• 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
January  2015, 35(1): 583-591. doi: 10.3934/dcds.2015.35.583

On global existence for the Gierer-Meinhardt system

1. 

Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, United States

Received  December 2013 Revised  May 2014 Published  August 2014

We consider the Gierer-Meinhardt system (1.1), shown below, on a bounded smooth domain $\Omega\subset\mathbb{R}^n$ ($n\ge1$) with a homogeneous Neumann boundary condition. For suitable exponents $a$, $b$, $c$ and $d$, we establish certain sufficient conditions for global existence. Theorem 1.1 here, combined with Theorem 1.2 of [6], implies a classical phenomenon on the effect of the initial data on global existence and finite time blow-up. This work is a continuation of our earlier result [6] for the Gierer-Meinhardt system.
    The Gierer-Meinhardt system was introduced in [1] to model activator-inhibitor systems in pattern formation in ecological systems.
Citation: Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583
References:
[1]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic (Berlin), 12 (1972), 1087.   Google Scholar

[2]

H. Jiang, Global existence of solutions of an activator-inhibitor system,, Discrete Contin. Dyn. Syst., 14 (2006), 737.  doi: 10.3934/dcds.2006.14.737.  Google Scholar

[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.  doi: 10.1007/BF03167754.  Google Scholar

[4]

W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system,, J. Diff. Equations, 229 (2006), 426.  doi: 10.1016/j.jde.2006.03.011.  Google Scholar

[5]

F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Lecture Notes in Math., (1072).   Google Scholar

[6]

H. Zou, Finte time blow-up and blow-up rates for the Gierer-Meinhardt system,, submitted., ().   Google Scholar

show all references

References:
[1]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic (Berlin), 12 (1972), 1087.   Google Scholar

[2]

H. Jiang, Global existence of solutions of an activator-inhibitor system,, Discrete Contin. Dyn. Syst., 14 (2006), 737.  doi: 10.3934/dcds.2006.14.737.  Google Scholar

[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.  doi: 10.1007/BF03167754.  Google Scholar

[4]

W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system,, J. Diff. Equations, 229 (2006), 426.  doi: 10.1016/j.jde.2006.03.011.  Google Scholar

[5]

F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Lecture Notes in Math., (1072).   Google Scholar

[6]

H. Zou, Finte time blow-up and blow-up rates for the Gierer-Meinhardt system,, submitted., ().   Google Scholar

[1]

Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437

[5]

Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192

[6]

Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115

[7]

Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651

[8]

Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443

[9]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721

[10]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519

[11]

Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805

[12]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817

[13]

Hua Nie, Sze-Bi Hsu, Feng-Bin Wang. Global dynamics of a reaction-diffusion system with intraguild predation and internal storage. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 877-901. doi: 10.3934/dcdsb.2019194

[14]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083

[15]

Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033

[16]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[17]

Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200

[18]

Congming Li, Eric S. Wright. Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics. Communications on Pure & Applied Analysis, 2002, 1 (1) : 77-84. doi: 10.3934/cpaa.2002.1.77

[19]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[20]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]