Let $Q=(0,T)\times \Omega$, $\Omega$ being a bounded subset of $\mathbb R^d$, sufficiently smooth. Consider three sequences $ u_n$, $ v_n$, and $w_n$ such that:

- $ u_n$ is bounded in $ L^\infty(Q)$ and converges strongly to $ u $ in $ L^2(0, T; H^1(\Omega)) $.
- $ v_n $ converges strongly to $ v $ in $ L^2(Q)$.
- $ w_n$ converges weakly to $w $ in $ L^2(Q)$.

Could us expect that: $$\lim_{n \to \infty} \int_Q u_n v_n w_n \, dx \, dt = \int_Q u v w \, dx \, dt?$$

I was trying to show that the sequence $u_nv_n$ converges strongly to $uv$ in $L^2(Q)$ and to use the weak strong convergence for the rest of the proof. I proceed as follows: $u_nv_n-uv=u_n(v_n-v)+v(u_n-u).$ But the fact is that by taking the $L^2$-norm on both sides, I need $v$ to be in $L^\infty(Q)$. Could you please give me some insights on how to proceed or suggest an alternative approach?

"such that"has to be replaced by"for which we want to prove that"$\endgroup$