Vanishing diffusion limits for planar fronts in bistable models with saturation

Abstract

We deal with heteroclinic planar fronts for parameter-dependent reaction-diffusion equations with bistable reaction and saturating diffusive term like $$u_t=\epsilon \text{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u{|}^2}}\right)+f(u),u=u(x,t),x\in {\text{R}}^n,t\in \text{R},$$ 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.