Attraction Repulsion Chemotaxis System Research Articles

Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system

In this paper, we consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-parabolic type \begin{equation*}\left\{\begin{split}&u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\qquad & x\in\Omega,\,\, t>0,\\&v_t=\Delta v+\alpha u-\beta v,\qquad &x\in\Omega, \,\,t>0,\\&w_t=\Delta w+\gamma u-\delta w,\qquad &x\in\Omega,\,\, t>0\end{split}\right.\end{equation*}under homogeneous Neumann boundary conditions, where $D(u)\geq c_D (u+\varepsilon)^{m-1}$ and $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. It is shown that whenever $m>1$, for any sufficiently smooth nonnegative initial data, the system admits a global bounded classical solution for the case of non-degenerate diffusion (i.e., $\varepsilon>0$), while the system possesses a global bounded weak solution for the case of degenerate diffusion (i.e., $\varepsilon=0$).

Global existence and convergence to steady states for an attraction–repulsion chemotaxis system

This paper deals with a Neumann initial–boundary value problem in the two-dimensional space for a coupled chemotaxis model {ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0, which describes the competition between attractive and repulsive signals produced by the same species. The parameters χ,ξ,α,β,γ and δ are positive.Relying on a new entropy-type inequality, we obtain the global existence and boundedness of solutions to this model if the initial data u0 satisfies ‖u0‖L1(Ω)<1kχα, where k is a constant depending only on Ω. The convergence to steady states is also given through this paper.

A quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source

This study considers the following quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source{ut=∇⋅(D(u)∇u)−∇⋅(χu∇v)+∇⋅(ξu∇w)+f(u),x∈Ω,t>0,0=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥2) with smooth boundary, where D(u)≥cD(u+σ)m−1 with m≥1, σ≥0, and cD>0, and f(u)≤a−buη with a≥0, b>0, and η>1. In the case of non-degenerate diffusion (i.e., σ>0), we show that the system admits a unique global bounded classical solution provided that the repulsion prevails over the attraction in the sense that ξγ−χα>0, or the logistic dampening is sufficiently strong, or the diffusion is sufficiently strong, while in the case of degenerate diffusion (i.e., σ=0), we show that the system admits a global bounded weak solution at least under the same assumptions. Finally, we obtain the large-time behavior of solutions for a specific logistic source.

Blow-up of nonradial solutions to attraction–repulsion chemotaxis system in two dimensions

This paper deals with the attraction–repulsion chemotaxis system {ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain in R2. We study the finite-time blowup of nonradial solutions in the parameter values χα−ξγ>0 and β≠δ.

Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type

This paper deals with the global existence and boundedness of solutions to the following quasilinear attraction-repulsion chemotaxis system: $$ \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot (\xi u\nabla w), & x\in\Omega, 0=\Delta v+\alpha u-\beta v,& x\in\Omega, 0=\Delta w+\gamma u-\delta w,& x\in\Omega, t>0, \end{array}\displaystyle \right . $$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^{n}$ ( $n\geq2$ ) with smooth boundary, where $D(u)\geq c_{D} u^{m-1}$ with $m\geq1$ and some constant $c_{D}>0$ . It is proved that if $\xi\gamma-\chi\alpha>0$ or $m>2-\frac{2}{n}$ , then for any sufficiently regular initial data, this system possesses a unique global bounded classical solution for the case of nondegenerate diffusion (i.e., $D(u)>0$ for all $u\geq 0$ ), whereas for the case of degenerate diffusion (i.e., $D(u)\geq0$ for all $u\geq0$ ), it is shown that there exists a global bounded weak solution under the same assumptions.

Global boundedness in quasilinear attraction–repulsion chemotaxis system with logistic source

This paper studies the quasilinear attraction–repulsion chemotaxis system with logistic source ut=∇⋅(D(u)∇u)−χ∇⋅(Φ(u)∇v)+ξ∇⋅(Ψ(u)∇w)+f(u), τvt=Δv+αu−βv, τ∈{0,1}, 0=Δw+γu−δw, in bounded domain Ω⊂RN, N≥1, subject to the homogeneous Neumann boundary conditions, D,Φ,Ψ∈C2[0,+∞) nonnegative, with D(s)≥(s+1)p for s≥0, Φ(s)≤χsq, ξsr≤Ψ(s)≤ζsr for s>1, and f smooth satisfying f(s)≤μs(1−sk) for s>0, f(0)≥0. It is proved that if the attraction is dominated by one of the other three mechanisms with max{r,k,p+2N}>q, then the solutions are globally bounded. Under more interesting balance situations, the behavior of solutions depends on the coefficients involved, i.e., the upper bound coefficient χ for the attraction, the lower bound coefficient ξ for the repulsion, the logistic source coefficient μ, as well as the constants α and γ describing the capacity of the cells u to produce chemoattractant and chemorepellent respectively. Three balance situations (attraction–repulsion balance, attraction–logistic source balance, and attraction–repulsion–logistic source balance) are considered to establish the boundedness of solutions for the parabolic–elliptic–elliptic case (with τ=0) and the parabolic–parabolic–elliptic case (with τ=1) respectively.

