I'm trying to understand the compatibility conditions for regularity of second order parabolic equations. We need the assumptions that $\frac{d^{m-1}f}{d^{m-1}}(0) - Lg_{m-1} \in H_0^1(\Omega)$ for the $L^2$ theory to work. As far as I can tell this is required so that we don't have the following situation:

$u_t - u_xx = 0$ on $[0,1] \times [0,1]$ with $u(0,x)=1$ and $u(t,0)=u(t,1)=0$. Here we clearly can't have differentiability up to the boundary since $1 \neq 0$. However after time $t = 0$ it would appear to me that this problem goes away, ie. that the solutions are smooth up to the boundary except at $t=0$. Is this the case? Can people perhaps share insightful examples?