For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and $f$ is given and $g$ is a nonlinear function from $\mathbb{R}$ to $\mathbb{R}$.
Define $T(w) = u$, the solution operator. Suppose we have $w_n \rightharpoonup w$ in $W:=L^2(0,T;H^1)\cap H^1(0,T;L^2)$ (weakly). $$\int Tw_n'(t)v(t) + \int g(w_n(t))\nabla Tw_n(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ So $w_n \to w$ in $L^2(0,T;L^2)$ by compact embedding, so $w_{n_m} \to w$ a.e. for a subsequence of $w_n$. Assume that $g$ is nice enough so that we can use DCT to deduce that $g(w_{n_m}) \to g(w)$ in $L^2(0,T;L^2)$.
I know that $Tw_n$ is bounded in $W$. Thus there is a subsequence $Tw_{n_l} \to \eta$ in $L^2(0,T;L^2)$ because bounded sequences have a weakly convergent subsequence, and we have a compact embedding as I wrote above.
Using this information, how do I pass to the limit in the equation above to deduce that $Tw = \eta$? My only problem is that I have two different subsequences $n_m$ and $n_l$ and I don't know how to "put them together" in the equation to pass to the limit properly. Help please! I did read this thread (Weak limits and subsequences) but didn't understand it.
Attempted proof: As in the OP, we have $Tw_{n_l}\to η$ in $L^2(L^2)$. Since we know $w_n\to w$ in $L^2(L^2)$, we have $w_{n_l}\to w$ also, and therefore $w_{n_{l_k}}\to w$ a.e. for another subsequence by the same compactness argument as before. DCT implies $g(w_{n_{l_k}})\to g(w)$. Since $Tw_{n_l}\to \eta$, we must also have $Tw_{n_{l_k}}\to \eta$. So we have a subsequence $n_{l_k}$ that does the job. Is this correct?
