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Research article Special Issues

Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system

  • Chameleon systems are dynamical systems that exhibit either self-excited or hidden oscillations depending on the parameter values. This paper presents a comprehensive investigation of a quadratic chameleon system, including an analysis of its symmetry, dissipation, local stability, Hopf bifurcation, and various chaotic dynamics as the control parameters (μ,a,c) vary. Here, μ serves as the dissipation parameter in the y direction. Bifurcation analysis for four scenarios with μ=0 was performed, revealing the emergence of various dynamical phenomena under different parameter settings. Offset boosting means introducing a constant into one of the state variables of the system for boosting the variable to a different level. Additionally, hidden chaotic bistability with offset boosting was exhibited by varying μ. The parameter μ serves as both the Hopf bifurcation parameter and the offset boosting parameter, while the other parameters (a,c) also play critical roles as control parameters, resulting in period-doubling routes to self-excited or hidden chaotic attractors. These findings enrich our understanding of nonlinear dynamics in quadratic chameleon systems.

    Citation: Jie Liu, Bo Sang, Lihua Fan, Chun Wang, Xueqing Liu, Ning Wang, Irfan Ahmad. Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system[J]. AIMS Mathematics, 2025, 10(3): 4915-4937. doi: 10.3934/math.2025225

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  • Chameleon systems are dynamical systems that exhibit either self-excited or hidden oscillations depending on the parameter values. This paper presents a comprehensive investigation of a quadratic chameleon system, including an analysis of its symmetry, dissipation, local stability, Hopf bifurcation, and various chaotic dynamics as the control parameters (μ,a,c) vary. Here, μ serves as the dissipation parameter in the y direction. Bifurcation analysis for four scenarios with μ=0 was performed, revealing the emergence of various dynamical phenomena under different parameter settings. Offset boosting means introducing a constant into one of the state variables of the system for boosting the variable to a different level. Additionally, hidden chaotic bistability with offset boosting was exhibited by varying μ. The parameter μ serves as both the Hopf bifurcation parameter and the offset boosting parameter, while the other parameters (a,c) also play critical roles as control parameters, resulting in period-doubling routes to self-excited or hidden chaotic attractors. These findings enrich our understanding of nonlinear dynamics in quadratic chameleon systems.



    Baló's concentric sclerosis (BCS) was first described by Marburg [1] in 1906, but became more widely known until 1928 when the Hungarian neuropathologist Josef Baló published a report of a 23-year-old student with right hemiparesis, aphasia, and papilledema, who at autopsy had several lesions of the cerebral white matter, with an unusual concentric pattern of demyelination [2]. Traditionally, BCS is often regarded as a rare variant of multiple sclerosis (MS). Clinically, BCS is most often characterized by an acute onset with steady progression to major disability and death with months, thus resembling Marburg's acute MS [3,4]. Its pathological hallmarks are oligodendrocyte loss and large demyelinated lesions characterized by the annual ring-like alternating pattern of demyelinating and myelin-preserved regions. In [5], the authors found that tissue preconditioning might explain why Baló lesions develop a concentric pattern. According to the tissue preconditioning theory and the analogies between Baló's sclerosis and the Liesegang periodic precipitation phenomenon, Khonsari and Calvez [6] established the following chemotaxis model

    ˜uτ=DΔX˜udiffusion ofactivated macrophagesX(˜χ˜u(ˉu˜u)˜v)chemoattractant attractssurrounding activated macrophages+μ˜u(ˉu˜u)production of activated macrophages,˜ϵΔX˜vdiffusion of chemoattractant=˜α˜v+˜β˜wdegradationproduction of chemoattractant,˜wτ=κ˜uˉu+˜u˜u(ˉw˜w)destruction of oligodendrocytes, (1.1)

    where ˜u, ˜v and ˜w are, respectively, the density of activated macrophages, the concentration of chemoattractants and density of destroyed oligodendrocytes. ˉu and ˉw represent the characteristic densities of macrophages and oligodendrocytes respectively.

