Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity

Abstract

We consider a chemotaxis system with singular sensitivity and logistic-type source: \begin{document}$ u_t = \Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k $\end{document} , \begin{document}$ v_t = \epsilon\Delta v-v+u $\end{document} in a smooth bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} with \begin{document}$ \chi,r,\mu,\epsilon>0 $\end{document} , \begin{document}$ k>1 $\end{document} and \begin{document}$ n\ge 2 $\end{document} . It is proved that the system possesses a globally bounded classical solution when \begin{document}$ \epsilon+\chi . This shows that the diffusive coefficient \begin{document}$ \epsilon $\end{document} of the chemical substance \begin{document}$ v $\end{document} properly small benefits the global boundedness of solutions, without the restriction on the dampening exponent \begin{document}$ k>1 $\end{document} in logistic source.