Properties Of Weak Solutions For The Keller-Segel Model With The General Potential

Degenerate Keller-Segel equations,as a class of the chemotaxis models,were widely used to describe biological,physical and chemical phenomena.In this thesis,we consider high-dimensional degenerate Keller-Segel equations with a general potential.For various classifications of the diffusion exponent m and the exponent k of the singular potential,global existence,blow-up and the uniform boundedness of L? norm of weak solutions are proved.The main results are divided into three aspects:First,for the weak singular potential and the strong diffusion(2-n<k<1,max{1,1-k/n)}<m<2),we prove global existence and the L? uniform bound of weak solutions for any initial data.The main method is to regularize the equation,use a series of analytical techniques to give the uniform estimates of solutions to regularized problem,and finally utilize the Lions-Aubin lemma to make a compactness argument in order to finish the proof on global existence of weak solutions.The L? bound of weak solutions is obtained by the Moser iteration.Secondly,we study existence and blow-up of weak solutions for the weak singular potential and the weak diffusion(2-n<k<1,2n/2n+k<m<1-k/n).In this case,by constructing an appropriate energy functional,boundedness of the L2n/2n+k norm of weak solutions is given for the initial data satisfying some conditions.Based on the upper and lower bounds of ???(x,t)?L2n/2n+k.we prove global existence and blow-up of weak solutions.Finally,existence of weak solutions for any initial data is proved in the case of the strong singular potential and the strong diffusion(1-n<k<2-n,m>2(1/n-k/n)).The difficulty of this part is to deal with the stronger singularity of the potential function.Instead of the estimate of ?Sk,we use estimate of ?Sk to discuss the problem and give the proof of global existence of weak solutions.