235
Proof. From (40) we deduce that if x E B, then Ix -xii::::: f P - 1 j K. Thus IB,I : : : : : (Nj K)E P - I
where N is the number of points of 8A.
(ii) It is clear that (41) implies that meas IAI = 0. Let [a,b] C J a connected component
of B, = {x E I : Iw(x) + 101 ::::: f}. Assume that wx(x) ~ Jon (a,b) [the other case
wx(x)::::: -J on (a,b) is treated in a similar way]. Then w(a} = -10 - f, w(b) = -10 + f
and there exists Xo E (a, b) such that w(xo) = -10. Then for any x E [xo, b] we have
f ~ w(x) + 10 = l
x wx(s)ds ~ J(x - xo).
xo
Analogously, for any x E [a, xo),
f ~ -10 - w(x) = [0 wx(s)ds ~ J(xo - x)
and thus (40), with p = 2, holds.
•
Remark 4. The nondegeneracy properties of the solutions of (P) can be obtained under
some additional assumptions on the initial datum. Let Uo E Clel) such that
and
Ao = {x E I: Uo(x) = -10} has a finite number of connected components,
(42)
3£0 ::: ° and K > ° such that \If E (0, Eo)and any )
x E B"o = {x E I: ° < luo(x) + 101::::: f}
we have luo(x) + 101> Klx - xiI1/{p-l) \Ix E 8A
(43)
Then there exists a Tu E (0, T] such that u(t) satisfies the weak non-degeneracy property
for any t E [0, Tu) where u is any continuous weak solution of (P). In particular if u
and u are continuous weak solutions of (P) there exists a T* E (0, T] such that u = u
on [0, T*) x I. Indeed; let u, u be continuous bounded weak solutions of (P), by the
continuity near t = ° we deduce that there exist Tu, Tu E (0, T] such that u(t), u(t)
satisfy (40) and that the set where they take the value -10 has the same (finite) nomber
of connected components for any t E [0, Tu), [0, Tu) respectively. Taking T* = min{Tu, Tu}
the conclusion follows from part (ii) of Theorem 4.
Remark 5. Let Uo E CI(J) such that Uo is an even function, uox(x) > ° for any
x E (-1,0), uo(O) > -10, u(-l) < -10. Then (42) and (43) holds for p = 2. Moreover,
if u is the solution built in the section 3.2 for p = 2 then u(t) satisfies the strong 2nondegeneracy property for any t E [0, T]. Finally, if p = 2 problem (P) has a unique
bounded solution on [0, T] x I. Indeed; it is an easy modification of Lemma 6.2 and
Corollary 6.3 of Feireisl-Norbury [1991]. For some other criterium on Uo see Diaz - Tello
[1996].
Remark 6. It should be interesting to know if the techniques on non-degeneracy properties for the parabolic obstacle problem (see,e.g., Pietra-Verdi [1985]) can be applied to
obtain the p-nondegeneracy properties for the solutions of (P).
Proof. From (40) we deduce that if x E B, then Ix -xii::::: f P - 1 j K. Thus IB,I : : : : : (Nj K)E P - I
where N is the number of points of 8A.
(ii) It is clear that (41) implies that meas IAI = 0. Let [a,b] C J a connected component
of B, = {x E I : Iw(x) + 101 ::::: f}. Assume that wx(x) ~ Jon (a,b) [the other case
wx(x)::::: -J on (a,b) is treated in a similar way]. Then w(a} = -10 - f, w(b) = -10 + f
and there exists Xo E (a, b) such that w(xo) = -10. Then for any x E [xo, b] we have
f ~ w(x) + 10 = l
x wx(s)ds ~ J(x - xo).
xo
Analogously, for any x E [a, xo),
f ~ -10 - w(x) = [0 wx(s)ds ~ J(xo - x)
and thus (40), with p = 2, holds.
•
Remark 4. The nondegeneracy properties of the solutions of (P) can be obtained under
some additional assumptions on the initial datum. Let Uo E Clel) such that
and
Ao = {x E I: Uo(x) = -10} has a finite number of connected components,
(42)
3£0 ::: ° and K > ° such that \If E (0, Eo)and any )
x E B"o = {x E I: ° < luo(x) + 101::::: f}
we have luo(x) + 101> Klx - xiI1/{p-l) \Ix E 8A
(43)
Then there exists a Tu E (0, T] such that u(t) satisfies the weak non-degeneracy property
for any t E [0, Tu) where u is any continuous weak solution of (P). In particular if u
and u are continuous weak solutions of (P) there exists a T* E (0, T] such that u = u
on [0, T*) x I. Indeed; let u, u be continuous bounded weak solutions of (P), by the
continuity near t = ° we deduce that there exist Tu, Tu E (0, T] such that u(t), u(t)
satisfy (40) and that the set where they take the value -10 has the same (finite) nomber
of connected components for any t E [0, Tu), [0, Tu) respectively. Taking T* = min{Tu, Tu}
the conclusion follows from part (ii) of Theorem 4.
Remark 5. Let Uo E CI(J) such that Uo is an even function, uox(x) > ° for any
x E (-1,0), uo(O) > -10, u(-l) < -10. Then (42) and (43) holds for p = 2. Moreover,
if u is the solution built in the section 3.2 for p = 2 then u(t) satisfies the strong 2nondegeneracy property for any t E [0, T]. Finally, if p = 2 problem (P) has a unique
bounded solution on [0, T] x I. Indeed; it is an easy modification of Lemma 6.2 and
Corollary 6.3 of Feireisl-Norbury [1991]. For some other criterium on Uo see Diaz - Tello
[1996].
Remark 6. It should be interesting to know if the techniques on non-degeneracy properties for the parabolic obstacle problem (see,e.g., Pietra-Verdi [1985]) can be applied to
obtain the p-nondegeneracy properties for the solutions of (P).