇xS0).

### 2.3. The convergence result

As a consequence of the a priori estimates in Theorem 2.1 and the discussion in section 2.2, we may also extract subsequences (still denoted by ) such that, weakly in for all , as , we have

 fε(n)→f0(t,x,v)=ρ0(t,x)F(v),Jε(n)(t,x):=1ε(n)∫Vvfε(n)dv→J0(t,x) (2.10)

where is a uniform distribution on :

 F(v)=\mathbbm1{v∈V}|V|. (2.11)

With the symmetry assumption of , it satisfies , and In the limit we infer from (2.9) that

 ∂ρ0∂t+∇x⋅J0=0. (2.12)

The flux can be identified and we are going to show in the next subsection the following

###### Theorem 2.2 (Derivation of the FLKS system).

Assuming (2.2), the above limit satisfies the FLKS system (1.1) with initial condition and

 D=1λ0|V|2∫Vv⊗vdv,ϕ(u)=−1λ0|V|u∫Vv1Ψ(v1u)dv>0 for u≥0,

where is the first component of the vector field .

Notice that is well defined by continuity and .

### 2.4. Asymptotic analysis

In order to complete the proof of Theorem 2.2, we proceed to find the flux term in (2.12). Multiplying (2.6) by and integrating, we get

 ε∂∂tJε(t,x)+∇x⋅∫Vv⊗vfεdv =λ0ε∫Vv∫V(f′ε−fε)dv′dv+χ∫Vv∫V(Ψ∗ε[Sε]f′ε−Ψε[Sε]fε)dv′dv =−λ0|V|Jε−χ|V|∫VvΨε[Sε]fεdv

using the definition of in (2.8), the definition of in (2.6), and the symmetry of .

We may pass to the weak limit and find, based on the above mentioned strong compactness for and its derivatives as well as (2.10) and (2.11), that

 ∫Vv⊗vdv∇xρ0=−λ0|V|J0−χ|V|ρ0∫VvΨ(v⋅∇S0)dv.

In other words, we have identified the flux term

 J0=−∇xρ01λ0|V|2∫Vv⊗vdv−χλ0|V|ρ0∫VvΨ(v⋅∇S0)dv. (2.13)

Using (2.13), the leading order terms of (2.9) and (2.5) lead to the following drift-diffusion equations:

 {∂tρ0=div(D∇ρ0−χρ0u[S]),τ∂tS0=ΔS0+ρ0−αS0, (2.14)

where

 D=1λ0|V|2∫Vv⊗vdv,u[S]=−1λ0|V|∫VvΨ(v⋅∇S0)dv. (2.15)

By rotational symmetry of , is proportional to and hence yields the expression of in Theorem 2.2. Due to the hypothesis (H) on , the drift velocity term is uniformly bounded in time and space . This is the main feature of the macroscopic limit model resulting from the stiff response postulated in the kinetic models. ∎

### 2.5. Example

We consider a specific form of signal response function as follows

 Ψ(Y/ε)=−Y√ε2+Y2  or  Ψ(z)=−z√1+z2

and derive an explicit flux-limited Keller-Segel system. When , which is a sign function reflecting the stepwise stiff response. However, as , is smooth and .

By substitution, we have from (2.4) that

 Ψε[S](v′,v)=−ε∂tS+v⋅∇S√1+(ε∂tS+v⋅∇S)2.

Then the limit equations of (2.3)-(2.5) read as (see (2.14)-(2.15))

 {∂tρ=div(D∇ρ−ρϕ(|∇S|)∇S),τ∂tS=ΔS+ρ−αS, (2.16)

where we have recovered by for brevity and, by rotational symmetry of V,

 D=1λ0|V|2∫Vv⊗vdv,ϕ(|∇S|)=χλ0|V|∫Vv21√1+v21|∇S|2dv, (2.17)

where is the first component of colinear to : Clearly both and are bounded for all , which implies that the chemotactic (drift) velocity is limited. The system (2.16) with (2.17) gives a specific example of the FLKS system (1.1).

