Asymptotic expansions of traveling wave solutions for a quasilinear parabolic equation

Abstract

In this paper, we investigate so-called slowly traveling wave solutions for a quasilinear parabolic equation in detail. Over the past three decades, the motion of the plane curve by the power of its curvature with positive exponent \(\alpha\) has been intensively investigated. For this motion, blow-up phenomena of curvature on cusp singularity in the plane curve with self-crossing points have been studied by several authors. In their analysis, particularly in estimating the blow-up rate, the slowly traveling wave solutions played a significantly important role. In this paper, aiming to clarify the blow-up phenomena, we derive an asymptotic expansion of the slowly traveling wave solutions with respect to the parameter \(\kappa\), which is proportional to the maximum of the curvature of the curve, as \(\kappa\) goes to infinity. We discovered that the result depends discontinuously on the parameter \(\delta = 1+ 1/\alpha\). It suggests that the blow-up phenomenon may also drastically change according to parameter \(\delta\).