Bulletin of the Australian Mathematical Society

Research Article

ON THE STABILITY OF STATIONARY LINE AND GRIM REAPER IN PLANAR CURVATURE FLOW

XIAOLIU WANGa1 c1 and WEIFENG WOa2

a1 Department of Mathematics, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China (email: xlwang.seu@gmail.com)

a2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong (email: weifengwo@gmail.com)

Abstract

The asymptotic stability of two types of invariant solutions under a curvature flow in the whole plane is studied. First, by extending the work of others, we prove that the stationary line with nonzero slope will attract the graphical curves which surround it. Then a similar property is obtained for the grim reaper.

(Received June 07 2009)

(Online publication February 07 2011)

2010 Mathematics subject classification

  • primary 35K55; secondary 35B35;
  • 35K15;
  • 53C44

Keywords and phrases

  • curvature flow;
  • asymptotic behavior;
  • stability

Correspondence:

c1 For correspondence; e-mail: xlwang.seu@gmail.com

Footnotes

This work is supported by the Grant for New PhD Teacher Program of Southeast University.