We consider the asymptotic derivation of a parabolic equation with nonlocal diffusivity and with a semi-linear nonlocal term that may also be discontinuous. From a reaction-diffusion system where the diffusivity of the second equation is arbitrarily large, using only energy estimates, we obtain a shadow system as an intermediate step for the limit equation. In particular, we obtain the existence of weak solutions and we give a rigorous derivation of a class of diffusion equations that have been used in the literature to model threshold phenomena, for instance, in porous-medium combustion or in localized patterns of excitable media.
"On a nonlocal diffusion equation with discontinuous reaction." Adv. Differential Equations 5 (4-6) 657 - 680, 2000. https://doi.org/10.57262/ade/1356651343