Evolution problem for the $1$-Laplacian with mixed boundary conditions

This paper deals with evolution problem for the $1$-Laplacian with mixed boundary conditions on a bounded open set $\mathrm{\Omega }$ of ${\mathbb{R}}^N$. We prove existence and uniqueness of strong solutions for data in $L^2(\mathrm{\Omega })$ by mean 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 \cite{MP}. We give explicit examples of solutions.