I'm reading the section on the Heat equation in Evans' book and there is something I don't quite understand.
He defines the heat ball as$$E(x, t, r) = \left\{(y, s) \in \mathbb R^n \times \mathbb R~\big|~s \le t,~ \Phi(t - s, x - y) \ge \frac{1}{r^n}\right\},$$with $\Phi(t, x)$ being the heat kernel in dimension $n$, i.e.$$\Phi(t, x) = \frac{1}{(4\pi t)^{n/2}} \exp\left(-\frac{|x|^2}{4t}\right).$$He gives an interpretation of this set, displayed on the figure below.
I am wondering how does he get the shape. It seems to be some kind of ellipsoid but I don't quite get it. I tried to get an intuition by shifting and rescaling to $E(0, 0, 1)$ and by fixing time, i.e. for some $s_0 \le 0$ and$$E(0,0,1) \cap (\mathbb R^2 \times \{s_0\}) = \left\{y \in \mathbb R^2 ~\bigg|~ \frac{1}{(4\pi |s_0|)} \exp\left(-\frac{|y|^2}{4 s_0}\right) \ge 1\right\},$$where I considered $n = 2$ for the sake of simplicity. According to the picture in Evans, this set should be some kind of ellipse (right?), but I don't see how one should get that. Any idea on how to interpret this set ?