A ONE-STEP OPTIMAL ENERGY DECAY FORMULA FOR INDIRECTLY NONLINEARLY DAMPED HYPERBOLIC SYSTEMS COUPLED BY VELOCITIES

被引：27

作者：

Alabau-Boussouira, Fatiha ^{[1
,2
]}

Wang, Zhiqiang ^{[3
,4
]}

Yu, Lixin ^{[5
]}

机构：

[1] Univ Lorraine, IECL, F-57045 Metz, France

[2] CNRS, UMR 7502, Delegat CNRS, LJLL,UMR 7598, F-57045 Metz, France

[3] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China

[4] Fudan Univ, Shanghai Key Lab Contemporary Appl Math, Shanghai 200433, Peoples R China

[5] Yantai Univ, Sch Math & Informat Sci, Yantai 264005, Peoples R China

来源：

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS | 2017年 / 23卷 / 02期

基金：

中国国家自然科学基金; 美国国家科学基金会;

关键词：

Energy decay; nonlinear damping; wave equation; plate equation; weighted nonlinear integral inequality; optimal-weight convexity method; INDIRECT BOUNDARY STABILIZATION; SEMILINEAR WAVE-EQUATION; INTEGRAL-INEQUALITIES; TIMOSHENKO BEAM; RATES; DISSIPATION;

D O I：

10.1051/cocv/2016011

中图分类号：

TP [自动化技术、计算机技术];

学科分类号：

0812 ;

摘要：

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in [F. Alabau, C.R. Acad. Sci. Paris S' er. I Math. 328 (1999) 10151020; F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127-150; F. Alabau, SIAM J. Control Optim. 41 (2002) 511-541; F. Alabau-Boussouira and M. Leautaud, ESAIM: COCV 18 (2012) 548-582] for the linear damping case, and in [F. Alabau-Boussouira, NoDEA 14 (2007) 643-669] for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim. 51 (2005) 61-105; F. Alabau-Boussouira, J. Differ. Equ. 248 (2010) 1473-1517].

引用

页码：721 / 749

页数：29