Quasilinear parabolic problems with nonlinear boundary conditions

Using (2.19) for aω and bω yields |gω(λ)| ≤ C2|λ + ω| β−α ≤ C2|λ| β−α , Reλ > 0, where
C2 > 0 is independent of λ. In view of (2.21) we thus obtain
|λg ′ ω(λ)| + |gω(λ)| ≤ C
, Re λ > 0, (2.22)
|λ| α−β
with C > 0 not depending on λ. Proposition 2.1.1 then allows us to define uω ∈
C(0, ∞) ∩ L1, loc(R+) by means of ûω(λ) = gω(λ), Re λ > 0. Estimate (2.2) implies
that uω is of subexponential growth. So we have û(λ) ˆ bω(λ) = âω(λ), λ ∈ C+, i.e.
(uω ∗ bω)(t) = aω(t), t > 0. With u(t) := uω(t)e ωt , t > 0, we thus arrive at u ∗ b = a.
Observe that the construction of u is independent of the chosen ω > 0. Further, u
is of subexponential growth, for uω possesses this property for each ω > 0. Since
û(λ) = â(λ)/ ˆ b(λ), Reλ > 0, it is clear that u is (θa + θb)-sectorial, even max{θa, θb}sectorial
provided that Im â(λ)·Im ˆ b(λ) ≥ 0 for all λ ∈ C+.
We now show min{r, s}-regularity of u. To this purpose we put m = min{r, s} and
ˆh(λ) = 1/ ˆ b(λ), λ ∈ C+. Then 1-regularity of b yields an estimate
|λ kˆ h (k) (λ)| ≤ C| ˆ h(λ)|, λ ∈ C+, (2.23)
for k = 1. Here the constant C does not depend on λ. Let us now assume that (2.23)
holds true for all k ∈ N with 1 ≤ k ≤ n, where n ∈ N and n < m. If we then differentiate
(n + 1) times both sides of the equation ˆ b ˆ h = 1 and use Leibniz’ formula it becomes
apparent that
ˆh (n+1) (λ) = − 1
ˆ b(λ)
i=1
n+1 �
� �
n + 1
i
i=1
ˆ b (i) (λ) ˆ h (n+1−i) (λ), Re λ > 0.
Thus, by hypothesis and m-regularity of b,
|λ n+1ˆ
n+1 �
� �
(n+1) n + 1
h (λ)| ≤
i
� �
��� λiˆ �
b (i) (λ)
� �
� �
� �λ
ˆb(λ) �
n+1−iˆ
�
(n+1−i) �
h (λ) � ≤ C1| ˆ h(λ)|
for all λ ∈ C+, with C1 not depending on λ. So, induction over n ≤ m establishes (2.23)
for all k ≤ m. Using this fact and m-regularity of a we can argue as in the proof of (iii)
(cp. (2.20)) to see that u is m-regular.
The function u also fulfills property (K3) with exponent α − β. In fact, we have
lim infµ→0 |û(µ)| > 0 by assumption. Moreover, owing to a1 ∈ Kr (α, θa), b1 ∈ Ks (β, θb),
and (2.19), we have
� �
�
�
lim sup
µ→∞
�
�â(µ)
�
�
� �
�ˆb(µ)
� µα−β �
�â(µ
+ 1)(µ + 1)
= lim sup �
µ→∞ �
α �
�
� < ∞.
ˆb(µ + 1)(µ + 1) β �
Likewise we see lim infµ→∞ |û(µ)|µ α−β > 0. Hence, the kernel c := u possesses all
properties claimed in (iv). Uniqueness follows from the unique inverse of the Laplace
transform. The proof of (iv) is complete.
Our next aim is to show (v). Suppose a ∈ K r (α, θa), ω, α > 0, and θa < π. Further
fix an arbitrary η > 0. Due to the last assumption there exists a constant c > 0 not
depending on λ such that
|1 + ωâ(λ)| ≥ c, Re λ > 0. (2.24)
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