{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:22:32Z","timestamp":1772295752789,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2015,12,29]],"date-time":"2015-12-29T00:00:00Z","timestamp":1451347200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Syst Sci Complex"],"published-print":{"date-parts":[[2016,6]]},"DOI":"10.1007\/s11424-015-4233-7","type":"journal-article","created":{"date-parts":[[2015,12,29]],"date-time":"2015-12-29T07:46:08Z","timestamp":1451375168000},"page":"657-680","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Decay rates of the hyperbolic equation in an exterior domain with half-linear and nonlinear boundary dissipations"],"prefix":"10.1007","volume":"29","author":[{"given":"Yuxiang","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pengfei","family":"Yao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,12,29]]},"reference":[{"issue":"5","key":"4233_CR1","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.1137\/0330055","volume":"30","author":"C Bardos","year":"1992","unstructured":"Bardos C, Lebeau G, and Rauch J, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM Journal on Control and Optimization, 1992, 30(5): 1024\u20131065.","journal-title":"SIAM Journal on Control and Optimization"},{"key":"4233_CR2","volume-title":"Exact Controllability and Stabilization. RAM: Research in Applied Mathematics","author":"V Komornik","year":"1994","unstructured":"Komornik V, Exact Controllability and Stabilization. RAM: Research in Applied Mathematics, Masson, Paris, 1994."},{"issue":"3","key":"4233_CR3","doi-asserted-by":"crossref","first-page":"507","DOI":"10.57262\/die\/1370378427","volume":"6","author":"I Lasiecka","year":"1993","unstructured":"Lasiecka I and Tataru D, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, 1993, 6(3): 507\u2013533.","journal-title":"Differential and Integral Equations"},{"issue":"1","key":"4233_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/1030001","volume":"30","author":"J L Lions","year":"1988","unstructured":"Lions J L, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 1988, 30(1): 1\u201368.","journal-title":"SIAM Review"},{"issue":"2","key":"4233_CR5","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF01182480","volume":"25","author":"I Lasiecka","year":"1992","unstructured":"Lasiecka I and Triggiani R, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Applied Mathematics and Optimization, 1992, 25(2): 189\u2013224.","journal-title":"Applied Mathematics and Optimization"},{"issue":"2","key":"4233_CR6","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.na.2005.11.026","volume":"66","author":"M Nakao","year":"2007","unstructured":"Nakao M and Bae J J, Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations, Nonlinear Analysis: Theory, Methods & Applications, 2007, 66(2): 301\u2013323.","journal-title":"Nonlinear Analysis: Theory, Methods & Applications"},{"issue":"3","key":"4233_CR7","doi-asserted-by":"crossref","first-page":"851","DOI":"10.2969\/jmsj\/06430851","volume":"64","author":"M Nakao","year":"2012","unstructured":"Nakao M, Energy decay for a nonlinear generalized Klein-Gordon equation in exterior domains with a nonlinear localized dissipative term, Journal of the Mathematical Society of Japan, 2012, 64(3): 851\u2013883.","journal-title":"Journal of the Mathematical Society of Japan"},{"issue":"2","key":"4233_CR8","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1006\/jdeq.2000.3933","volume":"174","author":"G Todorova","year":"2001","unstructured":"Todorova G and Yordanov B, Critical exponent for a nonlinear wave equation with damping, Journal of Differential Equations, 2001, 174(2): 464\u2013489.","journal-title":"Journal of Differential Equations"},{"issue":"2","key":"4233_CR9","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1137\/0328025","volume":"28","author":"E Zuazua","year":"1990","unstructured":"Zuazua E, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM Journal on Control and Optimization, 1990, 28(2): 466\u2013477.","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"5","key":"4233_CR10","doi-asserted-by":"crossref","first-page":"1568","DOI":"10.1137\/S0363012997331482","volume":"37","author":"P F Yao","year":"1999","unstructured":"Yao P F, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM Journal on Control and Optimization, 1999, 37(5): 1568\u20131599.","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"2","key":"4233_CR11","doi-asserted-by":"crossref","first-page":"900","DOI":"10.1016\/j.jmaa.2006.05.070","volume":"328","author":"M M Cavalcanti","year":"2007","unstructured":"Cavalcanti M M, Khemmoudj A, and Medjden M, Uniform stabilization of the damped Cauchy- Ventcel problem with variable coefficients and dynamic boundary conditions, Journal of Mathematical Analysis and Applications, 2007, 328(2): 900\u2013930.","journal-title":"Journal of Mathematical Analysis and Applications"},{"issue":"6","key":"4233_CR12","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1007\/s10114-004-0394-3","volume":"20","author":"S J Feng","year":"2004","unstructured":"Feng S J and Feng D X, Nonlinear internal damping of wave equations with variable coefficients, Acta Mathematica Sinica, 2004, 20(6): 1057\u20131072.","journal-title":"Acta Mathematica Sinica"},{"issue":"12","key":"4233_CR13","doi-asserted-by":"crossref","first-page":"5961","DOI":"10.1016\/j.na.2009.05.018","volume":"71","author":"B Z