# Global Existence to an Attraction-Repulsion Chemotaxis Model with Fast Diffusion and Nonlinear Source

Authors: Yingjie Zhu; Fuzhong Cong
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper deals with the global existence of solutions to a strongly coupled parabolic-parabolic system of chemotaxis arising from the theory of reinforced random walks. More specifically, we investigate the attraction-repulsion chemotaxis model with fast diffusive term and nonlinear source subject to the Neumann boundary conditions. Such fast diffusion guarantees the global existence of solutions for any given initial value in a bounded domain. Our main results are based on the method of energy estimates, where the key estimates are obtained by a technique originating from Moser’s iterations. Moreover, we notice that the cell density goes to the maximum value when the diffusion coefficient of the cell density tends to infinity.


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## Body

## 1. Introduction

Chemotaxis is known as the active orientation of moving organisms along the chemical gradient. It is observed in many natural systems. For example,myxobacteria produce so-called slime trails on which their cohorts can move more readily. The mathematical models of chemotaxis were introduced by Patlak in [1] and Keller and Segel in [2]. During the past four decades, chemotaxis models have been studied extensively (see, e.g., [3–10] and the rich references therein). For instance, Othmer and Stevens in [3] modeledmyxobacteria as individual random walkers and proposed the microscopic model based on the velocity jump process. By taking the parabolic limit of microscopic model, we can obtain the macroscopic chemotaxis model which is the well-known Keller-Segel system:(1)ut=∇·μu∇u-∇·χu∇Θv+fu,x∈Ω,t>0,τvt=dΔv+ϵu-βv,x∈Ω,t>0,where Ω⊂Rn is a bounded connected domain with a smooth boundary ∂Ω, ϵ>0, β>0, and χ>0 are positive constants, and τ=0,1. The function u=u(x,t) denotes the cell density and v=v(x,t) represents the chemotactic concentration, for example, the oxygen. The constant χ is called the chemosensitive coefficient, and the sign of χ corresponds to chemoattraction if χ>0, and chemorepulsion if χ<0. The function μ(u) and the constant d are the diffusion coefficient of the cell motility and the chemical, respectively. The function f(u) represents the kinetic function describing production and degradation of chemicals, and Θ(v) is commonly referred to the chemotactic potential function.In the absence of logistic source (i.e.,f(u)≡0), there have been extensive studies to system (1). The main feature of solution to the Keller-Segel model is the possibility of blowup in finite time in [3, 8, 11, 12]. For instance, the first result on finite-time blowup for a radially symmetric solutions was shown in [13, 14] when Ω is a ball in R2 under the condition that ∫Ωu0dx>8π/(ϵχ) and ∫Ωu0(x)|x|2dx is sufficiently small. For a general domain, Nagai in [15] further showed the finite-time blowup of nonradial solutions provided that ∫u0|x-y|2dx is sufficiently small and ∫Ωu0dx>8π/(ϵχ) if y∈Ω or ∫Ωu0dx>4π/(ϵχ) if y∈∂Ω with τ=1. Winkler in [16] studied finite-time blowup of radially symmetric solutions to the full parabolic system with logistic sources f(u)=λu-ζuκ in a ball Ω⊂Rn with parameters λ≥0, ζ≥0, and κ≥1. Moreover, Winkler in [8] gave the set of blowup to the system (1) by enforcing initial data with respect to the topology of Lp(Ω)×W1,2(Ω) for any p∈(1,2n/n+2), where Ω is a ball in Rn with n≥3. In summary, the solution of system (1) never blows up when n=1, whereas there is finite-time or infinite-time blowup when n≥3. Moreover, recent results in [17] confirmed that the attraction-repulsion is a plausible mechanism to regularize the classical Keller-Segel system (2) with f(u)=0 whose solutions may blow up in higher dimensions. Authors in [5, 9, 10, 18, 19] proved the global existence of solutions.However, in many biology progresses, the cells usually interact with not only the attractive combination but also repulsive signalling. Therefore, it is necessary to study the attraction-repulsion chemotaxis model:(2)ut=Δu-∇·χu∇v+∇·ξu∇w+fu,x∈Ω,t>0,τvt=Δv+ϵu-βv,x∈Ω,t>0,τwt=Δw+κu-δv,x∈Ω,t>0.System (2) with f(u)=0 was proposed in [20] to describe the aggregation of microglia observed in Alzheimer’s disease. Here u(x,t) represents the cell density and v(x,t) denotes the concentration of the chemoattractant and w(x,t) is the concentration of the chemorepulsion. The constants satisfy χ≥0 and ξ≥0, where χ and ξ measure the strength of chemotactic signal of attraction and repulsion, respectively. The constants ϵ, β, κ, and δ are positive, and they denote the production and degradation rates of the two chemicals, respectively. The first cross-diffusive term in the first equation of (2) means that the orientation movement of the cell is directed to the chemorepulsion, whereas the second cross-diffusive term implies that cells move down the chemoattraction. The second and third equations in (2) elaborate that the two chemicals of chemoattraction and chemorepulsion are released by cells and go through decay. For the case of f(u)≡0, the theorem of competing effects has been established in [17] with τ=0. Moreover, the global existence, asymptotic behavior, and steady states of classical solution were studied in [21] for one-dimensional case with τ=1.


