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Let $f_n \to f$ on compact subsets of the real line.

If $u_m \rightharpoonup u$ in $L^2(0,T;H^1) \cap L^p(0,T;L^p)$ and $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$.

What result is this? Please can you refer me?

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There can be no such result, because the statement is fundamentally false. It will stay false in case $f_n=f\;\,\forall\,n\,$, and it will stay false even if $f_n=f=I\;\,\forall\,n$ with the identity mapping $I\colon\,\mathbb{R}\to\mathbb{R}$. To see this, consider the following counterexample. Take a sequence $$ u_m(t,x)=\varphi(x)\sin{\frac{m\pi t}{T}}\,,\quad m\geqslant 1, $$ with some fixed nonzero $\varphi\in H^1\cap L^p$. It is clear that $\,u_m\rightharpoonup\,0=u=w\,$ in $\,L^2(0,T;L^2)$, as well as in $\,L^2(0,T;H^1) \cap L^p(0,T;L^p)$, while strong convergence $\,u_m\to\,0\,$ in $\,L^2(s,T;H^{-1})\,$ for any $\,s\in (0,T)\,$ is excluded by the fact that $$ \begin{align} \|u_m\|^2_{L^2(s,T;H^{-1})}=\frac{1}{2}\Bigl[T-s+\frac{T}{2\pi m} \sin{\Bigl(\frac{2\pi ms}{T}\Bigr)}\Bigr]\|\varphi\|^2_{H^{-1}}\\ \geqslant \frac{1}{2}\Bigl[T-s-\frac{T}{2\pi m}\Bigr]\|\varphi\|^2_{H^{-1}} \geqslant \frac{1}{2}\Bigl[T-s-\frac{T}{2\pi N_s}\Bigr] \|\varphi\|^2_{H^{-1}}\\ \geqslant \frac{1}{2}\Bigl[T-s-\frac{T-s}{2}\Bigr] \|\varphi\|^2_{H^{-1}}=\frac{T-s}{4}\|\varphi\|^2_{H^{-1}}>0 \quad\forall\,s\in (0,T),\;\forall m\geqslant N_s\,, \end{align} $$ with $N_s$ defined as the least natural number such that $$ N_s\geqslant\frac{T}{\pi(T-s)}\,. $$ To achieve what you want, you need some positive fractional order smoothness, no mattter how small, w.r.t. variable $t\in (0,T)$, or its generic substitute of the form $\frac{\partial\,}{\partial t}u_m\rightharpoonup \frac{\partial}{\partial t}u\,$ in $\,L^2(0,T;H^{-1})$, to say nothing of the more significant assumptions on $f_n$ and $f$.

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  • $\begingroup$ Thanks. If you want, could you take a look at this paper : link.springer.com/article/10.1007/BF01049490#page-1 On page 103, they write what I wrote in my OP. They do not prove anything about time derivatives so I don't know what they did. $\endgroup$
    – LapLace
    Jun 2, 2014 at 9:36

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