具有 Cauchy-Ventcel边界条件的有界域中亚临界半线性波方程的稳定与控制

doi:10.1007/s10483-008-0610-x

Applied Mathematics and Mechanics (English Edition) ›› 2008, Vol. 29 ›› Issue (6): 787-800 .doi: https://doi.org/10.1007/s10483-008-0610-x

具有 Cauchy-Ventcel边界条件的有界域中亚临界半线性波方程的稳定与控制

A.卡努里， N.麦希迪

贝加亚大学应用数学实验室，06000 贝加亚, 阿尔及利亚

收稿日期:2007-04-19 修回日期:2008-04-30 出版日期:2008-06-18 发布日期:2008-06-18
通讯作者: A.卡努里

Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions

A. Kanoune,N. Mehidi

Laboratory of Applied Mathematics, Department of Mathematics, University of Bejaia, 06000 Bejaia, Algeria

Received:2007-04-19 Revised:2008-04-30 Online:2008-06-18 Published:2008-06-18
Contact: Aomar Kanoune

摘要/Abstract

摘要： 分析R^N的有界域中半线性波方程解的指数衰减特性，有界域具有Cauchy-Ventcel型边界条件，并且球体外部作用着阻尼项。在对非线性作出适当又自然的假设后，倘若非线性在无穷大处为亚临界时，有限能量解的指数衰减性满足局部一致性。粗略地说，亚临界性意味着，在无穷大处非线性增长率次数不大于5。B.Dehman、G.Lebeau和E.Zuazua得到了R³和R^N中的经典能量（用于估计局限于球体外部以能量形式表示的解的总能量）不等式和Strichartz估计的结果，使得研究R^N有界域（域内及其边界上是亚临界非线性，边界为Cauchy-Ventcel型连续性）中半线性波方程的稳定性与可控性成为可能。

Abstract: We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy
solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p<5. The results obtained in R³ and R^N by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy (which estimate the total
energy of solutions in terms of the energy localized in the exterior of a ball) and on Strichartz's estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^Nwith a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.

中图分类号:

A.卡努里;N.麦希迪. 具有 Cauchy-Ventcel边界条件的有界域中亚临界半线性波方程的稳定与控制[J]. Applied Mathematics and Mechanics (English Edition), 2008, 29(6): 787-800 .

A. Kanoune;N. Mehidi. Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions[J]. Applied Mathematics and Mechanics (English Edition), 2008, 29(6): 787-800 .

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具有 Cauchy-Ventcel边界条件的有界域中亚临界半线性波方程的稳定与控制

Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions

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