## 2. Motivation

In this paper, we consider the following attraction-repulsion chemotaxis system including three parabolic equations:(3)ut=∇·μu∇u-∇·χvu,vu∇v+χwu,wu∇w+fu,x∈Ω,t>0,γ≥2,vt=dvΔv+gu,v,x∈Ω,t>0,wt=dwΔw+hu,w,x∈Ω,t>0,with the initial-boundary value conditions:(4)ux,0=u0,x∈Ω,vx,0=v0,x∈Ω,wx,0=w0,x∈Ω,∂u∂ν∂Ω=∂v∂ν∂Ω=∂w∂ν∂Ω=0,x∈∂Ω,t>0,where Ω⊂Rn is a bounded C2+η domain for some 0<η<1 and ν denotes the unit outer normal vector field on ∂Ω. The function of the cell density u=u(x,t) is the fraction of volume occupied by cells, whereas the fast diffusion coefficient of the cell is described as(5)μu=11-uγ,γ≥2.It is easy to see that the function 1/(1-u)γ is a monotonically increasing function of u which guarantees that the solution will not blow up. The functions v=v(x,t) and w=w(x,t) denote the concentration of the chemoattraction and chemorepulsion, respectively. The constants dv>0 and dw>0 are the diffusion coefficients of the chemoattraction and chemorepulsion, respectively. χv and χw denote the chemosensitive coefficients, and f(u) describes the kinetic function.In particular, the global existence of solutions was obtained by increasing the diffusion coefficientμ(u): for example, μ(u)=1/(1-u)γ, γ≥2 in [11], and μ(u)=(1+u)/(1-u+log⁡u) in [5]. Liu and Tao in [22] established global boundedness of classical solutions to the parabolic attraction-repulsion chemotaxis system (2) when f(u)≡0. For the case of including logistic source, global existence and boundedness of classical solution were studied in [23, 24]. For the case γ=1, the existence of families of traveling impulses and fronts was analyzed in [25].In this paper, we will prove the existence of global classical solutions to the generalized system (3) with the initial-boundary conditions and the no-flux boundary condition (4). In addition, we need the following assumptions.(A1)
LetD:=[0,1]×[0,+∞). Suppose that functions χv(u,v),χw(u,w):D→[0,+∞) are of C2 class, and χv(0,v)=0 for v≥0 and χw(0,w)=0 for w≥0.(A2)
Assume that functionsf,g,h:D→R are of C2 class and that f(u)≤Mf(1-u), g(u,v)≤Mg(1-u), and h(u,w)≤Mh(1-u) for all (u,v),(u,w)∈D, where Mf,Mg,Mh≥0.(A3)
Letdv,dw:[0,+∞)→[d0,d1) be of C2 class, where 0<d0<d1<∞.(A4)
u0,v0,w0∈C(Ω¯), 0≤u0(x)<1, and v0(x),w0(x)≥0 for x∈C(Ω¯) and ∂u0/∂ν=∂v0/∂ν=∂w0/∂ν=0 for x∈∂Ω.The main result is the following.Theorem 1.
Let(u0,v0,w0)∈[H2(Ω)]3. Then there exists a unique global triple solution of (u,v,w) to system (3) satisfying initial conditions (4) and (6)u,v,w∈C0Ω¯×0,+∞;R33∩C2,1Ω¯×0,+∞;R33.Furthermore, there exist two constants Cv≥0 and Cw≥0 such that (7)0≤vx,t≤Cv,0≤wx,t≤Cw,0≤ux,t<1.Throughout this paper, we introduce some notations.Ω is a bounded open interval in R=(-∞,+∞) and C denotes a general constant which may have different values in different place. Lp=Lp(Ω)(1≤p<∞) is the usual Lebesgue space with the norm fLp=∫Ω|f(x)|p1/p for 1≤p<∞ and fL∞=esssupx∈Ω⁡|f|. When p=2, we denote fL2=f for convenience. Hk is the kth-order Sobolev space Wk,2 with the norm fHk=∑α=0k∂xαf(x)21/2. For notational convenience, we write f(·,t)Lp and f(·,t)Hk as f(·,t)p and f(·,t)k, respectively. Moreover, we denote (8)f,g,hp=fp+gp+hp,1≤p≤+∞,f,g,hk=fk+gk+hk,k=1,2,3,….The rest of this paper is organized as follows. In Section2, we establish the local existence and give some preliminary lemmas. Some necessary priori estimates will be established in Section 3. We will complete the proof of Theorem 1 in Section 4.