### 2.6. Global existence for the macroscopic system

Finally, we state the existence result for system (1.1) under the assumption (1.2). A specific example of function satisfying this set of assumptions has been given in (2.17). Under the assumption (1.2), the chemotactic (or drift) velocity term is bounded and hence the global existence of classical solutions of (1.1) can be directly obtained.

###### Theorem 2.3 (Global existence).

Let with and . Let such that (1.2) holds. Then the Cauchy problem (1.1) has a unique solution such that

 ∀t>0,∥ρ(t,⋅)∥L∞(Rd)≤C,d≥2,

where is a constant independent of . Moreover cell mass is conserved: .

###### Proof.

The proof consists of two steps. The first step is the local existence of solutions which can be readily obtained by the standard fixed point theorem (cf. [3, 2]). The second step is to derive the a priori bound of in order to extend local solutions to global ones. This can be achieved by the method of Nash iterations as it is well described in [19, Lemma 1]. Although the procedure therein was shown for bounded domain with Neumann boundary conditions, the estimates directly carry over to the whole space . ∎

Since it is proved in section 5 that when diffusion takes the advantage over attraction implying the time decay towards zero of the solutions to system (1.1), we are only interested in the case . The stationary problem for system (1.1) is non-trivial due to the nonlinearity. Below we explore a simpler case: existence of radial symmetric stationary solutions. The stationary system of (1.1) when written in radial coordinates for reads

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−1rd−1(rd−1S(r)′)′=ρ(r),r>0,1rd−1[rd−1(−ρ(r)′+ρ(r)S(r)′ϕ(|S(r)′|))]′=0,S′(0)=0,ρ′(0)=0. (3.1)

Notice that there is another relation, at infinity, expressing that the mass is given by

 M|Sd−1|=∫+∞0rd−1ρ(r)dr=−∫+∞0(rd−1S(r)′)′dr=−limr→∞rd−1S(r)′. (3.2)

We are going to prove the following result.

###### Theorem 3.1.

There are no positive radially symmetric steady state solutions with finite mass to system (1.1) with in dimension . In dimension 2 (i.e. ), system (1.1) with has radially symmetric steady states if and only if .

###### Proof.

We use the unknown in order to carry out the analysis. The equation on in (3.1) now reads , and we obtain

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩v′=rd−1ρ(r)≥0,ρ′=−ρv(r)rd−1ϕ(v(r)rd−1),v(0)=0, ρ(0)=a>0. (3.3)

From (1.2) and the second equation of (3.3), we get and hence for all . Furthermore from the first equation of (3.3), we know that is non-decreasing and has a limit as which determines the total mass according to (3.2). Hence for finite mass , has a finite limit and thus for large enough, say for some , we have

 (lnρ)′=−v(r)rd−1ϕ(v(r)rd−1)≥−brd−1, (3.4)

for some positive constant .

If , integrating (3.4) yields

 ρ(r)≥Cebd−21rd−2.

This is incompatible with finite mass in (3.2).

In dimension , we recover a phenomenon similar to the multiple solutions for the critical mass in the Keller-Segel system but explicit solutions are not available. The system (3.3) reduces to

 {rv′′=v′(1−vϕ(vr)),r>0,v(0)=v′(0)=0. (3.5)

By the boundary conditions in (3.3), we see that positive solutions behave as for and some constant . Then, the proof of Theorem 3.1 is a consequence of Lemma 3.1 and Proposition 3.2 below. Lemma 3.1 states that a necessary condition of existence of radial solution is . Proposition 3.2 shows that for any finite mass larger than the critical mass , there exist radial solutions with mass to system (1.1) with . ∎

###### Lemma 3.1.

Let be a positive solution to (3.5). Then is increasing. If is bounded, then

 limr→+∞v(r)>4ϕ(0).
###### Proof.

We split the proof into three steps:

1. From the behaviour near , we know that for small enough. If we had for some , then the unique solution of (3.5) is which is a contradiction. Therefore for all .

