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?
I want to solve a parabolic obstacle problem, written as a variational inequality: For almost all $t\in [0,T]$
\begin{align*}
\langle u'(t), v - u(t)\rangle +a(u(t),v-u(t)) \geq \langle f(t),v-u(t)\rangle \quad \forall v \in K
\end{align*}
with $K = \{v \in H^1_0(\Omega) ~\vert ~ v \geq \chi ~ ...
I want to solve a parabolic obstacle problem, written as a variational inequality: For almost all $t\in [0,T]$
\begin{align*}
\langle u'(t), v - u(t)\rangle +a(u(t),v-u(t)) \geq \langle f(t),v-u(t)\rangle \quad \forall v \in K
\end{align*}
with $K = \{v \in H^1_0(\Omega) ~\vert ~ v \geq \chi ~ ...
