Spatial and time localization of solutions of the Boussinesq system with nonlinear thermal diffusion (Q1582114)

The author studies a quasilinear system of equations which generalizes the Boussinesq approximation. He concentrates on the properties of solutions to the scalar equation \((\theta^q)_t + (u\cdot \nabla) \theta^q - \Delta \theta = 0\) with \(u\) a given function (sufficiently smooth) and \(q\) a positive number. For \(q<1\) and \(u \in L^\infty((0,T); C^{0,1}(\overline \Omega))\) he proves the finite speed of propagation and the waiting time property along the characteristics, and for \(u \in L^\infty(Q_T)\) (i.e. the characteristics are not well defined) he shows a weaker version of the finite speed of propagation. Finally, for a weak solution \((u,\theta)\) and \(q>1\) he also proves that \(\theta\) has the extinction in finite time property, i.e. the solution is identically zero after some finite time. The proofs are based on the energy method.