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On global existence for the Gierer-Meinhardt system

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  • 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.
    Mathematics Subject Classification: Primary: 35K51, 35K57, 35K58.

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  • [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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