Spatial and time localization of solutions of the Boussinesq system with nonlinear thermal diffusion

The Boussinesq system arises in Fluid Mechanics when motion is governed by density gradients caused by temperature or concentration differences. In the former case, and when thermodynamical coefficients are regarded as temperature dependent, the system consists of the Navier-Stokes equations and the non linear heat equation coupled through the viscosity, bouyancy and convective terms. In this paper we show that for certain types of nonlinearities in the diffusion term of the heat equation solutions of Boussinesq system can exhibit two kinds of behaviours: the finite speed of propagation of the support and the extinction in finite time, both for the temperature component. Because of the lack of a comparison principle we use energy methods to proof the ocurrence of these properties