# $L^p$ regularity for weak solutions of non homogeneous heat equation

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!