Entire Solutions Of Reaction-diffusion Systems In Homogoneous Or Periodic Media

The spatio-temporal propagation of reaction-diffusion equation(system)has attracted worldwide concern and research,since it can well describe and explain many natural phenomena such as invasion of species and spread of disease.The traveling wave and entire solutions are important part of the spatio-temporal propagation theory,and the research of these solutions has more theoretical value and practical significance.It should be pointed out that there are many results on entire solutions originating from two monotone fronts.However,the research results on the entire solutions originating from three or more traveling wave fronts are very limited.In particular,there has been no results on the entire solutions originating from three fronts for reaction-diffusion system in homogeneous media and diffusion problems in periodic media.Therefore,it is very necessary to study these problems.The dissertation contains two parts.In the first part,we mainly study the traveling wave and entire solutions of reaction-diffusion system in homogeneous environments.The second part will be devoted to considering the traveling wave and entire solutions of reaction-diffusion problems in heterogeneous environments.See below for the details.1.We study the entire solutions originating from multiple wave profiles for a nonlocal epidemic model in homogeneous environments.Firstly,we prove the uniqueness,Lyapunov stability and continuous dependence on shift parameters of annihilating-front entire solutions obtained in [81].Secondly,we establish the existence results of two types of new entire solutions originating from three fronts.Finally,the occurrence of the nonlocal dispersal operator and zero diffusion coefficient leads to the lack of sufficient smoothness for these entire solutions,hence we show that they are global Lipschitz continuous with respect to the spatial variable .2.We study the invasion entire solutions for a three species Lotka-Volterra competitiondiffusion system in homogeneous environments.Firstly,we apply the standard asymptotic theory to establish the precise asymptotic behavior of the traveling wave solution at infinity.Then,with the help of a pair of appropriate super-and sub-solutions,we prove the existence of invasion entire solution originating from two fronts by using the comparison principle,and show some qualitative properties of this entire solution.Such entire solution provides another invasion way of the stronger species to the weak ones other than traveling wave solution.3.We study the pulsating type entire solutions originating from multiple fronts for a class of multidimensional reaction-advection-diffusion equation in periodic media.The existence results of entire solutions originating from three traveling fronts for reaction-diffusion problem in homogeneous environments are extended to the space periodic reaction-diffusion equations.Firstly,we construct a pair of appropriate sub-and super-solutions by a priori estimates of the asymptotic behavior of traveling wave front at infinity,and then applying sub-and super-solutions method and the comparison principle,we obtain new entire solutions originating from three pulsating traveling fronts for this bistable equation in periodic media.4.We study the pulsating traveling waves and entire solutions of a lattice dynamical system in periodic media.Firstly,we establish the exponential upper and lower bounds of the monostable pulsating profiles at minus infinity.Then we prove the uniqueness result and derive the exact asymptotic behavior of all non-critical monostable pulsating traveling waves.In contrast to the known result,our uniqueness result does not require any additional assumption on the pulsating traveling waves.Secondly,through a very delicate analysis relating to the auxiliary functions for linking three fronts,we establish two types of new entire solutions originating from three different pulsating traveling waves and show some qualitative properties of these entire solutions.