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

Abstract

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 $\square +\frac{\mu }{t}{\partial }_t$ 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 \[ \partial_t^2u-Δu+\frac{\partial_tu}{t}=|u|^p,~~~t>t_0\geq 0 \] is obtained in ${\mathbb{R}}^n(n\ge 4)$.