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.
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?????