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PACS numbers: 98.80.-k, 04.20.-q, 95.36.+x, 95.35.+d Introduction.From a phenomenological and empiric point of view, cosmological observations are usually tested as a first attempt by fitting them to the so-called Λ-CDM model, which is a FLRW spacetime with flat spacelike sections, whose source is dust (cold dark matter CDM) and a Λ field (dark energy) (see [1,2] for comprehensive reviews). Although the universe may be nearly homogeneous at scales larger than the so-called homogeneity scale (over 150-300 Mpc), thus justifying the use of linear perturbations in its dynamical study, it is clearly inhomogeneous at smaller scales in which structure formation mostly involving CDM has taken place. Hence, the study of inhomogeneous sources made of dust and a nonzero "Λ field" is a very relevant topic. In particular, spacetimes of this type of source with spherical symmetry provide simple but nontrivial inhomogeneous generalizations of the Λ-CDM model.

This Letter considers the generalized second law of gravitational thermodynamics in two scenarios featuring a phantom dominated expansion plus a black hole. The law is violated in both scenarios. *

The generalized second law of gravitational thermodynamics is applied to the present era of accelerated expansion of the Universe. In spite of the fact that the entropy of matter and relic gravitational waves inside the event horizon diminish, the mentioned law is fulfilled provided the expression for the entropy density of the gravitational waves satisfies a certain condition. *

Abstract. Quasi-local scalar variables approach is applied to a spherically symmetric inhomogeneous Lemaître-Tolman-Bondi metric containing a mixture of non-relativistic cold dark matter and coupled dark energy with constant equation of state. The quasi-local coupling term considered is proportional to the quasi-local cold dark matter energy density and a quasi-local Hubble factor-like scalar via a coupling constant α. The autonomous numerical system obtained from the evolution equations is classified for different choices of the free parameters: the adiabatic constant of the dark energy w and α. The presence of a past attractor in a non-physical region of the energy densities phase-space of the system makes the coupling term non physical when the energy flows from the matter to the dark energy in order to avoid negative values of the dark energy density in the past. On the other hand, if the energy flux goes from dark energy to dark matter, the past attractor lays in a physical region. The system is also numerically solved for some interesting initial profiles leading to different configurations: an ever expanding mixture, a scenario where the dark energy is completely consumed by the nonrelativistic matter by means of the coupling term, a scenario where the dark energy disappears in the inner layers while the outer layers expand as a mixture of both sources, and, finally, a structure formation toy model scenario, where the inner shells containing the mixture collapse while the outer shells expand.PACS numbers: 98.80.-k, 04.20.-q, 95.36.+x, 95.35.+d