Quasilinear parabolic problems with nonlinear boundary conditions

Lemma 4.2.1 Let 1 < p < ∞, s1, s2 ∈ (0, 1), 0 < T2, 0 ≤ T1 ≤ T2, and Ω be an
arbitrary domain in R n . Then
|mf| Y T 2 ≤ |m| Y T 1 ∩L∞ |f| Y T 1 ∩L∞ + 2|m| Y T 1 , T 2 ∩L∞ |f| Y T 2 ∩L∞ (4.26)
for all m, f ∈ Y T2 ∩ L∞([0, T2] × Ω).
We turn now to products where one factor might be unbounded. Such a constellation
arises for example when s1 < 1/p. We confine ourselves to the case Ω = R n . By means
of extension and restriction, the subsequent multiplication property can be transferred
to domains with sufficiently smooth boundary.
Suppose f ∈ Y T2 and m ∈ C r1 ([0, T2]; C(R n )) ∩ C([0, T2]; C r2 (R n )) =: M T2 for some
si < ri < 1, i = 1, 2. Letting J be a subinterval of [0, T2], we then estimate
[mf] Lp(J;B s 2
pp(R n ))
where
� �
I(m, f) ≤ (
J
R n
�
= (
�
�
J Rn Rn ≤ |m| s
L∞(J×Ω)[f]
Lp(J;B 2
pp(Rn ))
�
B1(y)
=: I1(m, f) + I2(m, f).
|m(t, x)f(t, x) − m(t, y)f(t, y)| p
|m(t, x) − m(t, y)| p |f(t, y)| p
|x − y| n+s2p
|x − y| n+s2p
+ I(m, f),
dx dy dt) 1
�
p + (
�
J Rn �
dx dy dt) 1
p
· · ·) 1
p
R n \B1(y)
We put [m] C(J; Cr2 (Rn )) = supt∈J, x, y∈Rn |m(t, x) − m(t, y)| |x − y| −r2 . By hypothesis
1 − s2/r2 > 0. If η ∈ [0, 1 − s2/r2), then (1 − η)r2 − s2 > 0, and we obtain
I1(m, f) ≤ (2|m| L∞(J×Rn )) η [m] 1−η
C(J; Cr2 (Rn )) (
� � �
|f(t, y)| p dx dy dt 1
) p
|x − y| n−((1−η)r2−s2)p
Further,
Hence
[mf] Y T 2
2
J Rn B1(y)
≤ C(p, r2, s2, n)|m| η
L∞(J×Rn ) [m]1−η
C(J; Cr2 (Rn )) |f| Lp(J×Rn ).
� �
I2(m, f) ≤ 2|m| L∞(J×Rn )(
J Rn �
Rn \B1(y)
≤ C(p, s2, n)|m| L∞(J×Rn )|f| Lp(J×Rn ).
|f(t, y)| p dx dy dt
|x − y| n+s2p
≤ [mf] Lp([0,T1];B s2 pp(Rn )) + [mf] Lp([T1,T2];B s2 pp(Rn ))
≤ C(|m| L∞([0,T1]×Rn ) + [m] C([0,T1]; Cr2 (Rn )) )(|f| Lp([0,T1]×Rn ) + [f] T
Y 1 )
2
+|m| L∞([T1,T2]×Rn �
) [f] T
Y 2 + C(1 + [m] C([T1,T2]; C
2
r2 (Rn )) )|f| Lp([0,T2]×Rn )
Let now [m] Cr1 (J; C(Rn )) = supt, τ∈J, x∈Rn |m(t, x) − m(τ, x)| |t − τ| −r1 . To estimate
, we take (4.23) with Ω = Rn as starting point, apply Fubini’s theorem, and
[mf] T
Y 2
1
use the same estimation techniques as for [mf] T
Y 2
2
[mf] Y T 2
1
to the result
≤ |m| L∞([0,T1]×Rn )[f] T
Y 1 + C1[m] Cr1 ([0,T1]; C(R
1
n )) |f| Lp([0,T1]×Rn )
+2|m| L∞([T1,T2]×Rn )([f] T
Y 2 + C2[m] Cr1 ([T1,T2]; C(R
1
n )) |f| Lp([0,T2]×Rn )).
66
) 1
p
�
.