Global Existence And Asymptotic Behavior Of Solutions To The Reaction-diffusion Equation With Nonlocal Diffusion

The classical reaction diffusion equation involves a diffusion process controlled by the Laplacian operator.The reaction diffusion equation is a fundamental mathematical model used in many scientific fields to describe the dynamics of diffusion processes,such as heat transfer and species migration.However,in recent years,scientists have discovered that some phenomena in nature can not be described by the classical reaction diffusion equation.For example,the diffusion of vascular mesenchymal cells[1]can not be described using the classical reaction diffusion equation,because the activities and interactions of vascular mesenchymal cells will be in a relatively large spatial range rather than limited to a small range,which leads to the occurrence of nonlocal actions in space.Therefore,many scientists introduce nonlocal diffusion operators(such as convolution operators or fractional Laplacian operators)to describe nonlocal actions in space and establish the reaction diffusion equation with nonlocal diffusion.Studying such equations can help us better understand some nonlocal phenomena in physical,chemical,and biological,and provide a new mathematical tool that enables us to more accurately describe and predict these phenomena.Therefore,the equations with nonlocal diffusion have become one of the frontier problems in the field of partial differential equations,and studying such problems has important scientific value and application prospects.This article mainly focuses on the theme of nonlocal diffusion,considering two kinds of nonlocal diffusion models:free boundary model with nonlocal diffusion and chemotaxis model with nonlocal diffusion,and studying the global existence,uniqueness and asymptotic behavior of classical solutions of these two models.The specific research work is organized as follows:In Chapter 1,we introduce the background knowledge,research status,and latest research results of the research topic.In Chapter 2,we give some basic definitions and lemmas related to dissertation.In Chapter 3,we are devoted to the nonlocal diffusion model with free boundaries in a time-periodic environment.First,by the contraction mapping principle and fixed point theorem,we obtain the global existence and uniqueness of classical solution.Moreover,using the properties of the main eigenvalues of time-periodic parabolic eigenvalue problem and the attractiveness of periodic solutions in bounded domains,we obtain the long-time dynamical behavior of the solution is characterized by a spreading-vanishing dichotomy.Finally,we give the criteria to the case with free boundaries,and use these criteria to determine whether spreading or vanishing happens.In Chapter 4,we consider the two species nonlocal diffusion model with free boundaries.First,by introducing a parameterized ODE problem and using contraction mapping principle,the existence and uniqueness of the global solution of the model are proved.When invasive species and native species are in competition,according to the comparison principle and the properties of the principal eigenvalues of the corresponding problem,we determine the asymptotic behavior of the solution under three competitive cases,case(Ⅰ):invasive species is an inferior competitor,case(Ⅱ):invasive species is a superior competitor and case(Ⅲ):both species are in a weak competition case.In Chapter 5,we investigate the fractional parabolic-elliptic chemotaxis model with a logistic source om RN.First,by using the blow-up method in combination with the Liouville-type theorem,we establish the regularity of weak solutions of fractional parabolic equations with convection term.Then,using the semigroup method and the regularity of solutions,the local existence and uniqueness of solutions to the model are obtained when the order of fractional diffsion s ∈(1/2,1).Under the assumption that the chemotaxis coefficient satisfies certain conditions,we prove the global existence and boundedness of solutions.Finally,we obtain the asymptotic behavior of the global solutions with strictly positive bounded and uniformly continuous initial function.In Chapter 6,we study the nonlocal parabolic-elliptic chemotaxis model with time-space dependent logistic source,based on the model in Chapter 5.When the function in the time and space dependent logistic term is uniformly Holder continuous with an exponent of 2-2s<ν<1,we obtain the local existence and uniqueness of classical solutions to the problem.Furthermore,under certain chemotaxis coefficient conditions,the global existence and boundedness of classical solutions are proved.Finally,we obtain the pointwise and uniform persistence of solutions for any strictly positive bounded and uniformly continuous initial function.In Chapter 7,we briefly summarize the work of the entire thesis and make further prospects for future research directions.