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

TL;DR

This paper establishes the existence and uniqueness of solutions for an inhomogeneous 1-Laplacian evolution equation with data in certain function spaces, improving the theoretical understanding of such parabolic problems.

Contribution

It provides a rigorous proof of existence and uniqueness for the inhomogeneous 1-Laplacian evolution equation with broader data conditions, revisiting and completing previous results.

Findings

Existence of solutions for data in L^1(0,T;L^2(Ω))

Uniqueness of solutions under these conditions

Global existence for data in L^1_{loc}(0,+∞;L^2(Ω))

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.