Suppose $\Omega $ is bounded and
$$u^m\xrightarrow[]{w^*} u \quad \text{in} \,\, L^{\infty} (\mathbb{R}^+,H^2(\Omega)\cap H_0^1(\Omega))$$
and
$$u^m_t\xrightarrow[]{w^*} u_t \quad \text{in} \,\,L^\infty (\mathbb{R}^+,L^2(\Omega))\cap L^2 (\mathbb{R}^+,H_0^1(\Omega))$$
I want to prove
$$u^m\xrightarrow[]{strongly} u \quad \text{in} \,\,L^2 ((0,T),H_0^1(\Omega))$$
It is used in some papers and non of them have a proof. I know it comes from Aubin-Lions Lemma but the most general lemma I found doesn't work. The lemma works If we show:
$$u^m\xrightarrow[]{w^*} u \quad \text{in} \,\,L^2 ((0,T),H^2(\Omega)\cap H_0^1(\Omega))$$
But how? I will be thankful for any hint.
You already have $u^m \to u$ weak-* in $L^\infty(\mathbb{R}^+; H_0^1(\Omega))$. Hence, $u^m \to u$ weak-* in $L^\infty(0,T; H_0^1(\Omega))$. And this gives $u^m \to u$ weak in $L^2(0,T; H_0^1(\Omega))$. No need for Aubin-Lions here.