TY - JOUR

T1 - Asymptotic expansions of traveling wave solutions for a quasilinear parabolic equation

AU - Anada, Koichi

AU - Ishiwata, Tetsuya

AU - Ushijima, Takeo

N1 - Funding Information:
This work was supported by KAKENHI No. 21H01001, No. 19H05599 and No. 18K03427.
Publisher Copyright:
© 2022, The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature.

PY - 2022/12

Y1 - 2022/12

N2 - 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.

AB - 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.

KW - Asymptotic expansion

KW - Blow-up phenomena

KW - Curvature flow

KW - Quasilinear parabolic equation

KW - Traveling wave

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U2 - 10.1007/s13160-022-00532-z

DO - 10.1007/s13160-022-00532-z

M3 - Article

AN - SCOPUS:85135605154

SN - 0916-7005

VL - 39

SP - 889

EP - 920

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

IS - 3

ER -