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

On global existence for the Gierer-Meinhardt system

Henghui Zou 1,

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.
Keywords: reaction-diffusion system., Global existence, Gierer-Meinhardt system.
Mathematics Subject Classification: Primary: 35K51, 35K57, 35K5.
Citation: Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 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), 1087-1097.

[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.

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