Quasilinear parabolic problems with nonlinear boundary conditions

where we used the assumption s ≥ 2(1/p − γ) for the second summand. Furthermore,
Thus, Theorem 3.1.4 yields
1 + γ
(1−θ)s −
u(0) ∈ (Xθ, DA) 1− 1
1
1
(1−θ)ps = 1 − η .
η , p = (D(Aθ ), D(A)) 1
1− , p, η
which entails, by the reiteration theorem (cf. Amann [5, Section 2.8]),
u(0) ∈ (X, D(A)) 1 1
θ +(1− η η ), p = DA(1 − 1−θ
η , p) = DA(1 + γ 1
s − ps , p).
Hence, u(0) ∈ DA(1 + γ/s − 1/ps, p) is established for all s > 1/p.
Suppose now that s + γ > n + 1/p with n ∈ N. If u ∈ Z, then u k (0) exists for all
0 ≤ k ≤ n. Taking θ = 1 − (k − γ)/s in (3.18) shows that
u (k) ∈ H s+γ−k
p (J; X) ∩ Lp(J; D 1−
A k−γ ).
s
k−γ
1− So, with B = A s , the above mapping property of the trace operator implies that
u (k) (0) ∈ DB(1 −
1
(s+γ−k)p , p) = D k−γ (1 −
1−
A s
1
(s+γ−k)p , p) = DA(1 + γ k 1
s − s − ps , p),
(3.19)
using once more Theorem 2.2.2.
If we replace in (3.17) and (3.19) the operator A by A s , assuming A ∈ RS(X) and
φ R A < π/s, we see that the composition of Dk t and the trace operator tr
tr ◦ D k t : H s+γ
p
(J; X) ∩ H γ p (J; DAs) → DA(s + γ − k − 1
p , p) (3.20)
is bounded. Thus, by real interpolation, we obtain boundedness of
tr ◦ Dk t : B s+γ
pp (J; X) ∩ H γ p (J; DA(s, p)) → DA(s + γ − k − 1,
p). (3.21)
Strong continuity of the translation group then yields (3.22) and (3.23) in the following
Theorem 3.2.1 Let X be a Banach space of class HT , p ∈ (1, ∞), γ ∈ [0, 1/p), and
s + γ > n + 1/p with n ∈ N0. Let further J = [0, T ] or R+, and A be an R-sectorial
operator in X with R-angle φR A < π/s. Then for all 0 ≤ k ≤ n,
and
H s+γ
p
(J; X) ∩ H γ p (J; DAs) ↩→ BUCk (J; DA(s + γ − k − 1
p , p)) (3.22)
B s+γ
pp (J; X) ∩ H γ p (J; DA(s, p)) ↩→ BUC k (J; DA(s + γ − k − 1,
p)). (3.23)
The proof of the previous result is inspired by [43]. Theorem 3.2.1 is an extension of [65,
Proposition 3], where κ = 0 and A is assumed to have bounded imaginary powers.
45
p
p