Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

A non-local bistable reaction-diffusion equation with a gap

Pages: 685 - 723, Volume 37, Issue 2, February 2017      doi:10.3934/dcds.2017029

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Henri Berestycki - École des hautes études en sciences sociales, PSL Research University and CNRS, CAMS, 190-198 avenue de France, F-75244 Paris Cedex 13, France (email)
Nancy Rodríguez - UNC Chapel Hill, Department of Mathematics, Phillips Hall, CB#3250, Chapel Hill, NC 27599-3250, United States (email)

Abstract: Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as $t\rightarrow -\infty$, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

Keywords:  Entire solution, gap problem, non-local diffusion, comparison principle, propagation.
Mathematics Subject Classification:  35B08, 35B50, 35K57, 35R09.

Received: August 2015;      Revised: November 2016;      Available Online: November 2016.