Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $

Linglong Du; Min Yang

doi:10.3934/nhm.2020033

Article Contents

2021, Volume 16, Issue 1: 49-67. Doi: 10.3934/nhm.2020033

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Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $

Linglong Du^1, , and
Min Yang^2,

1.
Department of Mathematics and Institute for Nonlinear Science, Donghua University, Shanghai, China
2.
Department of Mathematics, Donghua University, Shanghai, China

^* Corresponding author: Linglong Du
^* Corresponding author: Linglong Du

Received: April 30, 2020

Revised: July 31, 2020

Early access: December 2020

Published: March 2021

The first author is supported by the Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China (No. 11671075, No.12001097) and Natural Science Foundation of Shanghai (No. 18ZR1401300)

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Abstract

In this paper, we investigate the long time behavior of the solution for the nonlinear wave equation with frictional and visco-elastic damping terms in $ \mathbb{R}^n_+ $. It is shown that for the long time, the frictional damped effect is dominated. The Green's functions for the linear initial boundary value problem can be described in terms of the fundamental solutions for the full space problem and reflected fundamental solutions coupled with the boundary operator. Using the Duhamel's principle, we get the pointwise long time behavior of the solution $ \partial_{{\bf{x}}}^{\alpha}u $ for $ |\alpha|\le 1 $.

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

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