    By numerical simulation, the authors in [6,7] indicated that model (1.1) only produces heterogeneous concentric demyelination and homogeneous demyelinated plaques as χ value gradually increases. In addition to the chemoattractant produced by destroyed oligodendrocytes, "classically activated'' M1 microglia also can release cytotoxicity [8]. Therefore we introduce a linear production term into the second equation of model (1.1), and establish the following BCS chemotaxis model with linear production term

    {˜uτ=DΔX˜uX(˜χ˜u(ˉu˜u)˜v)+μ˜u(ˉu˜u),˜ϵΔX˜v+˜α˜v=˜β˜w+˜γ˜u,˜wτ=κ˜uˉu+˜u˜u(ˉw˜w). (1.2)

    Before going to details, let us simplify model (1.2) with the following scaling

    u=˜uˉu,v=μˉu˜ϵD˜v,w=˜wˉw,t=μˉuτ,x=μˉuDX,χ=˜χ˜ϵμ,α=D˜α˜ϵμˉu,β=˜βˉw,γ=˜γˉu,δ=κμ,

    then model (1.2) takes the form

    {ut=Δu(χu(1u)v)+u(1u),xΩ,t>0,Δv+αv=βw+γu,xΩ,t>0,wt=δu1+uu(1w),xΩ,t>0,ηu=ηv=0,xΩ,t>0,u(x,0)=u0(x),w(x,0)=w0(x),xΩ, (1.3)

    where ΩRn(n1) is a smooth bounded domain, η is the outward normal vector to Ω, η=/η, δ balances the speed of the front and the intensity of the macrophages in damaging the myelin. The parameters χ,α and δ are positive constants as well as β,γ are nonnegative constants.

    If δ=0, then model (1.3) is a parabolic-elliptic chemotaxis system with volume-filling effect and logistic source. In order to be more line with biologically realistic mechanisms, Hillen and Painter [9,10] considered the finite size of individual cells-"volume-filling'' and derived volume-filling models

    {ut=(Du(q(u)q(u)u)uq(u)uχ(v)v)+f(u,v),vt=DvΔv+g(u,v). (1.4)

    q(u) is the probability of the cell finding space at its neighbouring location. It is also called the squeezing probability, which reflects the elastic properties of cells. For the linear choice of q(u)=1u, global existence of solutions to model (1.4) in any space dimension are investigated in [9]. Wang and Thomas [11] established the global existence of classical solutions and given necessary and sufficient conditions for spatial pattern formation to a generalized volume-filling chemotaxis model. For a chemotaxis system with generalized volume-filling effect and logistic source, the global boundedness and finite time blow-up of solutions are obtained in [12]. Furthermore, the pattern formation of the volume-filling chemotaxis systems with logistic source and both linear diffusion and nonlinear diffusion are shown in [13,14,15] by the weakly nonlinear analysis. For parabolic-elliptic Keller-Segel volume-filling chemotaxis model with linear squeezing probability, asymptotic behavior of solutions is studied both in the whole space Rn [16] and on bounded domains [17]. Moreover, the boundedness and singularity formation in parabolic-elliptic Keller-Segel volume-filling chemotaxis model with nonlinear squeezing probability are discussed in [18,19].

    Very recently, we [20] investigated the uniform boundedness and global asymptotic stability for the following chemotaxis model of multiple sclerosis

    {ut=Δu(χ(u)v)+u(1u),χ(u)=χu1+u,xΩ,t>0,τvt=Δvβv+αw+γu,xΩ,t>0,wt=δu1+uu(1w),xΩ,t>0,

    subject to the homogeneous Neumann boundary conditions.

    In this paper, we are first devoted to studying the local existence and uniform boundedness of the unique classical solution to system (1.3) by using Neumann heat semigroup arguments, Banach fixed point theorem, parabolic Schauder estimate and elliptic regularity theory. Then we discuss that exponential asymptotic stability of the positive equilibrium point to system (1.3) by constructing Lyapunov function.

