Nonlinear diffusion in the Keller-Segel model of parabolic-parabolic type

Abstract

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.