On Asymptotic Behaviors Of Some Singular Limit Equations

This paper is divided into three parts: Firstly,we consider the high-order expansion of the Lotka-Volterra competitive diffusion model with nonsymmetric nonlocal dispersal operators as the diffusion rate tends to zero.In this part,on the one hand,the higher order expansion of the syetem is discussed when the diffusivity of the two species tends to zero at the same rate and at different rates respectively,on the other hand,the higher order expansion of the diffusivity tends to zero for systems with a general reaction term is considered;Secondly,the nonlinear damped hyperbolic Allen-Cahn system is studied by the matching asymptotic expansion as the coefficient of time tends to zero.In this part,on the one hand,the outer expansion and inner expansion are performed and the approximate solution of the system is constructed,on the other hand,The energy method is used to estimate the error;Thirdly,we study the vanishing Darcy number limit of the three dimensional StokesBrinkman coupled system.In this part,on the one hand,the asymptotic expansion is carried out and the approximate solution of the system is constructed,on the other hand,the approximate solution is modified appropriately and the error estimation is completed.