Inward/outward Energy Theory of Wave Equation in Higher Dimensions

TL;DR

This paper extends inward/outward energy theory and Morawetz estimates to higher dimensions for a semi-linear wave equation, demonstrating scattering of solutions under specific energy decay conditions at infinity.

Contribution

It generalizes energy and Morawetz estimates to higher dimensions and applies these to prove scattering results for solutions with decaying initial energy.

Findings

Extended energy theory and Morawetz estimates to higher dimensions.

Proved scattering of solutions with decaying initial energy.

Established conditions for energy decay leading to scattering.

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 $|x| \rightarrow \infty$.