= Parabolic boundary
{title2=$\partial_P(\Omega\times(0,T])=(\overline\Omega\times\{0\})\cup(\partial\Omega\times[0,T])$}
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.
Back to article page
