I will talk about our recent work with N. Kitajima and J. Soda, where we initiated a study on various cosmological imprints of string axions whose scalar potentials
have plateau regions. In such cases, a delayed onset of oscillation generically leads to a parametric resonance instability. We show that a sustainable resonance can lead to a turbulence phase during which detectable gravitational waves by pulsar timing arrays can be emitted