## 3. Local Existence

To prove Theorem1, we need to establish local existence of solutions to system (3) and some priori estimates in this section.Theorem 2 (local existence).
Let assumptions (A1)–(A4) hold. Then(1)
There is a positive constantT0 depending on initial value (u0,v0,w0) such that system (3) has a unique maximal solution (u,v,w) in the space C1+q/2,2+q(Ω¯×[0,T0);R3) with u(x,t),v(x,t),w(x,t)≥0.(2)
There is a global classical solution of system (3) if u(x,t) is bounded away from 1 for each finite time t>0, which means T0=+∞.Proof.
LetU=(u,v,w)∈R3. System (3) can be written as(9)∂tU=∇·JU∇U+FU,
where(10)JU=du-χvuχwu0dv000dw,FU=fugu,vhu,w.System (9) satisfies the initial value(11)U·,0=u0,v0,w0,x∈Ω.For some θ>0 and given initial conditions u0<1-θ, it is clear that the eigenvalues of matrix J(U) are positive at t=0. Thus, system (9) is normally parabolic and there exists local solution by Theorem 7.3 in [26]; that is, there is a T0>0 such that the unique solution (u,v,w)∈C([0,T0)×Ω¯;R3)∩C1,2([0,T0)×Ω¯;R3).
Next we rewrite the first equation of (3) as (12)ut=11-uγΔu+γ1-uγ+1∇u-χvu∇v+χwu∇w·∇u-χvΔv+χwΔw+χvv∇v2+χww∇w2+fu.Recalling that assumption (A2) ensures that χv(0,v)=0 for all v≥0 and χw(0,w)=0 for all w≥0, this means that χvv(0,v)=χww(0,w)=0 for all v≥0 and w≥0, where χvv=∂χv/∂v and χww=∂χw/∂w. Hence, the source term χvΔv+χwΔw+χvv|∇v|2+χww|∇w|2+f(u)=0 is zero when u=0 at some (x0,t0). By applying the maximum principle to (1.1)a, therefore, we obtain that u≥0 whenever v≥0 and w≥0. Similarly, we infer v≥0 (or w≥0) from the second equation of (3) (or the third of (3)) whenever u≥0. The proof of (1) in Theorem 2 is finished.
The proof of (2) in Theorem2 is completed by applying the theorem of quasilinear parabolic equations in [26] since system (9) is an upper triangular system.


