# American Institute of Mathematical Sciences

July  2013, 33(7): 2885-2900. doi: 10.3934/dcds.2013.33.2885

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

 1 Department of Applied Mathematics, University Crete, P.O. Box 2208, 71409, Heraklion, Crete 2 Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531 3 Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku 169-8555, Tokyo

Received  April 2012 Revised  September 2012 Published  January 2013

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.
Citation: Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885
##### References:
 [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. Yamada, Transient 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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##### References:
 [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. Yamada, Transient 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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