I-method for Defocusing, Energy-Subcritical Non-linear Wave Equation

Abstract

In this paper the author considers the global existence and well-posedness of the non-linear wave equation ${\partial }_t^{2}u-\Delta u=-|u{|}^{p-1}u$ in 3-dimensional space, assuming that the initial data is in the space $({\dot{H}}^s\cap {\dot{H}}^{s_p})\times ({\dot{H}}^{s-1}\cap {\dot{H}}^{s_p-1})$, with $11/3<p<5$, $s_p=3/2-2/(p-1)$ and $s$ sufficiently close to 1. The main idea is to use the I-method and consider the almost conservation law of the energy.

15 pages, 1 figure