Nonlocal equations with regular varying decay solutions

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.