Let $f\in C^\infty([0,2\pi])$ be a smooth periodic function with mean zero ($\int f = 0$). Let $f(t,x)$ be the heat flow of $f(x)$, so that $f(t,x)$ solves $$ \partial_t f = f''. $$ Given a small $\varepsilon>0$, define the approximate zero set $$ Z_\varepsilon(t) = \{x\in[0,2\pi] \mid |f(t,x)|\leq \varepsilon\}. $$ Is it the case that the measure of $Z_\varepsilon$ is monotone? That is, does it hold that $$ |Z_\varepsilon(t)| \leq |Z_\varepsilon(t')| $$ when $t<t'$?
If it is not monotone, how may one construct a counterexample?