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

TL;DR

This paper proves global existence and well-posedness for a defocusing, energy-subcritical nonlinear wave equation in 3D using the I-method and almost conservation laws.

Contribution

It introduces an application of the I-method to establish global results for the nonlinear wave equation with specific initial data regularity.

Findings

Global existence for the nonlinear wave equation in 3D.

Use of the I-method to handle energy subcritical cases.

Almost conservation law of energy established.

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.