Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation

This paper is concerned with the time periodic solutions to the one-dimensional nonlinear wave equation with either variable or constant coefficients. By adjusting the basis of L ² function space, we can circumvent the difficulties caused by η _u  = 0 and obtain the existence of a weak periodic solution, which was posed as an open problem by Baubu and Pavel in (Trans Am Math Soc 349:2035–2048, 1997). Finally, an application to the forced Sine-Gordon equation is presented to illustrate the utility of this technique.