241
The obstruction function Yoo in (49) was constructed in Diaz [1994] such that
Yoo(t,x) = +00 on (O,T) x Ow
~(t, x) = 0
on (0, T) x an.
In consequence, condition (48) implies that problem (Pw ) is not (in general) approximate
controllable since if IYd(X)1 > Yoo(T, x) a.e. x in a positively measured subset D of 0. - w
then for any v E L2((0, T) x w)
and so, if ( > 0 is small enough, it is imposible to choose v satisfying the required
properties. We remark that a previous uniform estimate (independetly of the control) for
superlinear equations but when the control acts on the boundary was due to A. Bamberger
(see Henry [1978]). Due to the relevance of the Sellers model, a natural question arises:
is problem Pw approximate controllable in a smaller class of desired states Yd?
The main contribution of this work is to give a positive answer to the above question.
For the sake of the exposition we shall simplify, even more, the model under consideration
to
in (0, T) x 0.
on (0, T) x an
on n.
The extension of the following results to the case of problem (Pw ), assumed (48), is
merely a technical matter and can be carried out as in Diaz [1994].
The starting point of our approach consists in improving the estimate (49) by obtaining
some sharp obstruction functions. This is the objective of the next result
Proposition 5
Given Yo E £1(0.) there exist r 00' V 00 E C((O, T] x 0. - w) such that r 00 is a weak
solution to the problem
in (0, T) x (0. - w)
on (O,T) x Ow
on (O,T) x an
on 0.
and V 00 satisfies the same conditions except that V 00 = +00 on (0, T) x ow.
Idea of the proof. As in Bandle, G. Diaz and J.l. Diaz [1994], given N E IN we define
r N as the (unique) solution of the problem
r; - ~Y + ,\lYlp- 2 y = 0
Y=-N
&Y - 0
an -
Y(O,·) = Sup{(Yo)_(')' -N}
III
(0, T) x (0. - w)
on (0, T) x ow
on (0, T) x an
on n.