    Although, in the pathological mechanism of BCS, the initial data in model (1.3) satisfy 0<u0(x)1,w0(x)=0, we mathematically assume that

    {u0(x)C0(ˉΩ)with0,u0(x)1inΩ,w0(x)C2+ν(ˉΩ)with0<ν<1and0w0(x)1inΩ. (1.5)

    It is because the condition (1.5) implies u(x,t0)>0 for any t0>0 by the strong maximum principle.

    The following theorems give the main results of this paper.

    Theorem 1.1. Assume that the initial data (u0(x),w0(x)) satisfy the condition (1.5). Then model (1.3) possesses a unique global solution (u(x,t),v(x,t),w(x,t)) satisfying

    u(x,t)C0(ˉΩ×[0,))C2,1(ˉΩ×(0,)),v(x,t)C0((0,),C2(ˉΩ)),w(x,t)C2,1(ˉΩ×[0,)), (1.6)

    and

    0<u(x,t)1,0v(x,t)β+γα,w0(x)w(x,t)1,inˉΩ×(0,).

    Moreover, there exist a ν(0,1) and M>0 such that

    uC2+ν,1+ν/2(ˉΩ×[1,))+vC0([1,),C2+ν(ˉΩ))+wCν,1+ν/2(ˉΩ×[1,))M. (1.7)

    Theorem 1.2. Assume that β0,γ0,β+γ>0 and

    χ<{min{22αβ,22αγ},β>0,γ>0,22αβ,β>0,γ=0,22αγ,β=0,γ>0. (1.8)

    Let (u,v,w) be a positive classical solution of the problem (1.3), (1.5). Then

    u(,t)uL(Ω)+v(,t)vL(Ω)+w(,t)wL(Ω)0,ast. (1.9)

    Furthermore, there exist positive constants λ=λ(χ,α,γ,δ,n) and C=C(|Ω|,χ,α,β,γ,δ) such that

    uuL(Ω)Ceλt,vvL(Ω)Ceλt,wwL(Ω)Ceλt,t>0, (1.10)

    where (u,v,w)=(1,β+γα,1) is the unique positive equilibrium point of the model (1.3).

    The paper is organized as follows. In section 2, we prove the local existence, the boundedness and global existence of a unique classical solution. In section 3, we firstly establish the uniform convergence of the positive global classical solution, then discuss the exponential asymptotic stability of positive equilibrium point in the case of weak chemotactic sensitivity. The paper ends with a brief concluding remarks.

    The aim of this section is to develop the existence and boundedness of a global classical solution by employing Neumann heat semigroup arguments, Banach fixed point theorem, parabolic Schauder estimate and elliptic regularity theory.

    Proof of Theorem 1.1 (ⅰ) Existence. For p(1,), let A denote the sectorial operator defined by

    Au:=ΔuforuD(A):={φW2,p(Ω)|ηφ|Ω=0}.

    λ1>0 denote the first nonzero eigenvalue of Δ in Ω with zero-flux boundary condition. Let A1=Δ+α and Xl be the domains of fractional powers operator Al,l0. From the Theorem 1.6.1 in [21], we know that for any p>n and l(n2p,12),

    zL(Ω)CAl1zLp(Ω)forallzXl. (2.1)

    We introduce the closed subset

    S:={uX|uL((0,T);L(Ω))R+1}

    in the space X:=C0([0,T];C0(ˉΩ)), where R is a any positive number satisfying

    u0(x)L(Ω)R

    and T>0 will be specified later. Note F(u)=u1+u, we consider an auxiliary problem with F(u) replaced by its extension ˜F(u) defined by

    ˜F(u)={F(u)uifu0,F(u)(u)ifu<0.

