# Existence of a Nontrivial Steady-State Solution to a Parabolic-Parabolic Chemotaxis System with Singular Sensitivity

Authors: Yingjie Zhu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper establishes the existence of a nontrivial steady-state solution to a parabolic-parabolic coupled system with singular (or logarithmic) sensitivity and nonlinear source arising from chemotaxis. The proofs mainly rely on the maximum principle, the implicit function theorem, and the Hopf bifurcation theorem.


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## Body

## 1. Introduction

Chemotaxis is the biased movement of cells toward the concentration gradient of a chemical. It plays a critical role in a wide range of biological phenomena. For example, cells migrate toward resources of food and stay away from harmful substances. The first mathematical model of chemotaxis was introduced by Patlak in [1] and Keller and Segel in [2]. There are numerous works dedicated to the analysis of chemotaxis models. For example, Othmer and Stevens in [3] modeledmyxobacteria as individual random walkers and proposed a microscopic model based on a velocity jump process. By taking the parabolic limit of the microscopic model, they obtain the macroscopic chemotaxis model, which is the well-known Keller-Segel system(1)ut=∇·μ∇u-∇·χu∇Θv+fu,v,x∈Ω,t>0,τvt=dΔv+gu,v,x∈Ω,t>0,where Ω⊂Rn is a bounded connected domain with a smooth boundary ∂Ω. The function u=u(x,t) denotes the cell density and v=v(x,t) represents the chemical concentration, for example, oxygen. The constant χ is called the chemotactic coefficient, and the sign of χ corresponds to chemoattraction if χ>0 and chemorepulsion if χ<0. Parameters μ and d are the diffusion coefficients of the cells and the chemical, respectively. The function f(u) represents the kinetic function describing production and degradation of cells, and Θ(v) is commonly referred to as the chemotactic potential function. Function g(u,v) describes the production and degradation of the chemical.The existence of global solutions, blow-up, and traveling wave solutions to the chemotactic system (1) were extensively studied during the past four decades (see, e.g., [4–12] and references therein). The authors studied the roles of growth, death and random in promoting population persistence through band popation in [4], and the work was related to the significance of cell motility and chemotaxis in microbial ecology.X.F. Wang addressed the trivial and nontrivial steady states with smallχ and μ of the quasi-linear system (1) in [13]. In the absence of population dynamics (i.e., f(u)≡0), there have been extensive studies. The main feature of solutions to the Keller-Segel model is the possibility of blow-up in finite time in [9, 14, 15]. Moreover, recent results in [6, 10, 11, 16, 17] proved the global existence of solutions under some conditions. Moreover, the global existence, asymptotic behavior, and steady states of classical solutions were studied in [18] for the one-dimensional case.The purpose of this paper is to study the existence of the nontrivial steady-state solution to a parabolic-parabolic coupled system arising from chemotaxis with singular sensitivity. We consider the following system(2)ut=εuxx-χuln⁡vxx+μHv-σu,x∈Ω,t>0,vt=vxx-uHv,x∈Ω,t>0,with the initial-boundary value conditions(3)ux,0=u0x,vx,0=v0x,x∈Ω,u′x=0=0,εu′-χuln⁡v′x=1=0,x∈∂Ω,t>0,v′x=0=0,γv′+vx=1=1,x∈∂Ω,t>0,where Ω=[0,1] and “′”=∂/∂x. The function u=u(x,t) denotes the density of the cells and v=v(x,t) denotes the concentration of the chemical. The parameters χ,ε,μ, and σ are all positive constants, with ε being the diffusion coefficient of cells and χ the chemotactic sensitivity coefficient as above. The function H(v) is assumed to be smooth.The steady-state problem corresponding to system (2)-(3) is(4)εu′′=χuln⁡v′′-μHv-σu,x∈Ω,t>0,v′′=uHv,x∈Ω,t>0,ux,0=u0x,vx,0=v0x,x∈Ω,where Ω=[0,1]. The linear chemotactic potential Θ(v)=v was considered in [19], and X.F. Wang’s model in [13] is similar to ours.In this paper, we establish the existence of nontrivial steady-state solutions to the parabolic-parabolic coupled chemotactic system (2)-(3) with singular sensitivity. Here we assume that the function H(v) satisfies(5)H0=0,H′v>0,asv>0.Obviously, u≡0 and v≡1 are the trivial solutions to system (4).Our main results in this paper are presented in the following theorem.Theorem 1.
Consider system (4) with condition (5). Then the following alternative holds: (1)
Ifσ≥μH(1), then system (4) has a pair of nonnegative steady-state solutions u≡0 and v≡1.(2)
Ifσ<μH(1), then there exists at least a pair of nontrivial steady-state solutions (u(x),v(x)).The rest of this paper is organized as follows. In Section2, we give some preliminary lemmas. In Section 3, we complete the proof of Theorem 1.


