I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in Differential Equations, Springer,1984 (pages177 and 187).
In these results we do not know if they remain true for Neumann Conditions or other mixed boundary conditions (not Dirichlet conditions)
Is there any resource which gives more detailed statements and proofs for maximum results and comparison principles in parabolic equations (the general case, not only the heat equation and accepting some general boundary conditions, like Neumann). To be more precise I want to state these results for the problems of the form:
$$\begin{cases}u_t-Lu=0 \ \text{in} \ (0,T)\times\Omega \\ \partial_{\nu} u=0,\ \text{in}\ (0,T)\times\partial\Omega\\ u(0;x)=\rho(x)>0,\ \text{in}\ x\in\Omega\end{cases}$$
where $Lu=D\Delta u+c(t,x)\cdot u$. $u=u(t,x_1,x_2,\dots,x_n)$, and $\Omega$ is a bounded open set with $C^1$ boundary in $\mathbb{R}^n$, $c$ is a bounded function.
I'm interested in these types of results for proving the uniqueness and boundedness of the solution $u$. Is there any other method?
