Geodesic
Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection
Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 708-722.

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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.
Keywords: singularly perturbed parabolic problem, periodic contrast structures.
Mots-clés : reaction-advection-diffusion equations 
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     title = {Existence and {Asymptotic} {Stability} of {Periodic} {Two-Dimensional} {Contrast} {Structures} in the {Problem} with {Weak} {Linear} {Advection}},
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N. N. Nefedov; E. I. Nikulin. Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection. Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 708-722. http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a5/

[1] P. Hess, Periodic–Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., 247, Longman Sci. Tech., Harlow, 1991 | MR | Zbl

[2] H. Amann, “Periodic solutions of semilinear parabolic equations”, Nonlinear Analysis, Academic Press, New York, 1978, 1–29 | MR

[3] R. Kannan, V. Lakshmikantham, “Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions”, J. Math. Anal. Appl., 97:1 (1983), 291–299 | DOI | MR | Zbl

[4] N. N. Nefedov, L. Recke, 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 (2013), 90–103 | DOI | MR | Zbl

[5] E. N. Dancer, P. Hess, “Behaviour of a semi-linear periodic-parabolic problem when a parameter is small”, Functional-Analytic Methods for Partial Differential Equations, Lecture Notes in Math., 1450, Springer-Verlag, Berlin, 1990, 12–19 | DOI | MR

[6] N. D. Alikakos, P. W. Bates, X. Chen, “Periodic traveling waves and locating oscillating patterns in multidimensional domains”, Trans. Amer. Math. Soc., 351:7 (1999), 2777–2805 | DOI | MR | Zbl

[7] N. T. Levashova, O. A. Nikolaeva, A. D. Pashkin, “Modelirovanie raspredeleniya temperatury na granitse razdela voda-vozdukh s ispolzovaniem teorii kontrastnykh struktur”, Vestn. Mosk. un-ta. Ser. 3. Fiz. Astron., 2015, no. 5, 12–16

[8] N. N. Nefedov, E. I. Nikulin, “Suschestvovanie i asimptoticheskaya ustoichivost periodicheskogo resheniya s vnutrennim perekhodnym sloem v zadache so slaboi lineinoi advektsiei”, Model. i analiz inform. sistem, 25:1 (2018), 125–132 | DOI

[9] N. N. Nefedov, 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 (2018), 88–101 | DOI | MR | Zbl

[10] N. N. Nefedov, “Abstraktnye evolyutsionnye differentsialnye uravneniya s razryvnymi operatornymi koeffitsientami”, Differents. uravneniya, 31:7 (1995), 1132–1141 | MR | Zbl

[11] N. N. Nefedov, K. Sakamoto, “Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equations with balanced nonlinearity”, Hiroshima Math. J., 33:3 (2003), 391–432 | DOI | MR | Zbl

[12] N. N. Nefedov, M. A. Davydova, “Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations”, Differ. Equ., 49:6 (2013), 688–706 | DOI | MR | Zbl

[13] P. C. Fife, M. M. Tang, “Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances”, J. Differential Equations, 40:2 (1981), 168–185 | DOI | MR | Zbl

[14] A. B. Vasileva, M. A. Davydova, “O kontrastnoi strukture tipa stupenki dlya odnogo klassa nelineinykh singulyarno vozmuschennykh uravnenii vtorogo poryadka”, Zh. vychisl. matem. i matem. fiz., 38:6 (1998), 938–947 | MR | Zbl

[15] N. N. Nefedov, “Metod differentsialnykh neravenstv dlya nekotorykh singulyarno vozmuschennykh zadach v chastnykh proizvodnykh”, Differents. uravneniya, 31:4 (1995), 719–722 | MR | Zbl

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

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

[18] H. Amann, “Periodic solutions of semilinear parabolic equations”, Nonlinear Analysis, Academic Press, New York, 1978, 1–29 | MR

[19] M. A. Davydova, “Suschestvovanie i ustoichivost reshenii s pogranichnymi sloyami v mnogomernykh singulyarno vozmuschennykh zadachakh reaktsiya-diffuziya-advektsiya”, Matem. zametki, 98:6 (2015), 853–864 | DOI | MR | Zbl

[20] N. N. Nefedov, E. I. Nikulin, “Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem”, Russ. J. Math. Phys., 22:2 (2015), 215–226 | DOI | MR | Zbl