具变指数源项和强阻尼项的波动方程解的渐近稳定性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (1): 146-155.

具变指数源项和强阻尼项的波动方程解的渐近稳定性

廖梦兰^1,²,郭斌^1,*()

¹ 吉林大学数学学院长春 130012
² 密歇根州立大学数学系美国密歇根东兰辛 48824

收稿日期:2018-11-07 出版日期:2020-02-26 发布日期:2020-04-08
通讯作者: 郭斌 E-mail:bguo@jlu.edu.cn
基金资助:
吉林省教育厅"十三五"科学技术规划项目(JJKH20180111KJ);国家自然科学基金(11301211)

Asymptotic Stability of Weak Solutions to Wave Equation with Variable Exponents and Strong Damping Term

Menglan Liao^1,²,Bin Guo^1,*()

¹ School of Mathematics, Jilin University, Changchun 130012
² Department of Mathematics, Michigan State University, MI East Lansing 48824, USA

Received:2018-11-07 Online:2020-02-26 Published:2020-04-08
Contact: Bin Guo E-mail:bguo@jlu.edu.cn
Supported by:
the Scientific and Technological Project of Jilin Province's Education Department in Thirteenth-five-Year(JJKH20180111KJ);the NSFC(11301211)

摘要/Abstract

摘要：

该文主要讨论下列具强阻尼项的波动方程的初边值问题

$u_{tt}-{\rm div}(|\nabla{u}|^{p(x)-2}\nabla{u})-\Delta u_t=|u|^{q(x)-2}u$

解的渐近行为.通过构造一个新的控制函数和利用Sobolev嵌入不等式，建立了源项和能量泛函之间的定性关系.进而，利用Komornik不等式和能量估计，给出了衰减估计.最后，证明u（x，t）=0是渐近稳定的.

关键词: 阻尼项, 衰减估计, 渐进稳定性

Abstract:

This paper deals with the following wave equation with strong damping term:

$u_{tt}-{\rm div}(|\nabla{u}|^{p(x)-2}\nabla{u})-\Delta u_t=|u|^{q(x)-2}u$

under initial and Dirichlet boundary value condition. By constructing a new control function and applying the Sobolev embedding inequality, the authors establish the relationship between source term and energy functional, and then decay estimates are obtained by means of Komornik's inequality and energy estimates. At last, we prove that u(x, t)=0 is asymptotic stable.

Key words: Damping term, Decay estimates, Asymptotic stability

中图分类号:

廖梦兰,郭斌. 具变指数源项和强阻尼项的波动方程解的渐近稳定性[J]. 数学物理学报, 2020, 40(1): 146-155.

Menglan Liao,Bin Guo. Asymptotic Stability of Weak Solutions to Wave Equation with Variable Exponents and Strong Damping Term[J]. Acta mathematica scientia,Series A, 2020, 40(1): 146-155.

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