Quasilinear parabolic problems with nonlinear boundary conditions

Clearly, φA ∈ [0, π) and φA ≥ sup{|arg λ| : λ ∈ σ(A)}.
We turn now to the H ∞ -calculus. For φ ∈ (0, π], we define the space of holomorphic
functions on Σφ by H(Σφ) = {f : Σφ → C holomorphic}, and the space
H ∞ (Σφ) = {f : Σφ → C holomorphic and bounded},
which when equipped with the norm |f| φ ∞ = sup{|f(λ)| : |arg λ| < φ} becomes a Banach
algebra. We further let H0(Σφ) = �
α, β< 0 Hα, β(Σφ), where Hα, β(Σφ) := {f ∈ H(Σφ) :
|f| φ
α, β
< ∞}, and |f|φ
α, β := sup |λ|≤1 |λ α f(λ)| + sup |λ|≥1 |λ −β f(λ)|. Now suppose that
A ∈ S(X) and φ ∈ (φA, π). We select any ψ ∈ (φA, φ) and denote by Γψ the oriented
contour defined by Γψ(t) = −te iψ , −∞ < t ≤ 0, and Γψ(t) = te −iψ , 0 ≤ t < ∞. Then
the Dunford integral
f(A) = 1
2πi
�
Γψ
f(λ)(λ − A) −1 dλ, f ∈ H0(Σφ),
converges in B(X) and does not depend on the choice of ψ. Further, it defines via
ΦA(f) = f(A) a functional calculus ΦA : H0(Σφ) → B(X) which is an algebra homomorphism.
The following definition is in accordance with McIntosh [57].
Definition 2.2.1 A sectorial operator A in X admits a bounded H ∞ -calculus if there
are φ > φA and a constant Kφ < ∞ such that
|f(A)| ≤ Kφ|f| φ ∞, for all f ∈ H0(Σφ). (2.3)
The class of sectorial operators which admit an H ∞ -calculus will be denoted by H ∞ (X).
The H ∞ -angle φ ∞ A of A ∈ H∞ (X) is defined by
φ ∞ A = inf{φ > φA : (2.3) is valid}.
If A ∈ H ∞ (X), then the functional calculus for A on H0(Σφ) extends uniquely to
H ∞ (Σφ).
We consider next operators with bounded imaginary powers. This subclass of S(X)
has been introduced in Prüss and Sohr [69]. To justify the subsequent definition, we first
note that for any A ∈ S(X) one can define complex powers A z , where z ∈ C is arbitrary;
cf. Komatsu [48], Prüss [63, Section 8.1] or Denk, Hieber, Prüss [29, Section 2.2].
Definition 2.2.2 A sectorial operator A in X is said to admit bounded imaginary
powers if A is ∈ B(X) for each s ∈ R and there is a constant C > 0 such that |A is | ≤ C
for |s| ≤ 1. The class of such operators will be denoted by BIP(X).
Since A is has the group property (see e.g. Prüss [63, Proposition 8.1]), it is evident
that A admits bounded imaginary powers if and only if {A is : s ∈ R} forms a strongly
continuous group of bounded linear operators in X. The growth bound θA of this group,
that is
θA = lim sup
|s|→∞
1
|s| log |Ais |,
will be called the power angle of A. Owing to the fact that the functions fs defined
by fs(z) = z is belong to H ∞ (Σφ), for any s ∈ R and φ ∈ (0, π), we clearly have the
inclusions
H ∞ (X) ⊂ BIP(X) ⊂ S(X),
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