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 ‖ρε(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.