The Study Of Some Singularly Perturbed Parabolic Problem

In recent years,singularly perturbed systems have been widely applied in establishing relevant models in natural science and engineering techniques such as hydrodynamics,chemical reaction and biology and so on.However,these models are often discontinuous at a curve in the domain,which are discontinuous singularly perturbed systems,or contain multiple small parameters,which are multi-scale slow-fast systems.Therefore,more and more researchers pay attention to the study of discontinuous singularly perturbed systems and multi-scale fast-slow systems,and greatly enrich and develop the content of singular perturbation theory.This thesis mainly studies the existence and stability of a contrast structure type solution for two classes of discontinuous singularly perturbed parabolic equations with Dirichlet boundary condition and periodical condition and the multi-scale dynamics of two classes of Leslie-Gower slow-fast system with different function responses.The thesis contains six chapters as follows:In chapter one,the development history of singular perturbation theory and its application to discontinuous singularly perturbed problems and multi-scale biological models are briefly introduced.Furthermore,the basic theorems,the related concepts and the main content of this thesis are introduced in detail.In chapter two,the existence and stability of a contrast structure type solution for the boundary value problem of a singularly perturbed reaction-diffusion equation containing discontinuous righthand function is studied.According to the boundary function method and spatial contrast structure theory,the asymptotical approximation of a solution is constructed,which has boundary layers and internal layer.By the differential inequality method and lower and upper solutions method,the existence and stability of the solution and the high precision of the asymptotical approximation are proven.Furthermore,an example is introduced to explain the results.In chapter three,the existence and stability of a contrast structure type solution for the boundary value problem of a singularly perturbed reaction-diffusionadvection equation are studied,which has discontinuous reaction and convection.Using boundary function method,spatial contrast structure theory and differential inequality method,we prove the existence and stability of the solution with internal layer and give its high precision asymptotical approximation.Finally,an example is introduced to explain the results.In chapter four,the existence of traveling waves and periodic solutions for a Leslie-Gower reaction-diffusion model with weak diffusion is investigated.Assuming the diffusion rate of prey and predator is sufficiently small and the natural growth rate of prey is much greater than that of predators,then the dimensionless Leslie-Gower reaction-diffusion model is a singularly perturbed model which contains two different small parameters.Using travelling wave transformation,the model are firstly transformed into a four dimensional Multi-scale slow-fast system with two small parameters.Applying the geometric singular perturbation theory,the existence of heteroclinic orbit,canard explosion phenomenon and relaxation oscillation cycle for the slow-fast system are proven,which show the existence of travelling waves and periodic solutions for the original dimensionless model.Meanwhile,some numerical examples are given to verify our theoretical result and the biological meanings of these dynamical phenomenon are also introduced.In chapter five,the multi-scale dynamic of a Leslie-Gower slow-fast system with Monod-Haldane function is studied.Applying the geometrical singular perturbation theory,some new multi-scale dynamical phenomenons are obtained,for example: singular Hopf bifurcation,canard explosion phenomenon,relaxation oscillation cycle,heteroclinic and homoclinic orbits and the coexistence of the canard cycle and the relaxation oscillation cycle.Furthermore,some numerical examples are introduced to explain the theoretical conclusions in this chapter.In chapter six,the brief summary and future research prospects are introduced.