Bistable traveling waves around an obstacle

Abstract

We consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity. The goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space R N . As a first step, we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle. This causes disturbance on the shape of the front, but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction, leaving the obstacle behind. Whether the recovery is uniform in space is shown to depend on the shape of the obstacle.

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

“…Spatial transition waves solutions of the homogeneous multidimensional equation with a convex obstacle have been constructed by Berestycki, Hamel and Matano [6]. However, in other frameworks the bistability could produce new steady states which could block the propagation between 0 and 1 [6,8,17,25].…”

“…6 Wave-blocking phenomena and critical travelling waves Several papers [6,8,17,25] observed in various framework that heterogeneous bistable equations might admit non-trivial stationary solutions. In this case the monotonicity and the convergences as t → ±∞ of the critical travelling waves will strongly depend on the normalization of the wave.…”

An Invariance Principle for Random Traveling Waves in One Dimension

“…Berestycki and Hamel [3] have developed a very general framework for such solutions to reaction-diffusion equations in inhomogeneous media. For some other recent work on generalized transition fronts we refer to [4,24,31].…”

Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media

“…Some applications in biology were given by Aronson and Weinberger [1] and Fisher [12]. For the results in periodic excitable media, we refer to [2,4,5,8,14,17,24,35] and for more general classical results on the traveling wave front, we refer to [11,[29][30][31][32]36] and references therein.…”