A refined asymptotic result of one-dimensional flux limited Keller–Segel models

Abstract

In this paper, we consider a flux-limited Keller–Segel model derived in [16, 18] in a one-dimensional bounded domain and give a refined asymptotic result of the spiky steady state by using the Sturm oscillation theorem in a more meticulous way based on the existence result of spiky steady state in [4], showing a more accurate characterization of the cell aggregation phenomenon.

Keywords:

• Flux-limited Keller–Segel model
• spiky steady states
• Sturm oscillation theorem

1. Introduction

Chemotaxis is a fundamental cellular reaction for survival, which describes the directional movements towards or away from the chemical stimuli. It is a fundamental mechanism in many important biological phenomenon such as the healing of wound [17], pattern formation [2, 14] and so on. Mathematical models of chemotaxis were widely studied in recent years and the model proposed by Keller and Segel [9, 10] was well known and received widespread acceptance. A lot of outstanding work has been done on the Keller–Segel system and its variants to study the aggregation and wave propagation of cells, two important phenomena in chemotaxis, and we refer the reader to [5, 7, 8] for details. About the wave propagation phenomenon, we refer the reader to [12] and [19], which are the papers for the traveling waves and the periodic waves generated from the Keller–Segel model.

In this paper, we focus on the flux-limited Keller–Segel (FLKS) system, which reads as follows: (1) $\begin{array}{r}\{\begin{array}{ll}{\mathrm{\partial }}_t\rho =\mathrm{\Delta }\rho -\mathrm{\nabla }\cdot \left(\chi \right(\rho \psi \left(|\mathrm{\nabla }S|\right)\mathrm{\nabla }S),& x\in \mathrm{\Omega },\\ {\mathrm{\partial }}_{t}S=\mathrm{\Delta }S-\alpha S+\rho ,& x\in \mathrm{\Omega },\\ (\rho ,S)(0,x)=({\rho }_0,S_0)\left(x\right),& \end{array}\end{array}$(1) where $\rho (t,x)$ represents the density of cells and $S(t,x)$ the concentration of chemical substance at time t and position x; ψ is the function to describe the chemical response mechanism, which is so-called the chemotactic sensitivity function with the chemotactic coefficient $\chi >0$. ψ is assumed to be a smooth function and a typical form of ψ is $\psi \left(r\right)=\frac{1}{\sqrt{1+r^2}}$. The assumption of limited flux here means that there exists a positive constant $A_{\mathrm{\infty }}$ such that $\underset{r\in {\mathbb{R}}^{+}}{max}|r\psi \left(r\right)|=A_{\mathrm{\infty }},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\psi (|\cdot |)\in C^2(\mathbb{R},{\mathbb{R}}^{+})\mathrm{a}\mathrm{n}\mathrm{d}\psi \left(0\right)>0,$and ${\mathbb{R}}^{+}=\{r\in \mathbb{R}:r\ge 0\}$. So when the cells are in the large gradient environment ($\mathrm{\nabla }S$ is large), the chemotactic flux, $\psi \left(|\mathrm{\nabla }S|\right)\mathrm{\nabla }S$, is bounded to model velocity saturation.

The earliest relevant work about the FLKS system is in [15], where Patlak first used the kinetic theory to express the chemotactic velocity in term of the average of velocities and run times of individual cells. Later, this approach was boosted and developed by Alt et al. [1, 13] and so on. The derivation of the kinetic chemotaxis model in previous work gives the motivation to study the FLKS system. One can see details of the derivation of the FLKS system in [16, 18].

