A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion

Tetsuya Ishiwata; Shigetoshi Yazaki

doi:10.3934/dcds.2014.34.2069

Article Contents

2014, Volume 34, Issue 5: 2069-2090. Doi: 10.3934/dcds.2014.34.2069

This issue Previous Article The Fourier restriction norm method for the Zakharov-Kuznetsov equation Next Article On Hamiltonian flows whose orbits are straight lines

A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion

Tetsuya Ishiwata^1, and
Shigetoshi Yazaki^2,

1.
Mathematical Sciences, College of Systems Engineering and Science, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama 337-8570
2.
Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kanagawa 214-8571

Received: January 2013

Revised: July 2013

Published: May 2014

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Abstract

The asymptotic behavior of solutions to an anisotropic crystalline motion is investigated. In this motion, a solution polygon changes the shape by a power of crystalline curvature in its normal direction and develops singularity in a finite time. At the final time, two types of singularity appear: one is a single point-extinction and the other is degenerate pinching. We will discuss the latter case of singularity and show the exact blow-up rate for a fast blow-up or a type II blow-up solution which arises in an equivalent blow-up problem.

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Mathematics Subject Classification: Primary: 34A26, 34A34; Secondary: 53A04.

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