Quasilinear parabolic problems with nonlinear boundary conditions

Letting gη(λ) = âη(λ)/(1 + ωâη(λ)), Reλ > 0, we thus obtain |gη(λ)| ≤ C0|λ| −α for all
λ ∈ C+, where C0 > 0 is independent of λ. Here, we also made use of (2.19) for the
kernel aη. Observe that
λg ′ η(λ) = gη(λ) λâ′ η(λ)
âη(λ)
·
1
, Re λ > 0.
1 + ωâη(λ)
Thanks to (2.24) and 1-regularity of aη, we therefore get an estimate
|λg ′ η(λ)| + |gη(λ)| ≤ C
, Re λ > 0, (2.25)
|λ| α
which, together with α > 0 and holomorphy of gη in C+, implies existence of uη ∈
C(0, ∞) ∩ L1, loc(R+) satisfying ûη(λ) = gη(λ), Re λ > 0, by Proposition 2.1.1. (2.2)
entails that uη is of subexponential growth. By inversion of the Laplace transform we
then obtain uη + ωaη ∗ uη = aη, hence u + ωa ∗ u = a, where u(t) := uη(t)eηt , t > 0. As
in the proof of (iv), we see that the construction of u is independent of η > 0, and that
u is of subexponential growth. From û = â/(1 + ωâ) and ω > 0 one deduces that u is
θa-sectorial. Furthermore, the function h(λ) := 1 + ωâ(λ) defined on C+ satisfies (2.23)
with a suitable constant C > 0 for k = 1, . . . , r. In fact,
|λ kˆ
� �
�
(k) k k (k)
h (λ)| = ω |λ â (λ)| ≤ C|â(λ)| ≤ C| h(λ)| ˆ �
1 �
�
�ω
+ 1/â(λ) � ≤ C1| ˆ h(λ)|
for all λ ∈ C+ and k = 1, . . . , r, in virtue of θa < π and r-regularity of a. By the
considerations in the proof of (iv), that property of h is passed on to the function
¯h(λ) := 1/h(λ), λ ∈ C+, which is well-defined in view of (2.24). Then û = â ¯ h, and as
above, with the aid of Leibniz’ formula, we see that u is r-regular. Concerning (K3), the
assumption ζ := lim infµ→0 |â(µ)| > 0 implies lim infµ→0 |û(µ)| > 0, since
lim inf
µ→0
|û(µ)| ≥ lim inf
µ→0 (ω + |1/â(µ)|)−1 = (ω + 1/ζ) −1 .
Besides, by (2.24), lim sup µ→∞ |â(µ)|µ α < ∞, and (2.19) applied to a1, we have
as well as
lim sup
µ→∞
lim inf
µ→∞ |û(µ)|µα ≥ lim inf
µ→∞
|û(µ)|µ α ≤ c −1 lim sup |û(µ)|µ
µ→∞
α < ∞,
(µ + 1) α
≥ lim inf
ω + |1/â(µ + 1)| µ→∞
(µ + 1) α
> 0.
ω + C(µ + 1) α
Hence, the kernel ξ := u satisfies ξ + ωa ∗ ξ = a and belongs to K r (α, θa). Uniqueness
follows again from the unique inverse of the Laplace transform. Thus (v) is shown.
It remains to prove (vi). Suppose a ∈ K r (α, θa) ∩ L1(R+), ω > 0, and ɛ :=
ω|a| L1(R+) < 1. We proceed as in the previous part. Fix an arbitrary η > 0 and
define gη(λ) := âη(λ)/(1 − ωâη(λ)), Reλ > 0. This time the assumption ɛ < 1 ensures
that the denominator in the definition of gη is bounded away from zero. Using this and
1-regularity of aη yields (2.25) in a similar fashion as above. With the aid of Proposition
2.1.1, existence of u ∈ L1, loc(R+) with u − ωa ∗ u = a can then be seen. By the
same line of arguments as in the previous part one further shows that u is r-regular.
Validity of the conditions in (K3) with exponent α can be proved for u similarly as
above. By elementary trigonometry, |arg (1 − ωâ(λ))| ≤ arcsin(ɛ) for all λ ∈ C+, i.e.
u is (θa + arcsin(ɛ))-sectorial. Last but not least, u ∈ L1(R+) follows from ɛ < 1 and
a ∈ L1(R+) by the Paley-Wiener theorem. Hence, ξ := u possesses all properties claimed
in (vi), uniqueness being evident as above. �
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