Consider parabolic equation
$$
u_t = u_{xx} + f(x, t),
$$
where $x \in (0, 1)$, $t \in R^+$, $f \in L^{\infty}$ (but not necessarily continuous) and Neumann boundary condition.
Assume that initial data is smooth and not constant.

Is in possible that in finite time $t_0$ there exists an interval $(a, b) \subset (0, 1)$ such that
$$
u(x, t_0) \equiv C, \quad x \in (a, b).
$$

I have a feeling that there should be some contradiction to analiticity argument, but cannot prove this due to low regularity of $f$.