Traveling Wave Solutions And Entire Solutions For Nonlocal Dispersal Systems

In recent decades,nonlocal diffusion systems have received widespread atten-tion from scholars.The traveling wave solution of nonlocal dispersal equations is an important topic,since it can explain the propagation with finite speed and os-cillatory phenomena of nature.However,the traveling wave solution is not enough to understand the global dynamic behavior of nonlocal diffusion systems,while the entire solution can.Therefore,the entire solution has been studied extensively.Firstly,we deal with the existence of entire solutions of degenerate Fisher equa-tion with nonlocal dispersal.Based on the traveling wave solutions with the minimal wave speed c=c*(c*>0),the existence of entire solutions is proved by combining super-and sub-solutions method with the comparison principle.Secondly,we study the existence of entire solutions of a Lotka-Volterra weak competition system with nonlocal dispersal.By using the comparison principle and super-and sub-solutions method,we obtain the existence of entire solutions.Moreover,some properties of entire solutions are obtained.Finally,we investigate the traveling wave solutions of Lotak-Volterra weak com-petition systems with nonlocal dispersals and nonlocal nonlinear terms.The exis-tence of traveling wave solutions with the wave speed c>c*and connecting equi-librium(0,0)to some unknown positive steady state is proved by constructing an invariant cone in a bounded domain,and applying Schauder's fixed point theorem as well as a limiting argument,where c*is some positive constant.Moreover,the nonexistence of traveling wave solutions with the speed c<c*is obtained.Further-more,when the nonlocal delays only appear in the inter-specific competition terms,the existence and nonexistence of traveling wave solutions connecting equilibrium(0,0)to positive equilibrium are established.