Partielle physic equation

mathmai

can check the pdf :

EXISTENCE OF SOLUTIONS FOR QUASILINEAR PARABOLIC
EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS:
BAILI CHEN

-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?