In the FLKS system, since the global existence in time t of the solutions of (1(1) $\begin{array}{r}\{\begin{array}{ll}{\mathrm{\partial }}_t\rho =\mathrm{\Delta }\rho -\mathrm{\nabla }\cdot \left(\chi \right(\rho \psi \left(|\mathrm{\nabla }S|\right)\mathrm{\nabla }S),& x\in \mathrm{\Omega },\\ {\mathrm{\partial }}_{t}S=\mathrm{\Delta }S-\alpha S+\rho ,& x\in \mathrm{\Omega },\\ (\rho ,S)(0,x)=({\rho }_0,S_0)\left(x\right),& \end{array}\end{array}$(1) ) with flux limited assumption is proven in [4, 6], blow-up will not happen in finite time. In [16], the large-time behaviour of the FLKS system in ${\mathbb{R}}^n(n=2,3)$ was studied with a fixed initial mass ${\int }_{{\mathbb{R}}^n}{\rho }_0\left(x\right)\mathrm{d}x$, denoted as m, for $\alpha >0$. From [16], we know that the existence of positive non-constant radial steady states depends on the dimension n and the initial mass m, that is, for n = 2, radial symmetric solutions exist if and only if $m>\frac{8\pi }{\varphi \left(0\right)}$ while there is no positive non-constant radial steady state for any m>0 when n>2. For any bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, the FLKS system with Neumann boundary conditions and mass constraint has positive non-constant radial steady states when the chemotactic coefficient χ is large enough (see [20]), where the mass constraint means that ${\int }_{\mathrm{\Omega }}\rho (t,x)\mathrm{d}x={\int }_{\mathrm{\Omega }}{\rho }_0\left(x\right)\mathrm{d}x:=m>0,$which comes from the integration of (1(1) $\begin{array}{r}\{\begin{array}{ll}{\mathrm{\partial }}_t\rho =\mathrm{\Delta }\rho -\mathrm{\nabla }\cdot \left(\chi \right(\rho \psi \left(|\mathrm{\nabla }S|\right)\mathrm{\nabla }S),& x\in \mathrm{\Omega },\\ {\mathrm{\partial }}_{t}S=\mathrm{\Delta }S-\alpha S+\rho ,& x\in \mathrm{\Omega },\\ (\rho ,S)(0,x)=({\rho }_0,S_0)\left(x\right),& \end{array}\end{array}$(1) ) with Neumann boundary conditions. In [20], the existence of radial spiky steady states is proved, that is, when χ tends to be infinity, the non-constant radial steady states found in [20] tends to be like spikes, where finding spiky steady states is another way in mathematics to describe the aggregation phenomenon of chemotaxis models except blow-up.

In this paper, we consider the one-dimensional steady states of the FLKS system (1(1) $\begin{array}{r}\{\begin{array}{ll}{\mathrm{\partial }}_t\rho =\mathrm{\Delta }\rho -\mathrm{\nabla }\cdot \left(\chi \right(\rho \psi \left(|\mathrm{\nabla }S|\right)\mathrm{\nabla }S),& x\in \mathrm{\Omega },\\ {\mathrm{\partial }}_{t}S=\mathrm{\Delta }S-\alpha S+\rho ,& x\in \mathrm{\Omega },\\ (\rho ,S)(0,x)=({\rho }_0,S_0)\left(x\right),& \end{array}\end{array}$(1) ) with the mass constraint and Neumann boundary conditions in a bounded domain as follows: (2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) where α, M and L are fixed positive constants. We denote ML as the total mass of cells and denote $\psi \left(|r|\right)=\varphi \left(r\right)$ for convenience. Hence $\varphi \in C^2(\mathbb{R};{\mathbb{R}}^{+})$ satisfying (3) $\underset{r\in {\mathbb{R}}^{+}}{max}|r\varphi \left(r\right)|=A_{\mathrm{\infty }}\mathrm{a}\mathrm{n}\mathrm{d}\varphi \left(0\right)>0.$(3) The flux limited situation $\underset{r\in {\mathbb{R}}^{+}}{max}|r\varphi \left(r\right)|=A_{\mathrm{\infty }}$ and the smoothness of the function ϕ imply that there must exist a constant $a\in {\mathbb{R}}^{+}$ such that $\underset{r\in {\mathbb{R}}^{+}}{max}\varphi \left(r\right)=\varphi \left(a\right)$. Without loss of generality, we assume $\underset{r\in {\mathbb{R}}^{+}}{max}\varphi \left(r\right)=\varphi \left(0\right)$ in the following for simplicity.