## 4. A Priori Estimates

Next, we recall the Gagliardo-Nirenberg inequality for functions satisfying the boundary condition fort>0 (see [27, 28]).Lemma 3.
LetΩ be a bounded open domain satisfying the uniform cone property in Rn with n≥2:(13)p∈1,+∞,ifn=2,1,2nn-2,ifn≥3.Then there exists a positive constant Cp, which depends on n, p, and Ω, such that for all u∈W1,2: (14)up≤Cp∇u2au11-a+u1,where a=(1-1/p)/(1/2+1/n) and 0≤a<1.Lemma 4.
Let the conditions in Lemma3 hold: a0=n/n+2 andl≥γ. Then there exists a constant N1 independent ofl such that(15)u2l/l+γ≤N1∇u2a0l-γ/lu11-a0l-γ/l+u1∀u∈W1,2Ω.Proof.
Recalling1≤2l/l+γ<2 for l≥γ and employing Lemma 3 with a=(1-1/p)/(1/2+1/n) and p=2, we obtain a0=n/n+2; thus,(16)u2l/l+γ≤Cq∇u2a0u11-a0+u1l-γ/lu1γ/l≤N1∇u2a0l-γ/lu11-a0l-γ/l+u1,where we have used the inequality (c+d)m≤2m-1(cm+dm) for c,d,m≥0.Lemma 5.
Let a nonnegative numerical sequence{Si}i=0∞ satisfying Si+1≤aiSi1+bi with ai≥1 and bi≥0 for i≥0. Suppose S0≥1 and ∏i=1∞(1+bi)=N2<+∞. Then one has(17)Si≤∏i=0∞aiN2S0N2.Proof.
By the definition of the sequence{Si}i=0∞, we obtain that for N2≥1(18)Si≤ai-1Si-11+bi-1≤ai-1ai-2Si-21+bi-21+bi-1=ai-1ai-21+bi-1Si-21+bi-21+bi-1≤ai-1ai-21+bi-1ai-31+bi-11+bi-2Si-31+bi-31+bi-21+bi-1≤ai-1ai-2N2ai-3N2Si-31+bi-31+bi-21+bi-1≤⋯≤∏i=0∞aiN2S0N2.Lemma 6.
Let conditions (A1)–(A4) hold. Then there exists a unique global solution(u,v,w) to system (3) such that u, v, and w are in C1+η/2,2+η(Ω¯×[0,+∞);R3). Furthermore, there exists two positive constants Cv and Cw satisfying (19)0≤vx,t≤Cv,0≤wx,t≤Cw,0≤ux,t<1∀x∈Ω¯,∀t≥0.Proof.
After appropriate scaling, the system is limited in a regionΩ satisfying |Ω|=1. Next, we introduce the auxiliary scalar equation:(20)∂tu=∇·11-uα∇u-∇·b→x,t+∇·c→x,t+P→u,x,t∀x∈Ω×0,Twith the initial-boundary condition:(21)ux,0=u0x,∀x∈Ω∂u∂ν=0,b→·ν→∂Ω=0,c→·ν→∂Ω=0∀x∈∂Ω,where b→(x,t), c→(x,t), and P→∈L∞(Ω×(0,∞)) are given functions. To complete the proof of Lemma 6, we will establish the following lemma.Lemma 7.
Suppose0≤u0(x)≤1 for all x∈Ω. Let(22)b→L∞Ω×0,+∞=Mb,c→L∞Ω×0,+∞=Mc,P→u,·,·L∞Ω×0,+∞≤MP1-uforu∈0,1.Let u be a classical solution to system (20)-(21) satisfying 0≤u0(x)<1 for (x,t)∈QT=Ω¯×[0,T). Then, for any T>0, there exists a constant QT>0 such that u(x,t)<1-εTfor all (x,t)∈QT, where the constant εT depends only on Mb, Mc, θ=1-u0L∞(Ω), and T.Proof.
