Attraction-repulsion Chemotaxis Model Research Articles

وجود حل محلي لنظام الجذب الكيميائي والتنافر مع الانتشار غير الخطي والمصدر اللوجستي

In this paper, we are concerned with a model arising from biology, which is a coupled system of chemotaxis equations and viscous compressible fluid equations through transport and external forcing. The local existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for an attraction-repulsion chemotaxis model system over three space dimensions, we obtain local existence and uniqueness of convergence on classical solutions near constant states. We prove local existence of unique solutions in three dimensions by using energy estimates.

On an attraction-repulsion chemotaxis model involving logistic source

This paper is concerned with the attraction-repulsion chemotaxis system involving logistic source: u_{t}=Δu-χ∇⋅(u∇υ)+ξ∇⋅(u∇ω)+f(u), ρυ_{t}=Δυ-α₁υ+β₁u, ρω_{t}=Δω-α₂ω+β₂u under homogeneous Neumann boundary conditions with nonnegative initial data (u₀,υ₀,ω₀)∈ (W^{1,∞}(Ω))³, the parameters χ, ξ, α₁, α₂, β₁, β₂>0, ρ≥0 subject to the non-flux boundary conditions in a bounded domain Ω⊂ℝ^{N}(N≥3) with smooth boundary and f(u)≤au-μu² with f(0)≥0 and a≥0, μ>0 for all u>0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a globally bounded classical solution provided that χ+ξ0 is sufficiently small for all β₁, β₂

Global boundedness and asymptotic behavior in a fully parabolic attraction-repulsion chemotaxis model with logistic source

In this paper, we consider a fully parabolic attraction-repulsion chemotaxis model with logistic source. First of all, we obtain an explicit formula [Formula: see text] for the logistic damping rate [Formula: see text] such that the model has no blow-up when [Formula: see text]. In addition, the asymptotic behavior of the solutions is studied. Our results partially generalize and improve some results in the literature, and partially results are new.

Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption

This paper is framed in a series of studies on attraction–repulsion chemotaxis models combining different effects: nonlinear diffusion and sensitivities and logistic sources, for the dynamics of the cell density, and consumption and/or production impacts, for those of the chemicals. In particular, herein we focus on the situation where the signal responsible of gathering tendencies for the particles’ distribution is produced, while the opposite counterpart is consumed. In such a sense, this research complements the results in Frassu et al. (Math Methods Appl Sci 45:11067–11078, 2022) and Chiyo et al. (Commun Pure Appl Anal, 2023, doi: 10.3934/cpaa.2023047), where the chemicals evolve according to different laws.

Uniform in time <inline-formula><tex-math id="M1">$ L^\infty $</tex-math></inline-formula>-estimates for an attraction-repulsion chemotaxis model with double saturation

<p style='text-indent:20px;'>In this paper we focus on this attraction-repulsion chemotaxis model with consumed signals</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1"> \begin{document}$\begin{equation}\label{problem_abstract}\tag{$\Diamond$}\begin{cases}u_t = \Delta u - \chi \nabla \cdot (u \nabla v)+\xi \nabla \cdot (u \nabla w) & \text{ in }~~ \Omega \times (0, T_{max}), \\v_t = \Delta v- uv & \text{ in }~~ \Omega \times (0, T_{max}), \\w_t = \Delta w- uw & \text{ in }~~ \Omega \times (0, T_{max}), \end{cases}\end{equation}$ \end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>formulated in a bounded and smooth domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^n $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M4">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, for some positive real numbers <inline-formula><tex-math id="M5">\begin{document}$ \chi, \xi $\end{document}</tex-math></inline-formula> and with <inline-formula><tex-math id="M6">\begin{document}$ {T_{max}}\in (0, \infty] $\end{document}</tex-math></inline-formula>. Once equipped with appropriately smooth initial distributions <inline-formula><tex-math id="M7">\begin{document}$ u(x, 0) = u_0(x)\geq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ v(x, 0) = v_0(x)\geq 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ w(x, 0) = w_0(x)\geq 0 $\end{document}</tex-math></inline-formula>, as well as Neumann boundary conditions, we establish sufficient assumptions on its data yielding global and bounded classical solutions; these are functions <inline-formula><tex-math id="M10">\begin{document}$ u, v $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ w $\end{document}</tex-math></inline-formula>, with zero normal derivative on <inline-formula><tex-math id="M12">\begin{document}$ \partial \Omega\times (0, {T_{max}}) $\end{document}</tex-math></inline-formula>, satisfying pointwise the equations in problem <inline-formula><tex-math id="M13">\begin{document}$\Diamond$\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ {T_{max}} = \infty $\end{document}</tex-math></inline-formula>. This is proved for any such initial data, whenever <inline-formula><tex-math id="M15">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> belong to bounded and open intervals, depending respectively on <inline-formula><tex-math id="M17">\begin{document}$ \|v_0\|_{L^{\infty}(\Omega)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M18">\begin{document}$ \|w_0\|_{L^{\infty}(\Omega)} $\end{document}</tex-math></inline-formula>. Finally, we illustrate some aspects of the dynamics present within the chemotaxis system by means of numerical simulations.</p>

