Vanishing diffusion limits for planar fronts in bistable models with saturation

Abstract

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