We takeθ>0 satisfying u0≤1-θ for all x∈Ω¯ and notice that |Ω|=1 by scaling x and t in the governing equation. To complete the proof, we have to divide the proof into three steps.
Step
1. We will firstly prove that 1/1-u∈Lq(Ω) for any 1≤q<∞. Multiplying both sides of the first equation of (3) by q(1-u)-q-1 and integrating the resultant equation, we yield(23)ddt∫1-u-qdx=∫q1-u-q-1utdx=q∫1-u-q-1∇∇u1-uγ-b→x,t+c→x,tdx+q∫1-u-q-1P→u,x,tdx=-q1+q∫∇u21-uγ+P+2-∇u·b→x,t-c→x,t1-uq+2dx+q∫P→u,x,t1-uq+1dx≤-q1+q∫∇u21-uγ+P+2-∇u·b→x,t-c→x,t1-uq+2dx+q∫MP1-uqdx,where the last inequality follows from (A3); then (23) is rewritten as(24)ddt∫1-u-qdx-q∫MP1-uqdx≤q1+q∫-∇u1-uγ+q+2/22+∇u·b→x,t-c→x,t1-uγ+q+2/2·11-uq+2-γ/2dx.Noticing Wq=(1-u)-γ+q/2, we obtain the fact that(25)1-u-q=Wq2q/γ+q,∇Wq=γ+q21-u-γ+q+2/2∇u.Substituting (24) into (25), we have(26)ddt∫1-u-qdx-q∫MP1-uqdx≤q1+q∫-4γ+q2∇Wq2+2γ+q·∇Wq·b→x,t-c→x,t1-uq+2-γ/2dx=q1+qγ+q∫-4γ+q∇Wq2+2∇Wq·b→x,t-c→x,t1-uq+2-γ/2dx≤q1+qγ+q∫-4γ+q∇Wq2dx+2∫∇WqMb-Mc1-uq+2-γ/2dx≤q1+qγ+q∫-4γ+q∇Wq2dx+2∫12ɛ∇Wq2+ɛ2Mb+Mc21-uq+2-γ/2dx≤q1+qγ+q∫-2γ+q∇Wq2dx+γ+q2Mb+Mc2∫11-uq+2-γ/2dx,where we have applied Young’s inequality with ɛ=1/2(γ+q).
Combing with the assumption of Lemma6, 0≤u(x,t)<1 and γ≥2, we obtain the following inequality:(27)ddt∫1-u-qdx≤-2qq+1q+γ2∫∇Wq2dx+q+1qM02∫1-u-qdx,where M02=1/2(Mb+Mc)2+MP.
For convenience, we defineJq=∫(1-u)-qdx. Therefore, for some constant C>0, we have (28)ddtJq≤Cqq+1Jq,from which we yield that (29)Jq1/q≤Jq01/qecq+1t≤Ω1/qσecq+1t≤c~ecq+1t,for some positive constant c~ which is independent of q. Thus, for any 0≤T<+∞, we conclude that (30)11-uq≤c~ecq+1T,which means that 1/1-u∈Lq(Ω) for any 1≤q<+∞ and 0≤T<+∞. We notice that 1/1-uq depends only on M0≡1/2(Mb+Mc)2+MP and θ≡1-u0∞ and T.
Step
2. Choose a q satisfying q≥3γ and q(q+1)/(γ+q)2≥1/2. Then we obtain from (25) that(31)ddt∫Wq2q/q+γdx≤-∫∇Wq2dx+q+γ2M02∫Wq2q/q+γdx≤-∫∇Wq2dx+23q2M02∫Wq2q/q+γdx≤∇Wq22+23q2M02M12q/q+γ∇Wq2a0l-γ/lWq11-a0l-γ/l+Wq12q/q+γ.Recalling ɛ-Young’s inequality ab≤ɛ(ar/r)+ɛ-(s/r)bs/s, 1/r+1/s=1, we have(32)∇Wq2a0l-γ/lWq11-a0l-γ/l≤ɛr∇Wq2a0l-γ/lr+1sWq11-a0l-γ/lɛ1/rs=ɛr∇wq2a0l-γ/lr+1-a0l-γlɛ-s/rwq11-a0l-γ/ls=ɛa0l-γl∇wq2+1-a0l-γl1ɛa0l-γ/l1-a0l-γ/lwq1,where r=l/a0(l-γ) and s=1/(1-a0l-γ/l). Combining (31) with (32), we yield(33)ddt∫Wq2q/q+γdx≤-∇Wq22+23q2M02M12q/q+γa0l-γɛl∇Wq2+1-a0l-γlɛ-a0l-γ/l-a0l-γWq12q/q+γ≤-∇Wq22+c1q2a0l-γɛl∇Wq2+1+l-a0l-γlɛ-a0l-γ/l-a0l-γWq12q/q+γ≤-∇Wq22+2q-γ/q+γc1q2a0l-γlɛ2q/q+γ∇Wq22q/q+γ+1+l-a0l-γlɛ-a0l-γ/l-a0l-γ2q/q+γWq12q/q+γ,where