Nonlinear Analysis 42 (2000) 1271 – 1291 www.elsevier.nl/locate/na Exponential decay of the solutions of the beams system Milton de Lacerda Oliveira a ; * , Osmundo Alves de Lima b a Dept Matematica e Estatistica, Universidade Federal da Paraiba, 58109-970 Campina Grande PB, Brazil b Dept Matematica, Universidade Estadual da Paraiba, Campina Grande, Brazil Received 10 May 1998; accepted 3 October 1998 Keywords: Exponential decay; Beams system; Hilbertian integral 1. Introduction The equation u tt + u xxxx −   +  L 0 u 2  (;t )d  u xx =0; where  and  are constants, with  positive, introduced by Woinowsky and Krieger [15], represents a mathematical model to describe the transversal vibrations of a beam of length L whose extremes are maintained at a xed distance and for which the stress variation of the beam due to its stretching is considered. Theoretically and experimentally, Haight and King [6] studied the vibrating thin thickness beams, and they observed that, in some circumstances, its oscillations are non-planar. Ho et al. [7] studied oscillations non-planar of an elastic beam of length L; whose extremes are maintained at a xed distance and they proposed a mathematical model that in the multidimensional case is as follows: u tt + 2 u − M (|∇u| 2 + |∇v| 2 )u + u t = f; (1.1) v tt + 2 v − M (|∇u| 2 + |∇v| 2 )v + v t = g: (1.2) * Corresponding author. Tel.: +83-3101110. E-mail addresses: oliveira@dme.ufpb.br (M.d. Lacerda Oliveira), oalves@dme.ufpb.br (O.A. de Lima) 0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(99)00155-8