Asymptotic expansions of traveling wave solutions for a quasilinear parabolic equation

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 α 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 κ, which is proportional to the maximum of the curvature of the curve, as κ goes to infinity. We discovered that the result depends discontinuously on the parameter δ= 1 + 1 / α. It suggests that the blow-up phenomenon may also drastically change according to parameter δ.