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 $t_0$ $$ u(x, t_0) \equiv C, \quad x \in (0, 1). $$

I have a feeling that there should be some contradiction to analyticity argument, but cannot complete a proof.