The existence of monotone solution of (2(2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) ) is proved in [4] for $\chi >{\overline{\chi }}_1$ where ${\overline{\chi }}_1$ is a constant depends on L, α, M and ϕ. As χ goes to infinity, it is proved that the decreasing solution ρ concentrates at x = 0, that is $\rho \left(x\right)\to LM\delta \left(x\right)$ in the sense of distribution, and the decreasing solution S converges to an explicit function uniformly on $[0,L]$, by using Helly's compactness theorem and the Sturm oscillation theorem (see Theorem 5.4 in [4]), which implies the existence of the spiky steady state and hence the aggregation of cells. In this paper, we use the Sturm oscillation theorem in a more meticulous way to give a refined asymptotic result, that is, to give a more accurate characterization of the cell aggregation.

2. A priori estimate

Here we give the priori estimate first.

Lemma 2.1

If $\left(\rho \right(x),S(x\left)\right)$ is the positive classical solution of (2(2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) ), we have (4) $\begin{array}{rl}\frac{ML}{\sqrt{\alpha }\sinh \sqrt{\alpha }L}& \le S\left(x\right)\le \frac{M}{\alpha }+2M{L}^2,\end{array}$(4) (5) $\begin{array}{rl}|S^{\mathrm{\prime }}\left(x\right)|& \le 2ML,\end{array}$(5) and (6) $M\exp (-2M{L}^2\varphi (0\left)\chi \right)\le \rho \left(x\right)\le M\exp \left(2M{L}^2\varphi \right(0\left)\chi \right).$(6)

Proof.

We use the Green's function to give the lower bound of S. For any fixed $x_0\in (0,L)$, let $G(x;x_0)$ be the Green's function satisfying $\{\begin{array}{ll}-G_{xx}+\alpha G=\delta (x-x_0),& x\in (0,L),\\ G_x(0;x_0)=G_x(L;x_0)=0.& \end{array}$Observe that G can be explicitly given by $G(x;x_0)=\{\begin{array}{ll}{\displaystyle \frac{\cosh \sqrt{\alpha }(L-x_0)}{\sqrt{\alpha }\sinh \sqrt{\alpha }L}\cosh \sqrt{\alpha }x,}& \mathrm{i}\mathrm{f}0<x<x_0,\\ {\displaystyle \frac{\cosh \sqrt{\alpha }{x}_0}{\sqrt{\alpha }\sinh \sqrt{\alpha }L}\cosh \sqrt{\alpha }(L-x),}& \mathrm{i}\mathrm{f}{x}_0<x<L.\end{array}$Since $G(x;x_0)\ge \frac{1}{\sqrt{\alpha }\sinh \sqrt{\alpha }L}$, $\mathrm{\forall }x,x_0\in (0,L)$, we have that for any $x_0\in [0,L]$, $S\left(x_0\right)={\int }_0^{L}G(x;x_0)\rho \left(x\right)\mathrm{d}x\ge \frac{1}{\sqrt{\alpha }\sinh \sqrt{\alpha }L}{\int }_0^L\rho \left(x\right)\mathrm{d}x=\frac{ML}{\sqrt{\alpha }\sinh \sqrt{\alpha }L}.$Now we give the upper bound of v. Recall ${\int }_0^{L}S\left(x\right)\mathrm{d}x=\frac{1}{\alpha }{\int }_0^L\rho \left(x\right)\mathrm{d}x=\frac{ML}{\alpha }.$Since $S\left(x\right)\in C^2\left(\right[0,L\left]\right)$ and $S^{\mathrm{\prime }}\left(0\right)=0$, we can rewrite $S^{\mathrm{\prime }}\left(x\right)$ as $S^{\mathrm{\prime }}\left(x\right)={\int }_0^x{S}^{\mathrm{\prime }\mathrm{\prime }}\left(y\right)\mathrm{d}y$ for $x\in [0,L]$. Because the solution is positive, that is, $\rho \left(x\right),S\left(x\right)>0$ on $[0,L]$, we have $|S^{\mathrm{\prime }}\left(x\right)|=\left|{\int }_0^x{S}^{\mathrm{\prime }\mathrm{\prime }}\right(y\left)\mathrm{d}y\right|\le \alpha {\int }_0^{x}S\left(y\right)\mathrm{d}y+{\int }_0^x\rho \left(y\right)\mathrm{d}y\le 2ML,$where the first inequality comes from the second equality in (1(1) $\begin{array}{r}\{\begin{array}{ll}{\mathrm{\partial }}_t\rho =\mathrm{\Delta }\rho -\mathrm{\nabla }\cdot \left(\chi \right(\rho \psi \left(|\mathrm{\nabla }S|\right)\mathrm{\nabla }S),& x\in \mathrm{\Omega },\\ {\mathrm{\partial }}_{t}S=\mathrm{\Delta }S-\alpha S+\rho ,& x\in \mathrm{\Omega },\\ (\rho ,S)(0,x)=({\rho }_0,S_0)\left(x\right),& \end{array}\end{array}$(1) ). The mean value theorem of integration implies that there exists some $x_0\in [0,l]$ such that $S\left(x_0\right)=\frac{{\int }_0^{L}S\left(x\right)\mathrm{d}x}{L}=\frac{M}{\alpha }$. Similarly, S can be estimated by $S\left(x\right)=S\left(x_0\right)+{\int }_{x_0}^x{S}^{\mathrm{\prime }}\left(y\right)\mathrm{d}y\le \frac{M}{\alpha }+{\int }_{x_0}^x|S^{\mathrm{\prime }}\left(y\right)|\mathrm{d}y\le \frac{M}{\alpha }+2M{L}^2.$By the equation in the first line of (2(2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) ) and the boundary condition, we have ${\rho }^{\prime }=\chi \rho \varphi \left(S^{\prime }\right)S^{\prime },$which implies that $\left|\right(\ln \rho {)}^{\prime }|=\left|\frac{{\rho }^{\prime }}{\rho }\right|=\chi \varphi \left(S^{\prime }\right)|S^{\prime }|\le 2ML\varphi \left(0\right)\chi .$Take $\rho \left(y_0\right)=M$ for some $y_0\in (0,L)$. Then $|\ln \rho (x)-\ln M|\le 2M{L}^2\varphi \left(0\right)\chi ,$that is $M\exp (-2M{L}^2\varphi (0\left)\chi \right)\le \rho \left(x\right)\le M\exp \left(2M{L}^2\varphi \right(0\left)\chi \right).$

