This lecture deals with evolution problem for the 1-Laplacian with mixed boundary conditions on bounded open sets. We prove existence and uniqueness of strong solutions for square-integrable data means of the theory of maximal monotone operator. We also see that if the flux on the boundary is 1 (that is, the maximum possible) then these strong solutions can be seen as the large solutions introduced in [Journal Functional Analysis 262 (2012), 1566–1602 ]. We give explicit examples of solutions. Joint work with N. Igbida and J. Toledo