Can we say that the boundary integral conditions= the nonlocal boundary condition?!!

-When the generalized solution = the weak solution?

-What's the norm of the sobolev space H^{-r} and how we can say u_{m}^{p-2}u_{m} \in L^{q}(0,T;H^{-r})(\omega))?

-In all the lemma 4.4....4.7 they try to use the norm in H^{-r})(\omega) why?