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.
Categories: math.AP math.DS