Quasilinear parabolic problems with nonlinear boundary conditions

Theorem 2.8.1 (Prüss [63, Thm. 8.6]) Suppose X belongs to the class HT , p ∈
(1, ∞), and let a ∈ L1, loc(R+) be of subexponential growth. Assume that a is 1-regular
and θ-sectorial, where θ < π. Then there is a unique operator B ∈ S(Lp(R; X)) such
that
(Bf)˜(ρ) = 1
â(iρ) ˜ f(ρ), ρ ∈ R, ˜ f ∈ C ∞ 0 (R \ {0}; X). (2.28)
Moreover, B has the following properties:
(i) B commutes with the group of translations;
(ii) (µ + B) −1 Lp(R+; X) ⊂ Lp(R+; X) for each µ > 0, i.e. B is causal;
(iii) B ∈ BIP(Lp(R; X)), and θB = φB = θa, where
(iv) σ(B) = {1/â(iρ) : ρ ∈ R \ {0}}.
θa = sup{|arg â(λ)| : Re λ > 0}; (2.29)
The next theorem provides information about the domain of the operator B.
Proposition 2.8.1 (Prüss [63, Cor. 8.1]) Suppose X belongs to the class HT , p ∈
(1, ∞). Assume a ∈ K 1 (α, θ) with θ < π, and let B be defined by (2.28). Then D(B) =
H α p (R; X).
Here H α p (R; X) := D(B α/2
0 ), α ∈ R+, where B0 = −(d 2 /dt 2 ) ∈ BIP(Lp(R; X)), cf. [63,
p. 226].
Suppose the assumptions of Proposition 2.8.1 hold. Let J = [0, T ] or J = R+. We put
H α p (J; X) = {f|J : f ∈ H α p (R; X)} and endow this space with the norm |f| H α p (J;X) =
inf{|g| H α p (R;X) : g|J = f}. We further introduce the subspace 0H α p (J; X) by means
of 0H α p (J; X) = {f|J : f ∈ H α p (R; X) and supp f ⊆ R+}. Define then the operator
B ∈ S(Lp(J; X)) as the restriction of the operator B constructed in Theorem 2.8.1 to
Lp(J; X). This makes sense in virtue of causality. In fact, we have
D(B) = D(B| Lp(J;X)) = {f ∈ Lp(J; X) ∩ D(B) : Bf ∈ Lp(J; X)}
= {f ∈ Lp(J; X) ∩ H α p (R; X) : Bf ∈ Lp(J; X)}
= {f|J : f ∈ H α p (R; X), supp f ⊆ R+, Bf ∈ Lp(R; X), and supp Bf ⊆ R+}
= {f|J : f ∈ H α p (R; X) and supp f ⊆ R+}
= 0H α p (J; X),
the equals sign before the last following from the causality of B. Assuming in addition
a ∈ L1(R+) in case J = R+, by Young’s inequality, the operator B is invertible and
B −1 w = a ∗ w for all w ∈ Lp(J; X). From Theorem 2.8.1 we further see that B ∈
BIP(Lp(J; X)) and θB ≤ θB = θa. We summarize these observations in the subsequent
corollary.
Corollary 2.8.1 Let X be a Banach space of class HT , p ∈ (1, ∞), and J = [0, T ] be a
compact interval or J = R+. Suppose a ∈ K 1 (α, θ) with θ < π, and assume in addition
a ∈ L1(R+) in case J = R+. Then the restriction B := B| Lp(J;X) of the operator B
constructed in Theorem 2.8.1 to Lp(J; X) is well-defined. The operator B belongs to
the class BIP(Lp(J; X)) with power angle θB ≤ θB = θa and is invertible satisfying
B −1 w = a ∗ w for all w ∈ Lp(J; X). Moreover D(B) = 0H α p (J; X).
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