Multidimensional Parabolic-elliptic Keller-Segel Model With Food Sources

In this thesis,we prove existence,uniqueness and L? uniform bound of weak solutions to a multi-dimensional parabolic-elliptic Keller-Segel system with food sources.First,we prove that if the initial data (?),then this model has a global weak solution.The initial condition indicates two kinds of phenomena:(i)if the initial amount.of food sources is sufficient,and the initial amount of microorganisms is small,the solution can be global;(ii)if the initial amount of microorganism is very large,and the initial amount of food sources is small,the solution of this model can also exist.That implies the existence of the weak solution is co-determined by the initial amount of the microorganisms and the initial amount of food sources.Next,we give hyper-contractivity of weak solutions and use it to prove uniqueness of weak solutions.Because of the emergence of food sources,in the proof of uniqueness we cannot only consider a function p(x,t),but need to deal with the functions p(x,t)and f(x,t)simultaneously.Finally,we obtain the L? norm of the weak solution is uniformly bounded in time and space variables using iterative inequality.