Coupled reaction–diffusion equations on adjacent domains

We consider a reaction–diffusion system for two unknowns defined in adjacent domains of ℝN. We treat two spatial configurations: a cylinder and its complement, and two half-spaces. Each one of these unknowns is governed by a specific reaction–diffusion equation of Fisher–KPP type. The two quantities interact by an exchange through the separating boundary. We study the long-time behavior of this system. We first derive or characterize the existence and uniqueness of a positive steady state and show that positive solutions converge to it (when it exists). Then, we establish the existence of an asymptotic speed of propagation in the directions along the interface separating the domains. Moreover, we study the qualitative dependence of this speed with respect to various parameters of the model. In the case N = 2, we compare our results with those obtained in [H. Berestycki, J.-M. Roquejoffre and L. Rossi, The influence of a line with fast diffusion in Fisher–KPP propagation, J. Math. Biol. 66 (2013) 743–766; H. Berestycki, J.-M. Roquejoffre and L. Rossi, Fisher–KPP propagation in the presence of a line: Further effects, Nonlinearity 29 (2013) 2623–2640; H. Berestycki, J.-M. Roquejoffre and L. Rossi, The shape of expansion induced by a line with fast diffusion in Fisher–KPP equations, Commun. Math. Phys. 343 (2016) 207–232; H. Berestycki, J.-M. Roquejoffre and L. Rossi, Travelling waves, spreading and extinction for Fisher-KPP propagation driven by a line with fast diffusion, Nonlinear Anal. 137 (2016) 171–189] for a model with a line representing a road of fast diffusion at the boundary of a half-plane. That case can be viewed as a singular limit of the problem studied here.