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

Abstract

This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation withyZf (x, t) on (0, p)!R under the periodic or anti-periodic boundary conditions y(0,t)ZGy(p, t), y x (0, t)ZGy x (p, t) and the time-periodic conditions y(x, tCT )Z y(x, t), y t (x, tCT )Zy t (x, t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion 'weak solution' to be given in §2. For TZ2p/k(k2R), we establish the existence of time-periodic solutions in the weak sense by investigating 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.…”

Time periodic solutions of one-dimensional forced Kirchhoff equations with x -dependent coefficients

“…This models the forced vibrations of an inhomogeneous string in which the dependence of the tension on the deformation can be neglected (see [18][19][20][21][22][23][24][25][26][27][28][29][30] and references therein). Beginning from the work of Barbu & Pavel [20][21][22], the problem of finding time periodic solutions of the wave equation with x-dependent coefficients has started to gain more attention.…”

“…Then, Rudakov [30] demonstrated the existence of time periodic solutions for the nonlinearity having power-law growth under Dirichlet boundary conditions. Later, Ji and Li obtained some related results for the general Sturm-Liouville boundary value problem [23,27], and the periodic and anti-periodic boundary value problem [24,28]. Furthermore, in [29], they also considered the case in which the coefficients do not satisfy the condition…”

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].…”