QUARTERLY OF APPLIED MATHEMATICS VOLUME LXXI, NUMBER 1 MARCH 2013, PAGES 183–199 S 0033-569X(2012)01282-3 Article electronically published on August 28, 2012 ON THE OVERDAMPING PHENOMENON: A GENERAL RESULT AND APPLICATIONS By GIS ` ELE RUIZ GOLDSTEIN (Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152 ), JEROME A. GOLDSTEIN (Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152 ), and GUSTAVO PERLA MENZALA (National Laboratory of Scientific Computation (LNCC/MCT), Ave. Getulio Vargas 333, Quitandinha, Petropolis, RJ, CEP 25651-070, Brasil and Federal University of Rio de Janeiro, Institute of Mathematics, P.O. Box 68530, Rio de Janeiro, RJ, Brazil ) Abstract. We study the best possible energy decay rates for a class of linear second- order dissipative evolution equations in a Hilbert space. The models we consider are generated by a positive selfadjoint operator A having a bounded inverse. Our discussion applies to important examples such as the classical wave equation, the dynamical wave equation with Wentzell boundary conditions and many others. 1. Introduction. Let d 2 y dt 2 +2a dy dt + w 2 y =0 (1.1) Received May 26, 2011. 2010 Mathematics Subject Classification. Primary 35Q99, 35L99; Secondary 47D06, 35K10, 47N20. Key words and phrases. Overdamping phenomenon, the spectral theorem, classical wave equations, Wentzell boundary conditions. Part of this work was done during the visit of GG and JG to the National Laboratory of Scientific Computation (Brasil) during July 2008. The Goldsteins are most grateful for the gracious hospitality they received from G. Perla Menzala and Carlos Moura. The third author was partially supported by a Research Grant of CNPq (Proc. 301134/2009-0) and Project Universal (Proc. 47296/2008-3) from the Brazilian Government. The third author would like to express his gratitude for such important support. He is also very thankful to Prof. E. Zuazua. Several years ago he discussed with him the overdamping phenomenon ([13]) and its implications. E-mail address : ggoldste@memphis.edu E-mail address : jgoldste@memphis.edu E-mail address : perla@lncc.br c 2012 Brown University 183 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf