Entire Solutions Of Bistable Reaction-diffusion Systems In R~n

Reaction-diffusion system theory is a hotspot in modern mathematics research,and it is widely used in many fields.For example,the spread of epidemics and the growth of tumors in biology,the phenomenon of heat conduction and the problem of free boundaries in physics,the problem of changes in the burning temperature and concentration of substances in chemistry,and so on,can all be characterized by reactive diffusion systems.This paper mainly studies the existence of the entire solutions of the bistable reaction-diffusion system.Furthermore,some qualitative properties of the entire solutions are studied.The entire solutions which are mentioned here refers to a solution that has a definition for all space and time.The study of the global solution of time,on the one hand,helps us understand the transient dynamics from a mathematical point of view,on the other hand,it helps us determine the structure of the global attractorThis paper adopts a common method to prove the existence of the entire solutions,namely the upper and lower solution method.First,we studied the exact exponential asymptotic behavior of the traveling wave solution,and from this we obtained some important estimates.Next,by introducing auxiliary equations and detailed analysis of its solutions,a pair of upper and lower solutions of the system are constructed using the solutions of auxiliary equations and traveling wave solutions.Finally,we consider a class of initial value problems where the lower solutions are initial values.Combining the principle of comparison,we prove the existence and qualitative nature of the entire solution.When t?-?,the entire solution appears as a traveling wave solution with two columns traveling towards each other.