Morawetz type estimate for damped wave equation in $\mathbb{R}^n (n\geq 4)$ and its application

In this paper we establish a Morawetz type etimate for the linear inhomogeneous wave equation with time-dependent scale invariant damping in $\mathbb{R}^n (n\geq 4)$. The novelty is that we view the differential operator $\Box+\frac{\mu}{t}\partial_t$ as $n+1+\mu$ dimensional operator, then a well-matched multiplier is introduced. As an application, a sharp global existence result for the small data Cauchy problem of the semilinear wave equation $$\partial_t^2u-\Delta u+\frac{\partial_tu}{t}=|u|^p,~~~t>t_0\geq 0$$ is obtained in $\mathbb{R}^n (n\geq 4)$.