PM  >> Vol. 2 No. 3 (July 2012)

    Blow-Up and Asymptotic Behavior of Global Solution of Damped Wave Equation with Dynamic Boundary Conditions

  • 全文下载: PDF(202KB) HTML    PP.144-151   DOI: 10.12677/PM.2012.23023  
  • 下载量: 2,328  浏览量: 7,234  


靳 妞:河南工业大学数学系;

波动方程动力边界条件凸性引理解的爆破衰减估计Wave Equation; Dynamic Boundary Condition; Convexity Lemma; Blow-Up of Solution; Energy Decay



In this paper, the blow-up and asymptotic behavior of global solution of damped wave equation with dynamic boundary condition are discussed. By the convexity lemma and unstable set, the sufficient con- dition of the solution of the wave equation with negative and positive initial energy respectively are obtained. With the help of Nakao and stable set, the energy decay of the solution is given.

靳妞, 张宏伟, 呼青英. 动力边界条件的阻尼波动方程解的爆破性和渐近性[J]. 理论数学, 2012, 2(3): 144-151.


[1] M. Pellicer, J. Sola-Morales. Analysis of a viscoelastic spring-mass model. Mathematical Analysis and Applications, 2004, 294(2): 687-698.
[2] M. G. Van Dalsen. On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions. Appli- cable Analysis, 1994, 53(1-2): 41-54.
[3] O. Morgul, B. P. Rao and F. Conrad. On the stabilization of a cable with a tip mass. IEEE Transaction on Automatic Control, 1994, 39(10): 2140-2145.
[4] W. D. Zhu, B. Z. Guo. On hybrid boundary control of flexible systems. Transactions of the ASME, 1997, 119: 836-839.
[5] A Mifdal. Uniform stabilization of a hybrid system. Comptes Rendus de l’Acadrmie des Sciences, 1997, 324(1): 37-42.
[6] C. F. Baicu, C. D. Rahn and D. M. Dawson. Exponentially stabilizing boundary control of string-mass systems. Journal of Vibration and Control, 1998, 5(3): 491-502.
[7] B. Z. Guo, C. Z. Xu. On the spectrum-determined growth condition of a vibration cable with a tip mass. IEEE Transaction on Automatic Con- trol, 2000, 45(1): 89-93.
[8] F. Conrad, G. O’Dowd and F.-Z. Saouri. Asymptotic behavior for a model of flexible cable with tip mass. Asymptotic Analysis, 2002, 30(3-4): 313-330.
[9] B. P. Rao. Decay estimates of solutions for a hybrid system of flexible structures. European Journal of Applied Mathematics, 1994, 4(3): 303-319.
[10] E. Feireisl, G. O’Dowd. Stabilization of a hybrid system with a nonlinear nonmonotone feedback. ESAIM: Control, Optimisation and Calculus of Variations, 1999, 4: 133-135.
[11] B. d’Andrea-Novel, F. Boustany, F. Conrad and B. P. Rao. Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane. Mathematics of Control, Signals and Systems, 1994, 7(1): 1-22.
[12] J. Vancostenoble. Strong stabilization (via weak stabilization) of hybrid systems with a nonmonotone feedback. ESAIM: Proceedings, 2000, 8: 157-159.
[13] S. M. Shahruz. Boundary control of a nonlinear axially moving string. International Journal of Robust and Nonlinear Control, 2000, 10(1): 17-25.
[14] M. Pellicer, J. Sola-Morales. Spectral analysis and limit behaviours in a spring-mass system. Communications on Pure and Applied Analysis, 2008, 7(3): 563-577.
[15] 呼青英, 张宏伟. 混合Cable-Mass动力系统的一致稳定性[J]. 动力与控制学报, 2007, 5(1): 27-29.
[16] J. A. Burns, B. B. King. Optimal sensor location for robust control of distributed paramater systems. Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista, December 1994: 3967-3972.
[17] A. S. Ackleh, H. T. Banks and G. A. Pinter. Well-posedness results for models of elastomers. Journal of Mathematical Analysis and Applica- tion, 2002, 258: 440-456.
[18] M. Pellicer. Large time dynamics of a nonlinear spring-mass-damper model. Nonlinear Analysis, 2008, 69(9): 3110-3127.
[19] S. Gerbi, B. Said-Houari. Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Advances in Differential Equations, 2008, 13(11): 1051-1074.
[20] S. Gerbi, B. Said-Houari. Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions. Nonlinear Analysis, 2011, 74(17): 7137-7150.
[21] 李玉环, 刘盈盈, 穆春来. 动态边界下一类强阻尼波动方程解的爆破[J]. 西南大学学报, 2011, 33(7): 10-15.
[22] W. Littman, L. Markus. Stabilization of a hybrid system of elasticity by feedback boundary damping. Annali di Matematica Pura ed Applicata, 1998, 152: 281-330.
[23] K. T. Andrews, K. L. Kuttler and M. Shillor. Second order evolution equations with dynamic boundary conditions. Journal of Mathematical Analysis and Applications, 1996, 197: 781-795.
[24] Q. Y. Hu, C. K. Zhu and X. Z. Zhang. Energy decay estimates for an Euller-Bernoulli beam with a tip mass. Annals of Differential Equations, 2009, 25(2): 161-164.
[25] G. Autuori, P. Pucci. Kirchhoff systems with dynamic boundary conditions. Nonlinear Analysis, 2010, 73(7): 1952-1965.
[26] H. A. Levine. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM Journal on Mathematical Analysis, 1974, 5(1): 138-146.
[27] G. Todorova. Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Journal of Mathematical Analysis and Applications, 1999, 239(1): 213-226.
[28] L. Payne, O. Sattinger. Saddle points and instability on nonlinear hyperbolic equations. Israel Journal of Mathematics, 1973, 22(3-4): 273-303.
[29] M. Nakao, K. Ono. Global existence to the Cauchy problem of the semilinear evolution equations with a nonlinear with a nonlinear dissipation. Funkcialaj Ekvacioj, 1995, 38: 417-431.