Global existence to a chemotaxis-consumption model with nonlinear diffusion and singular sensitivity

ABSTRACT

This paper deals with the chemotaxis-consumption model with singular sensitivity and nonlinear diffusion $\begin{array}{rl}& u_t=\mathrm{\nabla }\cdot \left(D\right(u\left)\mathrm{\nabla }u\right)-\chi \mathrm{\nabla }\cdot \left(\frac{f\left(u\right)}{v}\mathrm{\nabla }v\right),x\in \mathrm{\Omega },t>0,\\ & v_t=ϵ\mathrm{△}v-g\left(u\right)v,x\in \mathrm{\Omega },t>0,\end{array}$ under the homogenous Neumann boundary condition in a smooth-bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 2)$, with positive parameters $\chi ,ϵ$ and $D\left(u\right)\ge \delta (u+\epsilon {)}^{m-1}$, $f\left(u\right)\le K(u+\epsilon {)}^{\alpha }$, $g\left(u\right)\le \sigma (u+\epsilon {)}^{\beta }$ for $\delta ,K,\sigma ,\epsilon ,\beta >0$, m>1 and $\alpha \in \mathbb{R}$. The main goal of this paper is to prove the existence of global classical solutions when $\alpha <N/4,\beta \le 1$ and $m>\alpha +N/4$.

Keywords:

• Keller–Segel
• global existence
• chemotaxis consumption
• nonlinear diffusion
• singular sensitivity

2010 Mathematics Subject Classification:

• 35K55

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Project supported by the National Natural Science Foundation of China [grant nos 11571093 and 11471164].