# Explicit solution of a free boundary problem for heat equation

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.