3. Refined asymptotic

Recalling in [4], the results that as $\chi \to \mathrm{\infty }$, $\rho \left(x\right)\to LM\delta \left(x\right)$ in the sense of distribution and $S\left(x\right)\to \frac{ML}{\sqrt{\alpha }\sinh \left(\sqrt{\alpha }L\right)}\cosh \left(\sqrt{\alpha }\right(x-L\left)\right)$ uniformly on $[0,L]$ were proved. These results are true in our paper, and we give a refined asymptotic result below with a method coming from [3]. Although both the functions ρ and S depend on the parameter χ, for convenience, in the following we still use the notations $\rho \left(x\right)$ and $S\left(x\right)$ other than $\rho (x{)}_{\chi }$ and $S(x{)}_{\chi }$.

Theorem 3.1

For the solutions of (2(2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) ) in Theorem 5.4 in [4], we have a further result, which is (7) $\underset{\chi \to \mathrm{\infty }}{lim}\underset{x\in [0,L]}{sup}|\frac{\rho \left(x\right)}{\chi }-\frac{M^2{L}^2\varphi \left(0\right)}{2}{\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)|=0$(7) and (8) $\underset{\chi \to \mathrm{\infty }}{lim}\underset{x\in [0,L]}{sup}\left|S^{\prime }\right(x)-\frac{ML\sinh \sqrt{\alpha }(x-L)}{\sinh \sqrt{\alpha }L}\tanh \left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)|=0.$(8)

