×

Periodic solutions of second-order wave equations. II. (English. Russian original) Zbl 0637.35054

Ukr. Math. J. 38, 617-623 (1986); translation from Ukr. Mat. Zh. 38, No. 6, 733-739 (1986).
[For part I see ibid. 38, 505-511 (1986); translation from Ukr. Mat. Zh. 38, No.5, 593-600 (1986; Zbl 0632.34035).]
The existence of classical periodic solutions of the problem \[ (1)\quad u_{tt}-u_{xx}=g(x,t),\quad 0<x<\pi,\quad t\in {\mathbb{R}}, \]
\[ (2)\quad u(0,t)=u(\pi,t)=0,\quad u(x,t+T)=u(x,t),\quad 0\leq x\leq \pi,\quad t\in {\mathbb{R}}, \] and of the corresponding nonlinear problem \[ (3)\quad u_{tt}-u_{xx}=f(x,t,u,u_ t,u_ x),\quad 0<x<\pi,\quad t\in {\mathbb{R}}, \] and the condition (2) is studied in this paper.
The following results are obtained:
1) If the function g(x,t) is bounded, continuous together with the derivative \(g_ t(x,t)\) in the domain \(\Pi =\{0\leq x\leq \pi\), \(t\in {\mathbb{R}}\}\), then in some class of three types t-periodic functions g(x,t) closely connected with periods of the form \(T=2\pi (2p-1)/q\), \(T=4s\pi /(2m-1)\) where p,q,s,m are natural numbers, there exist classical periodic solutions of the boundary value problem (1), (2).
2) If \(\phi\) (u) is a continuous nondecreasing function such that \(\lim_{| u| \to \infty}\phi (u)/u=0\) then the boundary value problem (3) with sufficiently large T has a non-trivial T-periodic weak solution \(u\in L^{\infty}.\)
The essential peculiarities of the existence of periodic solutions for linear and nonlinear wave equations of the form (1) and (3) as well as for the simplest linear equation \(u_{tt}-u_{xx}=\phi (u)\), \(u(0,t)=u(\pi,t)=0\) are shown.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B10 Periodic solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)

Citations:

Zbl 0632.34035
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] O. Veivoda and M. Shtedry, ?Existence of classical periodic solutions of the wave equation. Relation of the theoretical-numerical character of the period and geometric properties of solutions,? Differents. Uravn.,20, No. 10, 1733-1739 (1984).
[2] Haim Brezis and Jean-Michel Coron, ?Periodic solutions of nonlinear wave equations and Hamiltonian systems,? Am. J. Math.,103, No. 3, 559-570 (1981). · Zbl 0462.35004 · doi:10.2307/2374104
[3] J. M. Coron, ?Solutions périodiques non triviales d’une équation des ondes,? Commun. Part. Different. Equat.,6, No. 7, 829-848 (1981). · Zbl 0491.35011 · doi:10.1080/0360530810882194
[4] Haim Brezis, Jean-Michel Coron, and Louis Nirenberg, ?Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,? Commun. Pure and Appl. Math.,33, No. 5, 667-684 (1980). · Zbl 0484.35057 · doi:10.1002/cpa.3160330507
[5] Yu. A. Mitropol’skii and G. P. Khoma, ?Periodic solutions of second-order wave equations. I,? Ukr. Mat. Zh.,38, No. 5, 593-600 (1986). · Zbl 0632.34035
[6] G. P. Khoma, ?Periodic solutions of second-order differential wave equations,? Preprint Akad. Nauk Ukr. SSR Inst. Mat.; 86.05 (1986).
[7] G. P. Khoma, ?The structure of periodic solutions of second-order wave equations,? Preprint, Akad. Nauk Ukr. SSR, Inst. Mat.; 85.53 (1985).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.