RAC Rev. R. Acad. Cien. Serie A. Mat. VOL. 95 (1), 2001, pp. 155–160 Matem´ atica Aplicada / Applied Mathematics Comunicaci´ on Preliminar / Preliminary Communication On the asymptotic behavior for a damped oscillator under a sublinear friction. J. I. D´ ıaz and A. Li ˜ n´ an Abstract. We show that there are two curves of initial data for which the solutions of the corresponding Cauchy problem associated to the equation where vanish after a finite time. By using asymptotic and methods and comparison arguments we show that for many other initial data the solutions decay to , in an infinite time, as Sobre el comportamiento asint ´ otico de un oscilador amortiguado con un t´ ermino de fricci ´ on sublineal Resumen. Mostramos la existencia de dos curvas de datos iniciales para las cuales las solu- ciones correspondientes del problema de Cauchy asociado a la ecuaci´ on supuesto , se anulan identicamente despues de un tiempo finito. Mediante m´ etodos asint´ oticos y argumentos de comparaci´ on mostramos que para muchos otros datos iniciales las soluciones decaen a en un tiempo infinito, como 1. Introduction We study the asymptotic behavior of solutions of the equation (1) where and We shall work with the formulation (2) which is attaint by dividing by and by introducing the rescaling where and Notice that the -rescaling fails for . In that case there is no well defined scale for and the equation is reduced to with remaining as a parameter to characterize the dynamics. The limit case corresponds to the Coulomb friction equation (3) Presentado por Gregorio Mill´ an Recibido: 16 de Abril 2001. Aceptado: 10 de Octubre 2001. Palabras clave / Keywords: sublinear damped oscillator, Coulomb friction, extinction in a finite time. Mathematics Subject Classifications: 34D05, 70K05,70K99 c 2001 Real Academia de Ciencias, Espa˜ na. 155