Vanishing diffusion limits for planar fronts in bistable models with saturation

• Published in: Transactions of the American Mathematical Society 2021-06, Vol.374 (6), p.3999-4021
• Main Author: Garrione, Maurizio
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/8348

We deal with planar fronts for parameter-dependent reaction-diffusion equations with bistable reaction and saturating diffusive term like \displaystyle u_t=\varepsilon \, \mathrm {div}\, \left (\frac {\nabla u}{\sqrt ... ...}}\right ) + f(u), \quad u=u(x, t), \; x \in \mathbb{R}^n, \, t \in \mathbb{R}, 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.