Consumption Of Chemoattractant Research Articles

Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant \[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \] in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$ with smooth boundary $\partial \Omega$ . It is shown that if $m>\max \{1,\,\frac {3N-2}{2N+2}\}$ , for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$ in an appropriate sense as $t\rightarrow \infty$ , where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$ . This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys.58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys.66 (2015), 3159–3179) to $m>\frac {3N-2}{2N}$ in the case $N\geq 3$ , but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B22 (2017), 669–686) to $m>\frac {3N}{2N+2}$ when $N\geq 2$ .

Boundedness in a two-dimensional attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation

In this paper, we study an attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation: nt+u⋅∇n=Δn−χ∇⋅(n∇c)+ξ∇⋅(n∇v)+λn−μnθ; ct+u⋅∇c=Δc−nc; τvt+u⋅∇v=Δv−v+n; ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ; ∇⋅u=0, x∈Ω, t>0 in a bounded and smooth domain Ω⊂R2 with no-flux/Dirichlet boundary conditions, where χ,ξ are positive constants, τ∈{0,1} and θ∈(1,2). We present the global boundedness of attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation.

A new result for global solvability to a two-dimensional attraction-repulsion Navier-Stokes system with consumption of chemoattractant

In this paper, we study an attraction-repulsion Navier-Stokes system with consumption of chemoattractant: nt+u⋅∇n=Δn−χ∇⋅(n∇c)+ξ∇⋅(n∇v); ct+u⋅∇c=Δc−nc; τvt+u⋅∇v=Δv−v+n; ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ; ∇⋅u=0, x∈Ω, t>0 in a bounded and smooth domain Ω⊂R2 with no-flux/Dirichlet boundary conditions, where χ,ξ are positive constants, τ∈{0,1}. We present the global existence of attraction-repulsion Navier-Stokes system with τ=1 and global boundedness of attraction-repulsion Navier-Stokes system with τ=0, and our result generalize the result obtained in previously known ones and partly result is new.

Stabilization in a two-dimensional attraction–repulsion Stokes system with consumption of chemoattractant

This paper is concerned with the following attraction–repulsion Stokes system $$\begin{aligned} \left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta n-\chi \nabla \cdot (n\nabla c)+\xi \nabla \cdot (n\nabla v),&{}x\in \Omega ,~t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,&{}x\in \Omega ,~t>0,\\ v_t+u\cdot \nabla v=\Delta v-v+n,&{}x\in \Omega ,~t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi ,&{}x\in \Omega ,~t>0,\\ \nabla \cdot u=0,&{}x\in \Omega ,~t>0, \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^2$$ is a general bounded domain with smooth boundary. It is shown that for any properly regular initial data, the above system with homogeneous Neumann boundary condition admits a unique classical solution which is globally bounded and particularly, if the uniform $$L^{\infty }$$ -module of n fulfills $$\begin{aligned} \Vert n\Vert _{L^{\infty }(\Omega \times (0,\infty ))}<\frac{2}{K_{\Omega }m\xi ^2} \end{aligned}$$ with $$m:=\int _{\Omega }n_0$$ and $$K_{\Omega }>0$$ only depending on $$\Omega ,$$ then $$\begin{aligned} \Vert n(\cdot ,t)-{\bar{n}}_0\Vert _{L^{\infty }(\Omega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\Omega )}+\Vert v(\cdot ,t) -{\bar{n}}_0\Vert _{W^{1,\infty }(\Omega )}+\Vert u(\cdot ,t)\Vert _{L^{\infty }(\Omega )}\rightarrow 0\quad \text {as}\quad t\rightarrow \infty , \end{aligned}$$ where $${\bar{n}}_0:=\frac{1}{|\Omega |}\int _{\Omega }n_0.$$

A new result for global solvability and boundedness in the N-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

We consider the following chemotaxis model{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+μ(u−u2),x∈Ω,t>0,vt−Δv=−uv,x∈Ω,t>0,(D(u)∇u−χu⋅∇v)⋅ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω on a bounded domain Ω⊂RN(N≥1), with smooth boundary ∂Ω,χ and μ are positive constants. Here, D is supposed to be smooth positive function satisfying D(u)≥CD(u+1)m−1 for all u≥0 with some CD,m>0. Besides appropriate smoothness assumptions, in this paper it is only required thatm>{1−μ2χ[1+max1≤s≤2⁡λ0(s)‖v0‖L∞(Ω)23]ifN≤2,1ifN≥3, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where λ0 is a positive constant which is corresponding to the maximal Sobolev regularity. The results of this paper extends the results of Jin (J. Differential Equations 263 (9) (2017) 5759–5772), who proved the possibility of boundness of weak solutions, in the case m>1 and N=3. To the best of our knowledge, this is the first result which gives the relationship between m and μχ that yields to the boundedness of the solution.

