Asymptotic Behavior Of Reaction Diffusion Systems And Their Random Perturbations

Reaction-diffusion equations are important to a wide range of applied areas such as chemical reactions, cell evolution processes, drug release, ecological evolution, spread of diseases, transport of contaminants in the environment, etc. The study of reaction-diffusion equations not only provides us a powerful tool for the scientific development in these areas, but also promotes the development of the theory of partial differential equations itself. This Ph.D. thesis, taking the reaction-diffusion system with real backgrounds as research objects, describes the asymptote properties and random perturbations of the reaction-diffusion system from many aspects. It mainly includes the following several aspects of the work.In Chapter1, we introduce the physical background, the mathematical models and the current situation of the objects.In Chapter2, we discuss the stability of monotonic travelling wave solutions of a class of competitive-diffusion system connecting a non-coexistence state with a coexistence state. Base on the local linear stability of stationary solution of reaction-diffusion system under weak and strong competitive conditions, in stead of using Green’s function of the second order linear system to describe linearized operator, we first develop a new method to establish the asymptotic representation of the resolvent operator of the linearized operator, directly by a first order linear differential system with four components, and then we describe the spectrum distribution of the linearized operator in weighed function space Bw,k(R, R2) by the asymptotic representation of the corresponding derivative components of the solution, at last we obtain the stability property of monotonic travelling wave solutions of the reaction-diffusion system.In Chapter3, we consider a class of Holling-Tanner pray-predator system with cross-diffusion. We extend the results from the case of part cross-diffusion coefficients d3=0, d4>0to the case of d3>0, d4>0. By means of the maximum principle, a priori estimates and the topological degree theory, we give a sufficient condition for the existence of constant and non-constant positive solutions of the corresponding stationary pray-predator system with cross-diffusion.In Chapter4, we study the blowup property of positive mild solutions to the Cauchy problem of a fractional reaction diffusion system. With the nonlocality of fractional Lapla-cian, fractional reaction diffusion system has been applied in molecular biology, fluid dynam-ics, statistical physics, economic and financial, etc. Unlike the method developed by Perez and Villa, we focus on the properties of the fundamental solution of the fractional heat oper-ator (?)t+-(-â–³)β/2in the whole space. By using analytic property of the fundamental Solution developed by H. Yosida and some estimates of the fundamental Solution developed by L Caffarelli and A. Figalli, we first establish the estimates of the lower bound of nonnegative solutions to the initial value problem, then prove the unboundedness of the solution in large time, and finally we get a sufficient condition that the positive mild solution of the fractional reaction diffusion system blows up in finite time.In chapter5, we study zero mean white noise perturbations on reaction-diffusion waves. For traveling wavefronts of the classic Nagumo equation that connect two stable states, by using the heat kernel on the whole real line R, we analyze the random perturbations about the lower (upper) stable state by two-parameter zero mean white noise. Under the zero mean white noise perturbations, as tâ†'∞, the mean of the traveling wavefronts of the Nagumo equation ut=uxx+u(u-a)(1-u) is increased at the lower stable state and decreased at the upper stable state.In chapter6, we study non zero mean white noise perturbations on reaction-diffusion waves. Firstly, near two stable states which are two boundaries of the traveling wavefront to Nagumo equation, the asymptotic fluctuating behaviors are given by the heat kernel on the whole real line R. Secondly, the statistical properties of traveling wavefronts under the influence of random perturbations by two-parameter non zero mean white noise α+βWxt are described. Lastly, we reveal that the perturbation effects on the lower (upper) stable state are different between zero mean white noiseWxt and non zero mean white noise α+βWxt astâ†'+∞.