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

Abstract

In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients u(x)y tt − (u(x), 2) and the periodic conditions y(x, t + T ) = y(x, t), y t (x, t + T ) = y t (x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For T = 2π/k (k ∈ N), we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients.

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.…”

Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

“…For the case the nonlinear term having power-law growth, Rudakov [27] proved the existence of periodic solutions under the Dirichlet boundary conditions. Later, Ji and his collaborators obtained some related results for the general Sturm-Liouville boundary value problem [18,21], and periodic and anti-periodic boundary value problem [19,22]. In [31], by using topological degree methods, Wang and An obtained an existence result on periodic solution of the problem with resonance and the sublinear nonlinearity.…”

Quasi-Periodic Solutions of Nonlinear Wave Equation with x-Dependent Coefficients

“…As stated in [Barbu & Pavel, 1996, 1997a, 1997bJi, 2008Ji, , 2009Ji & Li, 2006, 2007, Eq. (1) describes the forced vibration of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media.…”

“…The problem of finding periodic solutions to the nonlinear wave equation with x-dependent coefficients was firstly considered by Barbu and Pavel [1996, 1997a, 1997b for the monotone nonlinearities f in u. Later, using variational methods, Ji and Li obtained the existence of periodic solutions for the general boundary value problem [Ji, 2008;Ji & Li, 2006], periodic and anti-periodic boundary value problem [Ji, 2009;Ji & Li, 2007]. Moreover, Ji and Li [2010] also considered the case in which the coefficients do not satisfy the condition ess inf{ 1 2 ρ ρ − 1 4 ( ρ ρ ) 2 } > 0 and obtained the existence of a unique weak periodic solution, which actually solves an open problem posed by Barbu and Pavel [1997b].…”