# American Institute of Mathematical Sciences

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

 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.
Citation: Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete and 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. [2] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin, 1979. [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. [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. [5] S.-C. Fu, The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction,, Quarterly Appl. Math., (). [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. [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. [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. [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. [10] J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, New York, 2004. [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. [12] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. [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. [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. [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. [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. [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.

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##### References:
 [1] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. [2] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin, 1979. [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. [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. [5] S.-C. Fu, The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction,, Quarterly Appl. Math., (). [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. [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. [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. [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. [10] J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, New York, 2004. [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. [12] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. [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. [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. [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. [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. [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.
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