Is the heat ball really a ball?
That is,
consider the heat ball set for $n=2$, then one gets points $(y,s) \in \mathbb{R}^2 \times \mathbb{R} = \mathbb{R}^3$.
Do these points also satisfy an equation similar to the $\mathbb{R}^2$ ball:
$$x^2+y^2 \leq r^2$$
?
Also, I'm not entirely sure about how to interpret $t$ for a ball. I mean, what does one need $t$ for in a ball? Or $s$ for that matter?