Let $\Omega \subset \mathbb{R}^n$ be a bounded set with boundary $\partial \Omega,$ and consider the following stochastic heat equation $$ \begin{cases} du(x,t)=(\Delta u(x,t)+f(x,t))dt+\sigma u(x,t)dB(t),\quad (x,t) \in \Omega \times \mathbb{R}^+, \\ u(x,0)=u_0 \in \mathbb{R}^n, \quad x\in \Omega,\\ u(x,t)=0, \quad (x,t) \in \partial\Omega \times \mathbb{R}^+. \end{cases} $$ where $B(t)$ is a Brownian motion and $f \in L^2(\Omega \times \mathbb{R}^+).$
I'm trying to apply Itô's formula to $\Vert \partial_tu(x,t) \Vert_{L^2(\Omega)}^2$ and I'm wondering if it even makes sense, given that I have no prior information about $\partial_tu(x,0)$?