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A short proof of the logarithmic Bramson correction in Fisher-KPP equations
The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system
1. | Graduate School of Advanced Mathematical Science, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan |
References:
[1] |
H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011. |
[2] |
A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507.
doi: 10.1512/iumj.2001.50.1873. |
[3] |
A. Doelman and H. van der Ploeg, Homoclinic stripe patterns, SIAM J. Appl. Dyn. Syst., 1 (2002), 65-104 (electronic).
doi: 10.1137/S1111111101392831. |
[4] |
S. Ei and J. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions, Japan J. Indust. Appl. Math., 19 (2002), 181-226.
doi: 10.1007/BF03167453. |
[5] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[6] |
P. Hartman, "Ordinary Differential Equations," Birkhäuser Boston, Mass., second edition, 1982. |
[7] |
D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[8] |
T. Kolokolnikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Syst., 5 (2006), 313-363 (electronic).
doi: 10.1137/050635080. |
[9] |
S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.
doi: 10.1038/376765a0. |
[10] |
S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620. |
[11] |
P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997), 3601-3610.
doi: 10.1039/a702602a. |
[12] |
H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, 1982. |
[13] |
J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[14] |
A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo, Interactions between zebrafish pigment cells responsible for the generation of Turing patterns, Proceedings of the National Academy of Sciences, 106, (2009), 8429-8434.
doi: 10.1073/pnas.0808622106. |
[15] |
Y. Nakamura, C. D. Tsiairis, S. Özbek and T. W. Holstein, Autoregulatory and repressive inputs localize Hydra Wnt3 to the head organizer, Proceedings of the National Academy of Sciences, 108, (2011), 9137-9142.
doi: 10.1073/pnas.1018109108. |
[16] |
Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit, in "Dynamics Reported (New Series)", 3 (1994), Springer, Berlin, 25-103. |
[17] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. Translated in J. Soviet Math., 45 (1989), 1205-1218.
doi: 10.1137/0518124. |
[18] |
H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems, J. Theoret. Biol., 224 (2003), 339-350.
doi: 10.1016/S0022-5193(03)00170-X. |
[19] |
I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249.
doi: 10.1016/0022-0396(86)90119-1. |
[20] |
M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Adv. Differential Equations, 8 (2003), 29-54. |
[21] |
M. Taniguchi and Y. Nishiura, Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 117-145.
doi: 10.1017/S0308210500030638. |
[22] |
M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems, SIAM J. Math. Anal., 25 (1994), 99-134.
doi: 10.1137/S0036141092233500. |
[23] |
A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[24] |
M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability, European J. Appl. Math., 13 (2002), 283-320.
doi: 10.1017/S0956792501004442. |
[25] |
J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl. (9), 83 (2004), 433-476.
doi: 10.1016/j.matpur.2003.09.006. |
[26] |
J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458.
doi: 10.1007/s00332-001-0380-1. |
show all references
References:
[1] |
H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011. |
[2] |
A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507.
doi: 10.1512/iumj.2001.50.1873. |
[3] |
A. Doelman and H. van der Ploeg, Homoclinic stripe patterns, SIAM J. Appl. Dyn. Syst., 1 (2002), 65-104 (electronic).
doi: 10.1137/S1111111101392831. |
[4] |
S. Ei and J. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions, Japan J. Indust. Appl. Math., 19 (2002), 181-226.
doi: 10.1007/BF03167453. |
[5] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[6] |
P. Hartman, "Ordinary Differential Equations," Birkhäuser Boston, Mass., second edition, 1982. |
[7] |
D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[8] |
T. Kolokolnikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Syst., 5 (2006), 313-363 (electronic).
doi: 10.1137/050635080. |
[9] |
S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.
doi: 10.1038/376765a0. |
[10] |
S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620. |
[11] |
P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997), 3601-3610.
doi: 10.1039/a702602a. |
[12] |
H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, 1982. |
[13] |
J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[14] |
A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo, Interactions between zebrafish pigment cells responsible for the generation of Turing patterns, Proceedings of the National Academy of Sciences, 106, (2009), 8429-8434.
doi: 10.1073/pnas.0808622106. |
[15] |
Y. Nakamura, C. D. Tsiairis, S. Özbek and T. W. Holstein, Autoregulatory and repressive inputs localize Hydra Wnt3 to the head organizer, Proceedings of the National Academy of Sciences, 108, (2011), 9137-9142.
doi: 10.1073/pnas.1018109108. |
[16] |
Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit, in "Dynamics Reported (New Series)", 3 (1994), Springer, Berlin, 25-103. |
[17] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. Translated in J. Soviet Math., 45 (1989), 1205-1218.
doi: 10.1137/0518124. |
[18] |
H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems, J. Theoret. Biol., 224 (2003), 339-350.
doi: 10.1016/S0022-5193(03)00170-X. |
[19] |
I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249.
doi: 10.1016/0022-0396(86)90119-1. |
[20] |
M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Adv. Differential Equations, 8 (2003), 29-54. |
[21] |
M. Taniguchi and Y. Nishiura, Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 117-145.
doi: 10.1017/S0308210500030638. |
[22] |
M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems, SIAM J. Math. Anal., 25 (1994), 99-134.
doi: 10.1137/S0036141092233500. |
[23] |
A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[24] |
M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability, European J. Appl. Math., 13 (2002), 283-320.
doi: 10.1017/S0956792501004442. |
[25] |
J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl. (9), 83 (2004), 433-476.
doi: 10.1016/j.matpur.2003.09.006. |
[26] |
J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458.
doi: 10.1007/s00332-001-0380-1. |
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