Global solutions to a chemotaxis model with consumption of chemoattractant

Abstract

This paper is devoted to the following chemotaxis system $$\left\{ \begin{array}{llll}u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v),\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0,\end{array} \right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) (\({n\geq2}\)), not necessarily being convex. There are some constants \({c_D > 0}\), \({c_S > 0}\), \({m\in\mathbb{R}}\) and \({q\in\mathbb{R}}\) such that $$D(u) \geq c_D(u+1)^{m-1} \quad\text{and} \quad S(u)\leq c_S(u+1)^{q-1}\quad for all \,\,\,u\geq0.$$ If \({q < m+\frac{n+2}{2n}}\), it is shown that the model possesses a unique global classical solution which is uniformly bounded; if \({q < \frac{m}{2}+\frac{n+2}{2n}}\), the global existence of solution is established.