Geodesic
On the existence of periodic solutions of a fourth-order nonlinear system of parabolic equations
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 98-104.

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In this paper, we consider a fourth-order system of two nonlinear parabolic equations with the double Laplace operator. We propose two approaches to constructing a solution, which allow separating components depending on time and spatial variables. The original problem is reduced to solving systems of algebraic and ordinary differential equations. We obtain explicit expressions for the exact solutions in terms of elementary or harmonic functions. Also, we identify the cases of solutions that are periodic in time and anisotropic in spatial variables.
Keywords: periodic solution, nonlinear system
Mots-clés : parabolic equation. 
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     title = {On the existence of periodic solutions of a fourth-order nonlinear system of parabolic equations},
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A. A. Kosov; È. I. Semenov. On the existence of periodic solutions of a fourth-order nonlinear system of parabolic equations. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 98-104. http://geodesic.mathdoc.fr/item/INTO_2021_196_a8/

[1] Galaktionov V. A., Posashkov S. S., “O novykh tochnykh resheniyakh parabolicheskikh uravnenii s kvadratichnymi nelineinostyami”, Zh. vychisl. mat. mat. fiz., 29:4 (1989), 497–506

[2] Gantmakher F. R., Teoriya matrits, Nauka, M., 1988 | MR

[3] Zaitsev V. F., Polyanin A. D., Spravochnik po nelineinym uravneniyam matematicheskoi fiziki, Fizmatlit, M., 2002

[4] Kosov A. A., Semenov E. I., “O tochnykh mnogomernykh resheniyakh sistemy uravnenii reaktsii-diffuzii so stepennymi nelineinostyami”, Sib. mat. zh., 58:4 (2017), 796–812 | MR | Zbl

[5] Kosov A. A., Semenov E. I., “O tochnykh mnogomernykh resheniyakh odnoi nelineinoi sistemy uravnenii reaktsii-diffuzii”, Differ. uravn., 54:1 (2018), 108–122 | MR | Zbl

[6] Nefedov N. N., Nikulin E. I., “Suschestvovanie i asimptoticheskaya ustoichivost periodicheskikh dvumernykh kontrastnykh struktur v zadache so slaboi lineinoi advektsiei”, Mat. zametki., 106:5 (2019), 709–722 | MR

[7] Polyanin A. D., Zaitsev V. F., Zhurov A. I., Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki, Fizmatlit, M., 2005

[8] Rudykh G. A., Semenov E. I., “Postroenie tochnykh reshenii mnogomernogo uravneniya nelineinoi teploprovodnosti”, Zh. vychisl. mat. mat. fiz., 33:8 (1993), 1228–1239 | MR | Zbl

[9] Rudykh G. A., Semenov E. I., “Tochnye neotritsatelnye resheniya mnogomernogo uravneniya nelineinoi diffuzii”, Sib. mat. zh., 39:5 (1998), 1131–1140 | MR | Zbl

[10] Rudykh G. A., Semenov E. I., “Neavtomodelnye resheniya mnogomernogo uravneniya nelineinoi diffuzii”, Mat. zametki., 67:2 (2000), 250–256 | Zbl

[11] Skubachevskii A. L., “Uravneniya Vlasova—Puassona dlya dvukhkomponentnoi plazmy v odnorodnom magnitnom pole”, Usp. mat. nauk., 69:2 (2014), 107–148 | MR | Zbl

[12] Titov S. S., Metod konechnomernykh kolets dlya resheniya nelineinykh uravnenii matematicheskoi fiziki, SGU, Saratov, 1988

[13] Galaktionov V. A., Svirshchevskii S. R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman, 2007 | MR | Zbl

[14] Kosov A. A., Semenov E. I., Tirskikh V. V., “On exact multidimensional solutions of a nonlinear system of first order partial differential equations”, Izv. Irkutsk. gos. un-ta. Ser. Mat., 28 (2019), 53–68 | MR | Zbl

[15] Markov Y., Rudykh G., Sidorov N., Sinitsyn A., Tolstonogov D., “Steady state solutions of the Vlasov–Maxwell system and their stability”, Acta Appl. Math., 28:3 (1992), 253–293 | DOI | MR | Zbl