Consider the free boundary problem

$$ \min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\ u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\ u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \text{ in } (0,T) $$

Can we obtain an explicit solution for it?


From the maximum principle, the solution $v$ of $$v_t=v_{xx}+1,$$ together with the data $v=0$ at the boundary and initial time, is positive. Therefore your $u$ is nothing but $v$. It turns out that it can be expressed explicitly, using the heat kernel.

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