## 2. Lemmas

To prove Theorem1, we need to establish boundary estimates of solutions to system (4). Since the state variables represent densities, we only consider nonnegative solutions, that is, u(x)≥0 and v(x)≥0 on [0,1]. Firstly, we state one result concerning the estimate of a solution (u,v).Lemma 2.
Assume the solution(u,v) is a nonnegative steady state. Then we obtain the following: (i)
v′≥0, and 0≤v(x)≤1.(ii)
u′≥0.Proof.
(i)
By the second equation of (4), we have v′′≥0 and, integrating once, we obtain v′≥0. Hence, v′(x)≥0 and v(x)≤v(1). (ii)
Integrating the first equation of (4) yields (6)u′x=1ε∫0xu′′xdx=1ε∫0xχuln⁡v′′-μHv-σudx=χεuln⁡v′-1ε∫0xμHv-σudx≥0.Lemma 3.
Letv(x) solve system (2) with the initial-boundary value condition of (3). Then we have the lower-bound estimate(7)1v1≤M,whereM=G-1∫01usds-1,and G is defined in (11) below.Proof.
From the second equation in (4), we obtain (8)v′′xHvx=ux.Integrating the above equation once, we have (9)∫0xv′′sHvsds=∫0xusds.Integrating by part and applying the condition v′0=0, we rearrange the above equation(10)v′xHvx≤∫0xusds≤∫01usds.
From the boundary value condition ofγv′+vx=1=1 in (3), we have v′(1)=1-v1/γ. Define the function(11)Gy≔1-yγHy,y∈0,1.From the definition of the function G, we easily get(12)Gv1=1-v1γHv1≤∫01utdt.Applying the implicit function theorem, we have the estimate (7).Lemma 4.
There are two positive constantsC1=χM/εγ and C2=μ/χM such that a nonnegative steady state of system (4) satisfies(13)u0≥u1exp⁡-C1-1-exp⁡-C1C2.Proof.
By the steady-state system (4), we know v′′>0, and for every x∈(0,1), we have(14)v′x=∫0xv′′xdx≤∫01v′′xdx=v′1.Applying the condition γv′(1)+v(1)=1, we obtain the following equation(15)v′x=1-v1γ.
Similarly, we also get(16)∫01Hvudx=∫01v′′xdx=v′1-v′0≤1γ.
On the other hand, integrating once the first equation of (4), we get(17)∫1xεu′′ds=∫1xχuln⁡v′′-μHv-σuds,which is equal to(18)εu′x-εu′1dx=χuln⁡v′-χuln⁡v′x=1-∫1xμHv-σuds.Then, applying the boundary condition εu′-χuln⁡v′x=1=0 and rewriting (18), we have the following estimate(19)u′x=1εχuln⁡v′-∫1xμHv-σudx=1εχuv′v-∫1xμHv-σudx≤1εχuv′v+∫01μHvudx.
