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

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.