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September  2016, 36(9): 4997-5010. doi: 10.3934/dcds.2016016

Finite-time blowup of solutions to some activator-inhibitor systems

Grzegorz Karch 1, , Kanako Suzuki 2, and  Jacek Zienkiewicz 1,

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

Received  July 2015 Revised  February 2016 Published  May 2016

We study a dynamics of solutions to a system of reaction-diffusion equations modeling a biological pattern formation. This model has activator-inhibitor type nonlinearities and we show that it has solutions blowing up in a finite time. More precisely, in the case of absence of a diffusion of an activator, we show that there are solutions which blow up in a finite time at one point, only. This result holds true for the whole range of nonlinearity exponents in the considered activator-inhibitor system. Next, we consider a range of nonlinearities, where some space-homogeneous solutions blow up in a finite time and we show an analogous result for space-inhomogeneous solutions.
Keywords: Reaction-diffusion equations, blowup of solutions., activator-inhibitor system.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B40, 35K5.
Citation: Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016
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).

[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.

[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).

[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.

[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).

[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.

[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).

[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.

[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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