Abstract
We study how reaction-diffusion fronts can be pinned by a defect for a
quadratic (monostable) and cubic (bistable) nonlinearity. In one
dimension, an approximate analysis and numerical simulations show that
bistable fronts can be trapped by a defect; we derive a pinning
criterion. Conversely, a monostable front can never be pinned and this
has important consequences for many applications.
This one dimensional analysis can be extended to describe a 2D bistable
front propagating in a waveguide connected to a cone of angle theta. The
model captures the influence of both geometry and nonlinearity, shows
good agreement with numerical simulations and complements the analysis
of the group of Berestycki. Finally, we extend the analysis to more
complex geometries, including checkerboard-like obstacles, and derive
simple heuristic rules governing front propagation.
References
[1] J.-G. Caputo , G. Cruz-Pacheco , J. Gatlik and Benoit Sarels,
Blocking of 2D bistable reaction-diffusion fronts by obstacles,
arXiv
[2] J.-G. Caputo, Gustavo Cruz-Pacheco and Benoit Sarels
Stopping a reaction-diffusion front, Physical Review E 00, 002200 (2021)
[3] H. Berestycki, J. Bouhours and G. Chapuisat, "Front blocking and
propagation in cylinders with varying cross section", Calculus of
Variations and Partial Differential Equations, Springer, (2016).