Nontrivial large-time behaviour in bistable reaction–diffusion equations

Abstract

Bistable reaction-diffusion equations are known to admit one-dimensional travelling waves which are globally stable to one-dimensional perturbations-Fife and McLeod [7]. These planar waves are also stable to two-dimensional perturbations-Xin [30], Levermore-Xin [19], Kapitula [16]-provided that these perturbations decay, in the direction transverse to the wave, in an integrable fashion. In this paper, we first prove that this result breaks down when the integrability condition is removed, and we exhibit a large-time dynamics similar to that of the heat equation. We then apply this result to the study of the large-time behaviour of conical-shaped fronts in the plane, and exhibit cases where the dynamics is given by that of two advection-diffusion equations.

Dynamique en grand temps pour une classe d'équations de type KPP en milieu périodique

“…En particulier, que se passe-t-il quand la donnée initiale est encadrée, au voisinage de −∞, par deux ondes ? Une motivation de cette question est un travail récent [11], dans lequel la dynamique issue d'une classe générale de données initiales est décrite pour une équation de type bistable en dimension deux d'espace : si la donnée initiale est comprise entre deux translatées d'une onde progressive, le profil de la solution converge localement vers celui de l'onde, mais en diffère d'un shift qui peut ne converger vers aucune constante. Nous allons voir qu'un comportement analogue est vrai pour KPP en dimension 1 d'espace.…”

Localized Solutions of a Semilinear Parabolic Equation with a Recurrent Nonstationary Asymptotics

“…It is also worthwhile to mention that the results of [30,31,32] concerning localized solutions have some parallels in theorems concerning wavelike solutions. For example, solutions whose ω-limit sets in a moving coordinate frame are formed by families of traveling fronts can be found in [18,40,35] and references therein. A Liouville-type theorem for entire solutions squeezed between two traveling waves was proved in [17].…”

Propagating speeds of bistable transition fronts in spatially periodic media

“…Fronts with non-axisymmetric shapes, such as pyramidal fronts, are also known to exist, see [34,36]. For qualitative properties of these traveling fronts, we refer to [15,16,17,25,26,29,35,36].…”