RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 19 scientific papers (total in 19 papers)

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:
\begin{gather*} u_{tt}-u_{xx}=\pm|u|^{p-2}u+h(t,x)\quad(0<x<\pi),\qquad u(t,0)=u(t,\pi)=0,
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.

Full text: PDF file (815 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.95
MSC: Primary 35L05, 35B10; Secondary 35L20, 35L70
Received: 31.08.1987

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}


Linking options:
  • http://mi.mathnet.ru/eng/msb1759
  • http://mi.mathnet.ru/eng/msb/v178/i4/p546

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  mathscinet  zmath  isi
    3. Kuzin I., “About Equations with Noncompact Indefinite Operators”, Differ. Equ., 29:3 (1993), 335–344  mathnet  mathscinet  zmath  isi
    4. I. A. Rudakov, “Nelineinye kolebaniya neodnorodnoi struny”, Fundament. i prikl. matem., 8:3 (2002), 877–886  mathnet  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  isi
    6. I. A. Rudakov, “Periodic Solutions of a Nonlinear Wave Equation with Nonconstant Coefficients”, Math. Notes, 76:3 (2004), 395–406  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. Rudakov I., “A Nontrivial Periodic Solution of the Nonlinear Wave Equation with Homogeneous Boundary Conditions”, Differ. Equ., 41:10 (2005), 1467–1475  mathnet  crossref  mathscinet  zmath  isi  elib
    8. I. A. Rudakov, “Periodic solutions of a quasilinear wave equation with homogeneous boundary conditions”, J. Math. Sci., 150:6 (2008), 2588–2597  mathnet  crossref  mathscinet  zmath  elib
    9. I. A. Rudakov, “Periodic solutions of a non-linear wave equation with homogeneous boundary conditions”, Izv. Math., 70:1 (2006), 109–120  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. I. A. Rudakov, “Nonlinear equations satisfying the nonresonance condition”, J. Math. Sci. (N. Y.), 135:1 (2006), 2749–2763  mathnet  crossref  mathscinet  zmath
    11. I. A. Rudakov, “Periodic solutions of a quasilinear wave equation with variable coefficients”, Sb. Math., 198:7 (2007), 993–1009  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet  zmath  elib
    13. Jean-Marcel Fokam, “Forced Vibrations via Nash-Moser Iteration”, Comm Math Phys, 283:2 (2008), 285  crossref  mathscinet  zmath  adsnasa  isi
    14. V. A. Kondrat'ev, I. A. Rudakov, “Periodic Solutions of a Quasilinear Wave Equation”, Math. Notes, 85:1 (2009), 34–50  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. I. A. Rudakov, “On time-periodic solutions of a quasilinear wave equation”, Proc. Steklov Inst. Math., 270 (2010), 222–229  mathnet  crossref  mathscinet  zmath  isi  elib
    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  crossref  zmath  isi
    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  mathnet  crossref  crossref  isi  elib  elib
    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  mathnet  crossref  mathscinet  zmath  isi  scopus
    19. I. A. Rudakov, “Periodic Solutions of the Quasilinear Equation of Forced Vibrations of an Inhomogeneous String”, Math. Notes, 101:1 (2017), 137–148  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:325
    Full text:73
    References:32
    First page:2

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019