Global existence, boundedness and asymptotic behavior to a chemotaxis model with singular sensitivity and logistic source

ABSTRACT

This article deals with the chemotaxis model with singular sensitivity and logistic source $\left\{\begin{array}{ll}{u}_t=\mathrm{△}u-\chi \mathrm{\nabla }\cdot \left(\frac{u}{v}\mathrm{\nabla }v\right)+\kappa u-\mu u^k,& x\in \mathrm{\Omega },t>0,\\ v_t=\mathrm{△}v-g\left(u\right)v,& x\in \mathrm{\Omega },t>0,\end{array}\right.$ under homogenous Neumann boundary condition in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 2)$, with $\chi ,\mu ,k>0,\kappa \in \mathbb{R}$ and $g\left(s\right)\simeq s^{\beta }$ for all s>0 with $\beta >0$. For $N\ge 2$, we show that the system admits a global classical solution that provides $\chi <\sqrt{2/N}$, $\beta <2k/N+2$ and $k>1+\left(2/N\right)$. Furthermore, for N=2, under the assumptions above, we obtain the boundedness and asymptotic behavior of solutions that provide $\kappa >0$, $\beta <1$, $k\in \mathbb{N}$ and μ sufficiently large.

KEYWORDS:

• Global existence
• asymptotic behavior
• chemotaxis
• singular sensitivity
• logistic source

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 No.11571093]; the Anhui Natural Science Foundation [No. 1808085QA15].