A counterexample to finite time stopping property for one-harmonic map flow

Yoshikazu Giga; Hirotoshi Kuroda

doi:10.3934/cpaa.2015.14.121

Article Contents

2015, Volume 14, Issue 1: 121-125. Doi: 10.3934/cpaa.2015.14.121

This issue Previous Article Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation Next Article Phragmén--Lindelöf theorem for infinity harmonic functions

A counterexample to finite time stopping property for one-harmonic map flow

Yoshikazu Giga^1, and
Hirotoshi Kuroda^2,

1.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914
2.
Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531

Received: January 31, 2014

Revised: March 31, 2014

Published: December 2014

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Abstract

For a very strong diffusion equation like total variation flow it is often observed that the solution stops at a steady state in a finite time. This phenomenon is called a finite time stopping or a finite time extinction if the steady state is zero. Such a phenomenon is also observed in one-harmonic map flow from an interval to a unit circle when initial data is piecewise constant. However, if the target manifold is a unit two-dimensional sphere, the finite time stopping may not occur. An explicit example is given in this paper.

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Mathematics Subject Classification: Primary: 35K67; Secondary: 35K51, 35K92, 58E20.

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