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

TL;DR

This paper derives a Morawetz type estimate for a damped wave equation with time-dependent damping in high dimensions, leading to a new global existence result for small data solutions of a semilinear wave equation.

Contribution

It introduces a novel Morawetz estimate by viewing the operator as an (n+1+μ)-dimensional operator, enabling sharp global existence results for semilinear wave equations.

Findings

Established Morawetz estimate for damped wave equations in $\,\mathbb{R}^n$ with $n\geq 4$

Proved sharp global existence for small data solutions of the semilinear wave equation with damping

Introduced a new multiplier technique based on viewing the operator as higher-dimensional

Abstract

In this paper we establish a Morawetz type etimate for the linear inhomogeneous wave equation with time-dependent scale invariant damping in $\mathbb{R}^n (n\geq 4)$. 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 $\mathbb{R}^n (n\geq 4)$.