Temporal Decay For The Chemotaxis System Involving Fractional Diffusion

Scientists have found that the formation and cdiffusion of tumors are related to the diffusion and accumulation of cells closely.The migration of invasive cells is directed either by a mechanism termed chemotaxis or by a mechanism rtermed haptotaxis.We consider the process of the formation and diffusion of cells by chemo-taxis system,and people have obtained many meaningful results about the classical chemotaxis system.Recently,going deep into the researches about the chemotaxis,people discover that the diffusion of some population in nature vilolates the clas-sical chemotaxiy law.Therefore,people find that the description of a population undergoing Levy flights is more correct than the description of a population un-dergoing Brownian motion.More precisely,the correct description of a population undergoing Levy flights is given by the substitution of the Laplacian diffusion by a fractional diffusion.The study of the anomalous duffusion has become one of the most popular topics in partial differential euations.It,has also attracted the attention of several famous scholars,such as the Wolf prize winner Caffarelli,the famous French mathematician Perthame,the famous Italy mathematician Terracini and so on.The researches about the chemotaxis system not only promote the devel-opment of partial differential equations,but also play a guiiding role in preventions and treatments of tumors.In this paper,we mainly concern the classical chemotaxis system and the frae-tional chemotaxis systems.Moreover,we study the temporal decay of these models.It is an innovation that we introduce a functional space which can simultaneously embody the energy estimation and temporal decay of the solutions to the problem as the basic iterative space.Then,using the fixed point theorem,we can prove the existence and the uniqueness of the classical solutions to the problems,and obtain the temporal decay estimates of the higher derivatives of the solutions.The outline of the paper is as follows.In Chapter 1,we introduce the background and the research status of chemo-taxis systems.In Chapter 2,we are concerned with a Keller-Segel model arising from biol-ogy,which is a coupled system of two parabolic equations.The global existence result and the optimal temporal decay estimates of the classical solution to the fully parabolic Keller-Segel model are obtained by pure energy method under smallness initial data conditions.To be more precise,we derive the optimal decay rates of the higher-order spatial derivatives of the solution.In Chapter 3m we study a.parabolic-parabolic chemotaxis system involving frac-tional Laplacian with the same order.We prove the temporal decay of the solutions to the fractional linear evolution equations.Therefore,with the help of the decay estimates obtained before,the existence and the uniqueness of global classical solu-tion are proved under the assumption that the initial data are small enough.During the proof,the asymptotic decay behaviors of the solution are also shown.In the last chapter,we consider a,parabolic-parabolic-parabolic fractional chemo-taxis system with different fractional Laplacian orders.The global existence result and the optimal temporal decay estimates of the classical solution to the fractional Keller-Segel system are obtained by pure energy method based on the assumption of smallness initial conditions.More precisely,we derive the optimal deeay rates of the higher-order spatial derivatives of the solution.