Let $$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is monotone $$(f(x)-f(y))(x-y) \geq 0\quad\text{for all $x, y$}$$ and we have $$f(u_m) \rightharpoonup f(v) \quad \text{in $L^2(0,T;H^1(\Omega))$}$$ for some $v \in L^2(0,T;H^1(\Omega))$ (assume that for $u \in L^2(0,T;H^1)$, $f(u)$ and $f^{-1}(u)$ are in $L^2(0,T;H^{1}(\Omega))$.)

Is it possible to show that indeed $v=u$, i.e. $f(u_m) \rightharpoonup f(u)$? I can't seem to do it by using the monotonicity method.