354
so Au· - 8;u· E L2(Q) and Au· E L2(R;H- 1 (a,b)). Consequently as in [L3], u· is also a
strong solution of the variational inequality (2.21) and, by uniqueness, we have u· = u .•
We are going to present now a result of convergence of the non-coincidence sets X {UV>O}l
when II -+ O.
Corollary 2.13. With the preceding assumptions, ifu is the solution of the variational
inequality (2.21), we have
Au - 8;u = I X{u>O} a.e. in Q
and, if
r ---+ I in L2(Q) when II -+ 0, with I::f. 0 a.e. ,
then
Xv ---+ X in U( Q), 1:$ P < +00, when II -+ 0 ,
where Xv = X{uv>O} and X = X{u>O}·
(2.26)
Proof: Due to the property 0:$ Xv:$ 1, for all II E]O,I[, there exists X· E LOO(Q),
0:$ X· :$ 1, such that Xv ---+ X· when II -+ 0 in LOO(Q)-weak*.
We know, by Theorem 2.12, that U V ---+ u when II -+ 0 in L2(R;HJ(a,b))-weak, and
so
r Xv = Au V -1I8;u v - 8;u v ---+ Au - 8;u in V'(Q).
Remarking f" Xv = (Iv - f) Xv + I Xv ~ I X., we have
Au - 8;u = Ix· a.e. in Q.
Since, by Proposition 2.11, u E L2(R;H2(a,b)) and Au E L 2 (Q), the equation (2.26)
holds and we conclude I X = I X·· If I ::f. 0 a.e., we have X· = X and so Xv ---+ X, when
II -+ 0, in LOO(Q)-weak* and, as they are characteristic functions, also strongly in V(Q),
Vp < +00 .•
2.3. The asymptotic behaviour in time
We are going to consider in this section T = +00, in order to study the asymptotic
behavior of the solution of the ultraparabolic variational inequality (2.21), when t -+ +00,
proving that u(t) ---+ U oo , when t -+ +00 (where U oo denotes the solution of the parabolic
problem (2.5) and u is the solution of the ultraparabolic problem (2.21)), as long as
I(t) ---+ 100·
Remark 2.14. The variational inequality (2.21) is equivalent to the following one:
{
u(s,O) = uO, u(O,t) = u l , u(s,t) E K for a.e. (s,t) E R,
l Au(s,t)(v - u(s,t)) + l 8"u(s,t)8,,(v - u(s,t)) ~
~ll(s,t)(v-u(s,t)), VvEK, fora.e.(s,t)ER.o
(2.27)