Inward/outward Energy Theory of Wave Equation in Higher Dimensions

Abstract

We consider the semi-linear, defocusing wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in $\mathbb{R}^d$ with $1+4/(d-1)\leq p < 1+4/(d-2)$. We generalize the inward/outward energy theory and weighted Morawetz estimates in 3D to higher dimensions. As an application we show the scattering of solutions if the energy of initial data decays at a certain rate as $|x| \rightarrow \infty$.