Quasilinear parabolic problems with nonlinear boundary conditions

Theorem 3.4.2 Suppose X is a Banach space of class HT , p ∈ (1, ∞). Let F ∈
BIP(X) be invertible with power angle θF < π/2, and let D j
F denote the domain D(F j )
of F j equipped with its graph norm, j = 1, 2.
Then (3.25) has a unique solution u in Z := H2 p(R+; X) ∩ Lp(R+; D2 F ) if and only if
the following two conditions are satisfied.
(i) f ∈ Lp(R+; X); (ii) φ ∈ DF (2 − 1
p , p).
If this is the case we have in addition u ∈ H 1 p(R+; D 1 F ).
This result has been obtained by Prüss, cf. [65, Theorem 3]. Recall that DF (2−1/p, p) =
{g ∈ D(F ) : F g ∈ DF (1 − 1/p, p)}.
There is a corresponding result for the abstract second order problem with abstract
Robin condition �
−u ′′ (y) + F 2u(y) = f(y), y > 0,
−u ′ (3.26)
(0) + Du(0) = ψ,
in Lp(R+; X). For D = 0, the Robin condition becomes the Neumann condition.
Theorem 3.4.3 Suppose X is a Banach space of class HT , p ∈ (1, ∞). Let F ∈
BIP(X) be invertible with power angle θF < π/2, and let D j
F denote the domain D(F j )
of F j equipped with its graph norm, j = 1, 2. Suppose that D is pseudo-sectorial in X,
belongs to BIP(R(D)), commutes with F , and is such that θF + θD < π.
Then (3.26) has a unique solution u in Z := H2 p(R+; X) ∩ Lp(R+; D2 F ) with u(0) ∈
D(D) and Du(0) ∈ DF (1−1/p, p) if and only if the following two conditions are satisfied.
(i) f ∈ Lp(R+; X); (ii) ψ ∈ DF (1 − 1
p , p).
If this is the case we have in addition u ∈ H 1 p(R+; D 1 F ).
This result is also due to Prüss, see [65, Theorem 4].
3.5 Parabolic Volterra equations on an infinite strip
We now study the vector-valued problem
� u − a ∗ ∂ 2 yu + a ∗ Au = f, t ∈ J, y > 0,
u(t, 0) = φ(t), t ∈ J,
(3.27)
in Lp(J; Lp(R+; X)). Here X is a Banach space which belongs to the class HT , J = [0, T ]
is a compact time-interval, A is a sectorial operator in X and the kernel a belongs to the
class K 1 (α, θa) with α ∈ (0, 2). The data f and φ are given. Our aim is to characterize
unique existence of a solution u in the maximal regularity class of type Lp, i.e.
u ∈ Z := H α p (J; Lp(R+; X)) ∩ Lp(J; H 2 p(R+; X)) ∩ Lp(J; Lp(R+; DA))
in terms of regularity classes for the data. Recall that DA denotes the space D(A)
equipped with the graph norm of A. The main result reads as follows.
48