I'm trying to understand the compatibility conditions for regularity of second order parabolic equations. Let's consider the equation $u_t - Lu = f$ with $u(0)=g$ on $\Omega \times [0,T)$ with $u = 0$ on $\Gamma_T$, the parabolic boundary of $\Omega \times [0,T)$.

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 to conclude smoothness up to the boundary (I'm presuming $f$ and $\Omega$ are smooth). 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?