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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, 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), 207-226. doi: 10.1070/RM2005v060n06ABEH004289.  Google Scholar

[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), 325-327. doi: 10.1006/jmaa.1997.5757.  Google Scholar

[3]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751. doi: 10.3934/dcds.2006.14.737.  Google Scholar

[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), 2885-2900. doi: 10.3934/dcds.2013.33.2885.  Google Scholar

[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, Vol. 23, American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[6]

F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differential Equations, 247 (2009), 1762-1776. doi: 10.1016/j.jde.2009.04.009.  Google Scholar

[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), 59-68. doi: 10.1007/BF02012623.  Google Scholar

[8]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in Hydra, J. Biol. Sys. 199 (2006), 97-119. doi: 10.1142/S0218339003000889.  Google Scholar

[9]

A. Marciniak-Czochra, G. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543. doi: 10.1016/j.matpur.2012.09.011.  Google Scholar

[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, arXiv:1511.02510, (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), 47-58. doi: 10.1007/BF03167754.  Google Scholar

[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), 189-213. doi: 10.1080/10273660600969091.  Google Scholar

[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), 90-114. doi: 10.1051/mmnp:2008043.  Google Scholar

[14]

H. Meinhardt and A. Gierer, A theory of biological pattern formation, Kybernetik (Berlin), 85 (1972), 30-39. 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), 619-638. doi: 10.1023/A:1022633226140.  Google Scholar

[16]

J. Morgan, On a question of blow-up for semilinear parabolic systems, Differential Integral Equations, 3 (1990), 973-978.  Google Scholar

[17]

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.  Google Scholar

[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), 54-71. doi: 10.1080/17513758.2011.590610.  Google Scholar

[19]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269. doi: 10.1137/S0036141095295437.  Google Scholar

[20]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106. doi: 10.1137/S0036144599359735.  Google Scholar

[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], Birkhäuser Verlag, Basel, 2007.  Google Scholar

[22]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer-Verlag, Berlin, 1984.  Google Scholar

[23]

K. Suzuki, Existence and Behavior of Solutions to a Reaction-Diffusion System Modeling Morphogenesis, PhD thesis, Tohoku University, Sendai, Japan, March 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), 237-274. doi: 10.1619/fesi.54.237.  Google Scholar

[25]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72. Google Scholar

[26]

H. Zou, Global existence for Gierer-Meinhardt system, Discrete Contin. Dyn. Syst., 35 (2015), 583-591. doi: 10.3934/dcds.2015.35.583.  Google Scholar

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), 207-226. doi: 10.1070/RM2005v060n06ABEH004289.  Google Scholar

[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), 325-327. doi: 10.1006/jmaa.1997.5757.  Google Scholar

[3]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751. doi: 10.3934/dcds.2006.14.737.  Google Scholar

[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), 2885-2900. doi: 10.3934/dcds.2013.33.2885.  Google Scholar

[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, Vol. 23, American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[6]

F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differential Equations, 247 (2009), 1762-1776. doi: 10.1016/j.jde.2009.04.009.  Google Scholar

[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), 59-68. doi: 10.1007/BF02012623.  Google Scholar

[8]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in Hydra, J. Biol. Sys. 199 (2006), 97-119. doi: 10.1142/S0218339003000889.  Google Scholar

[9]

A. Marciniak-Czochra, G. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543. doi: 10.1016/j.matpur.2012.09.011.  Google Scholar

[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, arXiv:1511.02510, (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), 47-58. doi: 10.1007/BF03167754.  Google Scholar

[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), 189-213. doi: 10.1080/10273660600969091.  Google Scholar

[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), 90-114. doi: 10.1051/mmnp:2008043.  Google Scholar

[14]

H. Meinhardt and A. Gierer, A theory of biological pattern formation, Kybernetik (Berlin), 85 (1972), 30-39. 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), 619-638. doi: 10.1023/A:1022633226140.  Google Scholar

[16]

J. Morgan, On a question of blow-up for semilinear parabolic systems, Differential Integral Equations, 3 (1990), 973-978.  Google Scholar

[17]

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.  Google Scholar

[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), 54-71. doi: 10.1080/17513758.2011.590610.  Google Scholar

[19]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269. doi: 10.1137/S0036141095295437.  Google Scholar

[20]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106. doi: 10.1137/S0036144599359735.  Google Scholar

[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], Birkhäuser Verlag, Basel, 2007.  Google Scholar

[22]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer-Verlag, Berlin, 1984.  Google Scholar

[23]

K. Suzuki, Existence and Behavior of Solutions to a Reaction-Diffusion System Modeling Morphogenesis, PhD thesis, Tohoku University, Sendai, Japan, March 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), 237-274. doi: 10.1619/fesi.54.237.  Google Scholar

[25]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72. Google Scholar

[26]

H. Zou, Global existence for Gierer-Meinhardt system, Discrete Contin. Dyn. Syst., 35 (2015), 583-591. doi: 10.3934/dcds.2015.35.583.  Google Scholar

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