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Existence and Stability of Solutions with Internal Transition Layer for the Reaction–Diffusion–Advection Equation with a KPZ-Nonlinearity

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Abstract

We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions.

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REFERENCES

  1. Vasil’eva, A.B. and Davydova, M.A., On a contrast steplike structure for a class of second-order nonlinear singularly perturbed equations, Comput. Math. Math. Phys., 1998, vol. 38, no. 6, pp. 900–908.

    MathSciNet  MATH  Google Scholar 

  2. Davydova, M.A., A spikelike solution and a critical steplike solution to a singularly perturbed second-order equation, Comput. Math. Math. Phys., 1998, vol. 39, no. 8, pp. 1252–1263.

    MathSciNet  MATH  Google Scholar 

  3. Nefedov, N.N., Nikulin, E.I., and Recke, L., On the existence and asymptotic stability of periodic contrast structures in quasilinear reaction–advection–diffusion equations, Russ. J. Math. Phys., 2019, vol. 26, no. 1, pp. 55–69.

    Article  MathSciNet  MATH  Google Scholar 

  4. Nefedov, N.N. and Nikulin, E.I., Existence and asymptotic stability of a periodic solution with an internal transition layer in a problem with weak linear advection, Model. Anal. Inf. Sist., 2018, vol. 25, no. 1, pp. 125–132.

    Article  MathSciNet  Google Scholar 

  5. Nefedov, N.N. and Nikulin, E.I., Existence and asymptotic stability of periodic two-dimensional contrast structures in the problem with weak linear advection, Math. Notes, 2019, vol. 106, no. 5, pp. 771–783.

    Article  MathSciNet  MATH  Google Scholar 

  6. Nefedov, N.N., Development of methods of asymptotic analysis of transition layers in reaction–diffusion–advection equations: Theory and applications, Comput. Math. Math. Phys., 2021, vol. 61, no. 12, pp. 068–2087.

  7. Yafei Pan, Mingkang Ni, and Davydova, M.A., Contrast structures in problems for a stationary equation of reaction–diffusion–advection type with discontinuous nonlinearity, Math. Notes, 2018, vol. 104, no. 5, pp. 735–744.

    MathSciNet  MATH  Google Scholar 

  8. Nefedov, N.N., Nikulin, E.I., and Orlov, A.O., Existence of contrast structures in a problem with discontinuous reaction and advection, Russ. J. Math. Phys., 2022, vol. 29, no. 2, pp. 214–224.

    Article  MathSciNet  MATH  Google Scholar 

  9. Nefedov, N.N., Nikulin, E.I., and Orlov, A.O., Contrast structures in the reaction–diffusion–advection problem in the case of a weak reaction discontinuity, Russ. J. Math. Phys., 2022, vol. 29, no. 1, pp. 81–90.

    Article  MathSciNet  MATH  Google Scholar 

  10. Grimson, M.J. and Barker, G.C., Continuum model for the spatiotemporal growth of bacterial colonies, Phys. Rev. E, 1994, vol. 49, no. 2, pp. 1680–1688.

    Article  Google Scholar 

  11. Davydova, M.A. and Zakharova, S.A., Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient, J. Comput. Appl. Math., 2022, vol. 400, no. 1, p. 113731.

    Article  MathSciNet  MATH  Google Scholar 

  12. Krug, J. and Spohn, H., Universality classes for deterministic surface growth, Phys. Rev. A, 1988, vol. 38, no. 8, p. 4271.

    Article  MathSciNet  Google Scholar 

  13. Pohožaev, S.I., On equations of the form \(\Delta u=f (x, u, Du) \), Math. USSR-Sb., 1982, vol. 41, no. 2, pp. 269–280.

  14. Muravnik, A.B., Decay of nonnegative solutions of singular parabolic equations with KPZ-nonlinearities, Comput. Math. Math. Phys., 2020, vol. 60, no. 8, pp. 1375–1380.

    Article  MathSciNet  MATH  Google Scholar 

  15. Muravnik, A.B., On qualitative properties of solutions to quasilinear parabolic equations admitting degenerations at infinity, Ufa Math. J., 2018, vol. 10, no. 4, pp. 77–84.

    Article  MathSciNet  MATH  Google Scholar 

  16. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie metody v teorii singulyarnykh vozmushchenii (Asymptotic Methods in Singular Perturbation Theory), Moscow: Vyssh. Shkola, 1990.

  17. Butuzov, V.F., Vasil’eva, A.B., and Nefedov, N.N., Asymptotic Theory of Contrasting Structures. A Survey, Autom. Remote Control, 1997, vol. 58, no. 7, pp. 1068–1091.

    MATH  Google Scholar 

  18. Nefedov, N.N., The method of differential inequalities for some classes of nonlinear singularly perturbed problems with internal layers, Differ. Equations, 1995, vol. 31, no. 7, pp. 1077–1085.

    MathSciNet  MATH  Google Scholar 

  19. Nefedov, N.N., The method of differential inequalities for nonlinear singularly perturbed problems with contrast structures of step type in the critical case, Differ. Equations, 1996, vol. 32, no. 11, pp. 1526–1534.

    MathSciNet  Google Scholar 

  20. Nefedov, N.N. and Nikulin, E.I., Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem, Russ. J. Math. Phys., 2015, vol. 22, pp. 215–226.

    Article  MathSciNet  MATH  Google Scholar 

  21. Nefedov, N.N. and Nikulin, E.I., Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem in the case of a balanced nonlinearity, Differ. Equations, 2017, vol. 53, no. 4, pp. 516–529.

    Article  MathSciNet  MATH  Google Scholar 

  22. Pao, C.V., Nonlinear Parabolic and Elliptic Equations, New York–London: Springer Sci. & Bus. Media, 1993.

    Book  Google Scholar 

  23. Henry, D., Geometric Theory of Semilinear Parabolic Equations, Berlin–Heidelberg: Springer-Verlag, 1981. Translated under the title: Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Moscow: Mir, 1985.

    Book  MATH  Google Scholar 

  24. Nefedov, N.N. and Orlov, A.O., On unstable contrast structures in one-dimensional reaction–diffusion–advection problems with discontinuous sources, Theor. Math. Phys., 2023, vol. 215, no. 2, pp. 716–728.

    Article  MathSciNet  MATH  Google Scholar 

  25. Lopez-Gomez, J., The strong maximum principle (Mathematical analysis on the self-organization and self-similarity), RIMS Kôkyûroku Bessatsu, 2009, vol. B15, pp. 113–123.

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Funding

This work was financially supported by the Russian Science Foundation, project 23-11-00069, https://rscf.ru/en/project/23-11-00069/.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    N. N. Nefedov & A. O. Orlov

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  1. N. N. Nefedov
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  2. A. O. Orlov
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Correspondence to N. N. Nefedov or A. O. Orlov.

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Translated by V. Potapchouck

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Nefedov, N.N., Orlov, A.O. Existence and Stability of Solutions with Internal Transition Layer for the Reaction–Diffusion–Advection Equation with a KPZ-Nonlinearity. Diff Equat 59, 1009–1024 (2023). https://doi.org/10.1134/S0012266123080013

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  • DOI: https://doi.org/10.1134/S0012266123080013

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