LONG-TIME ASYMPTOTIC BEHAVIOR OF FISHER-KPP EQUATION FOR NONLOCAL DISPERSAL IN ASYMMETRIC KERNEL

In this paper, we main consider the asymptotic spreading speeds and the long-time asymptotic behavior of a nonlocal with asymmetric kernel diffusion Fisher-KPP equation

$ u_{t}(t, x)=k\ast u(t, x)-u(t, x)+f\left(u(t, x)\right), \; t>0, \; x\in\mathbb{R}. $

On the basis of the spreading speeds $ c_r^*=c(\lambda_r^*) $ and $ c_l^*=c(\lambda_l^*) $, the long-time asymptotic behavior is given by constructing a suitable upper solution and lower solution and using the tool of comparison principle. In particular, the core difficulty and breakthrough point is the lower bounds part. In this regard, we improve the "forward-backward spreading" method which was first proposed by Xu et al. (J Funct Anal 280(2021)108957) to fit the corresponding lower solution so that the asymptotic behavior can be obtained for the initial values that decays within a certain range of asymptotic decay rate $ \lambda_1\in (0, \lambda^+) $ and $ \lambda_2\in (\lambda^-, 0) $.