On a parabolic-parabolic system with gradient dependent chemotactic coefficient and consumption

This paper investigates a parabolic-parabolic system with a gradient dependent chemotactic coefficient and consumption of chemoattractant under homogeneous boundary conditions of Neumann type, in a bounded domain Ω⊂Rn (n≥2) with a smooth boundary, 1 &amp;lt; p &amp;lt; 2. It is proved that if initial data satisfy u0∈C0(Ω¯), v0∈W1,q(Ω)∩(W2,n+2n(Ω),Ln+2n(Ω))nn+2,n+2n for some q &amp;gt; n and 0&amp;lt;‖v0‖L∞(Ω)&amp;lt;14K, then the model admits at least one global weak solution for n&amp;lt;8−2(p−1)p−1 and possesses at least one global renormalized solution for n≥8−2(p−1)p−1. Here, K≔supξ≥0ξ(1+ξ)2⁡ln(1+ξ) is positive and finite.

Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant

Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant

Blow-up for a three dimensional Keller–Segel model with consumption of chemoattractant

We investigate blow-up properties for the initial-boundary value problem of a Keller–Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller–Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4,5] from the whole space R3 to the case of bounded smooth domain Ω⊂R3. Lower global blow-up estimate on ‖n‖L∞(Ω) is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.

Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant{ut=d1Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=d2Δv−∇⋅(vχ2(w)∇w)+μ2v(1−a2u−v),x∈Ω,t>0,wt=d3Δw−(αu+βv)w,x∈Ω,t>0under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥1) with smooth boundary, where the initial data (u0,v0)∈(C0(Ω‾))2 and w0∈W1,∞(Ω) are non-negative and the parameters d1,d2,d3>0, μ1,μ2>0, a1,a2>0 and α,β>0. The chemotactic function χi(w) (i=1,2) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for i=1,2,(i) χi(w)=χ0,i>0 and‖w0‖L∞(Ω)<πdid3n+1χ0,i−2did3n+1χ0,iarctan⁡di−d32n+1did3;(ii) 0<‖w0‖L∞(Ω)≤d33(n+1)‖χi‖L∞[0,‖w0‖L∞(Ω)]min⁡{2didi+d3,1}.Moreover, we prove asymptotic stabilization of solutions in the sense that:• If a1,a2∈(0,1) and u0≠0≠v0, then any global bounded solution exponentially converge to (1−a11−a1a2,1−a21−a1a2,0) as t→∞;• If a1>1>a2>0 and v0≠0, then any global bounded solution exponentially converge to (0,1,0) as t→∞;• If a1=1>a2>0 and v0≠0, then any global bounded solution algebraically converge to (0,1,0) as t→∞.

Boundedness in a chemotaxis model with exponentially decaying diffusivity and consumption of chemoattractant

This paper is devoted to the following chemotaxis model ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω⊂Rn (n≥2).It is proved that if D and S are sufficiently smooth nonnegative functions on [0,∞) satisfying K0e−β−s≤D(s)≤K1e−β+sfor alls≥0with some K0>0, K1>0, β−≥β+ and β+>0, then under the condition that S(s)D(s)≤K2sαfor alls≥0with some K2>0 and α≥0, for the initial data (u0,v0) are sufficiently regular satisfying u0≥0 and v0>0, the classical solutions to the system are uniformly-in-time bounded.

Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant

In this paper, we study the chemotaxis system:{ut=∇⋅(ξ∇u−χu∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn,n≥1, with smooth boundary. Here, ξ and χ are some positive constants.We prove that the classical solutions to the above system are uniformly in-time-bounded provided that:‖v0‖L∞(Ω)<{1χξ2(n+1)[π+2arctan⁡((1−ξ)22(n+1)ξ)],if0<ξ<1,πχ2(n+1),ifξ=1,1χξ2(n+1)[π−2arctan⁡((ξ−1)22(n+1)ξ)],ifξ>1. In the case of ξ=1, the recent results show that the classical solutions are global and bounded provided that 0<‖v0‖L∞(Ω)≤16(n+1)χ. Because of 16(n+1)χ<πχ2(n+1), or more precisely, limn→∞⁡πχ2(n+1)16(n+1)χ=+∞, our results extend the recent results.

Boundedness in a quasilinear chemotaxis model with consumption of chemoattractant and logistic source

This paper is devoted to the chemotaxis system with logistic sourceendowed with homogeneous Neumann boundary conditions and suitable initial conditions, where () is a bounded and smooth domain. Here we assume thatwhere , , and . Moreover, we will require for all with some , and . Then if one of the following cases hold:(i) , where(ii) ;(iii) and b is sufficiently large, it is proved that the system possesses a unique global bounded classical solution.

Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions

We study the quasilinear chemotaxis system (1.1) in a bounded domain Ω ⊂ ℝn(n ≥ 3) with smooth boundary, where the diffusion function D(u) satisfies D(u) ≥ cDum−1 for all u &amp;gt; 0 with some cD &amp;gt; 0. Under the condition m&amp;gt;32−1n, we show that for all reasonably regular initial data, the corresponding initial-boundary value problem for (1.1) possesses global boundedness of solution, which converges to the spatially homogeneous equilibrium (ū0,0) in an appropriate sense as t → ∞, where ū0=1Ω∫Ωu0.

Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source

This paper deals with the following chemotaxis system: urn:x-wiley:mma:media:mma4264:mma4264-math-0002 under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Here, δ and χ are some positive constants and f is a smooth function that satisfies urn:x-wiley:mma:media:mma4264:mma4264-math-0003 with some constants a⩾0,b &gt; 0, and γ &gt; 1. We prove that the classical solutions to the preceding system are global and bounded provided that urn:x-wiley:mma:media:mma4264:mma4264-math-0004 Copyright © 2016 John Wiley &amp; Sons, Ltd.

A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

The Neumann boundary value problem for the chemotaxis system generalizing the prototype KS \begin{document}$\left\{ \begin{array}{l}{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (u\nabla v),\;\;\;\;x \in \Omega ,t is considered in a smooth bounded convex domain \begin{document}$Ω\subset \mathbb{R}^N(N≥2)$\end{document} , where \begin{document}$D(u)≥ C_D(u+1)^{m-1}~~ \mbox{for all}~~ u≥0~~\mbox{with some}~~ m > 1~~\mbox{and}~~ C_D>0.$ \end{document} If \begin{document}$m >\frac{3N}{2N+2}$\end{document} and suitable regularity assumptions on the initial data are given, the corresponding initial-boundary problem possesses a global classical solution. Our paper extends the results of Wang et al. ([ 24 ]), who showed the global existence of solutions in the cases \begin{document}$m>2-\frac{6}{N+4}$\end{document} ( \begin{document}$N≥3$\end{document} ). If the flow of fluid is ignored, our result is consistent with and improves the result of Tao, Winkler ([ 15 ]) and Tao, Winkler ([ 17 ]), who proved the possibility of global boundedness, in the case that \begin{document}$N=2,m>1$\end{document} and \begin{document}$N= 3$\end{document} , \begin{document}$m > \frac{8}{7}$\end{document} , respectively.

Global solutions to a chemotaxis model with consumption of chemoattractant

This paper is devoted to the following chemotaxis system $$\left\{ \begin{array}{llll}u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v),\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0,\end{array} \right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) (\({n\geq2}\)), not necessarily being convex. There are some constants \({c_D > 0}\), \({c_S > 0}\), \({m\in\mathbb{R}}\) and \({q\in\mathbb{R}}\) such that $$D(u) \geq c_D(u+1)^{m-1} \quad\text{and} \quad S(u)\leq c_S(u+1)^{q-1}\quad for all \,\,\,u\geq0.$$ If \({q < m+\frac{n+2}{2n}}\), it is shown that the model possesses a unique global classical solution which is uniformly bounded; if \({q < \frac{m}{2}+\frac{n+2}{2n}}\), the global existence of solution is established.

Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant

We consider chemotaxis system with consumption of chemoattractant {vt=Δv−uv,ut=Δu−χ∇⋅(u∇v), under homogeneous Neumann boundary conditions. It is proved that if either n ≤ 2 or 0&amp;lt;χ≤16(n+1)‖v(x,0)‖L∞(Ω), n ≥ 3, the global classical solution (u, v) of this problem converges to (ū0,0) exponentially as t → ∞, where ū0≔1|Ω|∫Ωu(x,0)dx.

Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with an initial-boundary value problem for the chemotaxis system $$\left\{\begin{array}{ll} u_t = \nabla \cdot (D (u) \nabla u)- \nabla \cdot (u \nabla v), \quad & x\in \Omega, \quad t > 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t > 0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a convex smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) with \({n\geq3}\), where the diffusion function D(u) satisfying $$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u > 0 \end{array}$$ with some cD > 0 and m > 1. The main goal of this paper was to extend a previous result on global existence of solutions by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014) under the condition that \({m > 2-\frac{2}{n}}\) can be relaxed to \({m > 2-\frac{6}{n+4}}\).

Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source

This paper deals with a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source under homogeneous Neumann boundary conditions in a smooth bounded domain . The functions χ and f are assumed to generalize the chemotactic sensitivity function and logistic source respectively. Under some conditions, we obtain that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded.

Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant

This paper deals with positive solutions of { u t = Δ u − ∇ ⋅ ( u ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in bounded convex domains Ω ⊂ R 3 with smooth boundary. It is shown that for arbitrarily large initial data, this problem admits at least one global weak solution for which there exists T > 0 such that ( u , v ) is bounded and smooth in Ω × ( T , ∞ ) . Moreover, it is asserted that such solutions approach spatially constant equilibria in the large time limit.