Existence and uniqueness of weak solutions to a parabolic nonlocal 1-Laplacian equation

TL;DR

This paper proves the existence and uniqueness of weak solutions for a nonlocal parabolic 1-Laplacian equation, showing that solutions are 1/2-Hölder continuous in time, extending previous local case results.

Contribution

It introduces a novel approach using Rothe time-discretization to establish solution properties for a nonlocal 1-Laplacian, including temporal regularity.

Findings

Existence and uniqueness of weak solutions are proven.

Weak solutions exhibit 1/2-Hölder continuity in time.

The method extends results from local to nonlocal equations.

Abstract

We consider a class of parabolic nonlocal $1$-Laplacian equation \begin{align*} u_t+(-\Delta)^s_1u=f \quad \text{ in }\Omega\times(0,T]. \end{align*} By employing the Rothe time-discretization method, we establish the existence and uniqueness of weak solutions to the equation above. In particular, different from the previous results on the local case, we infer that the weak solution maintains $\frac{1}{2}$-H\"{o}lder continuity in time.