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Existence and Asymptotic Representation of the Autowave Solution of a System of Equations

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Abstract

A singularly perturbed parabolic system of nonlinear reaction–diffusion equations is studied. Systems of this class are used to simulate autowave processes in chemical kinetics, biophysics, and ecology. A detailed algorithm for constructing an asymptotic approximation of a travelling front solution is proposed. In addition, methods for constructing an upper and a lower solution based on the asymptotics are described. According to the method of differential inequalities, the existence of an upper and a lower solution guarantees the existence of a solution to the problem under consideration. These methods can be used for asymptotic analysis of model systems in applications. The results can also be used to develop and justify difference schemes for solving problems with moving fronts.

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References

  1. S. R. Dunbar, “Travelling wave solutions of diffusive Lotka–Volterra equations,” J. Math. Biol. 17 (1), 11–32 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  2. R. FitzHugh, “Impulses and physiological states in theoretical model of nerve membrane,” Biophys. J. 1 (1), 445–456 (1961).

    Article  MathSciNet  Google Scholar 

  3. V. I. Zamitsina, F. I. Alaullakhanov, A. I. Lobanov, and O. L. Morozova, “Dynamics of spatially nonuniform patterning in the model of blood coagulation,” Chaos Interdiscipl. J. Nonlinear Sci. 57 (11), 57–67 (2001).

    Article  Google Scholar 

  4. R. R. Aliev and A. V. Panfilov, “A simple two-variable model of cardiac excitation,” Chaos. Solitons Fractals 7 (3), 293–301 (1983).

    Article  Google Scholar 

  5. J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Springer, Berlin, 2003).

    MATH  Google Scholar 

  6. A. Hagberg and E. Meron, “Pattern formation in non-gradient reaction-diffusion systems: The effects of front bifurcations,” Nonlinearily 7 (3), 805–835 (1994).

    Article  MATH  Google Scholar 

  7. A. E. Sidorova, N. T. Levashova, A. A. Melnikova, and L. V. Yakovenko, “A model of a human dominated urban ecosystem as an active medium,” Biophysics 60 (3), 446–473 (2015).

    Article  Google Scholar 

  8. A. A. Mel’nikova and R. L. Argun, “Asymptotic approximation of the stationary solution with internal layer for FitzHugh–Nagumo system,” Model. Anal. Inf. Sist. 23 (5), 559–567 (2016).

    Article  MathSciNet  Google Scholar 

  9. V. F. Butuzov and N. N. Nefedov, “Asymptotic theory of contrast structures (review),” Autom. Remote Control 58 (7), 1068–1091 (1997).

    MATH  Google Scholar 

  10. C. V. Pao, Nonlinear Parabolic and Elliptic Equations (Plenum, New York, 1992).

    MATH  Google Scholar 

  11. N. N. Nefedov, “The method of differential inequalities for some classes of nonlinear singularly perturbed problems with internal layers,” Differ. Equations 31 (7), 1077–1085 (1995).

    MathSciNet  MATH  Google Scholar 

  12. N. Levashova and A. A. Melnikova, “Step-like contrast structure in a singularly perturbed system of parabolic equations,” Differ. Equations 51 (3), 342–367 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Melnikova, N. Levashova, and D. Lukyanenko, “Front dynamics in an activator-inhibitor system of equations,” Lect. Notes Comput. Sci. 10187, 492–499 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  14. G. I. Shishkin, L. P. Shishkina, and P. W. Hemker, “A class of singularly perturbed convection-diffusion problems with a moving interior layer: An a posteriori adaptive mesh technique,” Comput. Methods Appl. Math. 4 (1), 105–127 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Kopieva and M. Stynes, “Stabilized approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem,” Numer. Math. 119 (4), 787–810 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM Studies Appl. Math., Philadelphia, 1995).

    Book  MATH  Google Scholar 

  17. P. C. Fife and J. B. McLeod, “The approach of solutions of nonlinear diffusion equations to travelling front solutions,” Arch. Ration. Mech. Anal. 65 (4), 335–361 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  18. Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosion (Nauka, Moscow, 1980; Plenum, New York, 1985).

    Book  Google Scholar 

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Authors and Affiliations

  1. Physical Faculty, Moscow State University, Moscow, 119991, Russia

    A. A. Melnikova & M. Chen

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  1. A. A. Melnikova
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  2. M. Chen
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Correspondence to A. A. Melnikova.

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Original Russian Text © A.A. Melnikova, M. Chen, 2018, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2018, Vol. 58, No. 5, pp. 705–715.

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Melnikova, A.A., Chen, M. Existence and Asymptotic Representation of the Autowave Solution of a System of Equations. Comput. Math. and Math. Phys. 58, 680–690 (2018). https://doi.org/10.1134/S0965542518050147

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  • DOI: https://doi.org/10.1134/S0965542518050147

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