362
3.3. The asymptotic stabilization in time
In this subsection we are going to let t --+ +00 and prove that the ultraparabolic
solution of the two-phase evolutionary Stefan problem converges to the parabolic solution
of the steady-state problem. For simplicity, we assume first that the boundary condition,
in the evolutionary case, is independent of time, i.e. h(x,z,s,t) = hoo(x,z,s), hoo E
Hl(O, S; L2(8w)), an assumption that is easily generalized in a final remark.
Recalling the definition of (j given by (3.12), for large time t, since S is fixed, we have
I(x, z, s, t) = b(8 1 (x, Z, t - s)) + oX X{/11(."z,t-s»O} , 'VO < s < S < t .
Theorem 3.4. Let u denote the solution of problem (3.14) and U oo the solution of
problem (3.4) with the assumptions of the previous sections.
Suppose that
(3.21)
and
r IIX{81(T»O} - X{8~>o}IIL2(wxJo,S() ~ 0 when t --+ +00 . (3.22)
J,-S
Then
Remark 3.5. Notice that the condition
clearly implies (3.21) but, in general, not (3.22) , which is a complementary assumption.
Proof of Theorem 3.4: Let v = u(t, s) in the problem (3.4) and v = uoo(s) in the
problem (3.14). Then, calling w(s,t) = u(s,t) - uoo(s), we have
b.lIAw(s, t)12 + 1 V'w(s, t) . V' Aw(s, t) + Jaw a w(s, t) A w(s, t) ::;
::; L[/(s,t) - loo(s)] Aw(s,t). (3.23)
Writing (3.23) in the variables T = S, ~ = t-s and integrating (along the characteristics
of A) with respect to T between (s - t)+ and s, we obtain the following inequality
b. r J 18Tw(T,T+(t-s))12dT+lIV'w(s,tW+ao r Iw(s,tW::;
(3.24)
1(8-1)+ w
w
law
::; lIv'w(s - t)+, (s - t)+ + (t - s))1
2
+ Jaw a Iw(s - t)+, (s - t)+ + (t - s))1 2
+ r 11 / (T,T+(t-s))-/oo(T)1 2 dT.
1(8-1)+ w