Coupled reaction-diffusion equations on adjacent domains

Abstract

We consider a reaction-diffusion system for two densities lying in adjacent domains of R N . We treat two configurations: either a cylinder and its complement, or two halfspaces. Diffusion and reaction heterogeneities for the two densities are considered, and an exchange occurs through the separating boundary.We study the long-time behavior of the solution, and, when it converges to a positive steady state, we prove the existence of an asymptotic speed of propagation in some specific directions. Moreover, we determine how such a speed qualitatively depends with respect to several parameters appearing in the model.In the case N = 2, we compare such properties to those studied in [6-9] for a model with a line representing a road of fast diffusion at the boundary of a half-plane, which can be seen as a singular limit of the problem studied here.

Propagation for KPP bulk-surface systems in a general cylindrical domain

“…Finally, we briefly mention that a variety of other road-field systems have been considered in the literature: with fractional diffusion on the road [3], ignition type nonlinearity [15][16][17], nonlocal exchange terms [25], and a SIR epidemiological model with an additional compartment of infected that travel on a road with fast diffusion [7]. We also mention the forthcoming work [12] where the large time behaviour of solutions of both road-field systems and systems of parabolic equations on adjacent domains is studied.…”