Asymptotic expansions of traveling wave solutions for a quasilinear parabolic equation
Asymptotic expansions of traveling wave solutions for a quasilinear parabolic equation
复制标题
拟线性抛物型方程行波解的渐近展开式
DOI：
10.1007/s13160-022-00532-z
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发表时间：
2022
影响因子：
0.9
通讯作者：
Ushijima Takeo
中科院分区：
文献类型：
--
作者：
Anada Koichi;Ishiwata Tetsuya;Ushijima Takeo
本文详细研究了一类拟线性抛物方程的慢行波解。在过去的三十年里，人们深入地研究了平面曲线以其正指数曲率的幂运动。对于这一运动，几位作者研究了具有自交点的平面曲线的尖点奇点的曲率爆破现象。在他们的分析中，特别是在估计爆破率时，慢行波解起着非常重要的作用。为了阐明爆破现象，我们得到了慢行波解关于参数的渐近展开式，它与曲线的曲率的最大值成正比，直到无穷远。我们发现，结果不连续地依赖于参数。这表明爆破现象也可能随着参数的不同而发生很大的变化。
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 exponenthas 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, which is proportional to the maximum of the curvature of the curve, asgoes to infinity. We discovered that the result depends discontinuously on the parameter. It suggests that the blow-up phenomenon may also drastically change according to parameter.