We prove the existence of a finite extinction time for the solutions of the Dirichlet problem for the total variation flow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time. The asymptotic profile of the solutions of the Dirichlet problem is also studied. It is shown that the profiles are nonzero solutions of an eigenvalue-type problem that seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations.

The first and fourth authors have been partially support by the Spanish DGICYT, Project PB98-1442, the third author by the REN2000-0766. The second author acknowledges partial support by the TMR European Project ‘‘Viscosity Solutions and their Applications’’, Reference FMRXCT98-0234 and the PNPGC Project BFM2000-0962-C02-01.