Morawetz type estimate for damped wave equation in ${\mathbb{R}}^n(n\ge 4)$ 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 ${\mathbb{R}}^n(n\ge 4)$. The novelty is that we view the differential operator $\mathrm{\square }+\frac{\mu }{t}{\mathrm{\partial }}_t$ 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 $${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u+\frac{{\mathrm{\partial }}_{t}u}{t}=\mathrm{\mid }u{\mathrm{\mid }}^p,t>t_0\ge 0$$ is obtained in ${\mathbb{R}}^n(n\ge 4)$.