Quasilinear parabolic problems with nonlinear boundary conditions

Robin type. Unique existence of solutions of these problems in certain spaces of optimal
regularity is characterized in terms of regularity and compatibility conditions on the
given data. The main result concerning (1.16), Theorem 3.1.4, is proven in Section 3.1.
To describe it for the case J = [0, T ], let 1 < p < ∞, κ ∈ [0, 1/p), X be a Banach space
of class HT , A an R-sectorial operator in X with R-angle φR A , and a a K-kernel (with
angle θa) of order α ∈ (0, 2) such that α + κ /∈ {1/p, 1 + 1/p}. Let further DA denote
the domain of A equipped with the graph norm of A. Assume the parabolicity condition
θa + φR A < π. Then (1.16) has a unique solution u in the space Hα+κ p (J; X) ∩ Hκ p (J; DA)
if and only if the function f satisfies the subsequent conditions:
(i) f ∈ H α+κ
p (J; X);
(ii) f(0) ∈ DA(1 + κ 1
α − pα , p), if α + κ > 1/p;
(iii)
f(0) ˙ ∈ DA(1 + κ 1 1
α − α − pα , p), if α + κ > 1 + 1/p.
Here, DA(γ, p) stands for the real interpolation space (X, DA)γ, p. In the special case
a ≡ 1 (i.e. α = 1) and κ = 0, by putting g = ˙ f and u0 = f(0), we recover the main
theorem on maximal Lp-regularity for the abstract evolution equation
˙u + Au = g, t ∈ J, u(0) = u0, (1.17)
stating that in the above setting, unique solvability of (1.17) in the space H1 p(J; X) ∩
Lp(J; DA) is equivalent to the conditions g ∈ Lp(J; X) and u0 ∈ DA(1 − 1/p, p). We
remark that the motivation for considering also the case κ > 0 comes from the problem
studied in Chapter 5 which involves two independent kernels.
The proof of Theorem 3.1.4 essentially relies on techniques developed in Prüss [64]
using the representation of the resolvent S for (1.16) via Laplace transform, as well as
on the Mikhlin theorem in the operator-valued version. With the aid of the latter result
and an approximation argument, we succeed in showing Lp(R; X)-boundedness of the
operator corresponding to the symbol M(ρ) = A((â(iρ)) −1 + A) −1 , ρ ∈ R \ {0}; this
operator is closely related to the variation of parameters formula.
After proving a rather general embedding theorem in Section 3.2, we continue the
study of (1.16), now focusing on the case κ ∈ (1/p, 1 + 1/p), and establish a result corresponding
to Theorem 3.1.4. This is done in Section 3.3. In Section 3.4 we collect some
known results on maximal Lp-regularity of abstract problems on the halfline. Among
others, we consider two abstract second order equations that play a crucial role in the
treatment of problems on a strip which are respectively of the form
� u − a ∗ ∂ 2 yu + a ∗ Au = f, t ∈ J, y > 0,
u(t, 0) = φ(t), t ∈ J,
� u − a ∗ ∂ 2 yu + a ∗ Au = f, t ∈ J, y > 0,
−∂yu(t, 0) + Du(t, 0) = φ(t), t ∈ J,
(1.18)
where a is a K-kernel of order α ∈ (0, 2), and A and D are sectorial resp. pseudo-sectorial
operators in a Banach space X with D A 1/2 ↩→ DD. The investigation of these problems
is pursued in Section 3.5. We prove results characterizing unique solvability of (1.18) in
the regularity class H α p (J; Lp(R+; X)) ∩ Lp(J; H 2 p(R+; X)) ∩ Lp(J; Lp(R+; DA)) in terms
of regularity and compatibility conditions on the data. Besides the results concerning
(1.16) and that from Section 3.4 we make here repeatedly use of the inversion of the
convolution, the Dore-Venni theorem, as well as properties of real interpolation.
Chapter 4 is devoted to the study of linear scalar problems of second order in the
space Lp(J ×Ω), J = [0, T ] and Ω a domain in R n , with general inhomogeneous boundary
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