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We present two initial graphs over the entire $\mathbb{R}^n$, $n \geq 2$ forwhich the mean curvature flow behaves differently from the heat flow. In thefirst example, the two flows stabilize at different heights. With our secondexample, the mean curvature flow oscillates indefinitely while the heat flowstabilizes. These results highlight the difference between dimensions $n \geq2$ and dimension $n=1$, where Nara-Taniguchi proved that entire graphs in$C^{2,\alpha}(\mathbb{R})$ evolving under curve shortening flow converge tosolutions to the heat equation with the same initial data.

Mean curvature flow of an entire graph evolving away from the heat flow