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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

 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

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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
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