Null Condition and Global Classical Solutions to the Cauchy Problem of Nonlinear Wave Equations

2017 ◽  
pp. 263-301
Author(s):  
Tatsien Li ◽  
Yi Zhou
Keyword(s):  
Cauchy Problem ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Classical Solutions ◽  
Null Condition ◽  
Global Classical Solutions ◽  
The Cauchy Problem

scholarly journals Global classical solutions to the Cauchy problem for a nonlinear wave equation

1998 ◽  
Vol 21 (3) ◽  
pp. 533-548 ◽  
Author(s):  
Haroldo R. Clark
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Classical Solution ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Global Classical Solutions ◽  
The Cauchy Problem

In this paper we consider the Cauchy problem{u″+M(|A12u|2)Au=0   in   ]0,T[u(0)=u0,       u′(0)=u1,whereu′is the derivative in the sense of distributions and|A12u|is theH-norm ofA12u. We prove the existence and uniqueness of global classical solution.

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.

A global existence theorem for the Cauchy problem of nonlinear wave equations

1988 ◽  
Vol 109 (3-4) ◽  
pp. 261-269 ◽  
Author(s):  
Jianmin Gao ◽  
Lichen Xu
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Lorentz Invariance ◽  
Nonlinear Wave Equations ◽  
Homogeneous Wave ◽  
The Cauchy Problem ◽  
Global Existence Theorem

SynopsisIn this paper we consider the global existence (in time) of the Cauchy problem of the semilinear wave equation utt – Δu = F(u, Du), x ∊ Rn, t > 0. When the smooth function F(u, Du) = O((|u| + |Du|)k+1) in a small neighbourhood of the origin and the space dimension n > ½ + 2/k + (1 + (4/k)2)½/2, a unique global solution is obtained under suitable assumptions on initial data. The method used here is associated with the Lorentz invariance of the wave equation and an improved Lp–Lq decay estimate for solutions of the homogeneous wave equation. Similar results can be extended to the case of “fully nonlinear wave equations”.

scholarly journals BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS

2002 ◽  
Vol 04 (02) ◽  
pp. 223-295 ◽  
Author(s):  
SERGIU KLAINERMAN ◽  
SIGMUND SELBERG
Keyword(s):  
Systematic Review ◽  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Large Data ◽  
Nonlinear Wave Equations ◽  
Well Posedness ◽  
Bilinear Estimates ◽  
The Cauchy Problem

We undertake a systematic review of results proved in [26, 27, 30-32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

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