Quasilinear parabolic problems with nonlinear boundary conditions

We come now to sufficiency. Suppose we are given the data f and φ which fulfill the
conditions (i)-(v). We proceed similarly as in the proof of Theorem 3.5.1.
Firstly, let u1 ∈ Z be a function which satisfies (3.39). Such a function was constructed
in the proof of Theorem 3.5.1. Put φ1 := −∂yu1|y=0 + Du1|y=0 and consider
the problem �
v − a ∗ ∂2 yv + a ∗ Av = 0, t ∈ J, y > 0,
(3.48)
−∂yv(t, 0) + Dv(t, 0) = φ(t) − φ1(t), t ∈ J.
Owing to the necessity part of Theorem 3.5.2 and the above remark concerning (3.46),
φ1 is well-defined and lies in the same regularity class as the data φ. In addition, we see
that (φ−φ1)|t=0 = 0, if α > 2/(p−1), by construction of φ1 and thanks to condition (iv).
That is, the compatibility condition is satisfied. To solve (3.48), we let Y1 = Lp(J; X),
D be the natural extension of D to this space, and define the operator F1 as in the
paragraph following problem (3.41). Then (3.48) can be rewritten as
−v ′′ (y) + F 2 1 v(y) = 0, y > 0, −v ′ (0) + Dv(0) = φ − φ1. (3.49)
Observe that D is pseudo-sectorial in Y1 and D ∈ BIP(R(D)) with power angle θD ≤
θD. In view of (3.43), φ − φ1 ∈ DF1 (1 − 1/p, p). Further, we have θD + θF1 ≤ θD +
max{θa, θA}/2 < π. By Theorem 3.4.3, problem (3.48) thus admits a unique solution u2
in the space
with
0H α p (J; Lp(R+; X)) ∩ Lp(J; H 2 p(R+; X)) ∩ Lp(J; Lp(R+; DA))
Du2(·, 0) ∈ DF1 (1 − 1/p, p) = 0B
1 1
α( − 2 2p )
pp (J; X) ∩ Lp(J; DA( 1 1
2 − 2p , p)).
Finally put u := u1 + u2. Then u clearly possesses the desired regularity and solves
(3.45).
Uniqueness follows by Theorem 3.4.3. In fact, rewrite (3.45) with zero data as (3.49)
with zeros on the right-hand side. Then it is apparent that the zero-function is the only
solution of (3.45) in Z. �
Remarks 3.5.1 (i) Theorem 3.5.1 and Theorem 3.5.2 are also true, if the kernel a is of
the form a = b+dk∗b, where b is like a above and k ∈ BVloc(R+) with k(0) = k(0+) = 0.
(ii) The proofs of the two foregoing theorems are inspired by Prüss [65]. In the case
a ≡ 1 Theorem 3.5.1 is equivalent with a version of [65, Thm. 5] for compact J, while
Theorem 3.5.2, roughly speaking, can be regarded as an extension of [65, Thm. 6], at
least in the case D A 1/2 ↩→ DD.
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