  We study the asymptotic behavior for nonlocal diffusion equations
$\partial_tu=\mathcal{J}u-\chi_0u$ in $\mathbb{R}^n\times(0,\infty)$ and obtain
a sufficient condition so that solutions of the Cauchy problem decay in time at
the rate of a regular varying function. In the sufficient condition, a sharp
bound of certain forms is required for the $k$-fold iterations
$\mathcal{J}^ku_0$ or the kernels $J_k$. We prove the desired decay rate by
analyzing the asymptotic behavior of a regular varying modified exponential
series. Then we verify that the sufficient condition is true for most of the
known radially symmetric kernels, and for some more general kernels, using the
sharp Young's convolution inequality and a Fourier splitting argument.
Classical results on the decay of solutions for these nonlocal diffusion
equations are re-established and generalized. Finally, using our framework, we
can exhibit a kernel having a prescribed regular varying decay solutions for a
wide class of regular varying functions.
