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

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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

Abstract
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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. 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. 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. 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. 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, (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. 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. 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), 199 (2006), 97. 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. 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, (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. 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. 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. doi: 10.1051/mmnp:2008043. Google Scholar
[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. Google Scholar
[16]	J. Morgan, On a question of blow-up for semilinear parabolic systems,, Differential Integral Equations, 3 (1990), 973. 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. 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. 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. 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. 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], (2007). Google Scholar
[22]	F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Lecture Notes in Mathematics, (1072). Google Scholar
[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. Google Scholar
[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. Google Scholar

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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. 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. 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. 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. 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, (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. 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. 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), 199 (2006), 97. 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. 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, (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. 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. 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. doi: 10.1051/mmnp:2008043. Google Scholar
[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. Google Scholar
[16]	J. Morgan, On a question of blow-up for semilinear parabolic systems,, Differential Integral Equations, 3 (1990), 973. 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. 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. 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. 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. 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], (2007). Google Scholar
[22]	F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Lecture Notes in Mathematics, (1072). Google Scholar
[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. Google Scholar
[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. Google Scholar

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