c1=2/3M02M12q/q+γ. We notice that (34)∇Wq22q/q+γ≤∇Wq22+1≤∇Wq22+Wq1.Hence, we choose a sufficiently small ɛ such that(35)ddt∫Wq2q/q+γdx≤-12∇Wq22+Cq2+nWq12q/q+γ.Furthermore, we have the estimate(36)Wq2q/q+γ≤2N1∇Wq2+Wq1whose proof is similar to Lemma 4. Thus, we omit it. Squaring (36), we obtain that (37)Wq2q/q+γ2≤4N12∇Wq2+Wq12≤8N12∇Wq22+Wq12.Thus, (38)∇Wq22≥18N12Wq2q/q+γ2-Wq12.That is,(39)-12∇Wq22≤-116N12Wq2q/q+γ2+12Wq12.Substituting the above inequality into (39), we have(40)ddt∫Wq2q/q+γdx≤-116N12Wq2q/q+γ2+12Wq12+C~q2+nWq12q/q+γ≤-C1Wq2q/q+γ2q/q+γ+C2qn+2Wq12for some constants C1,C2>0.
Step
3. We notice that |Ω|=1, q≥3γ and define (41)Vq=max⁡11-u·,0∞,sup0≤t<T⁡∫ΩWqdx2/q+γ=max⁡11-u·,0∞,sup0≤t<T⁡∫Ω11-uq+γ/2dx2/q+γ≥1.We also see that Vq is bounded and depends only on Mb, Mc, θ, and T. It is easy to observe that 1/|Ω|∫Ω|u|pdx1/p is nondecreasing in p with p≥1 and Vq is a nondecreasing function with respect to q. Combining the above analysis, we yield (42)ddt∫ΩWq2q/q+γdx≤-C1Wq2q/q+γ2q/q+γ+C2qn+2Vqq+γ.That is,(43)ddtec~t∫ΩWq2q/q+γdx≤C2qn+2ec~tVqq+γ.Integrating (43) in t, we have(44)∫ΩWq2q/q+γdx≤e-c~t11-u·,0∞q+C2c~q2+nVqq+γ≤Uqq+C2c~q2+nVqq+γ≤Cq2+nVqq+γ.Assuming q=2k-γ by the definition of Vq, we get(45)V2k-γ=max⁡11-u·,0∞,sup0≤t<T⁡∫Ω11-ukdx1/k,where(46)∫Ω11-ukdx=∫ΩWk2k/k+γdx.From (44)–(46), we obtain that(47)V2k-γ≤max⁡11-u·,0∞,Ck2+nVkk+γ1/k≤max⁡Vk,Ck2+nVkk+γ1/k≤Ck2+n1/kVk1+γ/k.By a similar argument as in [18], we immediately yield that 1/1-u∈L∞(Ω). Thus, the proof of Lemma 7 is completed.


## 5. Proof of Theorem1

In this section, we will complete the proof of Theorem1 by the local existence and some priori estimates as given below.Proof.
Suppose there exists aC1+q/2,2+q solution for the maximal time T0<∞. Then according to Lemma 6 we know 0≤u≤1. Thus, if we treat u as a source term in the second and third equations of (9), we can obtain that(48)vWq2,1QT≤Cq,T01LqΩ×0,T0≤Cq,T0′for  anyq>1,wWq2,1QT≤C~q,T01LqΩ×0,T0≤C~q,T0′for  anyq>1.By employing the Lq estimate for parabolic equation in [27] when q goes to infinity, we obtain the L∞ norm bound of v, |∇v|, and w, |∇w|. So we have T0 being infinite from the second conclusion in Theorem 1 which contradicts our original assumption. That is, the maximal time T0=∞.



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