  This article deals with the logistic Keller-Segel model \[
  \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu
u^2, \\ \\ v_t = \Delta v - v + u
  \end{cases} \]
  in bounded two-dimensional domains (with homogeneous Neumann boundary
conditions and for parameters $\chi, \kappa\in \mathbb{R}$ and $\mu>0$), and
shows that any nonnegative initial data $(u_0,v_0)\in L^1\times W^{1,2}$ lead
to global solutions that are smooth in $\bar{\Omega}\times(0,\infty)$.

Analysis of PDEs