The inhomogeneous total variation flow with L 1 -data

Abstract 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 u ′ - div ⁡ ( D ⁢ u | D ⁢ u | ) = f in ⁢ ( 0 , + ∞ ) × Ω , see text u^{\prime}-\operatorname{div}\biggl{(}\frac{Du}{|Du|}\biggr{)}=f\quad\text{in % }(0,+\infty)\times\Omega, where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} is a bounded open set with Lipschitz boundary and f ∈ L loc 1 ⁢ ( 0 , + ∞ ; L 1 ⁢ ( Ω ) ) {f\in L_{\rm loc}^{1}(0,+\infty;L^{1}(\Omega))} is the source term. The initial datum we consider belongs to L 1 ⁢ ( Ω ) L^{1}(\Omega) . 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 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.