Dynamics Of Nonlocal Diffusive Equations With Free Boundary

The biological invasion may break the original ecological balance and will change or undermine the ecological outlook,thus threatening the develop of hu-man society.Many biomatologists have used mathematical modeling to simulate various performance traits of new species to have a more clearly understanding and predict of them,not only for animals or plants,but also for epidemic diseases.We will use some kinds of nonlocal free boundary problems to discuss the dy-namics of some epidemic disease or invasion species,to have a much better under-standing of the strong dependence of initial data or initial new environment.The second chapter is concerned with a nonlocal SIS epidemic model,which involved in nonlocal incidence rate and free boundary.We get the sufficient and necessary conditions that ensure the spreading or vanishing of the disease.All those conclusions make us have a much detailed and profound description of this disease.Indeed,our results also showed that the nonlocal incidence rate speed up the spreading of the disease.The third chapter focus on a nonlocal Volterra model with free boundary.Com-pared with the classical reaction-diffusion models,the most difficult point in nonlocal problem is the lack of comparison principle.After establishing the suitable compar-ison principles in different parabolic domains,we acquire the long-term behaviors of the responding species,and the estimate of the spreading as well.Particularly,by the upper and lower solutions successively improve method,we obtain the global attract of the unique positive stationary state.In addition,we discuss the case of environment inhomogeneous.The forth chapter is concerned with the traveling waves of a nonlocal reaction-diffusion model.The spreading fronts of this model is determined by the Stefan condition,and the reaction term is non-quasi-monotone.By constructing two quasi-monotone auxiliary equations,we study the global positive solution and then the long time behaviors.Meanwhile,by the phase-plane method,we obtain the traveling wave of this problem,as well as the spreading speed.In the last chapter,we establish some results for the nonlocal diffusion problem.In the study of the corresponding reaction-diffusion models,the derivation of the free boundary condition is deeply based on the Fick law.Obviously,this tool does not work here since the space regularity is unknown for the nonlocal problems.We give the well-posedness by proposing a new free boundary condition and then a totally new method.