Qualitative Analysis Of Solutions To Keller-Segel Equations With High-dimensional Logarithmic Potential

This thesis considers the property of weak solutions to Keller-Segel equations with logarithmic potential in high-dimensional space,including existence and the L? uni-form bound of weak solutions.First,we prove that if non-negative initial data ?0 ?L1(Rn + |x|2)dx)()?Lp(Rn),there is a global weak solution.Compared to the previous works,this result indicates that if the diffusion exponent is stronger or the singularity of concentration potential is weaker,the property of weak solutions to the Keller-Segel equa-tions will occur qualitative change.The proof on existence of weak solutions is divided into three subsections.In Subsection 1,construct the corresponding regularization equa-tions to the model we considered.In Subsection 2,uniform estimates of solutions to the regularization equations are given by using various inequalities and analytical techniques.In Subsection 3,utilize the Aubin-Lions lemma to discuss compactness,and obtain ex-istence of weak solutions to the original model.Furthermore,we prove the L? uniform bound of solutions to the model using the Moser iteration.Here we need to overcome the difficulty from nonlinear diffusion and excavate deeply the property of the constant in the Hardy-Littlewood-Sobolev inequality.