Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions

Abstract

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients (π, t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the 'weak solution'. For T , the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

Time periodic solutions to a nonlinear wave equation with x-dependent coefficients

“…In this line, we showed the existence of time periodic solutions for the general boundary conditions in the case that the nonlinear term has sublinear growth and satisfies the global Lipschitz condition in [14]. In addition, we also studied the periodic and anti-periodic boundary value problem in [15]. This paper follows this line, using the methods and techniques in [19], to establish the existence of time periodic solutions to wave equation (1.1) under the general boundary conditions (1.2) and the periodic conditions (1.3).…”

Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions

“…In order to prove our main result, we need to use the following properties of wave operator A K that can be proved along the lines given in Ji & Li (2007).…”

“…Not very many results seem to be known for the periodic or antiperiodic boundary-value problem. In Ji & Li (2007), we treated them for the case in which the nonlinear term has sublinear growth and satisfies the global Lipschitz condition. In addition, we also studied the general boundary-value problem in Ji & Li (2006) and Ji (2007), respectively, for sublinear growth and power-law growth nonlinearity.…”

Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

“…where a, b are relatively prime positive integers. As stated in [4,5,6,15,16,17,18,19,20,25,26], equation (1.1) is a mathematical model to account for the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. More precisely, the vertical displacement u(t, z) at time t and depth z of a plane seismic wave is described by the equation where ρ = (ων) 1/2 denotes the acoustic impedance function.…”