Let $u_0 \in L^\infty(\Omega)$ and $u_0 \in C^2(U)$ for $U \subset \subset \Omega$.
Consider $$ \begin{cases} u_t - \Delta u = 0 & t>0, x \in \Omega \\ u|_{\partial\Omega} = 0 & t \ge 0\\ u(0,x) = u_0 & x \in \Omega. \end{cases} $$
How can I prove that $u$ is $C^2$ in space and $C^1$ in time up to time $t=0$ for $x \in U$?