Remark 3.1

Comparing Equation (53) in [4] and Equation (2(2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) ) in our paper, we denote $Q\left(u\right)=\varphi \left(u\right)u$. Observe that if the assumptions in (54) in [4] are satisfied, that is, $Q\left(0\right)=0$, $|Q|\le 1$ and $Q^{\prime }\left(u\right)<0,\mathrm{\forall }u$, then our assumptions in (3(3) $\underset{r\in {\mathbb{R}}^{+}}{max}|r\varphi \left(r\right)|=A_{\mathrm{\infty }}\mathrm{a}\mathrm{n}\mathrm{d}\varphi \left(0\right)>0.$(3) ) are satisfied by the facts $|Q\left(u\right)|=|u\varphi \left(u\right)|$ and $Q^{\prime }\left(u\right)={\varphi }^{\prime }\left(u\right)u+\varphi \left(u\right)$. So the assumptions of function ϕ in our paper give more relaxed restrictions of the FLKS system (2(2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) ) than the one, system (53) and (56), in [4]. Actually, using the same methods in [4] and our paper, we can also obtain the results that as $\chi \to \mathrm{\infty }$, $\rho \left(x\right)\to LM\delta \left(x\right)$ in the sense of distribution and $S\left(x\right)\to \frac{ML}{\sqrt{\alpha }\sinh \left(\sqrt{\alpha }L\right)}\cosh \left(\sqrt{\alpha }\right(x-L\left)\right)$ uniformly on $[0,L]$ and the result in Theorem 3.1 for system (2(2) $\begin{array}{r}\{\begin{array}{ll}{\rho }^{″}-\chi \left(\rho \varphi \right(S^{\prime })S^{\prime }{)}^{\prime }=0,& x\in (0,L),\\ S^{″}-\alpha S+\rho =0,& x\in (0,L),\\ {\rho }^{\prime }\left(0\right)={\rho }^{\prime }\left(L\right)=S^{\prime }\left(0\right)=S^{\prime }\left(L\right)=0,& \\ {\displaystyle {\int }_0^L\rho \left(x\right)\mathrm{d}x=ML,}& \end{array}\end{array}$(2) ) with ϕ defined in (3(3) $\underset{r\in {\mathbb{R}}^{+}}{max}|r\varphi \left(r\right)|=A_{\mathrm{\infty }}\mathrm{a}\mathrm{n}\mathrm{d}\varphi \left(0\right)>0.$(3) ).

Remark 3.2

From (7(7) $\underset{\chi \to \mathrm{\infty }}{lim}\underset{x\in [0,L]}{sup}|\frac{\rho \left(x\right)}{\chi }-\frac{M^2{L}^2\varphi \left(0\right)}{2}{\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)|=0$(7) ), we can see that as $\chi \to \mathrm{\infty }$, the function $\rho \left(x\right)$ tends to be 0 with the order $\chi {\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)$ on $(0,L]$ and tends to be infinity with the order χ at the point 0. So we have $\rho \left(x\right)=O(\chi {\cosh }^{-2}(\frac{ML\varphi \left(0\right)}{2}\chi x\left)\right)$ for $x\in (0,L]$ and $\rho \left(0\right)=O\left(\chi \right)$ as $\chi \to \mathrm{\infty }$ and Theorem 3.1 gives more detailed description of the profile of $\rho \left(x\right)$ when χ is large, that is, how the profile of ρ tends to be like a spike at the point 0 as $\chi \to \mathrm{\infty }$. Denote $\left[S^{\prime }\right(x){]}_{\chi }=\frac{ML\sinh \sqrt{\alpha }(x-L)}{\sinh \sqrt{\alpha }L}\tanh (\frac{ML\varphi \left(0\right)}{2}\chi x)$ and $S(x{)}_{\mathrm{\infty }}=\frac{ML}{\sqrt{\alpha }\sinh \left(\sqrt{\alpha }L\right)}\cosh (\sqrt{\alpha }(x-L))$, it is easy to check that $\left[S^{\prime }\right(x){]}_{\mathrm{\infty }}=[S(x{)}_{\mathrm{\infty }}{]}^{\prime }$, and we can see the change of the profile of $S\left(x\right)$ clearly by (8(8) $\underset{\chi \to \mathrm{\infty }}{lim}\underset{x\in [0,L]}{sup}\left|S^{\prime }\right(x)-\frac{ML\sinh \sqrt{\alpha }(x-L)}{\sinh \sqrt{\alpha }L}\tanh \left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)|=0.$(8) ) when χ is large.

Proof.

