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

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.