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Let $U$ be a compact $C^2-$manifold and suppose $v:U\times (0,T) \to \mathbb R$ is the weak solution of:

$\partial_t v= \Delta v+f$ where $f\in L^{\infty}(U\times (0,T))$

I 'm interested in the $L^p$ regularity of $v$, but I can't find any useful references. For example, if $v$ was the weak solution of the Poisson equation then I could use the $L^p$ estimates which are presented in Gilbarg and Trudinger's book.

There should be an analog but I can't find it no matter how much I've searched.

Any help is much appreciated. Thanks in advance!

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