Quasilinear parabolic problems with nonlinear boundary conditions

Further, K ∞ (α, θa) := {a ∈ L1, loc(R+) : a ∈ K r (α, θa) for all r ∈ N}. The kernel a is
called a K-kernel if there exist r ∈ N, θa > 0, and α ≥ 0, such that a ∈ K r (α, θa).
An example for a K-kernel is the so-called standard kernel:
Example 2.6.1 (standard kernel) Let a(t) = t α−1 /Γ(α), t > 0, with α > 0. Then
a ∈ L1, loc(R+), a is of subexponential growth, and its Laplace transform is given by
â(λ) = λ −α , Re λ > 0. Hence, a ∈ K ∞ (α, α π
2 ).
From [58, p. 4793] it follows that Kr (α, θ) �= ∅ entails the inequality θ ≥ α π
2 . In view of
Example 2.6.1 we thus get the following equivalence:
Remarks 2.6.1 Let α > 0. Then K r (α, θ) �= ∅ if and only if θ ≥ α π
2 .
The subsequent lemma collects some important algebraic properties of K-kernels.
Lemma 2.6.2 Suppose a ∈ K r (α, θa), b ∈ K s (β, θb), and ω > 0. Let aω be defined by
aω(t) = a(t)e −ωt , t ≥ 0. Then the following statements hold true.
(i) aω ∈ K r (α, θa) ∩ L1(R+), and there exist positive constants C1, C2 such that
C1
|λ + ω| α ≤ |âω(λ)| ≤
C2
, Re λ > 0. (2.19)
|λ + ω| α
(ii) If a and b satisfy (2.18), then a + b ∈ K min{r,s} (min{α, β}, max{θa, θb}).
(iii) a ∗ b ∈ K min{r,s} (α + β, θa + θb).
(iv) If α > β and lim infµ→0 |â(µ)/ ˆ b(µ)| > 0, then there exists a unique kernel c ∈
K min{r,s} (α − β, θa + θb) such that a = b ∗ c. If in addition Im â(λ)·Im ˆ b(λ) ≥ 0 for
all Re λ > 0, then c ∈ K min{r,s} (α − β, max{θa, θb}).
(v) If α > 0 and θa < π, then there is a unique kernel ξ ∈ K r (α, θa) such that
ξ + ωξ ∗ a = a.
(vi) If a ∈ L1(R+) and ɛ := ω|a| L1(R+) < 1, then there is a unique kernel ξ ∈ K r (α, θa +
arcsin(ɛ)) ∩ L1(R+) such that ξ − ωξ ∗ a = a.
Proof. Suppose a ∈ K r (α, θa) and ω > 0. Then it is evident that aω lies in L1(R+) and
is of subexponential growth. Further, âω(λ) = â(λ + ω), Re λ > 0. Thus, for all k ∈ N
with 1 ≤ k ≤ r and λ ∈ C+,
|λ k â (k)
ω (λ)| = |λ k â (k) (λ + ω)| = |(λ + ω) k â (k) (λ + ω)|
≤ C|â(λ + ω)| = C|âω(λ)|,
� �
�
�
λ �
�
�λ
+ ω �
where C > 0 is the constant of r-regularity of a, i.e. aω is r-regular. We easily see θasectoriality
of aω, too. Moreover, Lemma 2.6.1(v) implies |âω(λ)| ≤ c|âω(|λ|)| for all λ ∈
C+, with c not depending on λ. So, owing to continuity and the asymptotic behaviour
of â on R+ described in (K3), there exist two positive constants C1 and C2 such that
|âω(λ)(λ+ω) α | = |â(λ+ω)(λ+ω) α | ∈ [C1, C2] for all Re λ > 0. This shows (2.19). As for
property (K3), by Lemma 2.6.1(ii), we have lim infµ→0 | âω(µ)| = |â(ω)| > 0. The other
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