Quasilinear parabolic problems with nonlinear boundary conditions

kernel a. The operator B is invertible and we have B −1 w = a ∗ w for all w ∈ Lp(J; X).
Furthermore, D(B) = 0H α p (J; X) so that we may set g = Bf. Then g ∈ Lp(J; X) and
u(t) = d
(S ∗ (a ∗ g))(t), t ∈ J.
dt
Let EJ : Lp(J; X) → Lp(R; X) denote the operator of extension by 0, i.e.
(EJh)(t) = h(t), t ∈ J, (EJ)(t) = 0, t �∈ J,
let PJ : Lp(R; X) → Lp(J; X) be the restriction to J, and define the operator-valued
kernel K by means of K(t) = (S ∗ a)(t)χ [0, ∞)(t), t ∈ R. Then the solution u can be
written in terms of a convolution operator on Lp(R; X):
d
u = PJ
dt (K ∗ EJg). (3.2)
In order to show that Au ∈ Lp(J; X), we study the symbol of the operator (A d
dtK∗), which reads
� �−1 1
M(ρ) = A + A , ρ ∈ R, ρ �= 0. (3.3)
â(iρ)
By Lemma 2.6.1, for each ρ �= 0, â(iρ) := limλ→iρ â(λ) exists and does not vanish.
Besides, â(i·) ∈ W 1 ∞, loc (R\{0}), and the sectoriality of a implies |arg(â(iρ))| ≤ θa for all
ρ �= 0. The idea is to apply the Mikhlin multiplier theorem in the operator-valued version,
Theorem 2.5.1, to the symbol M. But it is not clear that M ∈ C 1 (R\{0}; B(Lp(J; X))),
so we introduce the sequence of symbols
Mn(ρ) := A
�
1
â((i + 1 + A
n )ρ)
� −1
, ρ ∈ R, n ∈ N.
Since A is R-sectorial with R-angle φ R A < π − θa, we deduce that R({Mn(ρ) : ρ ∈
R \ {0}}) ≤ κ < ∞ for all n ∈ N with κ not depending on n. From
ρM ′ n(ρ) =
1 (i + n )ρâ′ ((i + 1
n )ρ)
â((i + 1
n )ρ)2
�
1
â((i + 1 + A
n )ρ)
� −1
Mn(ρ), ρ ∈ R,
using 1-regularity of a, R-sectoriality of A, and Kahane’s contraction principle (see [29,
Lemma 3.5]), we obtain
R({ρM ′ n(ρ) : ρ ∈ R \ {0}}) ≤ Cκ(1 + κ),
for all n with C not depending on n. By Theorem 2.5.1, it follows that the operators Tn
defined by
Tnφ = F −1 (MnFφ), for all Fφ ∈ D(R; X),
are uniformly Lp -bounded, i.e.
| Tn| B(Lp(R;X)) ≤ κ0, n ∈ N.
Furthermore we have, for all ρ �= 0, limn→∞ Mn(ρ) = M(ρ) and |Mn(ρ) − M(ρ)| ≤
2κ. Thus, by Lebesgue’s dominated convergence theorem, we conclude Mn → M in
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