2. Since from (1.2), we deduce from (3.5) that, for all ,

 v′(1−vϕ(0))≤rv′′.

This inequality may be rewritten as

 v′(2−vϕ(0))≤(rv′)′. (3.6)

Integrating (3.6) from to and using boundary conditions in (3.5), we deduce that

 2v−ϕ(0)2v2≤rv′.

This inequality implies that

 limr→∞v(r)≥4ϕ(0)

because if it were smaller, we would have for some and is not integrable.

As a consequence, we know that as , and thus, for some nonnegative constant , it holds that

 r3v′(r)≤C,  as  t→∞. (3.7)
3. We may go further and write the first equation of (3.5) as

 (rv′)′=v′(2−vϕ(0))+vv′[ϕ(0)−ϕ(vr)].

Integrating it from to , we have

 rv′=2v−ϕ(0)v22+Q(r),Q(r)=∫r0v(s)v′(s)[ϕ(0)−ϕ(v(s)s)]ds>0.

Therefore as , using (3.7), there holds that

 ϕ(0)v2∞2−2v∞=Q(∞)>0.

It implies that .

###### Proposition 3.2.

Let the function satisfies (1.2). Then for any , there exists a solution to (3.5) such that .

###### Proof.

We want to prove that for any , there exists such that the solution to (3.5) verifying and . We first simplify the problem by introducing the change of variable . Setting , we deduce, from straightforward computations, that is a solution to the system

 ⎧⎪⎨⎪⎩u′′=2u′1−y(1−u4yϕ(√1−yyu)),y∈(0,1),u(0)=0,u′(0)=a2. (3.8)

We are left to use a shooting method to show there is a number such that (3.8) has a solution satisfying for any .

• By definition of and thanks to the above results, we have that and on .

• Since , we deduce from (3.8) that

 (y(1−y)u′)′≥−uu′2ϕ(0)+u′.

After integration we obtain

 y(1−y)u′≥−u24ϕ(0)+u.

Thus, when , we have

 u′(y)u(y)−u2(y)ϕ(0)/4≥1y(1−y).

Upon integration, we find a positive constant such that for all and , we have

 u(y)≥4λy1−y+λϕ(0)y → 4ϕ(0) as y→1.

Thus, by continuity and the fact is increasing, we have that .

• Let us prove that: For any , there exists a number small enough such that the solution to (3.8) satisfies .
In the vicinity of , we have . Then for small enough, there exists such that . (Indeed, if it is not true, we will have on , which is not possible for since ). The function being bounded on , let us denote . By the same token as above, we deduce from (3.8) that

 (y(1−y)u′)′≤−uu′2ϕm+u′.

Integrating above inequality over gives

 y(1−y)u′≤−u22ϕm+u. (3.9)

On one hand, integrating (3.9) from to , we get

 u(y)≤2u(12)y2(1−y)(1−ϕm4u(12))+ϕm2u(12)y≤Cmy.

On the other hand, integrating (3.9) between and , we obtain

 u(y)≤u(y0)y/y01−y1−y0(1−u(y0)4ϕm)+ϕm4u(y0)yy0=ay(1−y0)1−y+ϕm4a(y−y0).

Then,

 √1−yyu(y)≤ξ=min⎛⎜⎝Cm√y(1−y),a√y(1−y)(1−y0)1−y+ϕm4a(y−y0)⎞⎟⎠.

It is clear that as for all . Let small. Then by the continuity of the function , for small enough, we can deduce from the above estimate that . Then, we can redo the same estimate as above, replacing by , we arrive at

 u(y)≤ay(1−y0)1−y+ϕ(0)−ε4a(y−y0),

which implies by taking ,

 u(1)≤b=:4ϕ(0)−ε.
• Let us prove that .

By the second assumption on in (1.2), we know that for any and ,

 ϕ(√1−yyu)≤A∞u√y1−y.

Since , we get from (3.8) that

 u′′≥2u′1−y(1−A∞4√y(1−y)