    Notice that ˜F(u) is a smooth globally Lipshitz function. Given ˆuS, we define Ψˆu=u by first writing

    w(x,t)=(w0(x)1)eδt0˜F(ˆu)ˆuds+1,xΩ,t>0, (2.2)

    and

    w0w(x,t)1,xΩ,t>0,

    then letting v solve

    {Δv+αv=βw+γˆu,xΩ,t(0,T),ηv=0,xΩ,t(0,T), (2.3)

    and finally taking u to be the solution of the linear parabolic problem

    {ut=Δuχ(ˆu(1ˆu)v)+ˆu(1ˆu),xΩ,t(0,T),ηu=0,xΩ,t(0,T),u(x,0)=u0(x),xΩ.

    Applying Agmon-Douglas-Nirenberg Theorem [22,23] for the problem (2.3), there exists a constant C such that

    vW2p(Ω)C(βwLp(Ω)+γˆuLp(Ω))C(β|Ω|1p+γ(R+1)) (2.4)

    for all t(0,T). From a variation-of-constants formula, we define

    Ψ(ˆu)=etΔu0χt0e(ts)Δ(ˆu(1ˆu)v(s))ds+t0e(ts)Δˆu(s)(1ˆu(s))ds.

    First we shall show that for T small enough

    Ψ(ˆu)L((0,T);L(Ω))R+1

    for any ˆuS. From the maximum principle, we can give

    etΔu0L(Ω)u0L(Ω), (2.5)

    and

    t0etΔˆu(s)(1ˆu(s))L(Ω)dst0ˆu(s)(1ˆu(s))L(Ω)ds(R+1)(R+2)T (2.6)

    for all t(0,T). We use inequalities (2.1) and (2.4) to estimate

    χt0e(ts)Δ(ˆu(1ˆu)v(s))L(Ω)dsCt0(ts)lets2Δ(ˆu(1ˆu)v(s))Lp(Ω)dsCt0(ts)l12(ˆu(1ˆu)v(s)Lp(Ω)dsCT12l(R+1)(R+2)(β|Ω|1p+γ(R+1)) (2.7)

    for all t(0,T). This estimate is attributed to T<1 and the inequality in [24], Lemma 1.3 iv]

    etΔzLp(Ω)C1(1+t12)eλ1tzLp(Ω)forallzCc(Ω).

    From inequalities (2.5), (2.6) and (2.7) we can deduce that Ψ maps S into itself for T small enough.

    Next we prove that the map Ψ is a contractive on S. For ˆu1,ˆu2S, we estimate

    Ψ(ˆu1)Ψ(ˆu2)L(Ω)χt0(ts)l12[ˆu2(s)(1ˆu2(s))ˆu1(s)(1ˆu1(s))]v2(s)Lp(Ω)ds+χt0ˆu1(s)(1ˆu1(s))(v1(s)v2(s))Lp(Ω)ds+t0e(ts)Δ[ˆu1(s)(1ˆu1(s))ˆu2(s)(1ˆu2(s))]L(Ω)dsχt0(ts)l12(2R+1)ˆu1(s)ˆu2(s)Xv2(s)Lp(Ω)ds+χt0(R+1)(R+2)(βw1(s)w2(s)Lp(Ω)+γˆu1(s)ˆu2(s)Lp(Ω))ds+t0(2R+1)ˆu1(s)ˆu2(s)Xdsχt0(ts)l12(2R+1)ˆu1(s)ˆu2(s)Xv2(s)Lp(Ω)ds+2βδχt0(R+1)(R+2)tˆu1(s)ˆu2(s)Lp(Ω)+γˆu1(s)ˆu2(s)Lp(Ω)ds+t0(2R+1)ˆu1(s)ˆu2(s)Xds(CχT12l(2R+1)(β|Ω|1p+γ(R+1))+2βδχT(R2+3R+γ+2)+T(2R+1))ˆu1(s)ˆu2(s)X.

    Fixing T(0,1) small enough such that

    (CχT12l(2R+1)(β|Ω|1p+γ(R+1))+2βδχT(R2+3R+γ+2)+T(2R+1))12.

    It follows from the Banach fixed point theorem that there exists a unique fixed point of Ψ.

