Abstract
The paper deals with the study of time-periodic solutions of boundary layer type for a two-dimensional reaction-diffusion problem with a small parameter at the parabolic operator in the case of singularly perturbed boundary conditions of the second kind. The asymptotic approximation with respect to the small parameter for solutions with a nonmonotone boundary layer is constructed. It is shown that all such solutions are unstable. The proof of the instability of the solutions is based on the construction of an unordered pair of upper and lower solutions and on the application of a corollary of the Krein–Rutman theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Barenblatt, V. Entov, and V. Ryzhik, Theory of Fluid Flows Through Natural Rocks (Kluwer Academic Publishers, Dordrecht, 1990).
M. Burger and J.-F. Pietschmann, “Flow characteristics in a crowded transport model,” Nonlinearity 29 (11), 3528–3550 (2016).
N. N. Nefedov and E. I. Nikulin, “Periodic boundary layer solutions of a reaction-diffusion problem with singularly perturbed boundary conditions of the third kind,” Differential Equations 56 (12), 1594–1603 (2020).
N. N. Nefedov and E. I. Nikulin, “Existence and stability of periodic boundary layer solutions in a two-dimensional reaction-diffusion problem in the case of singularly perturbed boundary conditions of the second kind,” Vestnik Moskov. Univ. Ser. III Phys. Astron., No. 2, 15–20 (2020).
P. C. Fife, “Semilinear elliptic boundary value problems with small parameters,” Arch. Rational Mech. Anal. 52, 205–232 (1973).
K. Sakamoto, “Infinitely many fine modes bifurcating from radially symmetric internal layers,” Asymptot. Anal. 42 (1-2), 55–104 (2005).
M. del Pino, M. Kowalczyk, and J. Wei, “Resonance and Interior Layers in an Inhomogeneous Phase Transition Model,” SIAM J. Math. Anal. 38 (5), 1542–1564 (2007).
O. Omel’chenko and L. Recke, “Boundary layer solutions to singularly perturbed problems via the implicit function theorem,” Asymptot. Anal. 62 (3-4), 207–225 (2009).
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, “Singularly perturbed problems with boundary and internal layers,” Proc. Steklov Inst. Math. 268, 258–273 (2010).
N. N. Nefedov, L. Recke, and K. R. Schneider, “Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations,” J. Math. Anal. Appl. 405 (1), 90–103 (2013).
A. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, “Contrast structures in singularly perturbed problems,” Fundam. Prikl. Mat. 4 (3), 799–851 (1998).
P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, in Pitman Res. Notes Math. (Longman Sci. Tech., Harlow, 1991), Vol. 247.
Funding
This work was supported by the Russian Foundation for Basic Research under grant 19-01-00327.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 899–910 https://doi.org/10.4213/mzm13211.
Rights and permissions
About this article
Cite this article
Nefedov, N.N., Nikulin, E.I. On Unstable Solutions with a Nonmonotone Boundary Layer in a Two-Dimensional Reaction-Diffusion Problem. Math Notes 110, 922–931 (2021). https://doi.org/10.1134/S0001434621110286
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434621110286