Vanishing diffusion limits for planar fronts in bistable models with saturation

We deal with planar fronts for parameter-dependent reaction-diffusion equations with bistable reaction and saturating diffusive term like u_t = ε div(∇u/sqrt{1 + |∇u|^2}) + f(u), u= u(x, t), x ∈ R^n, t ∈ R, analyzing in particular their behavior for ε → 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 of the equation above; then, we investigate the asymptotic behavior of the monotone fronts for ε → 0, showing their convergence to suitable step functions. A remarkable feature of the considered diffusive term is that the fronts connecting 0 and 1 are necessarily discontinuous (and steady, namely with 0- speed) for small ε, so that in this case the study of the convergence concerns discontinuous steady states, differently from the linear diffusion case.