scholarly journals LOWER BOUND OF THE LIFESPAN OF SOLUTIONS TO SEMILINEAR WAVE EQUATIONS IN AN EXTERIOR DOMAIN

2013 ◽  
Vol 10 (02) ◽  
pp. 199-234
Author(s):  
SOICHIRO KATAYAMA ◽  
HIDEO KUBO
Keyword(s):  
Cauchy Problem ◽  
Lower Bound ◽  
Exterior Domain ◽  
Wave Equations ◽  
Symmetric Case ◽  
Classical Solutions ◽  
Semilinear Wave Equations ◽  
The Cauchy Problem ◽  
Three Space ◽  
Dimensional Domain

We consider the Cauchy–Dirichlet problem for semilinear wave equations in a three space-dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hörmander where the Cauchy problem was treated. We show that our estimate is sharp at least for radially symmetric case.

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.

scholarly journals Analyticity of solutions for quasilinear wave equations and other quasilinear systems

2014 ◽  
Vol 144 (6) ◽  
pp. 1155-1169 ◽  
Author(s):  
Sergei Kuksin ◽  
Nikolai Nadirashvili
Keyword(s):  
Cauchy Problem ◽  
Classical Solution ◽  
Wave Equations ◽  
Cauchy Data ◽  
Classical Solutions ◽  
Quasilinear Equations ◽  
Quasilinear Systems ◽  
The Cauchy Problem ◽  
Quasilinear Wave Equations

We prove the persistence of analyticity for classical solutions of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy–Kowalewski and Ovsiannikov–Nirenberg theorems, lasts until a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of the space variables, then a classical solution is also analytic in these variables. The approach applies to other quasilinear equations and implies the persistence of the space analyticity (and the partial space analyticity) of their classical solutions.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

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