On a nonlocal diffusion equation with discontinuous reaction

Abstract

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.

Article information

Source
Adv. Differential Equations Volume 5, Number 4-6 (2000), 657-680.

Dates
First available in Project Euclid: 27 December 2012

Mathematical Reviews number (MathSciNet)
MR1750114

Zentralblatt MATH identifier
0990.35058

Subjects
Primary: 35R35: Free boundary problems
Secondary: 35K50 35K57: Reaction-diffusion equations

Citation

Hilhorst, Danielle; Rodrigues, José-Francisco. On a nonlocal diffusion equation with discontinuous reaction. Adv. Differential Equations 5 (2000), no. 4-6, 657--680.https://projecteuclid.org/euclid.ade/1356651343