Let \( \Omega \subseteq \mathbb{R}^d \) be open and \( u \in C^{(2,1)}(\Omega_T) \cap C^0(\overline{\Omega_T}) \) solves

\[ \partial_t u - \Delta u = 0 \text{ in } \Omega_T. \]

Then \( u \in C^\infty(\Omega_T) \). [1, 2.3 Theorem 8]

Proof(sketch) Link to heading

For a given point \( (x,t) \in \Omega_T \), represent \( u \) with a mean value, i.e.,

\[ u(x,t) = \iint_C K(x,t; y,s) \, u(y,s) \, dy\,ds, \]

for a suitable neighborhood \( C \), where \( K \) is smooth.

References Link to heading

1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.