Guo","year":"2009","unstructured":"Guo B Z and Shao Z C, On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback, Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(12): 5961\u20135978.","journal-title":"Nonlinear Analysis: Theory, Methods & Applications"},{"issue":"1","key":"4233_CR14","doi-asserted-by":"crossref","first-page":"71","DOI":"10.3233\/ASY-2008-0867","volume":"57","author":"I Hamchi","year":"2008","unstructured":"Hamchi I, Uniform decay rates for second-order hyperbolic equations with variable coefficients, Asymptotic Analysis, 2008, 57(1): 71\u201382.","journal-title":"Asymptotic Analysis"},{"issue":"1","key":"4233_CR15","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1006\/jmaa.1999.6348","volume":"235","author":"I Lasiecka","year":"1999","unstructured":"Lasiecka I, Triggiani R, and Yao P F, Inverse\/observability estimates for second-order hyperbolic equations with variable coefficients, Journal of Mathematical Analysis and Applications, 1999, 235(1): 13\u201357.","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"4233_CR16","doi-asserted-by":"crossref","unstructured":"Li J, Chai S, and Wu J, Energy decay of the wave equation with variable coefficients and a localized half-linear dissipation in an exterior domain, Zeitschrift F\u00fcr Angewandte Mathematik und Physik, DOI: 10.1007\/s00033-013-0385-2.","DOI":"10.1007\/s00033-013-0385-2"},{"issue":"3","key":"4233_CR17","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1093\/imamci\/20.3.335","volume":"20","author":"S E Rebiai","year":"2003","unstructured":"Rebiai S E, Uniform energy decay of Schr\u00f6dinger equations with variable coefficients, IMA Journal of Mathematical Control and Information, 2003, 20(3): 335\u2013345.","journal-title":"IMA Journal of Mathematical Control and Information"},{"key":"4233_CR18","doi-asserted-by":"crossref","DOI":"10.1201\/b11042","volume-title":"Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach","author":"P F Yao","year":"2011","unstructured":"Yao P F, Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach, CRC Press, 2011."},{"issue":"1","key":"4233_CR19","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s11401-008-0421-2","volume":"31","author":"P F Yao","year":"2010","unstructured":"Yao P F, Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation, Chinese Annals of Mathematics, Series B, 2010, 31(1): 59\u201370.","journal-title":"Chinese Annals of Mathematics, Series B"},{"issue":"2","key":"4233_CR20","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s00245-009-9088-7","volume":"61","author":"P F Yao","year":"2010","unstructured":"Yao P F, Boundary controllability for the quasilinear wave equation, Applied Mathematics and Optimization, 2010, 61(2): 191\u2013233.","journal-title":"Applied Mathematics and Optimization"},{"issue":"1","key":"4233_CR21","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.jde.2007.06.014","volume":"241","author":"P F Yao","year":"2007","unstructured":"Yao P F, Global smooth solutions for the quasilinear wave equation with boundary dissipation, Journal of Differential Equations, 2007, 241(1): 62\u201393.","journal-title":"Journal of Differential Equations"},{"key":"4233_CR22","first-page":"371","volume-title":"Control Conference (ASCC)","author":"P F Yao","year":"2011","unstructured":"Yao P F, How to understand Riemannian geometry is a necessary tool for control of the wave equation with variable coefficients, Control Conference (ASCC), 2011 8th Asian, 2011, 371\u2013374."},{"key":"4233_CR23","volume-title":"An introduction to Riemannian Geometry","author":"H Wu","year":"1989","unstructured":"Wu H, Shen C L, and Yu Y L, An introduction to Riemannian Geometry, Beijing University Press, Beijing, 1989 (in Chinese)."},{"issue":"11\u201312","key":"4233_CR24","doi-asserted-by":"crossref","first-page":"2069","DOI":"10.1080\/03605309908821495","volume":"24","author":"I Lasiecka","year":"1999","unstructured":"Lasiecka I and Ong J, Global solvability and uniform decays of solutions to quaslinear equation with nonlinear boundary dissipation, Communications in Partial Differential Equations, 1999, 24(11\u201312): 2069\u20132107.","journal-title":"Communications in Partial Differential Equations"},{"issue":"5\u20136","key":"4233_CR25","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1080\/03605309608821210","volume":"21","author":"D Tataru","year":"1996","unstructured":"Tataru D, The X s \u03b8 spaces and unique continuation for solutions to the semilinear wave equations, Communications in Partial Differential Equations, 1996, 21(5\u20136): 841\u2013887.","journal-title":"Communications in Partial Differential Equations"}],"container-title":["Journal of Systems Science and Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-015-4233-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11424-015-4233-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-015-4233-7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T09:12:53Z","timestamp":1692177173000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11424-015-4233-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,29]]},"references-count":25,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,6]]}},"alternative-id":["4233"],"URL":"https:\/\/doi.org\/10.1007\/s11424-015-4233-7","relation":{},"ISSN":["1009-6124","1559-7067"],"issn-type":[{"value":"1009-6124","type":"print"},{"value":"1559-7067","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,12,29]]}}}