Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

2021 ◽  
Author(s):  
Giuseppe Viglialoro
Keyword(s):  
Global Solvability ◽  
Chemotaxis System

COMPETING EFFECTS OF ATTRACTION VS. REPULSION IN CHEMOTAXIS

2012 ◽  
Vol 23 (01) ◽  
pp. 1-36 ◽  
Author(s):  
YOUSHAN TAO ◽  
ZHI-AN WANG
Keyword(s):  
Blow Up ◽  
Neumann Boundary ◽  
Global Solvability ◽  
Stationary Solutions ◽  
Two Dimensions ◽  
High Dimensions ◽  
Plausible Mechanism ◽  
Chemotaxis System ◽  
Parameter Values ◽  
Well Posed

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 > 0, β > 0, γ > 0, δ > 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 > 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 > 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.

scholarly journals How far does logistic dampening influence the global solvability of a high-dimensional chemotaxis system?

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ke Jiang ◽  
Yongjie Han
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Space Dimension ◽  
Neumann Boundary ◽  
Global Solvability ◽  
High Dimensional ◽  
Smooth Bounded Domain ◽  
Existence Of Weak Solutions ◽  
Neumann Boundary Value Problem ◽  
Chemotaxis System

AbstractThis paper deals with the homogeneous Neumann boundary value problem for chemotaxis system $$\begin{aligned} \textstyle\begin{cases} u_{t} = \Delta u - \nabla \cdot (u\nabla v)+\kappa u-\mu u^{\alpha }, & x\in \Omega, t>0, \\ v_{t} = \Delta v - uv, & x\in \Omega, t>0, \end{cases}\displaystyle \end{aligned}$$ { u t = Δ u − ∇ ⋅ ( u ∇ v ) + κ u − μ u α , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}(N\geq 2)$ Ω ⊂ R N ( N ≥ 2 ) , where $\alpha >1$ α > 1 and $\kappa \in \mathbb{R},\mu >0$ κ ∈ R , μ > 0 for suitably regular positive initial data.When $\alpha \ge 2$ α ≥ 2 , it has been proved in the existing literature that, for any $\mu >0$ μ > 0 , there exists a weak solution to this system. We shall concentrate on the weaker degradation case: $\alpha <2$ α < 2 . It will be shown that when $N<6$ N < 6 , any sublinear degradation is enough to guarantee the global existence of weak solutions. In the case of $N\geq 6$ N ≥ 6 , global solvability can be proved whenever $\alpha >\frac{4}{3}$ α > 4 3 . It is interesting to see that once the space dimension $N\ge 6$ N ≥ 6 , the qualified value of α no longer changes with the increase of N.

On a fully parabolic chemotaxis system with nonlocal growth term

2021 ◽  
Vol 213 ◽  
pp. 112518
Author(s):  
M. Negreanu ◽  
J.I. Tello ◽  
A.M. Vargas
Keyword(s):  
Chemotaxis System
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