maximum principles for parabolic PDEs seem to be well-known, if the solution is a priori $C^2$ (cf. Protter, Weinberger: Maximum principles in differential equations). However, what about weak solutions? To be specific, are there any maximum principles on the nonnegativity of solutions $u\in W^{1,p}(0,T;L^p(\Omega))\cap L^p(0,T;W^{2,p}(\Omega))$, $p\in(1,\infty)$, where $\Omega\subset R^n$ is a bounded domain? For given nonnegative initial data, does the solution remain positive, as long as it exists?

I assume yes, since there are numerous authors that use the results from Weinberger/Protter just for weak solutions. I would appreciate any hints on this topic.

Best regards, Marc