Morawetz Estimates Method for Scattering of Radial Energy Sub-critical Wave Equation

Abstract

In this short paper we consider a semi-linear, energy sub-critical, defocusing wave equation ${\partial }_t^{2}u-\Delta u=-|u{|}^{p-1}u$ in the 3-dimensional space with $p\in (3,5)$. We prove that if the energy of radial initial data $(u_0,u_1)$ outside a ball of radius $r$ centred at the origin decays faster than a certain rate $r^{-\kappa (p)}$, then the corresponding solution $u$ must scatter in both two time directions. The main tool of our proof is a more detailed version of the classic Morawetz estimate.

8 pages

References in corpus (1)

• Scattering for the radial 3d cubic wave equation