Volume effects in the Keller-Segel model: energy estimates preventing blow-up

We obtain a priori estimates for the classical chemotaxis model of Patlak, Keller and Segel when a nonlinear diffusion or a nonlinear chemosensitivity is considered accounting for the finite size of the cells. We will show how entropy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solutions. (C) 2006 Elsevier SAS. All rights reserved.