Vanishing diffusion limits for planar fronts in bistable models with saturation

TL;DR

This paper studies the behavior of planar fronts in bistable reaction-diffusion models with saturating diffusion as the diffusion parameter approaches zero, revealing convergence to discontinuous solutions.

Contribution

It constructs and analyzes monotone and non-monotone traveling waves in saturating diffusion models, especially their asymptotic limits as diffusion vanishes.

Findings

Construction of monotone and non-monotone traveling waves

Analysis of asymptotic behavior as diffusion parameter tends to zero

Identification of convergence to discontinuous solutions

Abstract

We deal with heteroclinic planar fronts for parameter-dependent reaction-diffusion equations with bistable reaction and saturating diffusive term like $$ u_t=\epsilon \, \textrm{div}\, \left(\frac{\nabla u}{\sqrt{1+\vert \nabla u \vert^2}}\right) + f(u), \quad u=u(x, t), \; x \in \textbf{R}^n, \, t \in \textbf{R}, $$ analyzing in particular their behavior for $\epsilon \to 0$. First, we construct monotone and non-monotone planar traveling waves, using a change of variables allowing to analyze a two-point problem for a suitable first-order reduction; then, we investigate their asymptotic behavior for $\epsilon \to 0$, showing in particular that the convergence of the critical fronts to a suitable step function may occur passing through discontinuous solutions.