Bifurcation structure of steady-states for bistable equations with nonlocal constraint

Pages: 467 - 476, Issue special, November 2013

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Kousuke Kuto - Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan (email)
Tohru Tsujikawa - Department of Applied Physics, University of Miyazaki, Miyazaki, 889-2192, Japan (email)

Abstract: This paper studies the 1D Neumann problem of bistable equations with nonlocal constraint. We obtain the global bifurcation structure of solutions by a level set analysis for the associate integral mapping. This structure implies that solutions can form a saddle-node bifurcation curve connecting boundary-layer states with internal-layer states. Furthermore, we exhibit the applications of our result to a couple of shadow systems arising in surface chemistry and physiology.

Keywords:  Allen-Cahn equation, nonlocal constraint, saddle-node bifurcation, level set, shadow system.
Mathematics Subject Classification:  Primary: 34B18; Secondary: 34C23, 34E20, 37G10.

Received: September 2012;      Revised: April 2013;      Published: November 2013.