Show that $\lim_{a \to \infty} \sup_{n} \int_0^{T-a}||v_{n,r}(t+a)-v_{n,r}(t)||_{\mathbb{L}^2(\Omega_{2r})}^2dt=0.$

Question

Let $\ 0 \leq t \leq t+a \leq T$, with $$\lim_{a \to 0} \sup_{n} \int_0^{T-a}\left\|u_n(t+a)-u_n(t)\right\|_{\mathbb{L}^2(\Omega_{r})}^2dt=0,$$ where $\Omega_r=\Omega \cap \left\{x \in \mathbb{R}^2; |x|<r \right\}$, let $\phi \in C^1(\mathbb{R}^+)$ with $\phi(s)=1$ for $s \in [0,1]$ an $\phi(s)=0$ for $s \in [2,+\infty)$ and consider \begin{eqnarray} v_{n,r}: \Omega_{2r} \times [0,T] &\to& \mathbb{R}^2 \\ (x,t) &\to& \phi\left( \frac{|x|^2}{r^2}\right)u_n(x,t). \end{eqnarray} Show that $$\lim_{a \to 0} \sup_{n} \int_0^{T-a}\|v_{n,r}(t+a)-v_{n,r}(t)\|_{\mathbb{L}^2(\Omega_{2r})}^2dt=0.$$