Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $

Linglong Du; Caixuan Ren

doi:10.3934/dcdsb.2018319

Article Contents

2019, Volume 24, Issue 7: 3265-3280. Doi: 10.3934/dcdsb.2018319

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Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $

Linglong Du and
Caixuan Ren^,

Department of Applied Mathematics, Donghua University, Shanghai, China

^* Corresponding author: Caixuan Ren
^* Corresponding author: Caixuan Ren

Received: October 31, 2017

Early access: January 2019

Published: May 2019

Du is supported by Fundamental Research Funds for the Central Universities (No. 2232016D3-32), Natural Science Foundation of Shanghai (No. 18ZR1401300) and partly by National Natural Science Foundation of China (No. 11671075). Ren is supported by NSFC(No. 11601075) and the Fundamental Research Funds for the Central Universities (No.16D110910).

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Abstract

In this paper, the asymptotic wave behavior of the solution for the nonlinear damped wave equation in $ \mathbb{R}^n_+ $ is investigated. We describe the double mechanism of the hyperbolic effect and the parabolic effect using the explicit functions. With the absorbing and radiative boundary condition, we show that the Green's function for the half space linear problem can be described in terms of the fundamental solution for the Cauchy problem and the reflected fundamental solution coupled with a boundary operator. Using the Duhamel's principle, we see that due to the fast decay property of the Green's function and the high nonlinearity, the pointwise decaying rate for the nonlinear solution and extra time decaying rate for its first order derivative are obtained.

Keywords:

Initial-boundary value problem,
nonlinear damped wave equation,
absorbing and radiative boundary condition,
Green's function.

Mathematics Subject Classification: Primary: 35B40; Secondary: 35A08.

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