Abstract
This paper considers an initial-boundary value problem for a reaction–diffusion equation with a singularly perturbed Neumann boundary condition in a closed, simply connected two-dimensional domain. From a physical point of view, the problem describes processes with an intensive flow through the boundary of a given area. The existence of a stationary solution is proved, its asymptotic is constructed, and the Lyapunov stability conditions for it are established. The asymptotics of the solution are constructed by the classical Vasilieva algorithm using the Lusternik–Vishik method. The existence and stability of the solution are proved using the asymptotic method of differential inequalities.
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REFERENCES
C. Pao, Nonlinear Parabolic and Elliptic Equations (Springer, New York, 1993).
V. Volpert, Elliptic Partial Differential Equations (Birkhäuser, Boston, 2011).
K. V. Zhukovsky, Mosc. Univ. Phys. Bull. 73, 45 (2018).
K. V. Zhukovsky, Mosc. Univ. Phys. Bull. 71, 237 (2016).
O. V. Rudenko, Dokl. Math. 94, 708 (2016).
N. N. Nefedov and O. V. Rudenko, Dokl. Math. 97, 99 (2018).
N. Nefedov, Diff. Equat. 36, 298 (2000).
N. Nefedov, N. Levashova, and A. Orlov, Mosc. Univ. Phys. Bull. 73, 565 (2018).
V. Butuzov, N. Nefedov, L. Recke, and K. Schneider, Int. J. Bifurc. Chaos 24 (8) (2014).
N. Nefedov and E. I. Nikulin, Mosc. Univ. Phys. Bull. 75, 116 (2020).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansion of Solutions to Singularly Perturbed Equations (Nauka, Moscow, 1973) [in Russian].
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations (Vysshaya Shkola, Moscow, 1990) [in Russian].
A. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, Tr. Mat. Inst. im. V.A. Steklova RAN 268, 268 (2010).
N. Nefedov, L. Recke, and K. Schneider, J. Math. Anal. Appl. 405, 90 (2013).
N. Nefedov and K. Sakamoto, Hiroshima Math. J. 33, 391 (2003).
A. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, Fundam. Prikl. Mat. 4, 799 (1998).
N. T. Levashova, N. N. Nefedov, and A. O. Orlov, Comput. Math. Math. Phys. 59, 573 (2019).
N. Nefedov, Lect. Notes Comput. Sci. 8236, 62 (2013).
N. Nefedov and O. Rudenko, Dokl. Math. 97, 99 (2018).
Funding
This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00327).
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Translated by L. Trubitsyna
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Nefedov, N.N., Deryugina, N.N. The Existence of a Boundary-Layer Stationary Solution to a Reaction–Diffusion Equation with Singularly Perturbed Neumann Boundary Condition. Moscow Univ. Phys. 75, 409–414 (2020). https://doi.org/10.3103/S0027134920050185
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DOI: https://doi.org/10.3103/S0027134920050185