Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 209, pp. 1-9.

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

Authors: Liangchen Wang     (Chongqing Univ., Chongqing, China)
         Shahab Ud-Din Khan (Chongqing Univ., Chongqing, China)
         Salah Ud-Din Khan  (King Saud Univ., Riyadh, Saudi Arabia)

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.