We will use Sturm's oscillation theorem in a more meticulous way to get our result. Recall $(S^{″}{)}^{\prime }+(\chi \rho \varphi \left(S^{\prime }\right)-\alpha )S^{\prime }=0$ for $x\in (0,L)$. Using Sturm's oscillation theorem by a comparison between $\sin \left(\frac{\pi x}{z}\right)$ and $S^{\prime }\left(x\right)$, and using the fact that $S^{\prime }\left(x\right)<0$ on $x\in (0,z)$ for fixed $z\in (0,L]$, we obtain that $\chi \rho \left(x\right)\varphi \left(S^{\prime }\right(x\left)\right)-\alpha \le \frac{{\pi }^2}{z^2}$ for $x\in (0,z)$. By (5(5) $\begin{array}{rl}|S^{\mathrm{\prime }}\left(x\right)|& \le 2ML,\end{array}$(5) ) and the fact that ρ is decreasing, we have $\chi \rho \left(z\right)\underset{|r|\le 2ML}{min}\left\{\varphi \right(r\left)\right\}\le \chi \rho \left(x\right)\varphi \left(S^{\prime }\right(x\left)\right)$ for $x\in (0,z]$ and hence $A\chi \rho \left(z\right)\le \frac{{\pi }^2}{z^2}+\alpha ,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}z\in (0,L],$where $A=\underset{|r|\le 2ML}{min}\left\{\varphi \right(r\left)\right\}$ is well defined because $\varphi (\cdot )$ is $C^1$ smooth. Thus we have $\rho \left(x\right)\le \frac{1}{A\chi }(\frac{{\pi }^2}{x^2}+\alpha ),x\in (0,L],$which implies that (9) ${\int }_{\frac{K}{\chi }}^L\rho \left(x\right)\mathrm{d}x\le \frac{1}{A}(\frac{\alpha L}{\chi }+\frac{{\pi }^2}{K}),\mathrm{\forall }K\in (0,\chi L].$(9) Recall that $\frac{{\rho }^{\prime }}{\rho }=\chi \varphi \left(S^{\prime }\right)S^{\prime }$. Thus with Lemma 2.1 we have $\begin{array}{rl}\rho \left(x\right)& =\rho \left(0\right)\exp \left[\chi {\int }_0^x\varphi \right(S^{\prime }\left(x\right)\left)S^{\prime }\right(x\left)\mathrm{d}x\right]\\ & \ge \rho \left(0\right)\exp [-2\chi ML\varphi (0\left)x\right].\end{array}$Integrating the above equation over $[0,L]$, we obtain $ML\ge \rho \left(0\right)\frac{1-\exp [-2\chi M{L}^2\varphi (0\left)\right]}{2\chi ML\varphi \left(0\right)},$which is $\frac{\rho \left(0\right)}{\chi }\le \frac{2M^2{L}^2\varphi \left(0\right)}{1-\exp [-2\chi M{L}^2\varphi (0\left)\right]}.$Since $\frac{\rho \left(0\right)}{\chi }$ is uniformly bounded for all large χ, we can assume that there exists some constant B such that $\frac{\rho \left(0\right)}{\chi }\to B\in [0,2M^2{L}^2\varphi (0\left)\right]$after passing to a subsequence as $\chi \to \mathrm{\infty }$. Then we take $y=\chi x$ and $W\left(y\right)=\chi {\int }_0^x\varphi \left(S^{\prime }\right(x\left)\right)S^{\prime }\left(x\right)\mathrm{d}x.$Thus $\rho \left(x\right)=\rho \left(0\right)\exp \left(W\right(y\left)\right)$ and $\begin{array}{rl}{W}_y\left(y\right)& =\varphi \left(S^{\prime }\right(x\left)\right)S^{\prime }\left(x\right)<0\mathrm{f}\mathrm{o}\mathrm{r}x\in (0,L),\\ W_{yy}\left(y\right)& =\frac{1}{\chi }\left[{\varphi }^{\prime }\right(S^{\prime }\left(x\right)\left)S^{\prime }\right(x\left)S^{″}\right(x)+\varphi (S^{\prime }\left(x\right)\left)S^{″}\right(x\left)\right]\\ & =\frac{1}{\chi }\left[{\varphi }^{\prime }\right(S^{\prime }\left(x\right)\left)S^{\prime }\right(x)+\varphi (S^{\prime }\left(x\right)\left)\right]\left[\alpha S\right(x)-\rho (x\left)\right]\\ & =\frac{1}{\chi }\left[{\varphi }^{\prime }\right(S^{\prime }\left(x\right)\left)S^{\prime }\right(x)+\varphi (S^{\prime }\left(x\right)\left)\right]\left[\alpha S\right(x)-\rho (0)\exp (W\left(y\right)\left)\right]\end{array}$Since S and $S^{\prime }$ are uniformly bounded for all $\chi >0$, $W_{yy}\left(y\right)$ is uniformly bounded. So we can assume $W\left(y\right)\to \mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\overline{W}\left(y\right)$ locally uniformly on $[0,\mathrm{\infty })$ after further passing to a subsequence as $\chi \to \mathrm{\infty }$. For any fixed point $y\in [0,\mathrm{\infty })$, since $x=\frac{y}{\chi }\to 0$ as $\chi \to \mathrm{\infty }$, $S^{\prime }\left(x\right)\to 0$ as $\chi \to \mathrm{\infty }$, which implies that ${\overline{W}}_{yy}\left(y\right)=-B\varphi \left(0\right)\exp \left(\overline{W}\right)\mathrm{o}\mathrm{n}(0,\mathrm{\infty }),\overline{W}\left(0\right)=0,{\overline{W}}_y\left(0\right)=0,{\overline{W}}_y\le 0.$By the integration of ${\overline{W}}_y{\overline{W}}_{yy}=-B\varphi \left(0\right)\exp \left(\overline{W}\right){\overline{W}}_y$, we have ${\overline{W}}_y=-\sqrt{2B\varphi \left(0\right)(1-\exp (\overline{W}\left)\right)},$which is $\overline{W}\left(y\right)=-2\ln \cosh \left(\sqrt{\frac{B\varphi \left(0\right)}{2}}y\right).$Now we need to find the value of B. Observe that ${\int }_0^{\frac{K}{\chi }}\rho \left(x\right)\mathrm{d}x={\int }_0^{\frac{K}{\chi }}\rho \left(0\right){\mathrm{e}}^{W\left(y\right)}\mathrm{d}x=\frac{\rho \left(0\right)}{\chi }{\int }_0^K{\mathrm{e}}^{W\left(y\right)}\mathrm{d}y.$By (9(9) ${\int }_{\frac{K}{\chi }}^L\rho \left(x\right)\mathrm{d}x\le \frac{1}{A}(\frac{\alpha L}{\chi }+\frac{{\pi }^2}{K}),\mathrm{\forall }K\in (0,\chi L].$(9) ), we know $\begin{array}{rl}ML& =({\int }_0^{\frac{K}{\chi }}+{\int }_{\frac{K}{\chi }}^L)\rho \left(x\right)\mathrm{d}x=\underset{K\to \mathrm{\infty }}{lim}\underset{\chi \to \mathrm{\infty }}{lim}{\int }_0^{\frac{K}{\chi }}\rho \left(x\right)\mathrm{d}x\\ & =B\underset{K\to \mathrm{\infty }}{lim}{\int }_0^K{\mathrm{e}}^{\overline{W}\left(y\right)}\mathrm{d}y=\sqrt{\frac{2B}{\varphi \left(0\right)}},\end{array}$which is, $B=\frac{M^2{L}^2\varphi \left(0\right)}{2}$.

