Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case

We study global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case. We prove the global well-posedness for small data in the energy space for the whole energy-subcritical case. This result implies that small data blow-up does not occur in the overdamping case, different from the other cases, i.e. effective or non-effective damping.