Global asymptotic stability of constant equilibrium point in attraction-repulsion chemotaxis model with logistic source term

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for an attraction-repulsion chemotaxis model with logistic source term of Equation 1 in bounded convex domains \(\Omega\subset\mathbb{R}^{n},~ n\geq1\), with smooth boundary. It is shown that if the ratio \(\frac{\mu}{\chi \alpha-\xi \gamma}\) is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by \((u_{1},u_{2},u_{3})=(1,~\frac{\alpha}{\beta},~\frac{\gamma}{\eta})\) is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data \((u_{10},u_{20},u_{30})\) such that \(u_{10}\not\equiv0\), the above problem possesses uniquely determined global classical solution \((u_{1},u_{2},u_{3})\) with \((u_{1},u_{2},u_{3})|_{t=0}=(u_{10},u_{20},u_{30})\) which satisfies $$\|u_{1}(\cdot,t)-1\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{2}(\cdot,t)-\frac{\alpha}{\beta}\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{3}(\cdot,t)-\frac{\gamma}{\eta}\|_{L^{\infty}(\Omega)}\rightarrow{0}.$$ \(\mathrm{as}~t\rightarrow{\infty}\).

Boundedness and stabilization in the 3D minimal attraction–repulsion chemotaxis model with logistic source

Boundedness and stabilization in the 3D minimal attraction–repulsion chemotaxis model with logistic source

Boundedness of solutions to an attraction-repulsion chemotaxis model in [formula omitted

We consider an attraction-repulsion chemotaxis model in two dimensions. When the chemoattractant is dominant, it is known that the solutions to the system exist globally, provided that the total mass is less than or equal to some critical value. In this article, we prove the boundedness of solutions with subcritical mass.

Optimal Control of a Fully Parabolic Attraction-Repulsion Chemotaxis Model with Logistic Source in 2D

Optimal Control of a Fully Parabolic Attraction-Repulsion Chemotaxis Model with Logistic Source in 2D

Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis model with nonlinear secretion

In this paper, we study the large time behavior of a quasilinear attraction–repulsion chemotaxis model with nonlinear secretion: ut = ∇ · (D(u)∇u − χΦ(u)∇v + ξΨ(u)∇w) + λu − μuϵ; 0=Δv−α1v+β1uγ1; 0=Δw−α2w+β2uγ2, x ∈ Ω, t > 0. We show that the global-in-time bounded smooth solution of the system converges exponentially/algebraically to steady state in the large time limit. Those results generalize some of our previous results [G. Ren and B. Liu, Math. Models Methods Appl. Sci. 30(13), 2619–2689 (2020) and G. Ren and B. Liu, J. Differ. Equations 268(8), 4320–4373 (2020)].

Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

We study this zero-flux attraction–repulsion chemotaxis model, with linear and superlinear production g for the chemorepellent and sublinear rate f for the chemoattractant: (♢)ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)in Ω×(0,Tmax),vt=Δv−f(u)vin Ω×(0,Tmax),0=Δw−δw+g(u)in Ω×(0,Tmax).In this problem, Ω is a bounded and smooth domain of Rn, for n≥1, χ,ξ,δ>0, f(u) and g(u) reasonably regular functions generalizing the prototypes f(u)=Kuα and g(u)=γul, with K,γ>0 and proper α,l>0. Once it is indicated that any sufficiently smooth u(x,0)=u0(x)≥0 and v(x,0)=v0(x)≥0 produce a unique classical and nonnegative solution (u,v,w) to (1), which is defined in Ω×(0,Tmax), we establish that for any such (u0,v0), the life span Tmax=∞ and u,v and w are uniformly bounded in Ω×(0,∞), (i) for l=1, n∈{1,2}, α∈(0,12+1n)∩(0,1) and any ξ>0, (ii) for l=1, n≥3, α∈(0,12+1n) and ξ larger than a quantity depending on χ‖v0‖L∞(Ω), (iii) for l>1, α∈(0,12+1n)∩(0,1), any ξ>0, and in any dimensional settings. Finally, an illustrative analysis about the effect by logistic and repulsive actions on chemotactic phenomena is proposed by comparing the results herein derived for the linear production case with those in Lankeit and Wang (2017).

Global boundedness and asymptotic behavior in a quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source

In this work, we consider the quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source. We present the global existence of classical solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature.

Global Existence for an Attraction–Repulsion Chemotaxis-Fluid System in a Framework of Besov–Morrey type

We consider an attraction-repulsion chemotaxis model in the whole space $${\mathbb {R}}^3$$ with logistic source coupled with the incompressible Navier–Stokes equations. By means of a contraction argument, we obtain the existence and uniqueness of global mild solutions in a framework of Besov type, namely Besov spaces based on Morrey spaces. In comparison with previous results for the system dealt with, that framework provides new solutions and allows us to consider larger initial data and force classes for global existence and uniqueness. To carry out our results, we prove some essential lemmas and estimates related to the heat semigroup and continuity properties for the Bony decomposition in our setting.

Global dynamics for an attraction-repulsion chemotaxis model with logistic source

In this paper, we investigate attraction-repulsion chemotaxis system with logistic source under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, by some Lp-estimate techniques, using different methods, we show that the system possesses global classical solution or at least one global weak solution. In addition to, we also consider large time behavior of solutions and steady solutions to an attraction-repulsion chemotaxis system. These results generalize and improve previously known ones, and partially results are new.

A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source

In this paper we study the quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source: ut=∇⋅(D(u)∇u−S(u)∇v)+f(u), vt=Δv−a1v+b1w, wt=Δw−a2w+b2u, under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥1), where ai,bi>0 (i=1,2), D,S∈C2([0,∞)) and f:R→R is a smooth function generalizing the logistic source f(s)=b−μsr for all s≥0 with b≥0, μ>0 and r≥1. We obtain the global boundedness of solutions in four cases: (i) the self-diffusion dominates the cross-diffusion; (ii) the logistic source suppresses the cross-diffusion; (iii) the logistic dampening balances the cross-diffusion with μ>0 suitably large; (iv) the self-diffusion and the logistic source both balance the cross-diffusion to some extent with μ>0 arbitrary. As corollaries, we also consider the global boundedness of solutions for the quasilinear attraction-repulsion chemotaxis model with logistic source: u˜t=∇⋅(D(u˜)∇u˜)−χ∇⋅(u˜∇z)+ξ∇⋅(u˜∇w˜)+f(u˜), zt=Δz−ρz+ηu˜, w˜t=Δw˜−δw˜+γu˜, where χ,η,ξ,γ,ρ,δ>0.

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of \begin{document}$\mathbb{R}^N,$\end{document} \begin{document}$N = 2, 3,$\end{document} for small initial data in \begin{document}$L^p$\end{document} -spaces.

A hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions

This paper studies the hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions and logistic source: $$u_{t}=-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+\mu u(1-u^k)$$ , $$0=\Delta v+\alpha u^q-\beta v$$ , $$ 0=\Delta w+\gamma u^r-\delta w$$ , in a bounded domain $$\Omega \subset \mathbb {R}^n$$ , $$n\ge 1$$ , subject to the non-flux boundary condition. We at first establish the local existence of solutions (the so-called strong $$W^{1,p}$$ -solutions, satisfying the hyperbolic equation weakly and solving the elliptic ones classically) to the model via applying the viscosity vanishing method and then give criteria on global boundedness versus finite- time blowup for them. It is proved that if the attraction is dominated by the logistic source or the repulsion with $$\max \{r,k\}>q$$ , the solutions would be globally bounded; otherwise, the finite-time blowup of solutions may occur whenever $$\max \{r,k\}<q$$ . Under the balance situations with $$q=r=k$$ , $$q=r>k$$ or $$q=k>r$$ , the boundedness or possible finite-time blowup would depend on the sizes of the coefficients involved.

Global weak solution and boundedness in a three-dimensional competing chemotaxis

We consider an initial-boundary value problem for a parabolic-parabolic-elliptic attraction-repulsion chemotaxis model \begin{document}$ \left\{ \begin{array}{l}u_t = Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w),x'>in a bounded domain \begin{document}$Ω\subset \mathbb{R}^3$\end{document} with positive parameters \begin{document}$χ, ξ, α, β, γ$\end{document} and \begin{document}$δ$\end{document} . It is firstly proved that if the repulsion dominates in the sense that \begin{document}$ξγ>χα$\end{document} , then for any choice of sufficiently smooth initial data \begin{document}$(u_0, v_0)$\end{document} the corresponding initial-boundary value problem is shown to possess a globally defined weak solution. To the best of our knowledge, this situation provides the first result on global existence of the above system in the three-dimensional setting when \begin{document}$ξγ>χα$\end{document} , and extends the results in Lin et al. (2016) [ 19 ] and Jin and Xiang (2017) [ 14 ] to more general case. Secondly, if the initial data is appropriately small or the repulsion is enough strong in the sense that \begin{document}$ξγ$\end{document} is suitable large as related to \begin{document}$χα$\end{document} , then the classical solutions to the above system are uniformly-in-time bounded.

Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

We consider the following attraction-repulsion Keller-Segel system: \begin{document}$\begin{equation*}\begin{cases}u_t=\nabla· (D(u) \nabla u)-χ\nabla·( u\nabla v)+ξ\nabla·( u\nabla w), x'>with homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n(n>2)$ with smooth boundary. Here all the parameters \begin{document}$χ, ξ, α, β, γ$\end{document} and \begin{document} $δ$\end{document} are positive. The smooth diffusion \begin{document}$D(u)$\end{document} satisfies \begin{document}$D(u)≥ d u^θ, u>0$\end{document} for some \begin{document}$d>0, θ∈\mathbb{R}$\end{document} . It is recently known from [ 25 ] that boundedness of solutions is ensured whenever \begin{document}$θ>1-\frac{2}{n}$\end{document} . Here, it is shown, if repulsion dominates or cancels attraction in the sense either \begin{document}$\{ξγ> χα\}$\end{document} or \begin{document}$\{ξγ=χα, β≥ δ\}$\end{document} , the corresponding initial-boundary value problem possesses a unique global classical solution which is uniformly-in-time bounded for large initial data provided \begin{document}$θ>1-\frac{4}{n+2}$\end{document} . In this way, the range of \begin{document}$θ>1-\frac{2}{n}$\end{document} of boundedness is enlarged and thus the repulsion effect on boundedness is exhibited.

Global existence of solutions to an attraction-repulsion chemotaxis model with growth

An attraction-repulsion chemotaxis model with nonlinear chemotactic sensitivity functions and growth source is considered. The global-in-time existence and boundedness of solutions are proved under some conditions on the nonlinear sensitivity functions and growth source function. Our results improve the earlier ones for the linear sensitivity functions.