Quasilinear parabolic problems with nonlinear boundary conditions

Chapter 3
Maximal Regularity for Abstract
Equations
3.1 Abstract parabolic Volterra equations
In this section we study the abstract Volterra equation
u + a ∗ Au = f, t ≥ 0, (3.1)
on a Banach space X. Here A is an R-sectorial operator in X, the kernel a belongs to the
class K1 (α, θa) with α ∈ (0, 2), and we assume the parabolicity condition θa + φR A < π.
Our aim is to find conditions on the given function f which are necessary and sufficient
for the existence of a unique solution u of (3.1) in the space
H α+κ
p (J; X) ∩ H κ p (J; DA),
where J is R+ or a compact time-interval [0, T ], DA denotes the domain of A equipped
with the graph norm of A, and κ is a real parameter belonging to the interval [0, 1/p).
We begin with the special case of vanishing traces at t = 0.
Theorem 3.1.1 Let X be a Banach space of class HT , p ∈ (1, ∞), and A an R-sectorial
operator in X with R-angle φR A . Further let J be R+ or a compact time-interval [0, T ].
Suppose that a belongs to K1 (α, θa) with α ∈ (0, 2) and that in addition a ∈ L1(R+)
in case J = R+. Further let κ ∈ [0, 1/p), α + κ /∈ {1/p, 1 + 1/p}, and suppose the
parabolicity condition θa + φR A < π.
Then (3.1) has a unique solution in Z := 0H α+κ
p (J; X) ∩ Hκ p (J; DA) if and only if
f ∈ 0H α+κ
p (J; X).
Proof. Suppose that u ∈ Z is a solution of (3.1). This clearly implies Au ∈ Hκ p (J; X) =
0H κ p (J; X). From Corollary 2.8.1 we then deduce that a ∗ Au ∈ 0H α+κ
p (J; X). This,
together with u ∈ 0H α+κ
p (J; X), entails f ∈ 0H α+κ
p (J; X). Hence, the necessity part is
established.
To prove the converse, we first consider the case κ = 0. Suppose f ∈ 0H α p (J; X)
is given. From Section 2.7 we know that equation (3.1) is parabolic and admits a
resolvent S(·), with the aid of which the mild solution u of (3.1) can be represented
by the variation of parameters formula (2.27). According to Corollary 2.8.1 it makes
sense to define B ∈ S(Lp(J; X)) as the inverse convolution operator associated with the
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