Inward/outward energy theory of non-radial solutions to 3D semi-linear wave equation

Abstract

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation ∂t2u−Δu=−|u|p−1u in the 3-dimensional space with 3≤p<5. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if 3<p<5 and κ>5−p2, then the solution scatters as long as the initial data (u0,u1) satisfy∫R3(|x|κ+1)(12|∇u0|2+12|u1|2+1p+1|u0|p+1)dx<+∞. If p=3, we can also prove the scattering result if initial data (u0,u1) are contained in the critical Sobolev space and satisfy the inequality∫R3|x|(12|∇u0|2+12|u1|2+14|u0|p+1)dx<+∞. These assumptions on the decay rate of initial data as |x|→∞ are weaker than previously known scattering results in the non-radial case.