Quasilinear parabolic problems with nonlinear boundary conditions

Definition 2.6.1 ([63, Def. 3.2]) Let a ∈ L1, loc(R+) be of subexponential growth and
suppose â(λ) �= 0 for all Re λ > 0. a is called sectorial with angle θ > 0 (or merely
θ-sectorial) if
|arg â(λ)| ≤ θ for all Re λ > 0. (2.16)
Here, arg â(λ) is defined as the imaginary part of a fixed branch of log â(λ), and θ in
(2.16) is allowed to be greater than π. In case a is sectorial, we always choose that
branch of log â(λ) which yields the smallest angle θ; in particular, if â(λ) is real for real
λ we choose the principal branch.
The next definition introduces an appropriate notion of regularity of kernels.
Definition 2.6.2 ([63, Def. 3.3]) Let a ∈ L1, loc(R+) be of subexponential growth and
k ∈ N. a is called k-regular if there is a constant c > 0 such that
|λ n â (n) (λ)| ≤ c |â(λ)| for all Re λ > 0, 0 ≤ n ≤ k. (2.17)
It is not difficult to see that convolutions of k-regular kernels are again k-regular. Furthermore,
k-regularity is preserved by integration and differentiation, while sums and
differences of k-regular kernels need not be k-regular. However, if a and b are k-regular
and
|arg â(λ) − arg ˆ b(λ)| ≤ θ < π, Re λ > 0, (2.18)
then a + b is k-regular as well (see Lemma 2.6.2(ii)).
Some important properties of 1-regular kernels are contained in the following lemma.
Lemma 2.6.1 ([63, Lemma 8.1]) Suppose a ∈ L1, loc(R+) is of subexponential growth
and 1-regular. Then
(i) â(iρ) := limλ→iρ â(λ) exists for each ρ �= 0;
(ii) â(λ) �= 0 for each Re λ ≥ 0;
(iii) â(i·) ∈ W ∞ 1, loc (R \ {0});
(iv) |ρâ ′ (iρ)| ≤ c|â(iρ)| for a.a. ρ ∈ R;
(v) there is a constant c > 0 such that
c|â(|λ|)| ≤ |â(λ)| ≤ c −1 |â(|λ|)|, Re λ ≥ 0, λ �= 0;
(vi) limr→∞ â(re iφ ) = 0 uniformly for |φ| ≤ π
2 .
With regard to Volterra operators in Lp (see Section 2.8), we now introduce the subsequent
class of kernels.
Definition 2.6.3 Let a ∈ L1, loc(R+) be of subexponential growth, and assume r ∈ N,
θa > 0, and α ≥ 0. Then a is said to belong to the class K r (α, θa) if
(K1) a is r-regular;
(K2) a is θa-sectorial;
(K3) lim sup µ→∞ | â(µ)| µ α < ∞, lim infµ→∞ | â(µ)| µ α > 0, lim infµ→0 | â(µ)| > 0.
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