Quasilinear parabolic problems with nonlinear boundary conditions

We now assume α + κ > 1 + 1/p. Put this time u = x + 1 ∗ S ˙ h(0) + u1, where u1
solves (3.1) with right-hand side g(t) := h(t) − x − t˙ h(0) on J. Because of (ii), we have
g ∈ 0H α+κ
p (J; X). So it follows by Remark 3.1.1(ii) that u1 ∈ 0H α+κ
p (J; X)∩0H κ p (J; DA).
Further, the property ˙ h(0) ∈ DA(1 + κ/α − 1/α − 1/pα, p) implies 1 ∗ S ˙ h(0) ∈ Z, thanks
to Remark 3.1.2(i). Finally, as in the first case, condition (i) ensures that the constant
function w(t) = x, t ∈ J lies in Z. So we conclude that u ∈ Z. From
u(t) + (a ∗ Au)(t) = (x + (1 ∗ a)(t)Ax) + t ˙ h(0) + (h(t) − x − t ˙ h(0)) = f(t), t ∈ J,
we see that u solves (3.1). �
It should be mentioned that, in the situation of Theorem 3.3.1, we have in general
1 ∗ a /∈ Hα+κ p (J). As illustration, we consider the following example.
Example 3.3.1 Let 0 ≤ κ �= 1/p, α ∈ (0, 1 − κ), and take a(t) = tα−1 /Γ(α), t > 0.
Further put b = 1 ∗ a. Since b(0) = 0, we see that b ∈ Hα+κ p (J) if and only if b ∈
(J). So, letting k(t) = t−(α+κ) /Γ(1 − (α + κ)), t > 0, we have
0H α+κ
p
b ∈ 0H α+κ
p (J) ⇔ d
dt (k ∗ b) ∈ Lp(J) ⇔ k ∗ a ∈ Lp(J).
But (k ∗ a)(t) = t−κ /Γ(1 − κ), t > 0, so that k ∗ a ∈ Lp(J) if and only if κ < 1/p. Hence,
1 ∗ a /∈ Hα+κ p (J) whenever κ > 1/p.
3.4 Abstract equations of first and second order on the
halfline
In this paragraph we collect some known results on maximal Lp-regularity of abstract
problems on the halfline.
The first theorem, which is due to Weis [81], concerns the abstract Cauchy problem
in a Banach space X.
˙u + Au = f, t > 0, u(0) = u0, (3.24)
Theorem 3.4.1 Let X be a Banach space of class HT , p ∈ (1, ∞), and A be an invertible
and R-sectorial operator in X with R-angle φR A < π/2.
Then (3.24) has a unique solution in Z := H1 p(R+; X) ∩ Lp(R+; DA) if and only if
(i) f ∈ Lp(R+; X); (ii) u0 ∈ DA(1 − 1/p, p).
Proof. The assertion follows immediately from Remark 3.1.1(iv), Remark 3.1.2(ii) and
Theorem 3.1.4 with a ≡ 1. �
In the remainder of this section, we consider two abstract second order problems, which
play an essential role in the treatment of abstract parabolic problems with inhomogeneous
boundary data. By the aid of the two subsequent key results, in Section 3.5, we
will succeed in finding the natural regularity classes for the data on the boundary.
The following theorem concerns the problem with Dirichlet condition
�
−u ′′ (y) + F 2u(y) = f(y), y > 0,
(3.25)
u(0) = φ,
in Lp(R+; X).
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