Geodesic
Existence, Asymptotics, and Lyapunov Stability of Solutions of Periodic Parabolic Problems for Tikhonov-Type Reaction--Diffusion Systems
Matematičeskie zametki, Tome 115 (2024) no. 2, pp. 276-285.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study a new class of time-periodic solutions of singularly perturbed systems of reaction–diffusion equations in the case of a fast and a slow equation, which are usually called Tikhonov-type systems. A boundary layer asymptotics of solutions is constructed, the existence of solutions with this asymptotics is proved, and conditions for the Lyapunov asymptotic stability of these solutions treated as solutions of the corresponding initial–boudary value problems are obtained.
Keywords: singularly perturbed problem, periodic parabolic boundary value problem, boundary and interior layers, asymptotic expansion, differential inequality, Lyapunov stability.
Mots-clés : reaction–diffusion equations 
@article{MZM_2024_115_2_a10,
     author = {N. N. Nefedov},
     title = {Existence, {Asymptotics,} and {Lyapunov} {Stability} of {Solutions} of {Periodic} {Parabolic} {Problems} for {Tikhonov-Type} {Reaction--Diffusion} {Systems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {276--285},
     publisher = {mathdoc},
     volume = {115},
     number = {2},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2024_115_2_a10/}
}
TY  - JOUR
AU  - N. N. Nefedov
TI  - Existence, Asymptotics, and Lyapunov Stability of Solutions of Periodic Parabolic Problems for Tikhonov-Type Reaction--Diffusion Systems
JO  - Matematičeskie zametki
PY  - 2024
SP  - 276
EP  - 285
VL  - 115
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2024_115_2_a10/
LA  - ru
ID  - MZM_2024_115_2_a10
ER  - 
%0 Journal Article
%A N. N. Nefedov
%T Existence, Asymptotics, and Lyapunov Stability of Solutions of Periodic Parabolic Problems for Tikhonov-Type Reaction--Diffusion Systems
%J Matematičeskie zametki
%D 2024
%P 276-285
%V 115
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2024_115_2_a10/
%G ru
%F MZM_2024_115_2_a10
N. N. Nefedov. Existence, Asymptotics, and Lyapunov Stability of Solutions of Periodic Parabolic Problems for Tikhonov-Type Reaction--Diffusion Systems. Matematičeskie zametki, Tome 115 (2024) no. 2, pp. 276-285. http://geodesic.mathdoc.fr/item/MZM_2024_115_2_a10/

[1] V. F. Butuzov, N. N. Nefedov, K. R. Schneider, “Singularly perturbed problems in case of exchange of stabilities”, J. Math. Sci. (N.Y.), 121:1 (2004), 1973–2079 | DOI | MR

[2] A. N.Tikhonov, “Sistemy differentsialnykh uravnenii, soderzhaschie malye parametry pri proizvodnykh”, Matem. sb., 31 (73):3 (1952), 575–586 | MR | Zbl

[3] A. B. Vasileva, V. F. Butuzov, Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vysshaya shkola, M., 1990 | MR

[4] A. B. Vasileva, V. F. Butuzov, N. N. Nefedov, “Singulyarno vozmuschennye zadachi s pogranichnymi i vnutrennimi sloyami”, Differentsialnye uravneniya i topologiya. I, Trudy MIAN, 268, Nauka, M., 2010, 268–283 | MR | Zbl

[5] N. N. Nefedov, “Razvitie metodov asimptoticheskogo analiza perekhodnykh sloev v uravneniyakh reaktsii-diffuzii-advektsii: teoriya i primenenie”, Zh. vychisl. matem. i matem. fiz., 61:12 (2021), 2074–2094 | DOI

[6] O. V. Rudenko, “Neodnorodnoe uravnenie Byurgersa s modulnoi nelineinostyu: vozbuzhdenie i evolyutsiya intensivnykh voln”, Dokl. RAN, 474:6 (2017), 671–674 | DOI | MR

[7] N. N. Nefedov, O. V. Rudenko, “O dvizhenii fronta v uravnenii tipa Byurgersa s kvadratichnoi i modulnoi nelineinostyu pri nelineinom usilenii”, Dokl. RAN, 478:3 (2018), 274–279 | DOI

[8] O. V. Rudenko, N. N. Nefedov, “O dvizhenii, usilenii i razrushenii frontov v uravneniyakh tipa Byurgersa s kvadratichnoi modulnoi nelineinostyu”, Dokl. RAN, 493:1 (2020), 26–31 | DOI | MR

[9] A. Olchev, K. Radler, A. Sogachev, O. Panferov, G. Gravenhorst, “Application of a three-dimensional model for assessing effects of small clear-cuttings on radiation and soil temperature”, Ecological Modelling, 220:21 (2009), 3046–3056 | DOI

[10] N. Levashova, A. Sidorova, A. Semina, M. Ni, “A spatio-temporal Autowave model of Shanghai territory development”, Sustainability, 11 (2019), 3658 | DOI

[11] A. Ya. Garaeva, A. E. Sidorova, V. A. Tverdislov, N. T. Levashova, “Model predposylok vidoobrazovaniya v predstavleniyakh teorii perkolyatsii i samoorganizovannoi kritichnosti”, Biofizika, 65:5 (2020), 932–948

[12] M. I. Budyko, “The effect of solar radiation variations on the climate of the Earth”, Tellus, 21:5 (1968), 611–619 | DOI

[13] J. I. Diaz, “Mathematical analysis of some diffusive energy balance models in climatology”, Mathematics, Climate and Environment (Madrid, 1991), RMA Res. Notes Appl. Math., 27, Masson, Paris, 1993, 28–56 | MR

[14] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., 247, Longman, New York, 1991 | MR

[15] C. V. Pao, “Periodic solutions of parabolic systems with nonlinear boundary conditions”, J. Math. Anal. Appl., 234:2 (1999), 695–716 | DOI | MR

[16] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992 | MR