Qualitative Analysis Of Robin Boundary Value Problems For Reaction-Diffusion Equations

The reaction diffusion equation is a kind of typical semilinear parabolic partial differential equation,which is widely used in physics,chemistry,biology and so on.The reaction-diffusion equation with free boundary conditions is one of its important research directions and has attracted the attention of many scientists.In this paper,we describe the evolution of two types of invasive species in nascent habitats using reaction diffusion equations with Robin boundary conditions and Stefan free boundary conditions as models,one is a Lotka-Volterra weak competition model for simultaneous invasion of two species in a homogeneous environment,and the other is a Lotka-Volterra competition model for invasion of native species by alien species in a non-homogeneous time-period environment.The objective is to investigate the long-time asymptotic behavior of the solutions of these two types of models,as well as the spreading and vanishing phenomena.It shows that different boundary conditions have a significant effect on the asymptotic behavior of the solutions.The models with Robin boundary conditions can better explain the dynamics of some natural phenomena.The structure of this thesis is as follows:In Chapter 1,we briefly introduce the research background and development status of the subject.In Chapter 2,we give some preliminary knowledge such as embedding theorem,maximum principle and Banach compression mapping principle.In Chapter 3,we establish a Lotka-Volterra competition model for two invasive species in a homogeneous environment,and study the dynamics of two competitive species’ outward diffusion through a free boundary.The long time asymptotic behavior is analyzed by constructing the upper and lower solutions and comparing the principle,and the sufficient conditions of diffusion and disappearance are given.In Chapter 4,we develop a Lotka-Volterra competition model under the environment of non-uniform time period.In the initial stage,the distribution range of native species is relatively wide,while the invasive species is locally distributed in the region,and the spatial variables of the intrinsic growth rate of the two species can be changed.Based on the properties of the principal eigenvalues of the time periodic eigenvalue problem and the initial boundary value problem on the half line,the alternative properties of diffusion and disappearance are obtained.Finally,the sufficient conditions of diffusion and disappearance are given.In chapter 5,we summarize the research content and look forward to the next research work.