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Finite-time blowup of solutions to some activator-inhibitor systems
1. | Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław |
2. | College of Science, Ibaraki University, 2-1-1 Bunkyo, Mito 310-8512, Japan |
References:
[1] |
M. Fila and K. Ninomiya, "Reaction-diffusion'' systems: Blow-up of solutions that arises or vanishes under diffusion,, Uspekhi Mat. Nauk, 60 (2005), 207.
doi: 10.1070/RM2005v060n06ABEH004289. |
[2] |
M. Guedda and M. Kirane, Diffusion terms in systems of reaction diffusion equations can lead to blow up,, J. Math. Anal. Appl., 218 (1998), 325.
doi: 10.1006/jmaa.1997.5757. |
[3] |
H. Jiang, Global existence of solutions of an activator-inhibitor system,, Discrete Contin. Dyn. Syst., 14 (2006), 737.
doi: 10.3934/dcds.2006.14.737. |
[4] |
G. Karali, T. Suzuki and Y. Yamada, Global-in-time behavior of the solution to a Gierer-Meinhardt system,, Discrete Contin. Dyn. Syst., 33 (2013), 2885.
doi: 10.3934/dcds.2013.33.2885. |
[5] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, (1968).
|
[6] |
F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems,, J. Differential Equations, 247 (2009), 1762.
doi: 10.1016/j.jde.2009.04.009. |
[7] |
M. D. Li, S. H. Chen and Y. C. Qin, Boundedness and blow up for the general activator-inhibitor model,, Acta Math. Appl. Sinica (English Ser.), 11 (1995), 59.
doi: 10.1007/BF02012623. |
[8] |
A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in Hydra,, J. Biol. Sys. 199 (2006), 199 (2006), 97.
doi: 10.1142/S0218339003000889. |
[9] |
A. Marciniak-Czochra, G. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis,, J. Math. Pures Appl., 99 (2013), 509.
doi: 10.1016/j.matpur.2012.09.011. |
[10] |
A. Marciniak-Czochra, G. Karch, K. Suzuki and J. Zienkiewicz, Diffusion-driven blowup of nonnegative solutions to reaction-diffusion-ODE systems,, To appear in Differential Integral Equations, (2016). Google Scholar |
[11] |
K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation,, Japan J. Appl. Math., 4 (1987), 47.
doi: 10.1007/BF03167754. |
[12] |
A. Marciniak-Czochra and M. Kimmel, Dynamics of growth and signaling along linear and surface structures in very early tumors,, Comput. Math. Methods Med., 7 (2006), 189.
doi: 10.1080/10273660600969091. |
[13] |
A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: The effects of influx of mutated cells,, Math. Model. Nat. Phenom., 3 (2008), 90.
doi: 10.1051/mmnp:2008043. |
[14] |
H. Meinhardt and A. Gierer, A theory of biological pattern formation,, Kybernetik (Berlin), 85 (1972), 30. Google Scholar |
[15] |
N. Mizoguchi, H. Ninomiya and E. Yanagida, Diffusion-induced blowup in a nonlinear parabolic system,, J. Dynamics and Differential Equations, 10 (1998), 619.
doi: 10.1023/A:1022633226140. |
[16] |
J. Morgan, On a question of blow-up for semilinear parabolic systems,, Differential Integral Equations, 3 (1990), 973.
|
[17] |
W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system,, J. Differential Equations, 229 (2006), 426.
doi: 10.1016/j.jde.2006.03.011. |
[18] |
K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H. M.. Byrne, V. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy,, J. Biol. Dyn., 6 (2012), 54.
doi: 10.1080/17513758.2011.590610. |
[19] |
M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM J. Math. Anal., 28 (1997), 259.
doi: 10.1137/S0036141095295437. |
[20] |
M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM Rev., 42 (2000), 93.
doi: 10.1137/S0036144599359735. |
[21] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).
|
[22] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Lecture Notes in Mathematics, (1072).
|
[23] |
K. Suzuki, Existence and Behavior of Solutions to a Reaction-Diffusion System Modeling Morphogenesis,, PhD thesis, (2006). Google Scholar |
[24] |
K. Suzuki and I. Takagi, On the role of basic production terms in an activator-inhibitor system modeling biological pattern formation,, Funkcial. Ekvac., 54 (2011), 237.
