351
and u O E V, U O ~ 0, u 1 ~ 0 are given functions.
Theorem 2.10. Let Q = la, b[ X ]0, S[ X ]0, T[ and suppose that
r E L 2 (Q), u 1 E H1(0,T;HJ(a,b)).
Then the variational inequality (2.14) has a unique solution U V E Hl( Q), which satisfies
OtUV + o.u v - vcP.u" - a!u v = f" X{u">O}, a.e. in Q.
(2.16)
Proof: We consider the following penalized problem,
{
0 U V + 0 u" - V0 2 U" - 02U" + RE(U") = I"
t£
' £
8 £
Z£
,..,
£
,
u~(a) = u~(b) = 0, u~I.=o = u 1 , o.u~I.=s = 0 ,
u~lt=o = u O ,
(2.17)
where {JE is defined as in (2.7)-(2.8) now with {JE(X, 5, t, u) = {J(~ )[f"(x, 5, t)J-.
We easily prove that u~ is uniformly bounded (independently of e) in L2(0, T; Hl(f!)).
As in the proof of Proposition 2.2, by multiplication with (u~t we also deduce u~ ~ 0
a.e. in Q and then also
-(f"t ~ {JE(U~) ~ 0 a.e. in Q.
(2.18)
Multiplying the first equation of (2.17) by OtU~ - OtU1, we prove easily that OtU~ is
bounded in L2(Q) independently of e.
So u~ E L2(0, T; H?",,(f!)) and we can pass to the limit when e -> 0, as in the last
subsection. We have then
O,U V E L2( Q), u" E L2(0, T; H~(f!) ,
the subsets A = {(x, 5, t) E Q: u"(x, 5, t) > OJ, and I = {(x, 5, t) E Q: u"(x, s, t) = O}
are a.e. well defined and, consequently, u" satisfies the equation (2.16) a.e. in Q .•
Next we are going to formulate and prove the existence and the uniqueness of solution
of the limit ultraparabolic problem, as well as to show that, when v -> 0, the solutions
of the corresponding parabolic problems converge to the solution of the ultraparabolic one.
Let R = ]0, S[ x ]0, T[, Av = OtV + O. v and define
K:={VEL 2 (R;HJ(a,b)): v2:0 a.e. inQ},
(2.19)
and
w = {v E L2(Q): Av E L2(Q)} .
(2.20)
The limit ultraparabolic variational inequality is given by:
{
u[EK:nW' u~=o=uo, ul'=O~ul,
(2.21)
lQAu(v-u)+ lQo",uo:z;(v-u)2:1 Q I(v-u), VvEK:.