The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction
Author:
Sheng-Chen Fu
Journal:
Quart. Appl. Math. 72 (2014), 649-664
MSC (2000):
Primary 35K40; Secondary 34A34, 35Q80, 35K57
DOI:
https://doi.org/10.1090/S0033-569X-2014-01349-5
Published electronically:
October 28, 2014
MathSciNet review:
3291819
Full-text PDF Free Access
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Abstract: We investigate the existence of traveling wave solutions to the one-dimensional reaction-diffusion system $u_t=\delta u_{xx}-2uv/(\beta +u)$, $v_t=v_{xx}+uv/(\beta +u)$, which describes 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. We show that this system has a unique, up to translation, traveling wave solution with speed $c$ iff $c\geq 2/\sqrt {\beta +1}$.
References
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- Sheng-Chen Fu, Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction, Discrete Contin. Dyn. Syst. Ser. B 16 (2011), no. 1, 189–196. MR 2799547, DOI https://doi.org/10.3934/dcdsb.2011.16.189
- Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490
- I. Lengyel, G. Pota and G. Bazsa, Wave profile in the acidic nitrate-ferroin reaction, J. Chem. Soc. Faraday Trans. 87 (1991), 3613–3615.
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- G. Pota, I. Lengyel and G. Bazsa, Traveling waves in the acidic nitrate-iron(II) reaction: analytical description of the wave velocity, J. Phys. Chem. 95 (1991), 4379–4381.
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825
References
- P. C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, vol. 28, Springer-Verlag, Berlin, 1979. MR 527914 (80g:35001)
- H. Berestycki, F. Hamel, A. Kiselev, and L. Ryzhik, Quenching and propagation in KPP reaction-diffusion equations with a heat loss, Arch. Ration. Mech. Anal. 178 (2005), no. 1, 57–80. MR 2188466 (2006h:35135), DOI https://doi.org/10.1007/s00205-005-0367-4
- R. J. Field and M. Burger, Oscillations and traveling waves in chemical systems, Wiley, New York, 1985.
- S.-C. Fu, Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction, Discrete Contin. Dyn. Syst. Ser. B 16 (2011), no. 1, 189–196. MR 2799547 (2012g:35129), DOI https://doi.org/10.3934/dcdsb.2011.16.189
- P. Hartman, Ordinary differential equations, 2nd ed., Birkhäuser Boston, Mass., 1982. MR 658490 (83e:34002)
- I. Lengyel, G. Pota and G. Bazsa, Wave profile in the acidic nitrate-ferroin reaction, J. Chem. Soc. Faraday Trans. 87 (1991), 3613–3615.
- J. H. Merkin and M. A. Sadiq, Reaction-diffusion travelling waves in the acidic nitrate-ferroin reaction, J. Math. Chem. 17 (1995), no. 4, 357–375. MR 1351924 (96f:80009), DOI https://doi.org/10.1007/BF01165755
- J. D. Murray, Mathematical biology. I, 3rd ed., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, 2002. An introduction. MR 1908418 (2004b:92003)
- G. Pota, I. Lengyel and G. Bazsa, Traveling waves in the acidic nitrate-ferroin reaction, J. Chem. Soc. Faraday Trans. 85 (1989), 3871–3877.
- G. Pota, I. Lengyel and G. Bazsa, Traveling waves in the acidic nitrate-iron(II) reaction: analytical description of the wave velocity, J. Phys. Chem. 95 (1991), 4379–4381.
- M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825 (86f:35034)
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Additional Information
Sheng-Chen Fu
Affiliation:
Department of Mathematical Sciences, National Chengchi University, 64, S-2 Zhi-nan Road, Taipei 116, Taiwan
Email:
fu@nccu.edu.tw
Received by editor(s):
August 22, 2012
Published electronically:
October 28, 2014
Additional Notes:
This work was partially supported by the National Science Council of the Republic of China under the contract 100-2115-M-004-003
Article copyright:
© Copyright 2014
Brown University