    (ⅱ) Regularity. Since the above of T depends on u0L(Ω) and w0L(Ω) only, it is clear that (u,v,w) can be extended up to some maximal Tmax(0,]. Let QT=Ω×(0,T] for all T(0,Tmax). From uC0(ˉQT), we know that wC0,1(ˉQT) by the expression (2.2) and vC0([0,T],W2p(Ω)) by Agmon-Douglas-Nirenberg Theorem [22,23]. From parabolic Lp-estimate and the embedding relation W1p(Ω)Cν(ˉΩ),p>n, we can get uW2,1p(QT). By applying the following embedding relation

    W2,1p(QT)Cν,ν/2(ˉQT),p>n+22, (2.8)

    we can derive u(x,t)Cν,ν/2(ˉQT) with 0<ν2n+2p. The conclusion wCν,1+ν/2(ˉQT) can be obtained by substituting uCν,ν/2(ˉQT) into the formulation (2.2). The regularity uC2+ν,1+ν/2(ˉQT) can be deduced by using further bootstrap argument and the parabolic Schauder estimate. Similarly, we can get vC0((0,T),C2+ν(ˉΩ)) by using Agmon-Douglas-Nirenberg Theorem [22,23]. From the regularity of u we have wC2+ν,1+ν/2(ˉQT).

    Moreover, the maximal principle entails that 0<u(x,t)1, 0v(x,t)β+γα. It follows from the positivity of u that ˜F(u)=F(u) and because of the uniqueness of solution we infer the existence of the solution to the original problem.

    (ⅲ) Uniqueness. Suppose (u1,v1,w1) and (u2,v2,w2) are two deferent solutions of model (1.3) in Ω×[0,T]. Let U=u1u2, V=v1v2, W=w1w2 for t(0,T). Then

    12ddtΩU2dx+Ω|U|2dxχΩ|u1(1u1)u2(1u2)|v1||U|+u2(1u2)|V||U|dx+Ω|u1(1u1)u2(1u2)||U|dxχΩ|U||v1||U|+14|V||U|dx+Ω|U|2dxΩ|U|2dx+χ232Ω|V|2dx+χ2K2+22Ω|U|2dx, (2.9)

    where we have used that |v1|K results from v1C0([0,T],C0(ˉΩ)).

    Similarly, by Young inequality and w0w11, we can estimate

    Ω|V|2dx+α2Ω|V|2dxβ2αΩ|W|2dx+γ2αΩ|U|2dx, (2.10)

    and

    ddtΩW2dxδΩ|U|2+|W|2dx. (2.11)

    Finally, adding to the inequalities (2.9)–(2.11) yields

    ddt(ΩU2dx+ΩW2dx)C(ΩU2dx+ΩW2dx)forallt(0,T).

    The results U0, W0 in Ω×(0,T) are obtained by Gronwall's lemma. From the inequality (2.10), we have V0. Hence (u1,v1,w1)=(u2,v2,w2) in Ω×(0,T).

    (ⅳ) Uniform estimates. We use the Agmon-Douglas-Nirenberg Theorem [22,23] for the second equation of the model (1.3) to get

    vC0([t,t+1],W2p(Ω))C(uLp(Ω×[t,t+1])+wLp(Ω×[t,t+1]))C2 (2.12)

    for all t1 and C2 is independent of t. From the embedded relationship W1p(Ω)C0(ˉΩ),p>n, the parabolic Lp-estimate and the estimation (2.12), we have

    uW2,1p(Ω×[t,t+1])C3

    for all t1. The estimate uCν,ν2(ˉΩ×[t,t+1])C4 for all t1 obtained by the embedded relationship (2.8). We can immediately compute wCν,1+ν2(ˉΩ×[t,t+1])C5 for all t1 according to the regularity of u and the specific expression of w. Further, bootstrapping argument leads to vC0([t,t+1],C2+ν(ˉΩ))C6 and uC2+ν,1+ν2(ˉΩ×[t,t+1])C7 for all t1. Thus the uniform estimation (1.7) is proved.

