Keller-Segel Equation With Degenerate Diffusion And Nonlocal Aggregation

In this thesis, we study the Keller-Segel equation with degenerate diffusion and nonlocal aggregation the solution to the chemotaxis model, including the existence and uniqueness of global weak solutions, as well as the finite time blow-up of weak solutions.The thesis consists of five chapters:In Chapter 1, we briefly introduce the background of the Keller-Segel system with the current research status.Chapter 2 deals with the properties of solutions. We proved that for m?(1,2-?/d) there is Cd,m> 0 such that the model possesses global weak solutions provided the initial data ?u0?p< Cd,m with p=d(2-m)/?; the finite time blow-up occurs if the free energy F[u0]<0 (which implies ?u0?p>Cd,m).In addition, the decay estimates are obtained for the global weak solutions.Chapter 3 gives a more precise criterion for global existence and finite time blow-up of weak solutions in the subrange ( The proof depends on the relation between the free energy functional and the HLS inequality.The uniqueness of weak solution is considered in Chapter 4. By the optimal transportation method, we establish the stability and uniqueness for global weak entropy solutions in the sense of Wasserstein distance.Chapter 5 summarizes the main results of the thesis, and proposes some interesting prob-lems to be done.