Can you point out some references that deal with the *obstacle problem for the heat equation*?

$$(OP) \quad\begin{cases}
\max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \mathbb{R}^n \\
u(0,\cdot) = \varphi(0,\cdot) & \text{ in } \mathbb{R}^n
\end{cases}.$$

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Works on elliptic obstacle problems appear to be much easier to find (see [Wikipedia][1], for instance). 


  [1]: https://en.wikipedia.org/wiki/Obstacle_problem