Next, we review the conditionγv′+vx=1=1, which implies that v′(1)≤1. Combined with the condition of 1/v≤M in Lemma 3, inequality (19) can be rewritten as(20)u′x≤χMεγux+1ε∫01μHvudx≤χMεγux+μεγ.Integrating (20) once, we have(21)∫0xu′sds≤χMεγ∫0xus+μχMds≤C1∫0xus+C2ds.Here C1=χM/εγ and C2=μ/χM. Hence, we obtain (22)ux-u0≤C1∫0xus+C2dswhich implies that (23)ux≤u0+C1∫0xus+C2ds.Define the function S(x)=u(x)+C2. Then we have S(0)=u(0)+C2. Combining the above inequalities, we obtain the following bound(24)Sx≤S0+C1∫0xSsds.By Gronwall’s inequality, we get(25)Sx≤S0exp⁡C1x.Substituting the definition of S(x) into inequality (25), we obtain(26)ux+C2≤u0+C2exp⁡C1.Combining this result with u′(x)≥0 in Lemma 2(ii), we obtain estimate (13).Next, we suppose thatFu=max⁡0,u for any u(x)∈C0[0,1]. Given u∈C0[0,1], we consider the equation(27)v′′=FuHv,v′x=0=0,γv′+vx=1=1.Then, we prove the following lemma.Lemma 5.
For any smallσ_>0, we pick a sufficiently small ε>0 such that μH(ε)<σ_. Next, we choose(28)x0=2μH1-σ_2μH1-σ_-CμHε-σ_,0<x0<1and set(29)N=max⁡2DC,4CH11-x02.Finally, we denote by (u(x),v(x)) a nonnegative solution to system (4) when σ∈σ_,+∞. Then we have |u(x)|≤N.Proof.
If not, there exists someσ0∈σ_,+∞ such that u(x)>N. According to Lemma 2 and the definition of N, we have(30)u1≥ux>N>2DC.From (13), we can easily get(31)u0≥Cu1-D>Cu1-Cu12=Cu12.However, v(x0)<δ.
In fact, if not we have(32)v1-vx0=v′x01-x0+12v′′x01-x02+∘1-x0n≥12Hvδuδ1-x02≥12Hvx0u01-x02≥12Hδ12Cu11-x02≥14HδCu11-x02>1which contradicts the fact that v(x)≤1.
Next, we consider(33)0=∫01μHv-σudx=∫0x0μHv-σudx+∫x01μHv-σudx<μHε-σ_u0x0+μH1-σ_u11-x0<μHε-σ_u0x0+μH1-σ_2u0C1-x0=u0C2μH1-σ_1-x0+Cx0μHε-σ_=u0C2μH1-σ_-x02μH1-σ_-CμHε-σ_=0which is a contradiction. In other words, we have |u(x)|≤N.Lemma 6.
(i)
For eachu(x)∈C0[0,1], system (27) has a unique solution v=Au. (ii)
A : u → A u is continuous. (iii)
Ifu1(x)≥u2(x), then Au1≤Au2.Proof.
For any givenu(x)∈C0[0,1], it is obvious that v(x)=1 and v(x)=0 are a pair of sup-sub-solutions. According to the standard comparison theorem, we easily obtain the estimate of the solution to system (27). If v1(x) and v2(x) are solutions to system (27), then we have(34)v1′′=FuHv1,v2′′=FuHv2.The difference between the first and second equation of (34) is(35)-v1-v2′′+FuHvv1-v2=0,v1-v2′x=0=0,γv1-v2′+v1-v2x=1=0,with the notation Hv≔dH/dv.
By the maximum principle, we obtainv1≡v2. Moreover, Au is uniquely defined and 0≤Au≤1.
Next we give the proof of (ii) and (iii) in Lemma6, respectively. (i)
Assume that there exists a sequenceunxn=1∞, which converges to u(x). Then we only point out that there exists a subsequence Aunk of Aun such that Aunk→w(x) on C0[0,1]. By regularity theory, we know that w(x) is the solution to system (34) and w(x)≠Au, which contradicts the uniqueness of Au.(ii)
We can directly prove (iii) from the maximum principle.


