Front blocking and propagation in cylinders with varying cross section

Abstract

In this paper we consider a bistable reaction-diffusion equation in unbounded domains and we investigate the existence of propagation phenomena, possibly partial, in some direction or, on the contrary, of blocking phenomena. We start by proving the well-posedness of the problem. Then we prove that when the domain has a decreasing cross section with respect to the direction of propagation there is complete propagation. Further, we prove that the wave can be blocked as it comes to an abrupt geometry change. Finally we discuss various general geometrical properties that ensure either partial or complete invasion by 1. In particular, we show that in a domain that is "starshaped" with respect to an axis, there is complete invasion by 1.

On the mean speed of bistable transition fronts in unbounded domains

“…We call such a solution a front-like solution. In fact, a similar result holds if Ω is bent, that is, the "left" and "right" parts of Ω may not be parallel to the same direction, see [3]. Moreover, it was shown in [3] that, under some geometrical conditions on Ω (for instance, if the map x 1 → ω(x 1 ) is non-increasing), the propagation is complete, in the sense that the solution u of (1.1) satisfying (1.16) satisfies (1.11) too.…”

“…In fact, a similar result holds if Ω is bent, that is, the "left" and "right" parts of Ω may not be parallel to the same direction, see [3]. Moreover, it was shown in [3] that, under some geometrical conditions on Ω (for instance, if the map x 1 → ω(x 1 ) is non-increasing), the propagation is complete, in the sense that the solution u of (1.1) satisfying (1.16) satisfies (1.11) too. However, under some other geometrical conditions (for instance, if x 1 → ω(x 1 ) is non-decreasing and if ω(0) is contained in a small ball and ω(1) contains a large ball), blocking phenomena may occur, that is, the solution u of (1.1) satisfying (1.16) is such that…”

“…The condition of complete propagation of all these front-like solutions u i is actually necessary for the conclusion to hold in general. Indeed, as already mentioned [3], in some bilaterally straight cylinders of the type (1.15), there are solutions u satisfying (1.16) and (1.17), which then do not satisfy (1.11) and yet are transition fronts connecting 0 and 1 without global mean speed. In the general case of a domain Ω with m (≥ 2) cylindrical branches, consider now, if any, a solution u i satisfying (1.24) but not (1.11).…”

“…where e is the direction of propagation for very negative times, see [3,6,10,44]. We will come back to these solutions later.…”

Front blocking in the presence of gradient drift

“…The main observation is that (1.1) becomes variational if the drift term is encoded in some weight. With this 'trick' we are able to use ideas from [1], where the authors show that a neck can be introduced into a given tube in such a way that propagation gets blocked by constructing a stationary supersolution that vanishes behind the neck.…”