The Global Dynamics Of Reaction-Diffusion Equation With Nonlocal Effect

The nonlocal reaction-diffusion equation can be used to describe many phenomena in nature more accurately which has attracted the attention of many scholars.However,due to the introduction of nonlocal in the classical model that the maximum principle is invalid and the classical method is no longer applicable.Therefore,the research on nonlocal problems is very challenging.This paper is devoted to studying the dynamic behavior of a single reactiondiffusion equation and a reaction-diffusion system with nonlocal term corresponding to the Cauchy problem to explore the influence of non-local introduction on the solution dynamics.In recent years,most scholars who study non-local reaction-diffusion equations have focused on the monostable state and have done little research on the bistable state and the research on the corresponding initial value problem is still blank.Therefore,this paper first studies the reaction-diffusion equation with nonlocal bistable corresponding to the Cauchy problem,mainly studies the well-posedness of the solution.And explore the asymptotic behavior of the solution when the initial value has compact support.Secondly,study the nonlocal Belousov-Zhabotinsky reaction-diffusion system and mainly study the dynamic behavior of the solution,including the existence,uniqueness,and uniform bounds of the solution.And through numerical simulation to explore the essential difference with the classical system.This paper mainly uses the following methods to study nonlocal problems.First redefines the appropriate super-and sub-solutions and constructs monotonic iterative sequences and using the comparison principle to obtain the existence of the solution.Then,the fundamental solution and Gronwall's inequality are used to prove the uniqueness of the solution with any non-negative bounded initial values.The uniform boundedness of the solution is obtained with the aid of the auxiliary problem.Finally,investigate the asymptotic behavior of the solution of a single equation when the initial value has compact support.Under different initial values and different kernel functions,the state of the solution of the Belousov-Zhabotinsky reactiondiffusion system with changing parameters is studied through stability analysis and numerical simulation,and some interesting new phenomena are shown.