  In this paper we study the initial boundary value problem for the system
$u_t-\Delta u^m=-\mbox{div}(u^{q}\nabla v),\ v_t-\Delta v+v=u$. This problem is
the so-called Keller-Segel model with nonlinear diffusion. Our investigation
reveals that nonlinear diffusion can prevent overcrowding. To be precise, we
show that solutions are bounded as long as $m>q>0$, thereby substantially
generalizing the known results in this area. Furthermore, our result seems to
imply that the Keller-Segel model can have bounded solutions and blow-up ones
simultaneously.
