Existence of a~countable set of periodic solutions of the problem of forced oscillations for a~weakly nonlinear wave equation
Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 543-556
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In the strip $0$ 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\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.
@article{SM_1989_64_2_a16,
author = {P. I. Plotnikov},
title = {Existence of a~countable set of periodic solutions of the problem of forced oscillations for a~weakly nonlinear wave equation},
journal = {Sbornik. Mathematics},
pages = {543--556},
publisher = {mathdoc},
volume = {64},
number = {2},
year = {1989},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1989_64_2_a16/}
}
TY - JOUR AU - P. I. Plotnikov TI - Existence of a~countable set of periodic solutions of the problem of forced oscillations for a~weakly nonlinear wave equation JO - Sbornik. Mathematics PY - 1989 SP - 543 EP - 556 VL - 64 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/SM_1989_64_2_a16/ LA - en ID - SM_1989_64_2_a16 ER -
%0 Journal Article %A P. I. Plotnikov %T Existence of a~countable set of periodic solutions of the problem of forced oscillations for a~weakly nonlinear wave equation %J Sbornik. Mathematics %D 1989 %P 543-556 %V 64 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/SM_1989_64_2_a16/ %G en %F SM_1989_64_2_a16
P. I. Plotnikov. Existence of a~countable set of periodic solutions of the problem of forced oscillations for a~weakly nonlinear wave equation. Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 543-556. http://geodesic.mathdoc.fr/item/SM_1989_64_2_a16/