  We present two initial graphs over the entire $\mathbb{R}^n$, $n \geq 2$ for
which the mean curvature flow behaves differently from the heat flow. In the
first example, the two flows stabilize at different heights. With our second
example, the mean curvature flow oscillates indefinitely while the heat flow
stabilizes. These results highlight the difference between dimensions $n \geq
2$ and dimension $n=1$, where Nara-Taniguchi proved that entire graphs in
$C^{2,\alpha}(\mathbb{R})$ evolving under curve shortening flow converge to
solutions to the heat equation with the same initial data.
