Abstract
We consider the boundary-value singularly perturbed time-periodic problem for the parabolic reaction-advection-diffusion equation in the case of a weak linear advection in a two-dimensional domain. The main result of the present paper is the justification, under certain sufficient assumptions, of the existence of a periodic solution with internal transition layer near some closed curve and the study of the Lyapunov asymptotic stability of such a solution. For this purpose, an asymptotic expansion of the solution is constructed; the justification of the existence of the solution with the constructed asymptotics is carried out by using the method of differential inequalities. The proof of Lyapunov asymptotic stability is based on the application of the so-called method of contraction barriers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Hess, Periodic–Parabolic Boundary Value Problems and Positivity, in Pitman Res. Notes Math. Ser. (Longman Sci. & Tech., Harlow, 1991), Vol. 247.
H. Amann, “Periodic solutions of semilinear parabolic equations,” in Nonlinear Analysis (Academic Press, New York, 1978), pp. 1–29.
R. Kannan and V. Lakshmikantham, “Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions,” J. Math. Anal. Appl. 97 (1), 291–299 (1983).
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).
E. N. Dancer and P. Hess, “Behaviour of a semi-linear periodic-parabolic problem when a parameter is small,” in Functional-Analytic Methods for Partial Differential Equations, Lecture We notes in Math. (Springer-Verlag, Berlin, 1990), Vol. 1450, pp. 12–19.
N. D. Alikakos, P. W. Bates, and X. Chen, “Periodic traveling waves and locating oscillating pat- terns in multidimensional domains,” Trans. Amer. Math. Soc. 351 (7), 2777–2805 (1999).
N. T. Levashova, O. A. Nikolaeva, and A. D. Pashkin, “Modeling the temperature distribution at the interface “water–air” using the theory of contrast structures,” Vestnik Moskov. Univ. Ser. III Phys. Astron., No. 5, 12–16 (2015).
N. N. Nefedov and E. I. Nikulin, “Existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection,” Model. Anal. Inform. Sist. 25 (1), 125–132 (2018).
N. N. Nefedov and E. I. Nikulin, “Existence and asymptotic stability of periodic solutions of the reaction–diffusion equations in the case of a rapid reaction,” Russ. J. Math. Phys. 25 (1), 88–101 (2018).
N. N. Nefedov, “Abstract evolution differential equations with discontinuous operator coefficients,” Differ. Uravn. 31 (7), 1132–1141 (1995) [Differ. Equ. 31 (7), 1067–1076 (1995)].
N. N. Nefedov and K. Sakamoto, “Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equations with balanced nonlinearity,” Hiroshima Math. J. 33 (3), 391–432 (2003).
N. N. Nefedov and M. A. Davydova, “Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations,” Differ. Equ. 49 (6), 688–706 (2013).
P. C. Fife and M. M. Tang, “Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation disturbances,” J. Differ. Equ. 40 (2), 168–185 (1981).
A. B. Vasil’eva and M. A. Davydova, “On a contrast steplike structure for a class of second-order nonlinear singularly perturbed equations,” Zh. Vychisl. Mat. Mat. Fiz. 38 (6), 938–947 (1998) [Comput. Math. Math. Phys. 38 (6), 900–908 (1998)].
N. N. Nefedov, “The method of differential inequalities for some singularly perturbed partial differential equations,” Differ. Uravn. 31 (4), 719–722 (1995) [Differ. Equ. 31 (4), 668–671 (1995)].
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations (Vyssh. Shkola, Moscow, 1990) [in Russian].
A. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, “Singularly perturbed problems with boundary and internal layers,” in Trudy Mat. Inst. Steklova, Vol. 268: Differential Equations and Topology. I (MAIK Nauka/Interperiodica, Moscow, 2010), pp. 268–283 [Proc. Steklov Inst. Math. 268, 258–273 (2010)].
H. Amann, “Periodic solutions of semilinear parabolic equations,” in Nonlinear Analysis (Academic Press, New York, 1978), pp. 1–29.
M. A. Davydova, “Existence and stability of solutions with boundary layers in multidimensional singularly perturbed reaction-diffusion-advection problems,” Mat. Zametki 98 (6), 853–864 (2015) [Math. Notes 98 (6), 853–864 (2015)].
N. N. Nefedov and E. I. Nikulin, “Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem,” Russ. J. Math. Phys. 22 (2), 215–226 (2015).
Funding
This work was supported by the Russian Science Foundation under grant 18-11-00042.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 5, pp. 708–722.
Rights and permissions
About this article
Cite this article
Nefedov, N.N., Nikulin, E.I. Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection. Math Notes 106, 771–783 (2019). https://doi.org/10.1134/S0001434619110105
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434619110105