Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$