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 with x -dependent coefficients u ( x ) y tt − ( u ( x ) y x ) x + au ( x ) y +| y | p −2 y = f ( x , t ) on (0, π )× under the periodic or anti-periodic boundary conditions y (0, t )=± y ( π , t ), y x (0, t )=± y x ( π , t ) and the time-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 non-homogeneous 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 T =2 π / k ( k ∈ ), 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.