Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On global existence for the Gierer-Meinhardt system

Pages: 583 - 591, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.583

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Henghui Zou - Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, United States (email)

Abstract: 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:  Global existence, Gierer-Meinhardt system, reaction-diffusion system.
Mathematics Subject Classification:  Primary: 35K51, 35K57, 35K58.

Received: December 2013;      Revised: May 2014;      Available Online: August 2014.