Let $f^j(x)$ be a sequence of integrable functions on the circle such that $$\int_{-\pi}^{\pi}|f^j (x) |^2 dx = 1.$$
ALso, let $u^j(x,t)$ solve the heat equation on the circle with initial data $u^j(x,0)=f^j(x)$.
Prove that for any $t_0>0$ there exists a function $u_{t_0}(x)$ and a subsequence $u^{k_{j}}$ such that $\int_{-\pi}^{\pi}|u^{k_{j}}(x,t_0)-u_{t_0}|^2 dx \rightarrow 0$ as $j \rightarrow \infty$.
Does it necessarily exist a function f_0(x) such that $$\int_{-\pi}^{\pi}|f^{k_{j}}(x)-f_0(x)|^2 dx \rightarrow 0 ?$$
Please give me some hints so I can get started. As usual, after so much contemplation, I can't arrive at anything concrete until I see some starting point. Thanks!
Prove that the sequence $\{u^j(x,t_0)\}$ is equicontinuous on $[-\pi,\pi]$ and then apply Ascoli-Arzela's theorem. The result does not hold in general for $t_0=0$.