Energy decay of the wave equation with variable coefficients and a localized half-linear dissipation in an exterior domain

详细信息    查看全文

• 作者：Jing Li (1)
  Shugen Chai (1)
  Jieqiong Wu (1)
  
  1. School of Mathematical Sciences ; Shanxi University ; Taiyuan ; 030006 ; Shanxi ; People鈥檚 Republic of China
  
• 关键词：35L05 ; 58J45 ; Energy decay ; Exterior domain ; Localized half ; linear dissipation ; Riemannian geometry method ; Variable coefficients
• 刊名：Zeitschrift für angewandte Mathematik und Physik
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：66
• 期：1
• 页码：95-112
• 全文大小：279 KB
• 参考文献：1. Nakao M.: Energy decay for the linear and semilinear wave equation in exterior domains with some Localized dissipations. Math. Z. 238(4), 781鈥?97 (2001) CrossRef
  2. Nakao M., Jung I.H.: Energy decay for the wave equations in exterior domains with some half-linear dissipation. Differ. Integral Equ. 16(8), 927鈥?48 (2003)
  3. Nakao M., Bae J.J.: Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations. Nonlinear Anal. Theory Methods Appl. 66, 301鈥?23 (2007) CrossRef
  4. Nakao M.: Existence of global solutions for the Kirchhoff-type quasilinear wave equation in exterior domains with a half-linear dissipation. Kyushu J. Math. 58(2), 373鈥?91 (2004) CrossRef
  5. Nakao M., Ono K.: Global existence to the Cauchy problem for the semilinear dissipative wave equation. Math. Z. 214(1), 325鈥?42 (1993) CrossRef
  6. Nakao M.: Remarks on the energy decay for a nonlinear wave equation with a nonlinear localized dissipative term. Nonlinear Anal. Theory Methods Appl. 73, 2158鈥?169 (2010) CrossRef
  7. Nakao M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305(3), 403鈥?17 (1996) CrossRef
  8. Todorova G.: Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Nonlinear Anal. Ser. A 41(7-8), 891鈥?05 (2000) CrossRef
  9. Todorova G., Yordanov B.: Critical exponent for a nonlinear wave equation with damping. J. Differ. Equ. 174(2), 464鈥?89 (2001) CrossRef
  10. Yao P.F.: On the observability inequalities for the exact controllability of the wave equation with variable coefficients. SIAM. J. Control Optim. 37(5), 1568鈥?599 (1999) CrossRef
  11. Chai S.G., Guo Y.X.: Boundary stabilization of wave equation with variable coefficients and memory. Differ. Integral Equ. 17, 669鈥?80 (2004)
  12. Feng S.J., Feng D.X.: Nonlinear internal damping of wave equations with variable coefficients. Acta Math. Sin. Engl. Ser. 20(6), 1057鈥?072 (2004) CrossRef
  13. Cavalcanti M.M., Khemmoudj A., Medjden M.: Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions. J. Math. Anal. Appl. 328, 900鈥?30 (2007) CrossRef
  14. Wu J.Q., Li S.J.: Stabilization of wave equation with variable coefficients by nonlinear boundary feedback. J. Syst. Sci. Compl. Engl. Ser. 24, 875鈥?82 (2011) CrossRef
  15. Wu J.Q., Li S.J., Chai S.G.: Exact controllability of wave equations with variable coefficients coupled in parallel. Asian J. Control 12(5), 650鈥?55 (2010) CrossRef
  16. Wu J.Q., Li S.J., Chai S.G.: Uniform decay of the solution to a wave equation with memory conditions on the boundary. Nonlinear Anal. Theory Methods Appl. 73, 2213鈥?220 (2010) CrossRef
  17. Yao, P.F.: Modeling and Control in Vibrational and Structural Dynamics, A Differential Geometric Approach. Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press (2011)
  18. Yao P.F.: Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation. Chin. Ann. Math. Ser. B 31(1), 59鈥?0 (2010) CrossRef
  19. Wu H., Shen C.L., Yu Y.L.: An Introduction to Riemannian Geometry. Peking University Press, Beijing (1989)
  20. Lions J.L., Strauss W.A.: Some nonlinear evolution equations. Bull. Soc. Math. France 93, 43鈥?6 (1965)
  21. Tataru D.: The \({X_{\theta}^{s}}\) spaces and unique continuation for solutions to the semilinear wave equations. Commun. Partial Differ. Equ. 21, 841鈥?87 (1996) CrossRef
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

The wave equation of variable coefficients in an exterior domain with a localized half-linear dissipation is considered. The energy decay estimate for the system is established by Riemannian geometry method.