Quasilinear parabolic problems with nonlinear boundary conditions

and the inequalities
φ ∞ A ≥ θA ≥ φA ≥ sup{|arg λ| : λ ∈ σ(A)}.
Operators with bounded imaginary powers are of overriding importance in the context
of sums of commuting linear operators. This finds expression in the Dore-Venni
theorem, which is one of the fundamental results in this connection. We will state a
version of it in the next section.
Another important application of the class BIP(X) concerns the fractional power
spaces
Xα = XA α = (D(Aα ), | · |α), |x|α = |x| + |A α x|, 0 < α < 1,
where A ∈ S(X). If A belongs to BIP(X), one can derive a characterization of Xα in
terms of complex interpolation spaces.
Theorem 2.2.1 Let A ∈ BIP(X). Then
Xα = [X, DA]α, α ∈ (0, 1),
the complex interpolation space between X and DA ↩→ X of order α.
For a proof we refer to Triebel [78, pp. 103-104], or Yagi [82].
At this point let us state a very useful property of the real interpolation spaces
(X, Xα)β, p, 0 < α, β < 1, 1 ≤ p ≤ ∞, between X and the fractional power spaces Xα
associated with an operator A ∈ S(X), defined by, e.g. the K-method. Recall that for
A ∈ S(X), 1 ≤ p ≤ ∞, and γ ∈ (0, 1), the real interpolation space (X, DA)γ, p coincides
with the space DA(γ, p) which is defined by means of
where
[x] DA(γ,p) =
DA(γ, p) := {x ∈ X : [x] DA(γ,p) < ∞},
�
( � ∞
0 (tγ |A(t + A) −1x|X) supt>0 tγ |A(t + A) −1x|X 1
p d
dt ) p : 1 ≤ p < ∞
: p = ∞,
(2.4)
see e.g. [16, Prop. 3].
Suppose now that A ∈ BIP(X). By Theorem 2.2.1 and the reiteration theorem (see
e.g. Amann [5, Section 2.8]), we deduce that
(X, Xα)β, p = (X, [X, DA]α)β, p = (X, DA)αβ, p, 0 < α, β < 1, 1 ≤ p ≤ ∞. (2.5)
Since A ∈ S(X) implies A α ∈ S(X) for all α ∈ (0, 1), we conclude from (2.5) that
DA α(β, p) = DA(αβ, p). One can show that this relation is even valid for all A ∈ S(X),
cf. Komatsu [49, Thm. 3.2].
Theorem 2.2.2 Let A ∈ S(X). Then
DA α(β, p) = DA(αβ, p), α, β ∈ (0, 1), 1 ≤ p ≤ ∞.
We come now to R-sectorial operators. First we have to recall the definition of R-bounded
families of bounded linear operators.
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