Why are porous medium equations posed on connected domains? Shouldn't it be done on a domain with holes (or pores)? The porous medium equation is supposed to model gas flow through porous media (i.e. some object with holes in it). Why then, in theory of weak solutions, do people study the equation on a sufficiently smooth connected bounded domain (the domain like in every other PDE)? Answer: because it's easier. But then what is the link to the application and the physical motivation?

In general, most differential equation models involve an idealized version of the real situation: nature itself is way too messy. Part of the art of mathematical modeling is to consider enough detail to capture the important effects, while ignoring irrelevant complications. A model of continuous matter ignores the fact that matter is made up of molecules and atoms. In the case of a porous medium, although the actual physical object has a very complicated structure with small holes, you consider a model of the medium "in bulk" where that complicated structure has been smoothed out.