You can use Holmgren's dual method for this kind of problems. Changing variables as $\Phi=b^{-1}$, $v=b(u)$ you can rewrite the PDE as the Generalized Porous Media Equation
$$
\partial_t v=\Delta \Phi(v) +f,\hspace{2cm}(\text{GPME})
$$
which has been studied intensively and for which I recommend [Vazquez's book][1] [The Porous Media Equation - mathematical theory, Oxford science publications '07]. Your regularity assumptions give here $v\in L^2H^{-1}\cap L^2H^1$ and $\Phi(v)\in L^2H^1$, which means that your solution is a weak energy solution and in particular a very weak solution (see Vazquez's book for definitions). By monotonicity of $\Phi=b^{-1}$ your statement is equivalent to showing a comparison principle $v_1\geq v_2$ for (GPME). This is well known to hold even for very weak solutions: you'll find the precise statement in [Vazquez, Theorem 6.5], which also includes the case of ordered Dirichlet boundary conditions. Vazquez's proof strongly relies on the aforementioned Holmgren's duality method.

See also [this post][2] for a related discussion.


  [1]: http://books.google.fr/books?id=R7MrAAAAYAAJ&q=the%20porous%20medium%20equation&dq=the%20porous%20medium%20equation&hl=fr&sa=X&ei=O6adU5vlDcPX7AbP2oDwBw&ved=0CDQQ6AEwAA
  [2]: https://mathoverflow.net/questions/163574/a-comparison-principle-for-parabolic-equation/163946#163946