On the time fractional heat equation with obstacle

Abstract

We study a Caputo time fractional degenerate diffusion equation which we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any $\alpha \in (0,1)$ to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with $\alpha$ is the convergence speed. We also study the problem from the numerical point of view, comparing some finite different approaches, and showing the results of some tests. These results extend what recently proved in [1] for the case $\alpha =1$.