Some qualitative properties for the total variation flow

Andreu, Fuensanta; Caselles, Vicente; Díaz, Jesus Ildefonso; Mazón, José

doi:http://dx.doi.org/10.1006/jfan.2001.3829

Some qualitative properties for the total variation flow

Citation

Andreu F, Caselles V, Díaz JI, Mazón JM. Some qualitative properties for the total variation flow. J Funct Anal. 2002 Feb 1;188(2):516-47. DOI: 10.1006/jfan.2001.3829

Abstract

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.