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

TL;DR

This paper analyzes a nonlinear Keller-Segel model, demonstrating that nonlinear diffusion can prevent overcrowding and that bounded and blow-up solutions can coexist under certain conditions.

Contribution

It proves that solutions are bounded when the diffusion exponent exceeds the chemotactic sensitivity exponent, generalizing previous results and suggesting coexistence of bounded and blow-up solutions.

Findings

Solutions are bounded if m > q > 0.

Nonlinear diffusion prevents overcrowding.

Bounded and blow-up solutions can coexist.

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.