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 $$L^1(0,T;L^2(\Omega ))$$ L 1 ( 0 , T ; L 2 ( Ω ) ) for every $$T>0$$ T > 0 . As a consequence, global existence and uniqueness for data in $$L^1_{loc}(0,+\infty ;L^2(\Omega ))$$ L loc 1 ( 0 , + ∞ ; L 2 ( Ω ) ) is obtained. Our analysis retrieves previous results in a correct and complete way.