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

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

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

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

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

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

Since a bounty has been offered for this question, I'll write down what I feel is missing in the current (very nice) answer and that I'd like to see:

• complete argument for the existence (with references too)
• further details on the representation of solutions using the heat kernel
• references about numerical analysis of the problem (and Matlab/Mathematica codes)
• references on physical motivations for the problem.

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

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

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

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

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

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

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

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

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

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

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

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