doi: 10.1619/fesi.54.237. |
[25] |
A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Roy. Soc. B, 237 (1952), 37. Google Scholar |
[26] |
H. Zou, Global existence for Gierer-Meinhardt system,, Discrete Contin. Dyn. Syst., 35 (2015), 583.
doi: 10.3934/dcds.2015.35.583. |
show all references
References:
[1] |
M. Fila and K. Ninomiya, "Reaction-diffusion'' systems: Blow-up of solutions that arises or vanishes under diffusion,, Uspekhi Mat. Nauk, 60 (2005), 207.
doi: 10.1070/RM2005v060n06ABEH004289. |
[2] |
M. Guedda and M. Kirane, Diffusion terms in systems of reaction diffusion equations can lead to blow up,, J. Math. Anal. Appl., 218 (1998), 325.
doi: 10.1006/jmaa.1997.5757. |
[3] |
H. Jiang, Global existence of solutions of an activator-inhibitor system,, Discrete Contin. Dyn. Syst., 14 (2006), 737.
doi: 10.3934/dcds.2006.14.737. |
[4] |
G. Karali, T. Suzuki and Y. Yamada, Global-in-time behavior of the solution to a Gierer-Meinhardt system,, Discrete Contin. Dyn. Syst., 33 (2013), 2885.
doi: 10.3934/dcds.2013.33.2885. |
[5] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, (1968).
|
[6] |
F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems,, J. Differential Equations, 247 (2009), 1762.
doi: 10.1016/j.jde.2009.04.009. |
[7] |
M. D. Li, S. H. Chen and Y. C. Qin, Boundedness and blow up for the general activator-inhibitor model,, Acta Math. Appl. Sinica (English Ser.), 11 (1995), 59.
doi: 10.1007/BF02012623. |
[8] |
A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in Hydra,, J. Biol. Sys. 199 (2006), 199 (2006), 97.
doi: 10.1142/S0218339003000889. |
[9] |
A. Marciniak-Czochra, G. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis,, J. Math. Pures Appl., 99 (2013), 509.
doi: 10.1016/j.matpur.2012.09.011. |
[10] |
A. Marciniak-Czochra, G. Karch, K. Suzuki and J. Zienkiewicz, Diffusion-driven blowup of nonnegative solutions to reaction-diffusion-ODE systems,, To appear in Differential Integral Equations, (2016). Google Scholar |
[11] |
K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation,, Japan J. Appl. Math., 4 (1987), 47.
doi: 10.1007/BF03167754. |
[12] |
A. Marciniak-Czochra and M. Kimmel, Dynamics of growth and signaling along linear and surface structures in very early tumors,, Comput. Math. Methods Med., 7 (2006), 189.
doi: 10.1080/10273660600969091. |
[13] |
A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: The effects of influx of mutated cells,, Math. Model. Nat. Phenom., 3 (2008), 90.
doi: 10.1051/mmnp:2008043. |
[14] |
H. Meinhardt and A. Gierer, A theory of biological pattern formation,, Kybernetik (Berlin), 85 (1972), 30. Google Scholar |
[15] |
N. Mizoguchi, H. Ninomiya and E. Yanagida, Diffusion-induced blowup in a nonlinear parabolic system,, J. Dynamics and Differential Equations, 10 (1998), 619.
doi: 10.1023/A:1022633226140. |
[16] |
J. Morgan, On a question of blow-up for semilinear parabolic systems,, Differential Integral Equations, 3 (1990), 973.
|
[17] |
W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system,, J. Differential Equations, 229 (2006), 426.
doi: 10.1016/j.jde.2006.03.011. |
[18] |
K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H. M.. Byrne, V. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy,, J. Biol. Dyn., 6 (2012), 54.
doi: 10.1080/17513758.2011.590610. |
[19] |
M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM J. Math. Anal., 28 (1997), 259.
doi: 10.1137/S0036141095295437. |
[20] |
M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM Rev., 42 (2000), 93.
doi: 10.1137/S0036144599359735. |
[21] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).
|
[22] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Lecture Notes in Mathematics, (1072).
|
[23] |
K. Suzuki, Existence and Behavior of Solutions to a Reaction-Diffusion System Modeling Morphogenesis,, PhD thesis, (2006). Google Scholar |
[24] |
K. Suzuki and I. Takagi, On the role of basic production terms in an activator-inhibitor system modeling biological pattern formation,, Funkcial. Ekvac., 54 (2011), 237.
doi: 10.1619/fesi.54.237. |
[25] |
A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Roy. Soc. B, 237 (1952), 37. Google Scholar |
[26] |
H. Zou, Global existence for Gierer-Meinhardt system,, Discrete Contin. Dyn. Syst., 35 (2015), 583.
doi: 10.3934/dcds.2015.35.583. |
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