Suppose $\Omega $ is a bounded domain in $\mathbb R^n$ and let $u$ be the weak solution of the initial value problem for the nonhomogeneous heat equation: $\begin{cases} \partial_t u-\Delta u=f,\;\;\;t\in (0,T)\\ u(x,0)=u_0\\ \end{cases}$
If $f\in L^1(\Omega \times (0,T))$ then what assumption should we have on $u_0$ in order to deduce a bound for the solution $u$ in the parabolic Sobolev space?
To be honest, I'm not even sure if we can obtain any bound for the given $f$ but since we have one for the Poisson equation by Stampacchia(Proposition 4.3), I 'm motivated to think that it should exist also one for the heat equation (most of the times, parabolic pdes have analog results to the elliptic ones).
However I wasn't able to find anything, so this is why I'm asking here.
Any help or hint will be much appreciated.
Thanks in advance!