The inhomogeneous Total Variation Flow with $L^1$-data

This paper is devoted to the study of the Dirichlet problem for the parabolic equation driven by the $1$--Laplacian operator under minimal integrability assumptions. Specifically, we consider \begin{equation*} u'-\Div(Du/|D u|)=f\qquad\text{ in } (0,+\infty)\times\Omega\,, \end{equation*} where $\Omega\subset\R^N$ is a bounded open set with Lipschitz boundary, $u_0\in L^1(\Omega)$ is the initial datum, and $f\in L_{loc}^1(0,+\infty; L^1(\Omega))$ is the source term. We establish the existence and uniqueness of entropy solutions in this low-regularity setting. Our approach relies on an approximation scheme and an entropy formulation adapted to the \mbox{$1$--Laplacian} structure. Additional results include comparison between solutions, further regularity when data have higher integrability and an analysis of the long-time decay of solutions in the homogeneous case.