Parabolic boundary (${\partial }_P(\Omega \times (0,T])=(\overline{\Omega }\times \{0\})\cup (\partial \Omega \times [0,T])$)

For a forward heat equation or a related reaction-diffusion equation on a spatial domain $\Omega$, the parabolic boundary is the union of the initial face and the lateral spatial boundary. It excludes the interior of the final-time face. This is the boundary relevant to the forward maximum principle: values at a later time are controlled by earlier data and the lateral boundary. Writing $\Omega \times \{0\}$ instead of $\overline{\Omega }\times \{0\}$ in the union gives the same set, since the lateral term includes the initial rim.