Pointwise estimates on the fundamental solution to nonlocal diffusion equation

In this article we study the pointwise behavior of the fundamental solution S(t,x) to the nonlocal diffusion model ut=D(J∗u−u) on R, S(t,x) can be written as e−Dtδ(x)+S0(t,x), where S0(t,x) is called the regularizing part. Under some conditions on J and its Fourier transform J^, we obtain some new pointwise estimates on S0(t,x). Then we apply our results to study the spatial dynamics of the nonlocal diffusion population model in a shifting environment considered in Li et al. [Spatial dynamics of a nonlocal dispersal population model in a shifting environment. J Nonlinear Sci. 2018;28:1189–1219]. In our study the convolution kernel J can be assumed to be more general, i.e. not necessarily of compact support, also the proof is greatly simplified. It is our believe that these pointwise estimates possess their own interest in further studying nonlocal diffusion models.