Let $0 < \epsilon < 1$. Consider $\{a_n\}_{n \geq 1} \in l_2$ and $L(t) = 1+\epsilon t$. Let $x$ be fixed such that $0 < x < L(t)$. Does there exist $\tau \geq 0$ such that the following inequality cound be true \begin{equation}\label{e1} \int_{0}^{\tau}\bigl|\sum_{n=1}^{+\infty}a_n e^{-i\pi^2n^2\frac{t}{L(t)}}\sin\bigl(\frac{n\pi x}{L(t)}\bigr)\bigr|^2 dt \gtrsim \sum_{n=1}^{+\infty}|a_n|^2 \end{equation}

Let $0 < \epsilon < 1$. Consider $\{a_n\}_{n \geq 1} \in l_2$ and $L(t) = 1+\epsilon t$. Let $x$ be fixed such that $0 < x < L(t)$. Does there exist $\tau \geq 0$ such that the following inequality could be true?: $$ \int_{0}^{\tau}\bigl|\sum_{n=1}^{+\infty}a_n e^{-i\pi^2n^2\frac{t}{L(t)}}\sin\bigl(\frac{n\pi x}{L(t)}\bigr)\bigr|^2 dt \gtrsim \sum_{n=1}^{+\infty}|a_n|^2 $$