-
> - > -
on (O,T) x 8DG.
(61)
8n - 8n - 8n
Then, for any 'Po E V' we have
(62)
where f is the functional (independent of a) given by
+6 I I 'Po Ilv' - < Yd, 'Po >vxv' .
From (58) we deduce that
1 · . f f('Po: Yd) >
1m III
6.
l'Polv'~oo l'Polv' -
So, there exists M > ° (independent of a) such that
f("Po: Yd) 2 i I I 'Po Ilv' assumed I I 'Po Ilv,2 M.
This implies that if 0 independently
of a such that
I I (63)
Using (61), (63) and (54) we get (56). •
Proof of Theorem 10. From assumption (52) and the construction of Land Y 00 we
deduce that there exists No E IN such that
Let N 2 No large enough and define
if
s::;-N
if -N::; s::; N
if
s 2: N.
Since iN is a (globally) Lipschitz function and bounded, as in Theorem 1.2 of Fabre -
Puel - Zuazua [1992]' there exists u~ E C([O, T] x 8w) such that if y*( t, x : un denotes
the solution of
Y; - /1y* + iN(:!/) = 0
y* = u~
By' = °
on
y*(O,:r) = YO(X)
in (0, T) x (0- w)
on (0, T) x 8w
on (0, T) x 80
on 0- w