Russian Universities Reports. MathematicsRussian Universities Reports. MathematicsВестник российских университетов. Математика2686-96672782-3342Tambov State University - G.R. Derzhavin29730510.20310/1810-0198-2019-24-125-99-111ArticlesСтатьиResearch ArticleDecay of the solutions of the generalized Korteweg-de Vries equation at large timesУбывание решений обобщенного уравнения Кортевега-де Фриза при больших временахNikolayevArtyom A.НиколаевАртем Александрович

Post-Graduate Student, Nonlinear Analysis and Optimization Department

аспирант, кафедра нелинейного анализа и оптимизации

nicepeopleproject@gmail.comRUDN UniversityФГАОУ ВО «Российский университет дружбы народов»1512201924125VOL 24, NO125 (2019)ТОМ 24, №125 (2019)9911120062025Copyright ©; 2025, Nikolayev A.A.Copyright ©; 2025, Николаев А.А.2025Nikolayev A.A.Николаев А.А.https://creativecommons.org/licenses/by/4.0https://journals.rcsi.science/2686-9667/article/view/297305

In this paper the existence of weak solutions of the nonlinear generalized KdV equation is shown and conditions for which weak solutions decay to zero at large times are obtained.

В данной работе доказано существование слабых решений для нелинейного обобщенного уравнения Кортевега-де Фриза и найдены условия, при которых слабые решения убывают к нулю при больших временах.

Korteweg-de Vries equationinitial-boundary problemweak solutiondecayуравнение Кортевега-де Фризаначально-краевая задачаслабое решениеубываниеL. Rosier, B.-Y. Zhang, “Global stabilization of the generalized Korteweg-de Vries qquation posed on a finite domain”, SIAM J. Control and Optimization, 45:3 (2006), 927-956.G. Perla Menzala, C. F. Vasconcellos, E. Zuazua, “Stabilization of the Korteweg-de Vries equation with localized damping”, Quarterly of Applied Mathematics, 60:1 (2002), 111-129.F. Linares, A. F. Pazoto, “On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping”, Proceedings of the American Mathematical Society, 135:5 (2007), 1515-1522.A. V. Faminskii, N. A. Larkin, “Odd-order quasilinear evolution equations posed on a bounded interval”, Bol. Soc. Paranaense Mat., 28:1 (2010), 67-77.A. V. Faminskii, A. Nikolayev, “On stationary solutions of KdV and mKdV equations”, Differential and Difference Equations with Applications, 164 (2016), 63-70.A. V. Faminskii, “Weak solutions to initial-boundary-value problems for quasilinear evolution equations of an odd order”, Advances in Differential Equations, 17:5-6 (2012), 421-470.A. Pazy, Applied Mathematical Sciences. V. 44, Springer-Verlag, New York, Heidelberg, Tokyo, 1983.Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai, K. Takemura, “The best constant of Sobolev inequality corresponding to Dirichlet boundary value problem for (-1)M(d=dx)2M ”, Sciential Mathematical Japanical Online, 2008, 439-451.