Existence and uniqueness for the inhomogeneous 1-Laplace evolution equation revisited

Abstract

In this paper we deal with an inhomogeneous parabolic Dirichlet problem involving the 1-Laplacian operator. We show the existence of a unique solution when data belong to $L^1(0,T;L^2(\Omega))$ for every $T>0$. As a consequence, global existence and uniqueness for data in $L^1_{loc}(0,+\infty;L^2(\Omega))$ is obtained. Our analysis retrieves the results of \cite{SW} in a correct and complete way.