This question stems from the proof of Theorem A.1 on page 425 of this paper.
Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in $C^0([0,T];H^{-1}(\Omega))$. This strong convergence yields that $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$ for any $q < \infty$.
I don't understand how this conclusion holds. Does anyone know?
Coincidentally a similar question was asked on MSE, for which I placed a bounty but received no answers so I think it is OK to post here as I have had no luck.