363
Let, from now on, t > 5. Then (s - t)+ = 0 and
V"w((s - t)+,(s - t)+ + (t - s)) = V"u(O, t - s) - V"uoo(O) = 0 ,
w((s - t)+,(s - t)+ + (t - s)) = u(O,t - s) - uoo(O) = 0,
since uls=o = 0 and uools=o = O.
Integrating the inequality (3.24) in order to s, between 0 and 5, we have
1IV"w(t)lIi2(WX]O,SI) + Go IIw(t)lli2(awx]o,s[) ~ (Sl
S
11/(7,7 + (t - s)) - /00(7)1
2 d7ds .
Jo (s-I)+ w
Changing the order of integration, we have
Introducing the change of the variable s to ( = 7 + t - s and applying once more Fubini
theorem, we obtain
as t -+ 00, by the assumptions (3.21) and (3.22).
Consequently,
11V"w(t)lIi2(WX]0,SI) + Gollw(t)lli2(awx]O,sl) ~ Ls 11/(7) - /oolli2(WX]O,SI) , (3.25)
and, since by a Poincare type inequality we have
(Sll w(tW ~ c[ (SllV"w(tW + (s ( Iw(tW] ,
10 w
10 w
10 Jaw
when t -+ +00, we conclude
Remark 3.6. We can easily drop the unnecessary restriction on the time independence of h(x, z, s, t). For instance, by assuming that, as t -+ 00,
we still have the convergence u(t) -+ Uoo in L 2 (0, 5; Hl(w)), as t -+ 00.
In fact, arguing as in the previous proof, we may control the additional term
r G[g(s,t)-goo(s)]Aw(s,t)
law