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This work addresses the regularity of solutions for a nonlocal diffusionequation over the space of periodic distributions. The spatial operator for thenonlocal diffusion equation is given by a nonlocal Laplace operator with acompactly supported integral kernel. We follow a unified approach based on theFourier multipliers of the nonlocal Laplace operator, which allows the study ofregular as well as distributional solutions of the nonlocal diffusion equation,integrable as well as singular kernels, in any spatial dimension. In addition,the results extend beyond operators with singular kernels to nonlocalsuper-diffusion operators. We present results on the spatial and temporalregularity of solutions in terms of regularity of the initial data or thediffusion source term. Moreover, solutions of the nonlocal diffusion equationare shown to converge to the solution of the classical diffusion equation fortwo types of limits: as the spatial nonlocality vanishes or as the singularityof the integral kernel approaches a certain critical singularity that dependson the spatial dimension. Furthermore, we show that, for the case of integrablekernels, discontinuities in the initial data propagate and persist in thesolution of the nonlocal diffusion equation. The magnitude of a jumpdiscontinuity is shown to decay overtime.

Regularity of Solutions for the Nonlocal Diffusion Equation on Periodic Distributions