Chaotic Behavior Of Stochastic Perturbation And Chemotaxis Of The Fractional Heat Equation

Replacing the classical integer-order time derivative and space Laplacian operator with the time-fractional derivative and the fractional-order Laplacian operator,the fractional-order diffusion equation in fractal media is obtained,it plays an important role in the study of fractional anomalous diffusion and random walk theory.This paper mainly concerns about the chaotic behavior of stochastic fractional heat equation and the Cauchy problem of the drift-diffusion equation with high nonlinearity in critical Besov space and in noncritical Sobolev space respectively.In Chapter 1,we first briefly introduce the physical background and mathematical char-acteristics of fractional heat equation and drift-diffusion equation,we focus on the chaot-ic characteristics of stochastic perturbations in fractional heat equations and the effects of higher-order nonlinear chemotactic terms on drift and diffusion and research progress of the stochastic heat equations and the drift-diffusion equations.The we present some prelimi-nary work about the problems studied,including some notations,function spaces,common definition and some technique lemmas.We also introduce the Littlewood-Paley dyadic de-composition and several relevant lemmas are reviewed.In Chapter 2,we study the chaotic behavior of the mild solution to the Cauchy problem of the nonlinear stochastic fractional heat equation （?）tu=-v/2（-（?）xx）^α/2 u+σ（u）（?）（t,x） on real line R driven by space-time white noise with bounded initial data.We analyze the large-|x|fixed-t behavior of the solution u_t（x）for Lipschitz continuous functionσ:R→R under three cases.（1）σis bounded below away from 0.（2）σis uniformly bounded away from 0 and∞.（3）σ（x）=cx（the parabolic Anderson model）.From the sensitivity to the initial data of stochastic fractional heat equation,we describe that the solution to the Cauchy problem of stochastic fractional heat equation exhibits chaotic behavior at fixed time before the onset of intermittency.In Chapter 3,we consider the global existence,regularizing decay rate and asymptot-ic behavior of mild solutions to Cauchy problem of fractional drift-diffusion system with power-law nonlinearity.Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel,we first prove the global-in-time existence and unique-ness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings.Then we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev（H-L-S）inequality.In Chapter 4,we continue consider the fractional drift diffusion system as in Chapter2.Instead of the convention that people always focus on the properties of the solution in critical spaces,here we are interested in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces.For the initial data in these non-critical spaces,using the properties of fractional heat semigroup and the classical H-L-S inequality,we get the existence and uniqueness of the mild solution,together with the decaying rate estimates in terms of time variable.In the last chapter,the work and innovation of the whole paper are summarized,and the above problems are further discussed.