I'm looking for the following result:

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in L^2(0,T;L^2)$ where $f:\mathbb R \to \mathbb R$ is convex.

Does anyone know how to prove this, or a reference for this result? Thank you.

I managed to this for a subsequence (so if $u_n \to u$, I showed for $\liminf_{n_j \to \infty}$) but could not show it for the full sequence.

I asked this on StackExchange (link) but got not reply.