| In recent years,the free boundary problem of the reaction-diffusion equation has been widely used in ecology and infectious diseases,and has been paid continuous attention by many scholars.On the basis of the above,this paper consider a class of multistable reaction-diffusion equation of the free boundary problem,where the equation is ut=uxx+f(u),x∈(0,h(t),free boundary h(t)represents the expansion frontier of the species and satisfies the Stefan condition h’(t)=-μux(t,h(t))-α,f is a class of multistable nonlinear term.This paper focuses on the influence of α,a parameter representing environmental resistance,on the asymptotic behavior of solutions to this problem.In this paper,we first prove the existence,uniqueness and regularity of bounded solutions by using contraction mapping principle,Lp theory of parabolic equations,and Schauder estimation.Secondly,five types of asymptotic behavior of bounded solutions is discussed by using zero argument(big spreading,small spreading,vanishing,two kinds of transitions).In particular,some sufficient conditions for spreading or vanishing are given.Finally,an accurate estimation of the asymptotic spreading speed is given by using the comparison principle. |
