Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Huafei Di; Yadong Shang; Jiali Yu; Huafei Di; Yadong Shang; Jiali Yu

doi:10.3934/era.2020015

Electronic Research Archive

2020, Volume 28, Issue 1: 221-261. doi: 10.3934/era.2020015

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Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Huafei Di ^{1
,
,},
Yadong Shang ^{1
,},
Jiali Yu ^{2
,}

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
School of Science, Dalian Jiaotong University, Dalian 116028, China

Received: 01 October 2019 Revised: 01 January 2020
Primary: 35L35; Secondary: 35D30, 35D35, 35B40

In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ and internal source $ |u|^{\rho}u $. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function $ \phi(t) $ (that depends on the behaviors of the functions $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ near the origin), nonlinear integral inequality and the Multiplier method.
- Fourth order wave equation,
- boundary velocity feedbacks,
- interior source,
- potential well,
- multiplier method
Citation: Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source[J]. Electronic Research Archive, 2020, 28(1): 221-261. doi: 10.3934/era.2020015

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Abstract

In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ and internal source $ |u|^{\rho}u $. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function $ \phi(t) $ (that depends on the behaviors of the functions $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ near the origin), nonlinear integral inequality and the Multiplier method.

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