To simplify, let us work on $Q_T:=[0,T]\times\mathbb{T}^N$ where $\mathbb{T}^N$ is the $N$-th dimensionnal torus.
Consider $(S_n)_n$ a sequence of $L^1(Q_T)$ and $(z_n)_n$ the sequence of solutions to the heat equation $\partial_t z_n - \Delta z_n = S_n$ (with $0$ as initial data).
Is it true that if $(S_n)_n$ is bounded in $L^1(Q_T)$, then $(z_n)_n$ converges (up to a subsequence) a.e. ? If it is not sufficient, does $(S_n)_n$ weakly converging in $L^1(Q_T)$ sufficient ?
I expect some compactness for the operator $S\mapsto z$ but I did not manage to find some reference for the $L^1(Q_T)$ case.
Thanks for any help !