scholarly journals Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations

2009 ◽  
Keyword(s):  
Cauchy Problem ◽  
Wave Equations ◽  
Global Solutions ◽  
Coupled System ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
The Cauchy Problem

scholarly journals On Classical Global Solutions of Nonlinear Wave Equations with Large Data

2017 ◽  
Vol 2019 (19) ◽  
pp. 5859-5913 ◽  
Author(s):  
Shuang Miao ◽  
Long Pei ◽  
Pin Yu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Wave Equations ◽  
Global Solutions ◽  
Large Data ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Classical Solutions ◽  
Linear Wave ◽  
The Cauchy Problem

Abstract This article studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial energy. The choice of the large Cauchy initial data is inspired by Christodoulou's characteristic initial data in his work [2] on formation of black holes. The main innovation of the current work is that we discovered a relaxed energy ansatz which allows us to prove decay-in-time-estimate. Therefore, the new estimates can also be applied in studying the Cauchy problem for Einstein equations.

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