Transition Fronts Of Discrete Fisher-KPP Equations In Time Varying Environments

In recent years,more and more attention has been paid to the study of reaction-diffusion equations,especially dynamics,which plays a significant role in analyzing a number of mathematical models in physics,chemistry and biology.One of the important issues is to analyze the long-term behavior of solutions of reaction-diffusion equations,and traveling wave solutions are the most typical,which reflect that the solutions have the property of fluctuation and can be usually characterized as solutions invariant with respect to translation in space.In 1937,Fisher and Kolmogorov,Petrovsky,Piskunov first proposed the concept of traveling wave solutions of the reaction-diffusion equations in the study of gene propagation models.Since then,traveling wave solutions and their properties for various reaction-diffusion equations have been studied extensively.The present paper is devoted to the study of transition fronts of spatially discrete Fisher-KPP equations in time varying environments First,we construct the appropriate global-in-time super-solutions and sub-solutions.Secondly,we construct a sequence of approximating solutions of the equation,and the sequence is monotone.Finally,based on the above conclusions and the convergence of approximating solutions,we can obtain the existence of traveling wave solutions.And we point that these transition fronts have exact decaying rates as the space variable tends to infinity in the given direction.