Let $\Omega$ be a bounded Lipschitz domain (can be smooth if necessary). Consider the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ $$\text{some Robin boundary condition (B) for $u$}$$ We know that $u$ lies in some Bochner space, and the first equation holds a.e.
My question is, is the solution of the heat equation always in $C^2(\Omega)$ (for fixed time), no matter what the boundary condition (B) is? I am asking for $C^2$ only on $\Omega$, not its closure so I can imagine it may be true.