-
Previous Article
Corrigendum to: Thermodynamic formalism for random countable Markov shifts
- DCDS Home
- This Issue
-
Next Article
Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space
On global existence for the Gierer-Meinhardt system
1. | Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, United States |
  The Gierer-Meinhardt system was introduced in [1] to model activator-inhibitor systems in pattern formation in ecological systems.
References:
[1] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic (Berlin), 12 (1972), 1087-1097. |
[2] |
H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751.
doi: 10.3934/dcds.2006.14.737. |
[3] |
K. Masuda and K. Takashima, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58.
doi: 10.1007/BF03167754. |
[4] |
W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Diff. Equations, 229 (2006), 426-465.
doi: 10.1016/j.jde.2006.03.011. |
[5] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math., 1072, Springer, New York, 1984. |
[6] |
H. Zou, Finte time blow-up and blow-up rates for the Gierer-Meinhardt system, submitted. |
show all references
References:
[1] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic (Berlin), 12 (1972), 1087-1097. |
[2] |
H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751.
doi: 10.3934/dcds.2006.14.737. |
[3] |
K. Masuda and K. Takashima, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58.
doi: 10.1007/BF03167754. |
[4] |
W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Diff. Equations, 229 (2006), 426-465.
doi: 10.1016/j.jde.2006.03.011. |
[5] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math., 1072, Springer, New York, 1984. |
[6] |
H. Zou, Finte time blow-up and blow-up rates for the Gierer-Meinhardt system, submitted. |
[1] |
Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363 |
[2] |
Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks and Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291 |
[3] |
Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885 |
[4] |
Manuel del Pino, Patricio Felmer, Michal Kowalczyk. Boundary spikes in the Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437 |
[5] |
Jan-Phillip Bäcker, Matthias Röger. Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1139-1155. doi: 10.3934/cpaa.2022013 |
[6] |
Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192 |
[7] |
Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115 |
[8] |
Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651 |
[9] |
Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443 |
[10] |
Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure and Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 |
[11] |
Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 |
[12] |
Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805 |
[13] |
Mengxin Chen, Ranchao Wu, Yancong Xu. Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2275-2312. doi: 10.3934/dcdsb.2021132 |
[14] |
Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817 |
[15] |
Hua Nie, Sze-Bi Hsu, Feng-Bin Wang. Global dynamics of a reaction-diffusion system with intraguild predation and internal storage. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 877-901. doi: 10.3934/dcdsb.2019194 |
[16] |
Jia-Cheng Zhao, Zhong-Xin Ma. Global attractor for a partly dissipative reaction-diffusion system with discontinuous nonlinearity. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022103 |
[17] |
Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083 |
[18] |
Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033 |
[19] |
Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631 |
[20] |
Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022103 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]