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Global-in-time behavior of the solution to a Gierer-Meinhardt system

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  • Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.
    Mathematics Subject Classification: Primary: 35K57; Secondary: 35Q92.

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  • [1]

    N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225.doi: 10.1016/0022-0396(79)90088-3.

    [2]

    A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.

    [3]

    J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988.

    [4]

    D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Math., 840, Sprinver-Verlag, Berlin, 1981.

    [5]

    H. Hoshino and Y. Yamada, Sovability and smoothing effect for semilinear parabolic equations, Funkcialaj Ekvacioj, 34 (1991), 475-494.

    [6]

    D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62.doi: 10.1016/S0167-2789(00)00206-2.

    [7]

    H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete and Continuous Dynamical Systems, 14 (2006), 737-751.doi: 10.3934/dcds.2006.14.737.

    [8]

    H. Jiang and W.-M. Ni, A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732.doi: 10.1512/iumj.2007.56.2982.

    [9]

    A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanism to complex structure, Rev. Modern Physiscs, 66 (1994), 1481-1510.

    [10]

    E. Latos, T. Suzuki and Y. YamadaTransient and asymptotic dynamics of a prey-predator system with diffusion, to appear in; Math. Meth. Appl. Sci. doi: 10.1002/mma.2524.

    [11]

    F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differential Equations, 247 (2009), 1762-1776.doi: 10.1016/j.jde.2009.04.009.

    [12]

    K. Masuda and T. Takahashi, Reaction-diffusion systems in Gierer-Meinhardt theory in biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58.doi: 10.1007/BF03167754.

    [13]

    J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," third edition, Springer, New York, 2003.

    [14]

    W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Differential Equations, 229 (2006), 426-465.doi: 10.1016/j.jde.2006.03.011.

    [15]

    W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368.doi: 10.2307/2000473.

    [16]

    F. Rothe, "Global Solutions of Reaction-Diffusion Equations," Lecture Notes in Math., 1072, Springer-Verlag, 1984.

    [17]

    A. Turing, The chemical basis of morphogenesis, Philos. Transl. Roy. Soc. London, B237 (1952), 37-72.

    [18]

    J. Wei, Existence and stability of spikes for the Gierer-Meinhardt sytem, Handbook of Differential Equations, Stationary Partial Differential Equations, 5 (ed. M. Chipot), Elsevier, Amsterdam, 2008.doi: 10.1016/S1874-5733(08)80013-7.

    [19]

    E. Yanagida, Reaction-diffusion systems with skew-gradient structure, Meth. Appl. Anal., 8 (2001), 209-226.

    [20]

    E. Yanagida, Mini-maximizers for reaction-diffusion systems with skew-gradient structure, J. Differential Equations, 179 (2002), 311-335.doi: 10.1006/jdeq.2001.4028.

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