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 Mat. Sb. (N.S.), 1988, Volume 136(178), Number 4(8), Pages 546–560 (Mi msb1759)

Existence of a countable set of periodic solutions of the problem of forced oscillations for a weakly nonlinear wave equation

P. I. Plotnikov

Abstract: In the strip $0<x<\pi$ of the plane of the points $t$, $x$ the following boundary value problem is considered:
u(t+2\pi,x)=u(t,x). \end{gather*}
It is proved that for any $p>2$ and for an arbitrary $2\pi$-periodic function $h$ which is locally integrable with power $p(p-1)^{-1}$ this problem has a countable set of geometrically distinct generalized solutions.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 64:2, 543–556

Bibliographic databases:

UDC: 517.95
MSC: Primary 35L05, 35B10; Secondary 35L20, 35L70

Citation: P. I. Plotnikov, “Existence of a countable set of periodic solutions of the problem of forced oscillations for a weakly nonlinear wave equation”, Mat. Sb. (N.S.), 136(178):4(8) (1988), 546–560; Math. USSR-Sb., 64:2 (1989), 543–556

Citation in format AMSBIB
\Bibitem{Plo88} \by P.~I.~Plotnikov \paper Existence of a~countable set of periodic solutions of the problem of forced oscillations for a~weakly nonlinear wave equation \jour Mat. Sb. (N.S.) \yr 1988 \vol 136(178) \issue 4(8) \pages 546--560 \mathnet{http://mi.mathnet.ru/msb1759} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=965892} \zmath{https://zbmath.org/?q=an:0683.35054} \transl \jour Math. USSR-Sb. \yr 1989 \vol 64 \issue 2 \pages 543--556 \crossref{https://doi.org/10.1070/SM1989v064n02ABEH003327} 

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. A. Kuzin, “Existence of a countable set of periodic, spherically symmetric solutions of a nonlinear wave equation”, Math. USSR-Izv., 38:1 (1992), 107–129
2. Kuzin I., “On Some Embedding-Theorems for the Spaces of the Krein-Pontryagin Type and their Application to Periodic Problems”, Dokl. Akad. Nauk, 325:3 (1992), 434–437
3. Kuzin I., “About Equations with Noncompact Indefinite Operators”, Differ. Equ., 29:3 (1993), 335–344
4. I. A. Rudakov, “Nelineinye kolebaniya neodnorodnoi struny”, Fundament. i prikl. matem., 8:3 (2002), 877–886
5. Rudakov I., “A Time-Periodic Solution of the Equation of Forced Vibrations of a String with Homogeneous Boundary Conditions”, Differ. Equ., 39:11 (2003), 1633–1638
6. I. A. Rudakov, “Periodic Solutions of a Nonlinear Wave Equation with Nonconstant Coefficients”, Math. Notes, 76:3 (2004), 395–406
7. Rudakov I., “A Nontrivial Periodic Solution of the Nonlinear Wave Equation with Homogeneous Boundary Conditions”, Differ. Equ., 41:10 (2005), 1467–1475
8. I. A. Rudakov, “Periodic solutions of a quasilinear wave equation with homogeneous boundary conditions”, J. Math. Sci., 150:6 (2008), 2588–2597
9. I. A. Rudakov, “Periodic solutions of a non-linear wave equation with homogeneous boundary conditions”, Izv. Math., 70:1 (2006), 109–120
10. I. A. Rudakov, “Nonlinear equations satisfying the nonresonance condition”, J. Math. Sci. (N. Y.), 135:1 (2006), 2749–2763
11. I. A. Rudakov, “Periodic solutions of a quasilinear wave equation with variable coefficients”, Sb. Math., 198:7 (2007), 993–1009
12. I. A. Rudakov, “Periodic solutions of a nonlinear wave equation with Neumann and Dirichlet boundary conditions”, Russian Math. (Iz. VUZ), 51:2 (2007), 44–52
13. Jean-Marcel Fokam, “Forced Vibrations via Nash-Moser Iteration”, Comm Math Phys, 283:2 (2008), 285
14. V. A. Kondrat'ev, I. A. Rudakov, “Periodic Solutions of a Quasilinear Wave Equation”, Math. Notes, 85:1 (2009), 34–50
15. I. A. Rudakov, “On time-periodic solutions of a quasilinear wave equation”, Proc. Steklov Inst. Math., 270 (2010), 222–229
16. Kharibegashvili S.S., Dzhokhadze O.M., “Time-Periodic Problem For a Weakly Nonlinear Telegraph Equation With Directional Derivative in the Boundary Condition”, Differ. Equ., 51:10 (2015), 1369–1386
17. S. S. Kharibegashvili, O. M. Dzhokhadze, “On solvability of a periodic problem for a nonlinear telegraph equation”, Siberian Math. J., 57:4 (2016), 735–743
18. Rudakov I.A., “Periodic solutions of the wave equation with nonconstant coefficients and with homogeneous Dirichlet and Neumann boundary conditions”, Differ. Equ., 52:2 (2016), 248–257
19. I. A. Rudakov, “Periodic Solutions of the Quasilinear Equation of Forced Vibrations of an Inhomogeneous String”, Math. Notes, 101:1 (2017), 137–148
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