Supersolution

For a differential equation $L[u]=0$ with its comparison order fixed, a supersolution satisfies $L[w]\ge 0$. For example, if $L={\partial }_t-{\partial }_x^2+Q$ with $Q\ge 0$, a sufficiently regular $w$ satisfying $L[w]\ge 0$ and dominating the initial and boundary data lies above a solution by the parabolic comparison principle. An unbounded-domain barrier such as $w=\eta e^{3t}(1+x^2)$ can provide the missing boundary control at large distance. The sign convention belongs to the chosen differential operator and must be stated.