Let \( \Omega \subset \mathbb{R}^d \) be a bounded domain, \( g \in C^0(\partial \Omega_T) \) and \( f \in C^0(\Omega_T) \). Then
\[ \begin{cases} \partial_t u - \Delta u = f & \text{in } \Omega_T \\ u = g & \text{on } \partial \Omega_T \end{cases} \]has at most one solution in \( C^{(2,1)}(\Omega_T) \cap C^0(\overline{\Omega_T}) \).
Proof