Quasilinear parabolic problems with nonlinear boundary conditions

ψ(f, g) ≤ Mρ + |bξ(·, ·, w)|∞, m, f, g ∈ Σ,
M being independent of T because (f − g)|t=0 = 0. Hence the assertion follows with
µ(T ) = [w] 0,s
(Y 1
1 ) m + T
r−s1+ 1
p . �
By repeating the above considerations with the roles of J and Ω being reversed, one
obtains (under the corresponding assumptions, cf. (H4) in Section 6.1) the estimate
�
�
≤ C ρ + µ(T ) + |bξ(·, ·, w)|∞, m
[b(·, ·, f) − b(·, ·, g)] Y 0,s 2
2
We come now to exponents greater than 1.
|f − g| (Y 0,s 1
1
∩Y 0,s2 2 ) m, f, g ∈ Σ.
Lemma 6.2.2 Let w ∈ (Y 1,s1
1 ∩ Y 1,s2
2 ) m be a fixed K-valued function and ρ > 0. Let
further Σ be as described above. Suppose bξ, btξ ∈ L∞(J × Ω × K, Rm ), bξξ ∈ L∞(J ×
Ω × K, Rm×m ), and assume that there exist CHL > 0, r1 ∈ (s1, 1), and Cb ∈ Lp(Ω) such
that
|btξ(t, x, ξ) − btξ(τ, x, η)| ≤ CHL(Cb(x)|t − τ| r1 + |ξ − η|),
|bξξ(t, x, ξ) − bξξ(τ, x, η)| ≤ CHL(|t − τ| r1 + |ξ − η|),
for all t, τ ∈ J, ξ, η ∈ K, and a.a. x ∈ Ω. Then there exists a constant C > 0 not
depending on T and ρ such that
�
�
≤ C ρ + µ(T ) + |bξ(·, ·, w)|∞, m
[b(·, ·, f) − b(·, ·, g)] Y 1,s 1
1
|f − g| (Y 1,s 1
1
∩Y 1,s2 2 ) m, f, g ∈ Σ.
Proof. For brevity we set Y = Y 1,s1
1 ∩ Y 1,s2
2 . Remember that we have the embedding
Y ↩→ Cr (J; C( ¯ Ω)) for some r ∈ (s1, 1). Let now f, g ∈ Σ be arbitrary functions. Put
h1(t, τ, x) = bt(t, x, f(t, x)) − bt(t, x, g(t, x)) − bt(τ, x, f(τ, x)) + bt(τ, x, g(τ, x)),
h2(t, τ, x) = bξ(t, x, f(t, x)) · ft(t, x) − bξ(t, x, g(t, x)) · gt(t, x)
for t, τ ∈ J, and a.a. x ∈ Ω. Then
[b(·, ·, f) − b(·, ·, g)] 1,s
Y 1
1
� T � T �
≤
2�
(
i=1
−bξ(τ, x, f(τ, x)) · ft(τ, x) + bξ(τ, x, g(τ, x)) · gt(τ, x)
0
0
Ω
= (
� T � T
0
0
�
Ω
(|h1(t, τ, x) + h2(t, τ, x)|) p
|t − τ| 1+s1p
|hi(t, τ, x)| p
1
dx dτ dt) p =: I1 + I2.
|t − τ| 1+s1p
dx dτ dt) 1
p
Concerning I1, we may use the estimates from the proof of Lemma 6.2.1, thereby obtaining
�
I1 ≤ C (ρ + |btξ(·, ·, w)|∞, m)[f − g] 0,s
(Y 1
1 ) m �
+ (ρ + µ(T ))|f − g|∞, m
�
�
≤ C
(ρ0 + |btξ|∞, m)µ(T )[f − g] (Cr 1 ) m + M(ρ + µ(T ))|f − g|Y m
≤ C(ρ + µ(T ))|f − g|Y m.
The term I2 is more sophisticated. We employ the identity
aA − bB − cC + dD = (a − b − c + d)D + (−a + b + c)(A − B − C + D) +
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