Global Existence And Asymptotic Stability Of The Fractional Chemotaxis System

Chemotaxis is the directional movement of cells towards or away from certain irritating chemical substances in the environment.For example,cell slime molds themselves release a certain chemical substance and move towards areas with high concentrations of the substance.Regarding this biological phenomenon,Keller-Segel in 1970 established a chemotaxis model consisting of reaction-convection-diffusion equations,which has become one of the best-studied model for chemotaxis.Recently,many experiments showed that there may exist a more accurate description of a cellular population self-interacting chemotactically in nature in which Brownain motion does not represent an approximation for the population dispersal.The motivation for using fractional diffusion comes from the fact that occur in nature,organism adopt Levy process search strategies over continuous paths interspersed with Levy flight.Research results has shown that the diffusion of cells can be better simulated by replacing the Laplacian operator with a fractional one.The study of the anomalous diffusion has become one of the most popular topics in partial differential equations.It has also attracted the attention of many scholars.In nature,cells often live in a viscous fluid.This kind of cell-fluid becomes more complicated since it not only includes chemotaxis,but also consists of fluid dynamics.In order to describe the coupled phenomena,Tuval et al.proposed a classical chemotaxis-fluid system.In recent years,many mathematicians and biologists have deeply investigated this type of model,and used it to reveal the formation and metastasis of tumors.This is a frontier field of applied mathematics with important scientific value and application prospects.The dissertation focuses on the impact of signal-dependent sensitivity,singular sensitivity,attraction-repulsion as well as fluid dynamics on the global existence,uniqueness and asymptotic stability of the solutions of fractional chemotaxis-fluid systems.The specific research work is organized as follows.In Chapter 1,we give a brief introduction of the background,research status and latest results about our research topic.In Chapter 2,we give some basic definitions and lemmas related to dissertation.In Chapter 3,we deal with the fractional chemotaxis system with signaldependent sensitivity in Rn(n?2).The model consists of two fractional parabolic equations.We introduce a suitable functional space as the basic iteration space,under the smallness initial assumptions,we show the existence,uniqueness and temporal decay of the global classical solutions to the problem simultaneously by utilizing contraction mapping principle and Sobolev imbedding theorem.In Chapter 4,we consider the fractional chemotaxis system with singular sensitivity in R3.By constructing an appropriate approximate solution and introducing Riesz transform,the local existence of the solution to the system is obtained.Then with the help of combining the local existence and a priori estimates as well as continuity argument,we establish the existence and uniqueness of the global classical solutions with small initial data.Moreover,we derive the asymptotic decay estimates of the solutions and their higher-order spatial derivatives by introducing the negative Sobolev space H-s(R3)(0