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摘要
This paper deals with the Keller-Segel chemotaxis model of parabolic-elliptic type with the volume-filling effect studied by Burger et al (2006 The Keller-Segel model for chemotaxis with prevention of overcrowding: linear versus nonlinear diffusion SIAM J. Math. Anal. 38 1288-315). In their discussion on the large time asymptotic behaviour of solutions, the diffusion rate of rho (the density of cells) had to be assumed to be large with epsilon > 1/4. While for the nonlinear diffusion model, it was proved that the asymptotic behaviour of solutions is fully determined by the diffusion constant being larger or smaller than the threshold value epsilon = 1. The same 'large epsilon-restriction' (epsilon > 1/4) was also made for studying the parallel parabolic-parabolic model in Di Francesco and Rosado (2008 Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding Nonlinearity 21 2715-30), where it was pointed out that 'Whether this condition is necessary to have large time decay (and consequently a self-similar behaviour) for rho is still an open problem even in the (simpler) parabolic-elliptic case'. The aim of the paper is to answer this problem for the parabolic-elliptic model. We prove the mentioned time decay estimate without the restriction epsilon > 1/4. The main technique used in this paper is the L-p-L-q estimate method.
