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

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