Geodesic
Existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection
Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 1, pp. 125-132 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulation of the problem is investigated, with weaker sufficient conditions for the existence of a solution with an internal transition layer to be provided than in previous studies. For convenience, the known results are given, which ensure the fulfillment of the existence theorem of the contrast structure. The justification for the existence of a solution with an internal transition layer is based on the use of an asymptotic method of differential inequalities based on the modification of the terms of the constructed asymptotic expansion. Further, sufficient conditions are established to fulfill these requirements, and they have simple and concise formulations in the form of the algebraic equation $w(x_0,t) = 0$ and the condition $w_x(x_0,t) 0$, which is essentially a condition of simplicity of the root $x_0(t)$ and ensuring the stability of the solution found. The function w is a function of the known functions appearing in the reactive and advective terms of the original problem. The equation $w(x_0,t) = 0$ is a problem for finding the zero approximation $x_0(t)$ to determine the localization region of the inner transition layer. In addition, the asymptotic Lyapunov stability of the found periodic solution is investigated, based on the application of the so-called compressible barrier method. The main result of the paper is formulated as a theorem.
Keywords: singularly perturbed parabolic problems, periodic problems, weak advection, contrast structures, internal layers, asymptotic methods, differential inequalities, Lyapunov asymptotical stability.
Mots-clés : reactionadvection-diffusion equations, fronts 
@article{MAIS_2018_25_1_a11,
     author = {N. N. Nefedov and E. I. Nikulin},
     title = {Existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {125--132},
     year = {2018},
     volume = {25},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2018_25_1_a11/}
}
TY  - JOUR
AU  - N. N. Nefedov
AU  - E. I. Nikulin
TI  - Existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2018
SP  - 125
EP  - 132
VL  - 25
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MAIS_2018_25_1_a11/
LA  - ru
ID  - MAIS_2018_25_1_a11
ER  - 
%0 Journal Article
%A N. N. Nefedov
%A E. I. Nikulin
%T Existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection
%J Modelirovanie i analiz informacionnyh sistem
%D 2018
%P 125-132
%V 25
%N 1
%U http://geodesic.mathdoc.fr/item/MAIS_2018_25_1_a11/
%G ru
%F MAIS_2018_25_1_a11
N. N. Nefedov; E. I. Nikulin. Existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection. Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 1, pp. 125-132. http://geodesic.mathdoc.fr/item/MAIS_2018_25_1_a11/

[1] Nefedov N. N., Davydova M. A., “Periodic contrast structures in systems of the reaction-diffusion-advection type”, Differential Equations, 46:9 (2010), 1309–1321 | DOI | MR

[2] Vasil'eva A. B., Davydova M. A., “On a contrast steplike structure for a class of second-order nonlinear singularly perturbed equations”, Computational Mathematics and Mathematical Physics, 38:6 (1998), 900–909 | MR

[3] Vasil'eva A. B., “Periodic solutions of a parabolic problem with a small parameter at the derivatives”, Computational Mathematics and Mathematical Physics, 43:7 (2003), 932–944 | MR | MR

[4] Nefedov N. N., Nikulin E. I., “Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem”, Russian Journal of Mathematical Physics, 22:2 (2015), 215–226 | DOI | MR

[5] Nefedov N. N., Nikulin E. I., “Existence and Stability of Periodic Contrast Structures in the Reaction-Advection-Diffusion Problem in the Case of a Balanced Nonlinearity”, Differential equations, 53:4 (2017), 516–529 | DOI | DOI | MR