The Properties Of Solution To Keller-Segel Equations With Nonlinear Aggregation Term

In this thesis, we consider the following Keller-Segel model with a nonlinear aggre-gation term where n> 3, the aggregation exponent q> 1. We mainly prove the existence, uniqueness, L∞ uniform bound and long-time behavior of weak solutions to this system. In particular, we firstly give the global existence of weak solutions by using the priori estimates, regu-larized problem and compactness argument under the initial data satisfying the condition Then, we use the iterative method to prove that the weak solutions possess L∞ uniform bound on time and space when u0 ∈ L∞. Furthermore, the properties of the hyper-contractivity and the infinite decay of the LP norm of the solutions are given. At last, the uniqueness of the solution is proved in the L∞ norm sense by using the semi-group theory and the properties of the nonlinear aggregation term.