Global bifurcation structure on a shadow system with a source term - Representation of all solutions-

Hideaki Takaichi; Izumi Takagi; Shoji Yotsutani

doi:10.3934/proc.2011.2011.1344

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2011, Volume 2011: 1344-1350. Doi: 10.3934/proc.2011.2011.1344 open access

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Global bifurcation structure on a shadow system with a source term - Representation of all solutions-

1.
Information Media Center, Hyogo University, Hiraoka, Kakogawa, 675-0195
2.
Tohoku University, Sendai, 980-8578
3.
Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194

Received: July 2010

Revised: April 2011

Published: August 2017

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Abstract

As the first step to understand the Gierer-Meinhardt system with source term, it is important to know the global bifurcation diagram of a shadow system. For the case without source term, it is well-understood. However, for the case with source term, the shadow system has a nonlocal term. Thus standard methods do not work, and there are a few partial results even for one-dimensional case. We give explicit representations of all solutions in terms of elliptic functions. They play crucial roles to clarify the global bifurcation diagram.

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Mathematics Subject Classification: 35J55.

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