Global solvability in a singular chemotaxis system with logistic source and non-sublinear production

This paper deals with a singular chemotaxis system with logistic source and non-sublinear production under homogeneous boundary condition: $u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v}\nabla v)+ru-\mu u^k$, $v_t=\Delta v-v+u^{\beta }$ in a bounded convex domain $\Omega \subset R^n$ with $n\ge 1$, here $\chi ,\mu >0$, $r\in R$, $k>1$ and $\beta \ge 1$. It is proved that the system admits a global solution if $k>2$ with $\beta \in [1,k-1)$, or $k>1$ and $\beta \ge 1$ with $\chi \le \frac{4}{n{\beta }^2}$. Moreover, the solution is globally bounded for the second case with $r\le -\frac{1}{\beta }$. This means that the logistic source along with non-sublinear production indeed benefits to ensure the global existence-boundedness of classical solution to this chemotaxis system with singular sensitivity.