The Oberbeck–Boussinesq problem modified by a thermo-absorption term

We consider the Oberbeck–Boussinesq problem with an extra coupling, establishing a suitable relation between the velocity and the temperature. Our model involves a system of equations given by the transient Navier–Stokes equations modified by introducing the thermo-absorption term. The model involves also the transient temperature equation with nonlinear diffusion. For the obtained problem, we prove the existence of weak solutions for any N⩾2 and its uniqueness if N=2. Then, considering a low range of temperature, but upper than the phase changing one, we study several properties related with vanishing in time of the velocity component of the weak solutions. First, assuming the buoyancy forces field extinct after a finite time, we prove the velocity component will extinct in a later finite time, provided the thermo-absorption term is sublinear. In this case, considering a suitable buoyancy forces field which vanishes at some instant of time, we prove the velocity component extinct at the same instant. We prove also that for non-zero buoyancy forces, but decaying at a power time rate, the velocity component decay at analogous power time rates, provided the thermo-absorption term is superlinear. At last, we prove that for a general non-zero bounded buoyancy force, the velocity component exponentially decay in time whether the thermo-absorption term is sub or superlinear.