How to select a strongly convergent subsequence from a weak convergent sequence in $L^2$?

Question

Let $(p_n)_{n \in \mathbb N}$ be a sequence of probability density functions, which satisfies

i)$$ \partial_t p_n (t,x) = \partial_{xx} (a_n (t,x) p_n (t,x)), $$ where $a_n$ is a sequence upper bounded and lower bounded away from zero;

ii)The sequence $(p_n)_{n \in \mathbb N}$ is uniformly bounded in $L^2$ (and thus admits a weakly convergent subsequence $(p_{n_k})_{k \in \mathbb N}$ in $L^2$).

Here is the question: is there a subsequece of $(p_{n_k})_{k \in \mathbb N}$ converging strongly? If there is a chance, which additional conditions (on $a_n$) are needed?