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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 |
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. |
show all references
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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