Since the limit is unique, we have $\underset{\chi \to \mathrm{\infty }}{lim}W\left(y\right)=-2\ln \cosh \left(\frac{ML\varphi \left(0\right)}{2}y\right)\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y}\mathrm{o}\mathrm{n}[0,\mathrm{\infty }).$By the monotonicity of $W\left(y\right)$, we have (10) $\underset{\chi \to \mathrm{\infty }}{lim}\exp \left(W\right(y\left)\right)={\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}y\right)\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y}\mathrm{o}\mathrm{n}[0,\mathrm{\infty }).$(10) Then the Lebesgue's dominated convergence theorem gives us $\underset{\chi \to \mathrm{\infty }}{lim}[{\int }_0^{\chi L}|{\mathrm{e}}^{W\left(y\right)}-{\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}y\right)|\mathrm{d}y+|\frac{\rho \left(0\right)}{\chi }-\frac{M^2{L}^2\varphi \left(0\right)}{2}|]=0.$Recall that ${\mathrm{e}}^{W\left(y\right)}=\frac{\rho \left(x\right)}{\rho \left(0\right)}\le \frac{1}{A\chi \rho \left(0\right)}(\frac{{\pi }^2}{x^2}+\alpha )=\frac{\chi }{A\rho \left(0\right)}(\frac{{\pi }^2}{y^2}+\frac{\alpha }{{\chi }^2})\le \frac{C}{y^2},$for some positive constant C and $y\in [1,\chi L]$. Then we have $\begin{array}{rl}& \underset{\chi \to \mathrm{\infty }}{lim}\underset{x\in [0,L]}{sup}|\frac{\rho \left(x\right)}{\chi }-\frac{M^2{L}^2\varphi \left(0\right)}{2}{\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)|\\ & \le \underset{\chi \to \mathrm{\infty }}{lim}\underset{y\in [0,\mathrm{\infty })}{sup}|\frac{\rho \left(0\right)}{\chi }{\mathrm{e}}^{W\left(y\right)}-\frac{M^2{L}^2\varphi \left(0\right)}{2}{\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}y\right)|=0\end{array}$by (10(10) $\underset{\chi \to \mathrm{\infty }}{lim}\exp \left(W\right(y\left)\right)={\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}y\right)\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y}\mathrm{o}\mathrm{n}[0,\mathrm{\infty }).$(10) ) and the fact that $\frac{\rho \left(0\right)}{\chi }$ converges to $B=\frac{M^2{L}^2\varphi \left(0\right)}{2}$ as $\chi \to \mathrm{\infty }$. Thus we prove (7(7) $\underset{\chi \to \mathrm{\infty }}{lim}\underset{x\in [0,L]}{sup}|\frac{\rho \left(x\right)}{\chi }-\frac{M^2{L}^2\varphi \left(0\right)}{2}{\cosh }^{-2}\left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)|=0$(7) ).

