Qualitative Analysis Of Free Boundary Problems For A Class Of Reaction-diffusion Cooperative Systems

Reaction-diffusion equations are widely used in the research of ecology,epidemiology,biochemistry,et.A large number of reaction-diffusion models are established on the fixed region to study dynamics behaviour.Considering the species migration and transmission phenomena of infectious diseases,to investigate the problems of free boundary will be conducive to explain accurately many phenomena in nature,which will fill a gap of fixed boundary problems.This research has important theoretical and practical significance,which has attracted many researchers' attention.Herein,we study the reaction-diffusion cooperative systems and free boundary problem with nonlinearities under a high dimensional radical symmetry environment,which is based on the mosquito-borne infectious disease model.The main results are as follows:Firstly,the original free boundary problem is transformed into initial boundary value problem on a fixed region by functional transformation.Furthermore,the existence and uniqueness of local solution is evidenced by the above results combining with contraction mapping theorem,standard LP theory of initial boundary problem of parabolic equation,Sobolev imbedding theorem and Schauder estimates.And then the global solution is prolonged by the local solution.Finally,according to Hopf lemma and Stefan condition,the monotonicity of free boundary is described.Secondly,considering the corresponding eigenvalue problem of the equations,we get that the dependence of its principal eigenvalues about the region.The spreadingvanishing dichotomy of the system is discussed,and then the sufficient conditions for spreading and vanishing of infectious diseases are given by constructing suitable upper and lower solutions and combining comparison principle.Thirdly,the existence of semi-wave solutions associated to this model are established by constructing upper and lower solutions,and the uniqueness is studied by the sliding method techniques.In addition,in the case of successful spreading,the precise asymptotic spreading speed is determined by using the suitable semiwave solution to construct upper and lower solutions and combining the comparison principle.Finally,the influences of the initial region,transmission capability on the spreading and vanishing with the certain initial density are investigated,which obtained from the numerical simulation.To further verify the criteria for the occurrence of spreading and vanishing situations,the asymptotic spreading speed of free boundary is simulated.Our simulation is consistent with the theoretical results.