    Remark 2.1. Assume the initial data 0<u0(x)1 and w0(x)=0. Then the BCS model (1.3) has a unique classical solution.

    In this section we investigate the global asymptotic stability of the unique positive equilibrium point (1,β+γα,1) to model (1.3). To this end, we first introduce following auxiliary problem

    {uϵt=Δuϵ(uϵ(1uϵ)vϵ)+uϵ(1uϵ),xΩ,t>0,Δvϵ+αvϵ=βwϵ+γuϵ,xΩ,t>0,wϵt=δu2ϵ+ϵ1+uϵ(1wϵ),xΩ,t>0,ηuϵ=ηvϵ=0,xΩ,t>0,uϵ(x,0)=u0(x),wϵ(x,0)=w0(x),xΩ. (3.1)

    By a similar proof of Theorem 1.1, we get that the problem (3.1) has a unique global classical solution (uϵ,vϵ,wϵ), and there exist a ν(0,1) and M1>0 which is independent of ϵ such that

    uϵC2+ν,1+ν/2(ˉΩ×[1,))+vϵC2+ν,1+ν/2(ˉΩ×[1,))+wϵCν,1+ν/2(ˉΩ×[1,))M1. (3.2)

    Then, motivated by some ideas from [25,26], we construct a Lyapunov function to study the uniform convergence of homogeneous steady state for the problem (3.1).

    Let us give following lemma which is used in the proof of Lemma 3.2.

    Lemma 3.1. Suppose that a nonnegative function f on (1,) is uniformly continuous and 1f(t)dt<. Then f(t)0 as t.

    Lemma 3.2. Assume that the condition (1.8) is satisfied. Then

    uϵ(,t)1L2(Ω)+vϵ(,t)vL2(Ω)+wϵ(,t)1L2(Ω)0,t, (3.3)

    where v=β+γα.

    Proof We construct a positive function

    E(t):=Ω(uε1lnuϵ)+12δϵΩ(wϵ1)2,t>0.

    From the problem (3.1) and Young's inequality, we can compute

    ddtE(t)χ24Ω|vϵ|2dxΩ(uϵ1)2dxΩ(wϵ1)2dx,t>0. (3.4)

    We multiply the second equations in system (3.1) by vϵv, integrate by parts over Ω and use Young's inequality to obtain

    Ω|vϵ|2dxγ22αΩ(uϵ1)2dx+β22αΩ(wϵ1)2dx,t>0, (3.5)

    and

    Ω(vϵv)2dx2γ2α2Ω(uϵ1)2dx+2β2α2Ω(wϵ1)2dx,t>0. (3.6)

    Substituting inequality (3.5) into inequality (3.4) to get

    ddtE(t)C8(Ω(uϵ1)2dx+Ω(wϵ1)2dx),t>0,

    where C8=min{1χ2β28α,1χ2γ28α}>0.

    Let f(t):=Ω(uϵ1)2+(wϵ1)2dx. Then

    1f(t)dtE(1)C8<,t>1.

    It follows from the uniform estimation (3.2) and the Arzela-Ascoli theorem that f(t) is uniformly continuous in (1,). Applying Lemma 3.1, we have

    Ω(uϵ(,t)1)2+(wϵ(,t)1)2dx0,t. (3.7)

    Combining inequality (3.6) and the limit (3.7) to obtain

    Ω(vϵ(,t)v)2dx0,t.

    Proof of Theorem 1.2 As we all known, each bounded sequence in C2+ν,1+ν2(ˉΩ×[1,)) is precompact in C2,1(ˉΩ×[1,)). Hence there exists some subsequence {uϵn}n=1 satisfying ϵn0 as n such that

    limnuϵnuC2,1(ˉΩ×[1,))=0.

    Similarly, we can get

    limnvϵnvC2(ˉΩ)=0,

    and

    limnwϵnwC0,1(ˉΩ×[1,))=0.

