On the influence of a dissipative boundary on the energy decay for a porous elastic solid q Barbara Lazzari * , Roberta Nibbi Department of Mathematics, University of Bologna, 5 Piazza di Porta S. Donato, 40126 Bologna, Italy article info Article history: Received 12 November 2008 Received in revised form 23 December 2008 Available online 1 February 2009 Keywords: Porous-elasticity Dissipative boundary Exponential stability abstract In this paper we study the asymptotic behavior of a porous elastic solid with a dissipative boundary. We prove that the energy exponentially decays when the porosity viscosity is present. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Recently interesting results have been obtained in the study of the temporal decay behavior in porous-elasticity (Casas and Quintanilla, 2005, Magaña and Quintanilla, 2006, Muñoz-Rivera and Quintanilla, 2008, Pamplona et al., 2009, Quintanil- la, 2003, Soufyane, 2008). By limiting the attention to the one-dimensional models, Quintanilla (2003) showed that the dis- sipation due to the porous viscosity is not strong enough to obtain exponential decay of the energy, while Casas and Quintanilla (2005) and Magaña and Quintanilla (2006) obtained the exponential decay by coupling the porous dissipation with thermal and viscoelastic effects respectively. In this paper we focus our attention on the three-dimensional model and, instead of coupling the porous viscosity with other internal dissipative effects, we consider a dissipative porous elastic solid with a dissipative boundary. As in elasticity and thermo-elasticity (Bosello et al., 2007, Lazzari and Nibbi, 2007, 2008), the boundary dissipation turns out to be sufficient for the exponential decay of the energy. The outline of the paper is the following. In Section 2 the problem is set up. In Section 3 we prove existence, uniqueness and regularity of the solution. In Section 4, after developing the needed energy estimates, we prove the exponential decay of the solution. 2. Setup of the problem Let X be a bounded open set of R 3 with regular boundary @X. 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.01.010 q Research performed under the auspices of G.N.F.M. – I.N.d.A.M. and partially supported by Italian M.I.U.R. and the University of Bologna within the project ‘‘Modelli matematici di transizione di fase per sistemi complessi”. * Corresponding author. Tel.: +39 512094495; fax: +39 512094490. E-mail addresses: lazzari@dm.unibo.it (B. Lazzari), nibbi@dm.unibo.it (R. Nibbi). Mechanics Research Communications 36 (2009) 581–586 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom