Quasilinear parabolic problems with nonlinear boundary conditions

and which serves as a reference function in the following sense. For F = Y k1,s1
1 ∩ Y k2,s2
2 ,
and given ρ ∈ (0, ρ0], let Σ = Σ(ρ, w, F ) be the set of all K-valued f in F m such that
f − w ∈ 0F m and |f − w|F m ≤ ρ. The aim is to show that Bf ∈ F whenever f ∈ Σ and
that
|Bf − Bg|F ≤ C(ρ + µ(T ) + |bξ(·, ·, w)|∞, m)|f − g|F m, f, g ∈ Σ, (6.19)
where C > 0 is independent of ρ and T , and 0 < µ(T ) → 0 as T tends to zero. We
further need the property that Bf ∈ X1, s if f ∈ X m 1, s0 ∩ (C0 1 )m (s0 ∈ (s, 1)), and we
wish to have a Lipschitz estimate of the form
|Bf − Bg|X1, s ≤ µ(T )(|f − g|X m 1, s 0
+ |f − g|∞, m), (6.20)
for all (K-valued) f, g ∈ Σ ′ = Σ ′ (ρ, w) := {h ∈ X m 1, s0 ∩ (C0 1 )m : (h − w)|t=0 = 0 and |h −
w|X m 1, s0 + |h − w|∞, m ≤ ρ}, where again the constant µ(T ) > 0 vanishes as T → 0.
In both cases we shall only establish the Lipschitz estimate. The corresponding
mapping property of B follows by means of the same techniques; here the proof is
even simpler than for the Lipschitz estimate. When proving Lipschitz estimates we will
restrict ourselves to the seminorms of highest order, that is, e.g., if we are to estimate
Bf − Bg in the Y 1,s1
1
and [Bf − Bg] Y 1,s 2
2
∩ Y 1,s2
2
-norm, we shall consider only the seminorms [Bf − Bg] Y 1,s 1
1
. Having proved the desired estimate for these terms, it will then be
clear how to obtain it for the seminorm terms of lower order, which are much easier to
treat.
We begin now with the spaces
(Y k1,s1
1
∩ Y k2,s2
2
) m = B k1+s1
pp
(J; Lp(Ω, R m )) ∩ Lp(J; B k2+s2
pp (Ω, R m )),
ki ∈ {0, 1}, si ∈ (0, 1), i = 1, 2. Here, the cases (k1, k2) = (0, 0), (0, 1), (1, 1) have to
be studied. We assume that in each of these cases, p is large enough such that we
have the embedding Y 0,s1
1 ∩ Y 0,s2
2 ↩→ C(J; C( ¯ Ω)), Y 0,s1
1 ∩ Y 1,s2
2 ↩→ C(J; C1 ( ¯ Ω)), and
Y 1,s1
1 ∩ Y 1,s2
2 ↩→ Cr (J; C( ¯ Ω)) ∩ C(J; C1 ( ¯ Ω)), respectively, with some number r in (s1, 1).
More precisely, we make the assumption that p > p⋆(n, k1, s1, k2, s2), where
⎧
⎪⎨
p⋆(n, k1, s1, k2, s2) :=
⎪⎩
1
s1
1
s1
max
+ n
s2
(1 + 1
� 1
s2
s2
) + n
s2
( 1+s2
1+s1
+ n), 1 + n 1+s1
1+s2
�
: (k1, k2) = (0, 0)
: (k1, k2) = (0, 1)
: (k1, k2) = (1, 1).
Roughly speaking, this condition on the exponent p when translated to the situation of
Section 6.1 corresponds to the assumption (H2) therein.
The first lemma is concerned with the case (k1, k2) = (0, 0). Here and in the subsequent
estimates we denote by µ(T ) a positive constant depending on T such that
µ(T ) → 0 as T → 0. Further, M and C denote constants which may differ from line to
line, but which do not depend on T and ρ.
Lemma 6.2.1 Let w ∈ (Y 0,s1
1 ∩ Y 0,s2
2 ) m be a fixed K-valued function and ρ > 0. Let
further Σ be as described above. Suppose that there exist CHL > 0, r ∈ (s1, 1), and
Cb ∈ Lp(Ω) such that
|bξ(t, x, ξ) − bξ(τ, x, η)| ≤ CHL(Cb(x)|t − τ| r + |ξ − η|),
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