    Combining above limiting relations yields that (u,v,w) satisfies model (1.3). The conclusion (u,v,w)=(u,v,w) is directly attributed to the uniqueness of the classical solution of the model (1.3). Furthermore, according to the conclusion, the strong convergence (3.3) and Diagonal line method, we can deduce

    u(,t)1L2(Ω)+v(,t)vL2(Ω)+w(,t)1L2(Ω)0,t. (3.8)

    By applying Gagliardo-Nirenberg inequality

    zLCz2/(n+2)L2(Ω)zn/(n+2)W1,(Ω),zW1,(Ω), (3.9)

    comparison principle of ODE and the convergence (3.8), the uniform convergence (1.9) is obtained immediately.

    Since limtu(,t)1L(Ω)=0, so there exists a t1>0 such that

    u(x,t)12forallxΩ,t>t1. (3.10)

    Using the explicit representation formula of w

    w(x,t)=(w0(x)1)eδt0F(u)uds+1,xΩ,t>0

    and the inequality (3.10), we have

    w(,t)1L(Ω)eδ6(tt1),t>t1. (3.11)

    Multiply the first two equations in model (1.3) by u1 and vv, respectively, integrate over Ω and apply Cauchy's inequality, Young's inequality and the inequality (3.10), to find

    ddtΩ(u1)2dxχ232Ω|v|2dxΩ(u1)2dx,t>t1. (3.12)
    Ω|v|2dx+α2Ω(vv)2dxβ2αΩ(w1)2dx+γ2αΩ(u1)2dx,t>0. (3.13)

    Combining the estimations (3.11)–(3.13) leads us to the estimate

    ddtΩ(u1)2dx(χ2γ232α1)Ω(u1)2dx+χ2β232αeδ3(tt1),t>t1.

    Let y(t)=Ω(u1)2dx. Then

    y(t)(χ2γ232α1)y(t)+χ2β232αeδ3(tt1),t>t1.

    From comparison principle of ODE, we get

    y(t)(y(t1)3χ2β232α(3δ)χ2γ2)e(1χ2γ232α)(tt1)+3χ2β232α(3δ)χ2γ2eδ3(tt1),t>t1.

    This yields

    Ω(u1)2dxC9eλ2(tt1),t>t1, (3.14)

    where λ2=min{1χ2γ232α,δ3} and C9=max{|Ω|3χ2β232α(3δ)χ2γ2,3χ2β232α(3δ)χ2γ2}.

    From the inequalities (3.11), (3.13) and (3.14), we derive

    Ω(vβ+γα)2dxC10eλ2(tt1),t>t1, (3.15)

    where C10=max{2γ2α2C9,2β2α2}. By employing the uniform estimation (1.7), the inequalities (3.9), (3.14) and (3.15), the exponential decay estimation (1.10) can be obtained.

    The proof is complete.

    In this paper, we mainly study the uniform boundedness of classical solutions and exponential asymptotic stability of the unique positive equilibrium point to the chemotactic cellular model (1.3) for Baló's concentric sclerosis (BCS). For model (1.1), by numerical simulation, Calveza and Khonsarib in [7] shown that demyelination patterns of concentric rings will occur with increasing of chemotactic sensitivity. By the Theorem 1.1 we know that systems (1.1) and (1.2) are {uniformly} bounded and dissipative. By the Theorem 1.2 we also find that the constant equilibrium point of model (1.1) is exponentially asymptotically stable if

    ˜χ<2ˉw˜β2Dμ˜α˜ϵˉu,

    and the constant equilibrium point of the model (1.2) is exponentially asymptotically stable if

    ˜χ<22Dμ˜α˜ϵˉumin{1ˉw˜β,1ˉu˜γ}.

    According to a pathological viewpoint of BCS, the above stability results mean that if chemoattractive effect is weak, then the destroyed oligodendrocytes form a homogeneous plaque.

    The authors would like to thank the editors and the anonymous referees for their constructive comments. This research was supported by the National Natural Science Foundation of China (Nos. 11761063, 11661051).

    We have no conflict of interest in this paper.



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