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

In this paper we study the initial boundary value problem for the system u(t) - Delta u(m)= -div(u(q)del 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. (C) 2020 Elsevier Inc. All rights reserved.