Quasilinear parabolic problems with nonlinear boundary conditions

two properties in (K3) follow immediately from (2.19) and limµ→∞ |µ α /(µ + ω) α | = 1.
Hence, assertion (i) of Lemma 2.6.2 is proved.
To show (ii), suppose a ∈ K r (α, θa) and b ∈ K s (β, θb). Trivially, a + b ∈ L1, loc(R+),
and a+b is of subexponential growth. We then note that by (2.18) there exists a constant
c > 0 such that |â(λ)| + | ˆ b(λ)| ≤ c|â(λ) + ˆ b(λ)|, Re λ > 0. Therefore, for all k ∈ N with
1 ≤ k ≤ min{r, s} and λ ∈ C+,
|λ k (â(λ) + ˆ b(λ))| ≤ |λ k â(λ)| + |λ kˆ b(λ)| ≤ C(|â(λ)| + | ˆ b(λ)|) ≤ C c|â(λ) + ˆ b(λ)|,
where C only depends on the constants of min{r, s}-regularity of a and b. This proves
min{r, s}-regularity of a + b. From θa-sectoriality of a, θb-sectoriality of b, and (2.18)
it follows that |arg (â(λ) + ˆb(λ))| ≤ max{θa, θb} for all Re λ > 0. Thanks to (2.18) we
further have
lim inf
µ→0 |â(µ) + ˆb(µ)| ≥ c −1 lim inf
µ→0 (|â(µ)| + |ˆb(µ)|) > 0,
which shows the third condition in (K3) for a + b. W.l.o.g. we may then assume α ≤ β
and obtain the estimates
lim inf
µ→∞ |â(µ) + ˆ b(µ)|µ α ≥ c −1 lim inf
µ→∞ (|â(µ)| + |ˆ b(µ)|)µ α ≥ c −1 lim inf
µ→∞ |â(µ)|µα > 0,
lim sup
µ→∞
|â(µ) + ˆ b(µ)|µ α ≤ lim sup
µ→∞
|â(µ)|µ α + lim sup |
µ→∞
ˆb(µ)|µ β µ α−β < ∞.
So assertion (ii) is also established.
We now come to (iii). Suppose a ∈ K r (α, θa), b ∈ K s (β, θb). By Young’s inequality,
a ∗ b ∈ L1, loc(R+) and given ε > 0, we have
|(a ∗ b)e −ε· | L1(R+) = |(aε ∗ bε)| L1(R+) ≤ |aε| L1(R+) |bε| L1(R+) < ∞,
i.e. a ∗ b is of subexponential growth. Further, (a ∗ b)ˆ = â ˆ b due to the convolution
theorem. So a ∗ b is (θa + θb)-sectorial. Assuming k ∈ N, 1 ≤ k ≤ min{r, s} Leibniz’
formula in combination with r-regularity of a and b yields
|λ k (â(λ) ˆ b(λ)) (k) | ≤
k�
i=0
Thus a ∗ b is min{r, s}-regular. Finally,
lim sup
µ→∞
� �
k
|λ
i
i â (i) (λ)| |λ (k−i)ˆ(k−i) b (λ)| ≤ C < ∞, Re λ > 0. (2.20)
|â(µ) ˆ b(µ)|µ α+β ≤ (lim sup
µ→∞
|â(µ)|µ α )(lim sup |
µ→∞
ˆb(µ)|µ β ) < ∞.
The other two conditions in (K3) are shown similarly. Hence, a ∗ b satisfies (K3) with
exponent α + β, and so (iii) is proved.
Next we show (iv). Suppose a ∈ Kr (α, θa), b ∈ Ks (β, θb), and α > β. For any fixed
ω > 0 we know from (i) that aω ∈ Kr (α, θa) and bω ∈ Ks (β, θb), in particular ˆbω(λ) �= 0,
Reλ > 0, due to Lemma 2.6.1(ii). So we can define gω(λ) := âω(λ)/ ˆbω(λ), Reλ > 0. The
function gω is holomorphic in C+, and by 1-regularity of aω and bω we get
|λg ′ ω(λ)| =
�
�
�
�
�
λâ ′ ω(λ) ˆ bω(λ) − λâω(λ) ˆ b ′ ω(λ)
ˆ bω(λ) 2
�
�
�
�
� ≤
����� λâ ′ �
ω(λ) �
�
âω(λ) � +
�
�
�λ
�
�
ˆb ′ ��
�
ω(λ) �
� |gω(λ)|
ˆbω(λ) �
≤ C1|gω(λ)|, Re λ > 0. (2.21)
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