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 \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 $\varepsilon \to 0$. 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 $\varepsilon \to 0$, 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 $\varepsilon$, so that in this case the study of the convergence concerns discontinuous steady states, differently from the linear diffusion case.