American Institute of Mathematical Sciences

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September  2013, 33(9): 4041-4069. doi: 10.3934/dcds.2013.33.4041

Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction

Sheng-Chen Fu 1, and  Je-Chiang Tsai 2,

1. 

Department of Mathematical Sciences, National Chengchi University, 64, S-2 Zhi-nan Road, Taipei 116
2. 

Department of Mathematics & AIM-HI, National Chung Cheng University, Chiayi, Taiwan

Received  July 2012 Revised  December 2012 Published  March 2013

We consider the reaction-diffusion system $u_t=\delta u_{xx}-2uv/(\beta+u)$, $v_t=v_{xx}+uv/(\beta+u)$, which is used to model the acidic nitrate-ferroin reaction. Here $\beta$ is a positive constant, $u$ and $v$ represent the concentrations of the ferroin and acidic nitrate respectively, and $\delta$ denotes the ratio of the diffusion rates. The existence of travelling waves for this system is known. Using energy functionals, we provide a stability analysis of travelling waves.
Keywords: acidic nitrate-ferroin reaction., travelling wave, Stability, reaction-diffusion system.
Mathematics Subject Classification: Primary: 35K57, 35B35; Secondary: 35C0.
Citation: Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4041-4069. doi: 10.3934/dcds.2013.33.4041
References:
[1]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[2]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin, 1979.  Google Scholar

[3]

S. Focant and Th. Gallay, Existence and stability of propagating fronts for an autocatalytic a reaction-diffusion system, Physica D, 120 (1998), 346-368. doi: 10.1016/S0167-2789(98)00096-7.  Google Scholar

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S.-C. Fu, Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction, Discrete Contin. Dyn. Syst., 16 (2011), 189-196. doi: 10.3934/dcdsb.2011.16.189.  Google Scholar

[5]

S.-C. Fu, The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction,, Quarterly Appl. Math., ().   Google Scholar

[6]

I. Lengyel, G. Pota and G. Bazsa, Wave profile in the acidic nitrate-ferroin reaction, J. Chem. Soc. Faraday Trans., 87 (1991), 3613-3615. Google Scholar

[7]

Y. Li and Y. Wu, Stability of travelling waves with noncritical speeds for double degenerate Fisher-type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 149-170. doi: 10.3934/dcdsb.2008.10.149.  Google Scholar

[8]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521. doi: 10.1137/100814974.  Google Scholar

[9]

J. H. Merkin and M. A. Sadiq, Reaction-diffision travelling waves in the acidic nitrate-ferroin reaction, J. Math. Chem., 17 (1995), 357-375. doi: 10.1007/BF01165755.  Google Scholar

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J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, New York, 2004.  Google Scholar

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A. Pazy, "Semigroup of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, Berling, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.  Google Scholar

[13]

G. Pota, I. Lengyel and G. Bazsa, Travelling waves in the acidic nitrate-ferroin reaction, J. Chem. Soc. Faraday Trans., 85 (1989), 3871-3877. Google Scholar

[14]

G. Pota, I. Lengyel and G. Bazsa, Travelling waves in the acidic nitrate-iron(II) reaction: Analytical description of the wave velocity, J. Phys. Chem., 95 (1991), 4379-4381. Google Scholar

[15]

G. Raugel and K. Kirchgässner, Stability of fronts for a KPP-system. II. The critical case, J. Differential Equations, 146 (1998), 399-456. doi: 10.1006/jdeq.1997.3391.  Google Scholar

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Y. Wu, X. Xing and Q. Ye, Stability of travelling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66. doi: 10.3934/dcds.2006.16.47.  Google Scholar

[17]

Y. Wu and X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123-1139. doi: 10.3934/dcds.2008.20.1123.  Google Scholar

show all references

References:
[1]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[2]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin, 1979.  Google Scholar

[3]

S. Focant and Th. Gallay, Existence and stability of propagating fronts for an autocatalytic a reaction-diffusion system, Physica D, 120 (1998), 346-368. doi: 10.1016/S0167-2789(98)00096-7.  Google Scholar

[4]

S.-C. Fu, Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction, Discrete Contin. Dyn. Syst., 16 (2011), 189-196. doi: 10.3934/dcdsb.2011.16.189.  Google Scholar

[5]

S.-C. Fu, The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction,, Quarterly Appl. Math., ().   Google Scholar

[6]

I. Lengyel, G. Pota and G. Bazsa, Wave profile in the acidic nitrate-ferroin reaction, J. Chem. Soc. Faraday Trans., 87 (1991), 3613-3615. Google Scholar

[7]

Y. Li and Y. Wu, Stability of travelling waves with noncritical speeds for double degenerate Fisher-type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 149-170. doi: 10.3934/dcdsb.2008.10.149.  Google Scholar

[8]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521. doi: 10.1137/100814974.  Google Scholar

[9]

J. H. Merkin and M. A. Sadiq, Reaction-diffision travelling waves in the acidic nitrate-ferroin reaction, J. Math. Chem., 17 (1995), 357-375. doi: 10.1007/BF01165755.  Google Scholar

[10]

J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, New York, 2004.  Google Scholar

[11]

A. Pazy, "Semigroup of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, Berling, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.  Google Scholar

[13]

G. Pota, I. Lengyel and G. Bazsa, Travelling waves in the acidic nitrate-ferroin reaction, J. Chem. Soc. Faraday Trans., 85 (1989), 3871-3877. Google Scholar

[14]

G. Pota, I. Lengyel and G. Bazsa, Travelling waves in the acidic nitrate-iron(II) reaction: Analytical description of the wave velocity, J. Phys. Chem., 95 (1991), 4379-4381. Google Scholar

[15]

G. Raugel and K. Kirchgässner, Stability of fronts for a KPP-system. II. The critical case, J. Differential Equations, 146 (1998), 399-456. doi: 10.1006/jdeq.1997.3391.  Google Scholar

[16]

Y. Wu, X. Xing and Q. Ye, Stability of travelling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66. doi: 10.3934/dcds.2006.16.47.  Google Scholar

[17]

Y. Wu and X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123-1139. doi: 10.3934/dcds.2008.20.1123.  Google Scholar

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