Boundedness to a logistic chemotaxis system with singular sensitivity

TL;DR

This paper investigates the boundedness of solutions to a chemotaxis system with singular sensitivity and logistic growth, establishing conditions for global boundedness and existence of solutions in bounded domains.

Contribution

It provides new criteria for global boundedness and existence of solutions to a chemotaxis system with singular sensitivity and logistic source terms.

Findings

Global bounded classical solutions exist under certain parameter conditions.

Global generalized solutions are obtainable when parameters are within specific ranges.

Small initial data and parameter ratios ensure global boundedness of solutions.

Abstract

In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $ u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$, $\chi,r,\mu>0$, $k>1$ and $n\ge 2$. It is shown that the system possesses a globally bounded classical solution if $k>\frac{3n-2}{n}$, and $r>\frac{\chi^2}{4}$ for $0<\chi\le 2$, or $r> \chi-1$ for $\chi>2$. In addition, under the same condition for $r,\chi$, the system admits a global generalized solution when $k\in(2-\frac{1}{n},\frac{3n-2}{n}]$, moreover this global generalized solution should be globally bounded provided $\frac{r}{\mu}$ and the initial data $u_0$ suitably small.