Existence of solutions for quasilinear parabolic equations

http://ejde.math.txstate.edu/Volumes/2011/18/chen.pdf

try to enter to this link which is an article about [EXISTENCE OF SOLUTIONS FOR QUASILINEAR PARABOLIC
EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS], I have question in the page 3 section 3:
the expression:
[We need a priori estimates that permit us to extend the solution to the whole
domain [0, T].] what does it mean this expression mathematicaly?????

The formulas used require the assumption that " ". The "a-priori estimates" (initial estimates of the values of f) are to make sure that f satisfies that.

But the priori estimates in this articl (page 5) are all depend in u not f how this??

please tell me why u mention f here and what's the relation between these values u and f....?