I would like to know what the basic references to study non-linear parabolic problems, specifically problems of the type
\begin{equation*} \frac{\partial u}{\partial t} - \Delta u = f(x,u), \end{equation*} with the standard boundary conditions.
What the main techniques to study the existence and qualitative results related to the above problem?
I know about Galerkin and semigroups methods in the context of linear equations, however I don't know examples of these and other methods applied in the non-linear case.
Thanks in advance.