Properties Of Solutions To A Keller-Segel Model With Logistic Source In The Whole Space

In this thesis,we study some properties of weak solutions to a Keller-Segel model with Logistic sources in whole space with the dimension n?3.Namely,global existence,the L?-bound and uniqueness of weak solutions are obtained.Existence of weak solutions to this model is proved for the various case of the damping coefficient b:(?)when damping coefficient b is larger,i.e.,the parameter b>1-2/n weak solutions exist globally for arbitrary initial data and any birth rate a?0;(?)when damping coefficient is smaller,i.e,the parameter 0<b?1-2/n,we prove global existence of weak solutions for the small initial data and the small birth rate.There are essential differences on the proof of existence in the two cases.For the first case,the aggregation term can be controlled by the damping term and the diffusion term,hence we use the method from the classical Keller-Segel equations to deal with existence of weak solutions for any initial value and any birth rate.For the second case,since the aggregation term can not be controlled by the damping term and the diffusion term,the help of small birth rate and small initial conditions is necessary for completing the proof of existence.Furthermore,when the damping effect is stronger,we get the L?-bound of weak solutions by establishing the differential iteration inequality and using the Moser iteration steps,and uniqueness by using the hyper-contractivity of solutions.