Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source

Pan Zheng; Chunlai Mu; Xuegang Hu

doi:10.3934/dcds.2015.35.2299

Article Contents

2015, Volume 35, Issue 5: 2299-2323. Doi: 10.3934/dcds.2015.35.2299

This issue Previous Article Dynamics of hyperbolic meromorphic functions Next Article

Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source

1.
College of Sciences, Chongqing University of Posts and Telecommunications, Chongqing 400065
2.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331

Received: July 2014

Revised: August 2014

Published: May 2017

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Abstract

This paper deals with a parabolic-elliptic chemotaxis system with generalized volume-filling effect and logistic source \begin{eqnarray*} \left\{ \begin{split}{} &u_t=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(\psi(u)\nabla v)+f(u), &(x,t)\in \Omega\times (0,\infty), \\ &0=\Delta v-m(t)+u, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi>0$, $m(t)=\frac{1}{|\Omega|}\int_{\Omega}u(x,t)dx$, the nonlinear diffusivity $\varphi(u)$ and chemosensitivity $\psi(u)$ are supposed to extend the prototypes $$\varphi(u)=(u+1)^{-p},\quad \psi(u)=u(u+1)^{q-1}$$ with $p\geq0$, $q\in \mathbb{R}$, and $f(u)$ is assumed to generalize the standard logistic function $$f(u)=\lambda u-\mu u^{k},\;\text{with}\;\;\lambda\geq 0,\mu>0\;\text{and}\;k>1.$$ Under some different suitable assumptions on the nonlinearities $\varphi(u), \psi(u)$ and logistic source $f(u)$, we study the global boundedness and finite-time blow-up of solutions for the problem.

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Mathematics Subject Classification: Primary: 35K55, 35B35; Secondary: 35B40.

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