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

Pages: 1344 - 1350, Issue Special, September 2011

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Hideaki Takaichi - Information Media Center, Hyogo University, Hiraoka, Kakogawa, 675-0195, Japan (email)
Izumi Takagi - Tohoku University, Sendai, 980-8578, Japan (email)
Shoji Yotsutani - Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194, Japan (email)

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.

Keywords:  reaction di usion, stationary, parameter representation
Mathematics Subject Classification:  35J55

Received: July 2010;      Revised: April 2011;      Published: October 2011.