Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source

2019 ◽  
Vol 46 ◽  
pp. 545-582 ◽  
Author(s):  
Guoqiang Ren ◽  
Bin Liu
Keyword(s):  
Global Existence ◽  
Logistic Source ◽  
Bounded Solutions ◽  
Chemotaxis System

An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization

2020 ◽  
pp. 1-21 ◽  
Author(s):  
Wenbin Lv ◽  
Qingyuan Wang
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Neumann Boundary ◽  
Logistic Source ◽  
Global Classical Solution ◽  
Higher Dimensional ◽  
Image Position ◽  
Chemotaxis System

Abstract This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as $$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta v-v + u,\quad x\in \Omega ,\;t > 0.} $$ It is shown that whenever ρ ∈ ℝ, μ > 0 and $$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$ then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium $$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$ as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.

scholarly journals Global Existence and Blow-up of Classical Solution for an Attraction-repulsion Chemotaxis System with Logistic Source

2019 ◽  
pp. 1-16
Author(s):  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Finite Time ◽  
Blow Up ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Elliptic Type ◽  
Logistic Source ◽  
Chemotaxis System

We consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic-elliptic type with logistic source under homegeneous Neumann boundary conditions in a bounded domain `\Omega\subset R^{n}(n\geq2)` with smooth boundary, where`D(u)\geq c_{D}(u+1)^{m-1}` with `m\geq1`and `c_{D}>0`, `f(u)\leq a-bu^{\eta}` with `\eta>1`.{ We show two cases that the system admits a uniqueglobal bounded classical solution depending on `0\leq S(u)\leq C_{s}(u+1)^{q}, 0\leq F(u)\leq C_{F}(u+1)^{g}` by Gagliardo-Nirenberg inequality.For specific `D(u),S(u),F(u)` with logistic source for `\eta>1` and `n=2`, we establish the finite time blow-up conditions forsolutions that the finite time blow-up occurs at `x_{0}\in\Omega` whenever `\int_{\Omega}u_{0}(x)dx>\frac{8\pi}{\chi\alpha-\xi\gamma}`with `\chi\alpha-\xi\gamma>0`, under `\int_{\Omega}u_{0}(x)|x-x_{0}|^{2}dx` sufficiently small.

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