On the structure of the set of periods for periodic solutions of some linear integro-differential equations on the multidimensional sphere
HTML articles powered by AMS MathViewer
- by
Dang Khanh Hoi
Translated by: A. Plotkin - St. Petersburg Math. J. 18 (2007), 573-581
- DOI: https://doi.org/10.1090/S1061-0022-07-00961-2
- Published electronically: May 29, 2007
- PDF | Request permission
Abstract:
The problem of periodic solutions for the family of linear differential equations \begin{equation*} (L-{\lambda })u\equiv \Big (\frac {1}{i}\frac {\partial }{\partial t} - a\Delta - \lambda \Big ) u(x,t)=\nu G(u-f) \end{equation*} is considered on the multidimensional sphere $x\in S^n$ under the periodicity condition $u|_{t=0}=u|_{t=b}$. Here $a$ and $\lambda$ are given reals, $\nu$ is a fixed complex number, $G u(x,t)$ is a linear integral operator, and $\Delta$ is the Laplace operator on $S^n$. It is shown that the set of parameters $(\nu , b)$ for which the above problem admits a unique solution is a measurable set of full measure in ${\mathbb C} \times {\mathbb R}^+$.References
- I. P. Kornfel′d, Ya. G. Sinaĭ, and S. V. Fomin, Ergodicheskaya teoriya, “Nauka”, Moscow, 1980 (Russian). MR 610981
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
- M. A. Šubin, Psevdodifferentsial′nye operatory i spektral′naya teoriya, “Nauka”, Moscow, 1978 (Russian). MR 509034
- Dang Khanh Hoi, Periodic solutions for some nonlinear evolution systems of natural differential equations, Differential Equations and Related Problems (Moscow, 2004): Thesis, p. 48 (Russian)
- —, On periodic solutions for some nonlinear evolution natural differential equations on multidimensional torus, Vestnik Novgorod. Gos. Univ. Ser. Tekhn. Nauki No. 28 (2004), 77–79. (Russian)
Bibliographic Information
- Dang Khanh Hoi
- Affiliation: Division of Mathematical Analysis, Novgorod State University, Bol′shaya St.-Peterburgskaya Ulitsa 41, 173003, Velikiĭ Novgorod, Russia
- Email: dangkhanhhoi@yahoo.com
- Received by editor(s): December 1, 2005
- Published electronically: May 29, 2007
- © Copyright 2007 American Mathematical Society
- Journal: St. Petersburg Math. J. 18 (2007), 573-581
- MSC (2000): Primary 35K20
- DOI: https://doi.org/10.1090/S1061-0022-07-00961-2
- MathSciNet review: 2262584