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

中文翻译：

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饱和双稳态模型中平面前沿的消失扩散极限

我们处理具有双稳态反应和饱和扩散项的参数相关反应扩散方程的异斜平面前沿，如 $$ u_t=\epsilon \, \textrm{div}\, \left(\frac{\nabla u}{\sqrt {1+\vert \nabla u \vert^2}}\right) + f(u), \quad u=u(x, t), \; x \in \textbf{R}^n, \, t \in \textbf{R}, $$ 特别分析他们在 $\epsilon \to 0$ 时的行为。首先，我们构造单调和非单调平面行波，使用变量的变化来分析两点问题以获得合适的一阶归约；然后，我们研究了它们在 $\epsilon \to 0$ 的渐近行为，特别表明临界前沿收敛到合适的阶跃函数可能会通过不连续的解来发生。