The obstacle problem for fractional heat equations with memory

We study the obstacle problem for the evolution equation $\partial_t^αu+(-Δ)^s u=0$ in the whole space $\mathbb{R}^d$, where $\partial_t^α$ with $α\in(0,1)$ is the Caputo derivative in time and $(-Δ)^s$ with $s\in(0,1]$ is the fractional Laplacian in space ($s<1$) or the classical Laplacian ($s=1$). We prove that the problem has at most one bounded viscosity solution, and we obtain existence as the limit of an explicit monotone finite-difference scheme, which combines the L1 discretization of the Caputo derivative with powers of the discrete Laplacian. The scheme is also used to establish regularity estimates in space and time. We then study the large-time behaviour: solutions converge, along subsequences, to stationary solutions, and we identify the limit when the initial datum coincides with the obstacle (the minimal stationary solution) or has a limit $c_*$ at infinity. In the latter case the dimension plays a role, as a consequence of the recurrence of the underlying process: the limit is the unique stationary solution with limit $c_*$ at infinity if $d>2s$, and a constant (the maximum between $c_*$ and the supremum of the obstacle) if $d\le2s$. Numerical experiments in one dimension, based on an explicit stationary solution, illustrate the convergence of the scheme and the relaxation towards the stationary profile.