Convergence to equilibrium for a hyperbolic/elliptic system modelling the viscoelastic-gravitational deformation of a layered Earth

In this communication we prove the stabilization, as t goes to infinity, of a model (which is an adaptation of the one possed by A. E. H. Love in 1911, see [8] Love, A.E.H., Some problems in Geodynamics,Cambridge University Press, New York, 1911) for the study of the displacements due to internal sources of strain in layered linear viscoelasticgravitational continua. The existence and uniqueness of weak solutions has been obtained recently in [1] Arjona, A., Díaz, J.I., Fernández, J., Rundle, J.B.,On the mathematical analysis of an elasticgravitational layered Earth model for magmatic intrusion: The stationary case. To appear in Pure and Applied Geophysics, PAGEOPH, 2007 and [2] Arjona, A., Díaz, J.I., Fernández, J., Rundle, J.B.,On the mathematical analysis of an elasticgravitational layered Earth model for magmatic intrusion: The dynamic case. To appear, 2007. Here we prove that, under some additional conditions on the data, the difference of the respective solutions converges to zero, as t goes to infinity, in a suitable functional space. Our proof uses a reformulation of the hyperbolic/elliptic system in terms of a nonlocal hyperbolic system leading to which we apply the La Salle invariance principle for a Lyapounov function involving the nonlocal terms