By the representation of S via Green's function, which is $\begin{array}{rl}S\left(x\right)& ={\int }_0^{L}G(z;x)\rho \left(z\right)\mathrm{d}z\\ & =\frac{\cosh \sqrt{\alpha }(L-x)}{\sqrt{\alpha }\sinh \sqrt{\alpha }L}{\int }_0^x\rho \left(z\right)\cosh \sqrt{\alpha }z\mathrm{d}z\\ & +\frac{\cosh \sqrt{\alpha }x}{\sqrt{\alpha }\sinh \sqrt{\alpha }l}{\int }_x^l\rho \left(z\right)\cosh \sqrt{\alpha }(L-z)\mathrm{d}z,\end{array}$then we have $S^{\prime }\left(x\right)=\frac{\sinh \sqrt{\alpha }(x-L)}{\sinh \sqrt{\alpha }L}{\int }_0^x\rho \left(z\right)\cosh \sqrt{\alpha }z\mathrm{d}z+\frac{\sinh \sqrt{\alpha }x}{\sinh \sqrt{\alpha }L}{\int }_x^L\rho \left(z\right)\cosh \sqrt{\alpha }(L-z)\mathrm{d}z$With a same discussion of the estimate of $v^{\prime }$ in the proof of Theorem 3.2 in [11], we have (8(8) $\underset{\chi \to \mathrm{\infty }}{lim}\underset{x\in [0,L]}{sup}\left|S^{\prime }\right(x)-\frac{ML\sinh \sqrt{\alpha }(x-L)}{\sinh \sqrt{\alpha }L}\tanh \left(\frac{ML\varphi \left(0\right)}{2}\chi x\right)|=0.$(8) ) and we finish this proof.

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The author was partially supported by National Key Research and Development Program of China (2021YFA1002400).