Finite-time blow-up in the Cauchy problem of a Keller-Segel system with logistic source

The Cauchy problems of Keller-Segel system with logistic source seem much less throughly understood than associated initial-boundary problems in bounded domains. This paper is concerned with the Cauchy problem of Keller-Segel system with generalized logistic source given by$ \begin{eqnarray*} \left\{ \begin{array}{llc} \label{188} u_t = \Delta u - \nabla \cdot (u\nabla v)+\lambda u-\mu u^{k}, \\ 0 = \Delta v+u, \end{array} \right. \end{eqnarray*} $in \begin{document}$ \mathbb{R}^{n} $\end{document} for \begin{document}$ n\ge 3 $\end{document}, where \begin{document}$ \lambda\in \mathbb{R} $\end{document}, \begin{document}$ \mu >0 $\end{document}, \begin{document}$ k>1 $\end{document}.Under the assumption \begin{document}$ k<\frac{3}{2}-\frac{1}{n} $\end{document}, it is shown that there exists \begin{document}$ m_{*}>0 $\end{document} such that if the radially symmetric initial data satisfies \begin{document}$ \int_ {B_{\frac{1}{2}}(0)}u_{0}\geq m_{*} $\end{document}, then the problem admits a finite-time blowup solution.