Let \(x \in \mathbb{R}^d\), \(t \in \mathbb{R}\), \(r > 0\). The heat ball is defined by
\[ E(x,t;r) = \left\{ (y,s) \in \mathbb{R}^{d+1} : s \leq t,\ \phi(x - y, t - s) \geq r^{-d} \right\}, \]where \(\phi\) is the fundamental solution .
Remarks
- A heat ball lies in the past.