Pointwise long time behavior for the mixed damped nonlinear wave equation in <inline-formula><tex-math id="M1">$ \mathbb{R}^n_+ $</tex-math></inline-formula>

Linglong Du; Min Yang; Linglong Du; Min Yang

doi:10.3934/nhm.2020033

Networks and Heterogeneous Media

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
,
,},
Min Yang ^{2
,}

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

Received: 01 May 2020 Revised: 01 August 2020 Published: 08 December 2020
Primary: 35B40; Secondary: 35A08

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 $.
- Initial boundary value problem,
- mixed damped,
- nonlinear wave equation,
- Green's function method,
- pointwise estimate
Citation: Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $[J]. Networks and Heterogeneous Media, 2021, 16(1): 49-67. doi: 10.3934/nhm.2020033

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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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