Properties Of Solutions To Keller-Segel Model With Linear Diffusion Or Degenerate Diffusion

This thesis studies the.properties of solutions to Keller-Segel models,including the L?-estimate of solutions to parabolic-parabolic model and global smoothness of solutions to Keller-Segel equations with linear diffusion.First,we prove a L?-estimate of weak solutions to the parabolic-parabolic model with linear diffusion using boundedness of||?c||L?(Rn)and the Moser iteration.It is a feature of the thesis that we obtain the explicit coefficient in the up-bound of ||?c||L?(Rn).Next,we prove that the solution to the Keller-Segel model with linear diffusion is smooth globally based on the L?-bound.Finally,for the degenerate parabolic-parabolic Keller-Segel model with the diffusion exponent 2n+2+n<m<2-2/n,a uniform in time L? estimate of weak solutions is obtained by the Moser iteration.Compared with the parabolic-parabolic Keller-Segel model with linear diffusion,there are two different points.One is that we need to use degenerate diffusion to balance the nonlinear concentration;the second one is that the L?-bound is uniform in time and space,which is better than the conclusion on the linear parabolic-parabolic model.