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

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.

Keywords

• Chemotaxis
• global existence
• boundedness
• logistic source