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 $ \mathbb{R}^{n} $ for $ n\ge 3 $, where $ \lambda\in \mathbb{R} $, $ \mu >0 $, $ k>1 $.Under the assumption $ k<\frac{3}{2}-\frac{1}{n} $, it is shown that there exists $ m_{*}>0 $ such that if the radially symmetric initial data satisfies $ \int_ {B_{\frac{1}{2}}(0)}u_{0}\geq m_{*} $, then the problem admits a finite-time blowup solution. [ABSTRACT FROM AUTHOR]