Standing waves and global existence for the nonlinear wave equation with potential and damping terms

This paper is concerned with the Cauchy problem of the nonlinear wave equation with potential and damping terms. Firstly, by using intricate variational arguments, the existence of a standing wave with the ground state is obtained. Then the instability of the standing wave is shown by applying potential well arguments and concavity methods. Finally, we answer the question of how small the initial data are for the global solutions to exist.