Periodic solutions of second-order wave equations. II
[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.
scientific article; zbMATH DE number 4038353