The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds

We are interested in the one-dimensional nonlinear wave equations with multiple wave speeds by the energy method. By choosing different multipliers corresponding to the different wave speeds, we show that the one-dimensional nonlinear wave equations also have globally smooth solutions provided that the nonlinearities satisfy certain structural conditions when the initial data are small. Furthermore, we can prove that the global solutions will converge to the solutions of the linearized system based on the decay properties of the nonlinearities.