Global existence for semilinear wave equations with scaling invariant damping in 3-D

TL;DR

This paper proves global existence of solutions for small radial initial data in 3-D semilinear wave equations with scaling invariant damping, using novel weighted estimates.

Contribution

It introduces a new weighted $L^2-L^2$ estimate for inhomogeneous wave equations to establish global existence under specific damping conditions.

Findings

Global existence for small radial data with damping constant in [1.5, 2)

Development of a weighted $L^2-L^2$ estimate for inhomogeneous wave equations

Interpolation between energy and Morawetz estimates

Abstract

Global existence for small data Cauchy problem of semilinear wave equations with scaling invariant damping in 3-D is established in this work, assuming that the data are radial and the constant in front of the damping belongs to $[1.5, 2)$. The proof is based on a weighted $L^2-L^2$ estimate for inhomogeneous wave equation, which is established by interpolating between energy estimate and Morawetz type estimate.