Existence Of Solutions For Time-space Fractional Keller-Segel Equations

This paper investigates two kinds of the time-space fractional Keller-Segel equations with Caputo time derivatives and fractional Laplacian in Rd(d? 2).With regard to the generalized Keller-Segel model that de-scribes the biological phenomenon chemotaxis with both anomalous diffu-sion and memory effects,the local existence and uniqueness of mild so-lution in C((0,T];L d/??+?-2(Rd))?((0,T];W??-1,d/??+?-2(Rd))and global ex-istence and uniqueness in C((0,?);Ld/?+?-2(Rd))?C((0,?);Ld/??+?-2(Rd))?C((0,?);W??-1,d/??+?-2 are established.In addition,we construct the ex-istence and decay estimates of the weak solutions to the time-space fractional parabolic-elliptic Keller-Segel equation.The thesis is divided into five chapters as follows.In Chapter 1,we introduce the background with rich conclusions about Keller-Segel system and fractional derivatives.Then we will briefly describe the main results of this thesis.In Chapter 2,some notations,definitions and lemmas are listed.In Chapter 3,we consider the generalized Keller-Segel model in Rd.Based on the contraction properties that follow from the asymptotic behaviors of the Mittag-Leffler functions to obtain the Lp and Ws,p estimates of the solution,combined with the fixed point theorem,the local and global existence of mild solutions are established.In addition,with the help of comparison principle,we construct the uniqueness of mild solution.In Chapter 4,with regard to the time-space fractional parabolic-elliptic Keller-Segel equation in Rd,by the classical energy method,we get some a priori estimate of weak solution,then construct a corresponding regularized system,which has a strong solution.Combined with the generalized strong compactness criteria to time fractional PDEs,we address the existence of weak solution.And the proofs of decay estimates is based on the generalized comparison principle and the decay behavior of the solutions to the nonlinear fractional differential equation.In Chapter 5,we summarize the work of this paper,and propose some problems to be researched.