## 3. Proof of Theorem1

We dedicate this section to the proof of Theorem1.Assume there existsu(x)>0 such that (u,Au) satisfies the following system(36)εu′′=χuln⁡Auxx-μHAu-σu,x∈0,1,u′x=0=0,εu′-χuln⁡Au′x=1=0.It is easy to see that Au is the nonnegative solution to system (36) by the definition of A and H(Au).DefineRψ(u¯) to be the unique solution to the system(37)-εu′′+q-μH1u=0,x∈0,1,u′x=0=0,u′x=1=ψu¯,where q is a constant satisfying q>μH(1). For any given φ(x)∈C0[0,1], we define Kφ to be the unique solution to the following system(38)-εu′′+q-μH1u=φx,x∈0,1,u′x=0=0,u′x=1=0.Then we rewrite (36) and obtain the following operator equation(39)u=Kφ+Rψu=K-εu′′+q-μH1u+Rψu=K-χuln⁡Auxx+μHAu-σu+q-μH1u+Rψu.We can reformulate (39) as the following operator equation(40)u=q-σKu+K-χuln⁡Auxx+μHAu-H1u+Rψu.Next, we introduce some notation:(41)E=C10,1,F=ux∈E;ux>0,x∈0,1.It is easy to see that the operator K:E→E is compact and linear.Define the operatorL as the map(42)Lu≔-χuln⁡Auxx+μHAu-H1u.Similar to the proof of Lemma 5(i), we can show that the operator L is continuous and bounded from E to E. We also have(43)LuE=ouEasuE→0.Hence, K∘L is compact operator from E to E with the norm K∘LuE=o(uE). Moreover, the operator R is also compact from E to E and R∘ψuE=o(uE).From the above lemmas, we can show the proof of Theorem1.Proof of Theorem1.
( 1 ) Assume that (u,v)≠(1,0) is the nonnegative solution to system (2). Then we have (44)∫01μHv-σudx=∫01χuln⁡vxx-εuxxdx=0.However, from the proof of Lemma 3, we have (45)∫μHv-σudx<∫01μH1-σvdx≤0which is a contradiction. Thus, we obtain (u,v)≡(1,0).
( 2 ) Assume that there is a constant σ such that σ>μH(1).
Next, we choose a variableι=q-σ as the parameter of bifurcation. Furthermore, σ-μH(1) is a single-value characteristic root. Thus we know that system produces a bifurcation in a small neighbourhood of the point (σ-μH(1),0) according to the classical bifurcation theorem. Then there is a bifurcating solution (ι,ψ), represented as(46)ι=ιζ,ψζ=ζu0+νζ,where ζ≤ε and ε is sufficiently small. Furthermore, ι0=σ-μH1 and ψ(0)=0. Therefore, when ζ is sufficient small, the bifurcation solution is in the set of(47)S+=ιζ∣ζu0+ζψζ,0<ζ<ε,where u0 is the characteristic eigenvector of characteristic eigenvalue (σ-μH(1)) and u0 is a positive constant. There is a nontrivial connected set S+⊂S¯ satisfying the following alternative conclusion: (i)
S¯ connects the two points (σ-μH(1),0) and (μ,0), where ι1 is another characteristic eigenvalue of K.(ii)
S¯ is unbounded and S¯⊆R×E.
We firstly proveS¯⊆R×E. Otherwise, there exists a point (ι^,u^)∈S¯∩(R×∂E) such that (ι^,u^)≠(σ-μH(1),0) and(48)limn→∞⁡ιn,un=ι^,u^,ιn,un⊆​S¯∩R×∂E.Choose a sufficient large constant L. Then by (36), we can obtain (49)εu′′-χln⁡Auxu~′-Lu~=χln⁡Auxx-μHln⁡Au-σ-Lu~≤0.However, u~∈∂E which means that u~ is equal to zero at some point. We can directly have u~≡0 by the maximum principle. That is, (0,ι¯) is a bifurcation point of (40).
Denoteωn=un/unE. Because K is a compact operator, there exists a convergent subsequence of ωn. For the convenience, we still write it as ωn which satisfies Kωn→z. Next, we divide by unE both sides of (40) and let n go to infinity. Then we obtain (50)z~=ι~Kz~,where z~=K-1Z. By the definition of z we know that z~>0. However, K has a unique characteristic eigenvalue with positive characteristic function. Thus, we have ι~=σ-μH(1) which contradicts the assumption. So conclusion (i) is impossible. In other words, S¯ is unbounded and S¯⊆R×E.
According to (i) in Theorem1, we know that if (u,ι)∈S¯, then ι=q-σ>σ-μH(1), which is equivalent to the condition σ<μH(1). However, from Lemma 5, we have that u is bounded for any σ>0. By Lemma 4, we have that uxE is bounded. Furthermore, because the set S¯ is unbounded and connected, we can obtain that (σ-μH(1))⊂ι;u,ι∈S¯, which means that when σ<μH(1), system (4) has nontrivial steady-state solution (u,v).
Finally, we show that system (2)-(3) has a positive solution (u(x),v(x)) when x∈[0,1]. In fact, we know that u(x)>0 because u(x)∈E. Next we prove that v(x)>0. If v(x) is zero at some point, then the initial value problem v′′=uH(v),v(0)=0,v′(0)=0 has a solution v≡0, which contradicts the boundary condition γv′+vx=1=1.



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*Source: 8140380-2019-01-01.xml*