Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

Mohamed Ali Hamza; Yuta Wakasugi; Shuji Yoshikawa

doi:10.3934/dcds.2024027

Article Contents

2024, Volume 44, Issue 8: 2280-2308. Doi: 10.3934/dcds.2024027

This issue Previous Article Closest distance between iterates of typical points Next Article Optimal control of a stationary Navier-Stokes hemivariational inequality with numerical approximation

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

1.
Imam Abdulrahman Bin Faisal University, Dammam, KSA
2.
Hiroshima University, Japan
3.
Oita University, Japan

^*Corresponding author: Yuta Wakasugi
^*Corresponding author: Yuta Wakasugi

Received: October 2023

Revised: February 2024

Early access: March 22, 2024

Published online: March 22, 2024

The second author is supported by JSPS KAKENHI Grant Numbers JP18H01132, JP20K14346.

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Abstract

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients:

$ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u = \partial_x \left( N(\partial_x u) \right). $

In the authors' previous article [17], the asymptotic profile of solutions for linearized problem ($ N \equiv 0 $) was classified depending on the assumptions for the coefficients $ a(t) $ and $ b(t) $ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $ b(t) \partial_t u - a(t) \partial_x^2 u = 0 $ asymptotically as $ t \to \infty $.

Keywords:

Nonlinear damped beam equation,
asymptotic behavior.

Mathematics Subject Classification: Primary: 35G25; Secondary: 35B40, 35A01.

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References

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