Abstract
The asymptotic behavior of the solution with boundary layers in the time-independent mathematical model of reaction–diffusion–advection arising when describing the distribution of greenhouse gases in the surface atmospheric layer is studied. On the basis of the asymptotic method of differential inequalities, the existence of a boundary-layer solution and its asymptotic Lyapunov stability as a steady-state solution of the corresponding parabolic problem is proven. One of the results of this work is the determination of the local domain of the attraction of a boundary-layer solution.
Similar content being viewed by others
References
V. T. Volkov, N. N. Nefedov, N. E. Grachev, and D. S. Senin, “Parameter Estimation for in-situ combustion front under air injection,” Neft. Khoz., No. 4, 93–96 (2010).
N. T. Levashova, Yu. V. Mukhartova, M. A. Davydova, N. E. Shapkina, and A. V. Ol’chev, “The application of the theory of contrast structures for describing wind field in spatially heterogeneous vegetation cover,” Mosc. Univ. Phys. Bull. 70 (3), 167–174 (2015).
Yu. V. Bozhevol’nov and N. N. Nefedov, “Front motion in a parabolic reaction-diffusion problem,” Comput. Math. Math. Phys. 50 (2), 264–273 (2010).
N. N. Nefedov, A. G. Nikitin, and L. Recke, Preprint No. 2007-22 (Institut für Mathematik an der Mathematisch-Naturwissenschaftlichen Fakultät II der Humbolt-Universität zu Berlin, Berlin, 2007).
N. N. Nefedov and M. A. Davydova, “Contrast structures in multidimensional singularly perturbed reaction–diffusion–advection problems,” Differ. Equations 48 (5), 745–755 (2012).
N. N. Nefedov and M. A. Davydova, “Contrast structures in singularly perturbed quasilinear reaction–diffusion–advection equations,” Differ. Equations 49 (6), 688–706 (2013).
V. T. Volkov and N. N. Nefedov, “Asymptotic-numerical investigation of generation and motion of fronts in phase transition models,” Lect. Notes Comput. Sci. 8236, 524–531 (2013).
N. T. Levashova, N. N. Nefedov, and A. V. Yagremtsev, “Contrast structures in reaction–diffusion–advection equations in the case of balanced advection,” Comput. Math. Math. Phys. 53 (3), 273–283 (2013).
M. A. Davydova, “Existence and stability of solutions with boundary layers in multidimensional singularly perturbed reaction–diffusion–advection problems,” Math. Notes 98 (6), 909–919 (2015).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations (Vysshaya Shkola, Moscow, 1990) [in Russian].
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).
Yu. M. Romanovskii, N. V. Stepanova, and D. S. Chernavskii, Mathematical Biophysics (Nauka, Moscow, 1984) [in Russian].
Ya. I. Kanel’, “Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory,” Mat. Sb. 59 (101) (supplementary) 245–288 (1962).
A. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, “Singularly perturbed problems with boundary and internal layers,” Proc. Steklov Inst. Math. 268, 258–273 (2010).
C. V. Pao, Nonlinear Parabolic and Elliptic Equations (Plenum, New York, 1992).
N. N. Kalitkin and P. V. Koryakin, Methods of Mathematical Physics (Akademiya, Moscow, 2013) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M.A. Davydova, S.A. Zakharova, N.T. Levashova, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 9, pp. 1548–1559.
Rights and permissions
About this article
Cite this article
Davydova, M.A., Zakharova, S.A. & Levashova, N.T. On one model problem for the reaction–diffusion–advection equation. Comput. Math. and Math. Phys. 57, 1528–1539 (2017). https://doi.org/10.1134/S0965542517090056
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542517090056