Weak convergence of a nonlinear sequence.

Question

Let $\left( {{u^n},{u_t}^n,{u_{tt}}^n} \right)$ be a bounded sequences $${\left[ {{L^2}(0,T;H_0^2(0,1))} \right]^2} \times {L^2}(0,T;H_0^1(0,1))$$, then there exists a sebsequences that will be noted $\left( {{u^n},{u_t}^n,{u_{tt}}^n} \right)$ such that $$\left( {{u^n},{u_t}^n,{u_{tt}}^n} \right) \to \left( {u,{u_t},{u_{tt}}} \right)$$ weakly star in $${\left[ {{L^2}(0,T;H_0^2(0,1))} \right]^2} \times {L^2}(0,T;H_0^1(0,1))$$. Let ${\left( {{u_x^n}} \right)^2}$ be a bounded sequence in ${{L^2}(0,T;{L^2}(0,1))}$, how can I prove that $${\left( {u_x^n} \right)^2} \to {u_x}^2$$ in $${{L^2}(0,T;{L^2}(0,1))}$$ ?