American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 1344-1350. doi: 10.3934/proc.2011.2011.1344

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

Hideaki Takaichi 1, , Izumi Takagi 2, and  Shoji Yotsutani 3,

1. 

Information Media Center, Hyogo University, Hiraoka, Kakogawa, 675-0195, Japan
2. 

Tohoku University, Sendai, 980-8578, Japan
3. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194

Received  July 2010 Revised  April 2011 Published  October 2011

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: parameter representation, reaction di usion, stationary.
Mathematics Subject Classification: 35J55.
Citation: Hideaki Takaichi, Izumi Takagi, Shoji Yotsutani. Global bifurcation structure on a shadow system with a source term - Representation of all solutions-. Conference Publications, 2011, 2011 (Special) : 1344-1350. doi: 10.3934/proc.2011.2011.1344
[1]

Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519
[2]

Paul Deuring, Stanislav Kračmar, Šárka Nečasová. A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 237-253. doi: 10.3934/dcdss.2010.3.237
[3]

Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure and Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189
[4]

L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure and Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9
[5]

Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113
[6]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[7]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
[8]

El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $ L^1 $ revisited. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 455-464. doi: 10.3934/dcdss.2020355
[9]

Feng Qi, Bai-Ni Guo. Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1975-1989. doi: 10.3934/cpaa.2009.8.1975
[10]

Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046
[11]

Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery. Mathematical Biosciences & Engineering, 2005, 2 (4) : 719-741. doi: 10.3934/mbe.2005.2.719
[12]

Hengrui Luo, Alice Patania, Jisu Kim, Mikael Vejdemo-Johansson. Generalized penalty for circular coordinate representation. Foundations of Data Science, 2021, 3 (4) : 729-767. doi: 10.3934/fods.2021024
[13]

Monica Pragliola, Daniela Calvetti, Erkki Somersalo. Overcomplete representation in a hierarchical Bayesian framework. Inverse Problems and Imaging, 2022, 16 (1) : 19-38. doi: 10.3934/ipi.2021039
[14]

Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295
[15]

Gregory Beylkin, Lucas Monzón. Efficient representation and accurate evaluation of oscillatory integrals and functions. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4077-4100. doi: 10.3934/dcds.2016.36.4077
[16]

Tyrus Berry, Timothy Sauer. Consistent manifold representation for topological data analysis. Foundations of Data Science, 2019, 1 (1) : 1-38. doi: 10.3934/fods.2019001
[17]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7
[18]

Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems and Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135
[19]

L. Bakker. A reducible representation of the generalized symmetry group of a quasiperiodic flow. Conference Publications, 2003, 2003 (Special) : 68-77. doi: 10.3934/proc.2003.2003.68
[20]

Wael Bahsoun, Christopher Bose, Anthony Quas. Deterministic representation for position dependent random maps. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 529-540. doi: 10.3934/dcds.2008.22.529

 Impact Factor: 

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy