Consider parabolic equation
$$
u_t = u_{xx},
$$
where $x \in (0, 1)$, $t \in (0, T)$ with nonhomogenious Dirichlet boundary conditions.
$$
u(0, t) = \psi_0(t) \in C^0,
$$
$$
u(1, t) = \psi_1(t) \in C^0.
$$

Assume that initial data 
$$
u(x, 0) = \varphi(x)
$$
is smooth and not constant,

Is it possible that in finite time $t_0$ 
$$
u(x, t_0) \equiv C, \quad x \in (0, 1).
$$

ANSWER: Yes, it can