Synchronous and Asynchronous Solutions for Some Nonlocal Dispersal Equations

This paper is concerned with the nonlocal dispersal equation with synchronous and asynchronous kernel functions. We study the asymptotic limiting behavior of solution when the support set of kernel function is small. Our interesting result is that the synchronous kernel function always makes the diffusion occurs in the whole domain, whereas the asynchronous kernel function can lead to the diffusion only occurs in the low-dimensional spatial domains, depending on the asynchronism of nonlocal dispersal. Moreover, we find that the solution may exhibit a quenching phenomenon for diffusion. In this situation, it is shown that the solution converges to the solution of the ODE without dispersal or it vanishes in the whole domain.