232
w(x) :::; w(x) < -10 and x E B,. Consequent ely, inequality (33) follows from the strong
p-nondegeneracy assumption on w.
Let 10, W satisfying the weak p-nondegeneracy property. As before we can assume that
II 10 - w I/LOO(I):::: EO. Then remarking that
(z(x) - z(x))(1o(x) - w(x)) = 0 if x E A n A
and that if1O(x) -I- -10 (resp. w(x) -I- -10) and x E A (resp. x E A) we have that
x E {x E I: 0 < Iw(x)+ 101:::: E} (resp. {x E I: 0 < Iw(x) + 101:::: E})
we obtain (34). •
Proof of Theorem 4. Let u be any other bounded weak solution of (P). Then, as in the
proof of Theorem 2, using the monotonicity of Re
~ h lu(t) - u(tWdx + h P(x)(lu x(t)IP-2 ux (t) -lux(t)IP-2 ux (t))(ux(t) - ux(t))dx
:::: Q h(z(x,t) - z(x,t))(u(x,t) - u(x,t))dxdt
for some z,z E LOO(/ x (O,T)) with z(x,t) E j3(u(x,t)), z(x,t) E j3(u(x,t)) for a.e.
(x, t) E I x (0, T). Now assume p > 2. Then by (24) we obtain that
:t h lu(t) - u(tWdx+ II (u(t) - u(t))x IItp(I:p)::::
:::: Q II z(t) - z(t) 11£1 (I) II u(t) - u(t) IILOO(I) .
From Theorem 4 of Rakotoson-Simon [1993] we have the estimate
(36)
where
with Co > 0 independent of I and
Then by Lemma 3 and using (a + W :::; 2 P ( a P + bP) we get
Q II z(t) - z(t) 11£1(1)11 u(t) - u(t) I I Loo (I) -II (u(t) - u(t))x IItp(I:p)::::
::::11 u(t) - u(t) IIt"(I) (QC(aJ - ail - 2p~i) + (C1lllpt P I I u(t) - u(t) IItl(I:p)::::
I I u(t) - u(t) Iltoo(l) (QC(a J - ail - 2p~f) + C2 II u(t) - u(t) Ili2(1)