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

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