Quasilinear parabolic problems with nonlinear boundary conditions

L1, loc(R; B(X)) as n → ∞. It follows now by an approximation result, cp. Clément and
Prüss [24, p. 6] or [29, Proposition 3.18], that (A d
dtK∗), which corresponds to the symbol
M, is a bounded linear operator on Lp(R; X) with |A d
dtK∗)| B(Lp(R;X)) ≤ κ0. Since EJ
and PJ are bounded as well we see that Au ∈ Lp(J; X). As in the necessity part, this
implies a ∗ Au ∈ 0H α p (J; X), i.e. u = f − a ∗ Au ∈ 0H α p (J; X). Hence u ∈ Z.
We now consider the case κ ∈ (0, 1/p). Suppose that f ∈ 0H α+κ
p (J; X). Putting
b(t) = e−ttκ−1 , t > 0, yields b ∈ K1 (κ, κπ/2) ∩ L1(R+). Let Bκ be the operator constructed
in Corollary 2.8.1 associated with the kernel b. Then Bκf ∈ 0H α p (J; X). Sufficiency
being already established for κ = 0, we may define v ∈ 0H α p (J; X) ∩ Lp(J; DA)
as the solution of
v + a ∗ Av = Bκf, t ≥ 0.
Since Bκ commutes with both A and the Volterra operator corresponding to the kernel
a, we see that u := b ∗ v solves (3.1) and lies in Z. �
Remarks 3.1.1 (i) Although not explicitly stated in Theorem 3.1.1, we have an estimate
of the form
C −1 |f|
0H α+κ
p (J;X) ≤ |u|Z ≤ C|f|
0H α+κ
p (J;X) , f ∈ 0H α+κ
p (J; X),
where C is a positive constant not depending on f. This follows immediately from
the above proof. Note that the subsequent theorems on linear problems have to be
understood in the same sense: whenever necessary resp. sufficient conditions are stated
in terms of regularity classes, this, by convention, means that the corresponding a priori
estimates hold true.
(ii) Observe that the statement of Theorem 3.1.1 remains true if κ ≥ 1/p and Z is
defined by 0H α+κ
p (J; X) ∩ 0H κ p (J; DA).
(iii) In the case of a compact interval J, one can weaken the assumption on A. In
view of the transformation property of (3.1) discussed at the end of Section 2.8, it suffices
to know that µ + A ∈ RS(X) with θa + φR µ+A < π for some µ ≥ 0.
(iv) If κ = 0, equation (3.1) is equivalent to Bu+Au = Bf in the space Y := Lp(J; X),
where A stands for the natural extension of A to Y . If one additionally assumes that
A ∈ BIP(X) and θA + θa < π, the assertion of Theorem 3.1.1 can also be proved by
means of the Dore-Venni theorem, Theorem 2.3.1. This approach has been used in Prüss
[63, Thm. 8.7].
(v) In the case J = R+ there is a variant of Theorem 3.1.1 which does not need
the assumption a ∈ L1(R+). Instead one assumes that A is invertible and that f in
(3.1) is of the form f = a ∗ g. In this situation, existence of a unique solution of
(3.1) in Z is equivalent to the condition g ∈ 0H κ p (R+; X). In fact, if g ∈ Lp(R+; X),
then, as seen in the above proof, we have Au ∈ Lp(R+; X), which by invertibility of A
entails u ∈ Lp(R+; X). From Bu + Au = g, we then deduce that Bu ∈ Lp(R+; X). So
u ∈ 0H α p (R+; X) ∩ Lp(R+; DA). The converse direction is trivial, and the case κ > 0 can
be reduced to the case κ = 0 as above.
(vi) The idea to use Theorem 2.5.1 to show that the linear operator corresponding to
the symbol (3.3) is bounded in Lp(R; X) goes back to Clément and Prüss [24]. However,
they do not give a detailed proof including the approximation argument, by the aid of
which one can surmount the technical difficulty consisting in the fact that Theorem 2.5.1
cannot be applied directly to (3.3).
We now turn our attention to situations where the function f or its derivative ˙ f, if it
exists, has a non-vanishing trace at t = 0. If f ∈ Hα+κ p (J; X) and α + κ > 1/p, then
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