Electron. J. Diff. Equ., Vol. 2013 (2013), No. 209, pp. 1-9.

Boundedness in a chemotaxis system with consumption of chemoattractant and logistic source

Liangchen Wang, Shahab Ud-Din Khan, Salah Ud-Din Khan

Abstract:
In this article, we consider a chemotaxis system with consumption of chemoattractant and logistic source
$$\displaylines{ u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+f(u),\quad x\in \Omega,\; t>0,\cr v_t=\Delta v-uv,\quad x\in\Omega,\; t>0, }$$
under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^n$, with non-negative initial data $u_0$ and $v_0$ satisfying $(u_0,v_0)\in (W^{1,\theta}{(\Omega)})^2$ (for some $\theta>n$). $\chi>0$ is a parameter referred to as chemosensitivity and $f(s)$ is assumed to generalize the logistic function
$$ f(s)=as-bs^2,\quad s\geq0,\hbox{ with } a>0,\;b>0. $$
It is proved that if $\|v_0\|_{L^\infty(\Omega)}>0$ is sufficiently small then the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded.

Submitted July 1, 2013. Published September 19, 2013.
Math Subject Classifications: 35B35, 35K55, 92C17.
Key Words: Chemotaxis; global existence; boundedness; logistic source.

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Liangchen Wang
College of Mathematics and Statistics
Chongqing University, Chongqing 401331, China
email: liangchenwang324@126.com
Shahab Ud-Din Khan
College of Mathematics and Statistics
Chongqing University, Chongqing 401331, China
email: sudkhan@163.com
Salah Ud-Din Khan
College of Engineering, King Saud University
P.O. Box 800, Riyadh 11421, Kingdom of Saudi Arabia
email: salahudkhan@126.com

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