On an attraction–repulsion chemotaxis system with a logistic source

On an attraction–repulsion chemotaxis system with a logistic source

Boundedness in a two‐dimensional attraction–repulsion system with nonlinear diffusion

This paper is devoted to the attraction–repulsion chemotaxis system with nonlinear diffusion: urn:x-wiley:mma:media:mma3477:mma3477-math-0001 where χ &gt; 0, ζ &gt; 0, αi&gt;0, βi&gt;0 (i = 1,2) and f(s)≤κ − μsτ. In two‐space dimension, we prove the global existence and uniform boundedness of the classical solution to this model for any μ &gt; 0. Copyright © 2015 John Wiley &amp; Sons, Ltd.

A Hopf bifurcation of one dimensional attraction-repulsion chemotaxis system

A Hopf bifurcation of one dimensional attraction-repulsion chemotaxis system

Large-time behavior of an attraction–repulsion chemotaxis system

In this study, we consider an attraction–repulsion chemotaxis system{ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0 with homogeneous Neumann boundary conditions, where χ, ξ, α, β, γ and δ are positive parameters. Under the critical condition that χα−ξγ=0, we prove that the system possesses a unique global solution, which is uniformly bounded in the physical domain Ω⊂Rn (n=2,3). In addition, we assert that we can find ϵ0>0 such that for all of the initial data u0 that satisfy ∫Ωu0<ϵ0, the solution of the system approaches the steady state (u¯0,αβu¯0,γδu¯0) exponentially as time tends to infinity, where u¯0:=1|Ω|∫Ωu0.

Well-posedness for a model derived from an attraction–repulsion chemotaxis system

In this paper, we are interested in a model derived from an attraction–repulsion chemotaxis system in high dimensions:{∂tu−Δu=−β1∇⋅(u∇v)+β2∇⋅(u∇w),x∈Rn,t>0,λ1v−Δv=u,x∈Rn,t>0,λ2w−Δw=u,x∈Rn,t>0,u(x,0)=u0(x),x∈Rn, with the parameters β1≥0, β2≥0, λ1>0, λ2>0 and nonnegative initial data u0(x)∈L1(Rn)∩L∞(Rn). We prove that a global bounded solution exists when the repulsion prevails over the attraction in the sense of β1<β1. Moreover, we give the smoothness of the solution and obtain its decay rates in Ws,p(Rn), which coincide with the ones of the classical heat equation. Conversely, when n=2, β1>β2, we prove that the finite time blow-up may occur.

Global boundedness in a fully parabolic attraction–repulsion chemotaxis model

This paper deals with a fully parabolic attraction–repulsion chemotaxis system in two‐dimensional smoothly bounded domains. It is shown that the system admits global bounded classical solutions whenever the repulsion is dominated. The proof is based on an entropy‐like inequality and coupled estimate techniques. Copyright © 2014 John Wiley &amp; Sons, Ltd.

COMPETING EFFECTS OF ATTRACTION VS. REPULSION IN CHEMOTAXIS

We consider the attraction–repulsion chemotaxis system [Formula: see text] under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ ℝn with smooth boundary, where χ ≥ 0, ξ ≥ 0, α &gt; 0, β &gt; 0, γ &gt; 0, δ &gt; 0 and τ = 0, 1. We study the global solvability, boundedness, blow-up, existence of non-trivial stationary solutions and asymptotic behavior of the system for various ranges of parameter values. Particularly, we prove that the system with τ = 0 is globally well-posed in high dimensions if repulsion prevails over attraction in the sense that ξγ - χα &gt; 0, and that the system with τ = 1 is globally well-posed in two dimensions if repulsion dominates over attraction in the sense that ξγ - χα &gt; 0 and β = δ. Hence our results confirm that the attraction–repulsion is a plausible mechanism to regularize the classical Keller–Segel model whose solution may blow up in higher dimensions.