Quasilinear parabolic problems with nonlinear boundary conditions

Strict contractivity of Υ can be established in similar fashion. Let u, ū ∈ Σ(ρ, T, φ)
and put v = Υ(u) and ¯v = Υ(ū). Then ¯v − v satisfies
⎧
∂t(¯v − v) + dk ∗ A0 : ∇
⎪⎨
⎪⎩
2 (¯v − v) = F (ū) − F (u) + dk ∗ (G(ū) − G(u)) (J × Ω)
+dk ∗ (((A0−A(ū)):∇ 2ū)−(A0−A(u)):∇ 2u) B◦ D (φ)(¯v − v) = −(Rφ D (ū) − Rφ
D (u)) (J × ΓD)
B◦ N (φ)(¯v − v) = −(Rφ N (ū) − Rφ
N (u)) (J × ΓN)
(¯v − v)|t=0 = 0 (Ω),
whence
|¯v − v| Z T ≤ M1
�
|G(ū) − G(u) + (((A0−A(ū)):∇ 2 ū)−(A0−A(u)):∇ 2 u)| XT + |F (ū) − F (u)| XT + |R
1 φ
D (ū) − Rφ
D (u)| Y T D
≤ M(µ(T ) + ρ)|ū − u| Z T ,
+ |Rφ N (ū) − Rφ
N (u)| Y T N
by maximal regularity and the estimates from Theorem 6.1.1. Here the constants M
and µ(T ) are like those in (6.18). Hence Υ becomes a strict contraction, when ρ and T
are selected sufficiently small.
The assertion follows now by the contraction mapping principle and the fact that
fixed points of Υ correspond to solutions of (6.1) for small time-intervals J = [0, T ]. �
Remarks 6.1.1 (i) The statement of Theorem 6.1.2 is also true, if k is of the form
k = k1 + dl ∗ k1, where k1 is like k in (H1) and l ∈ BVloc(R+) with l(0) = l(0+) = 0.
(ii) One can further replace k on the right-hand side of (6.1) by an arbitrary kernel
k1 ∈ BVloc(R+) ∩ K 1 (1 + α1, θ1) with k(0) = 0, α1 ≥ α, θ1 < π, and the theorem still
holds true.
6.2 Nemytskij operators for various function spaces
This paragraph can be regarded as an appendix to Section 6.1. Its purpose is to complete
the proof of Theorem 6.1.1. We still have to show certain mapping properties and
Lipschitz estimates for the substitution operators which involve the nonlinear functions
f, b D and b N , cf. the beginning of Section 6.1. The Nemytskij operators under consideration
act on the function spaces which arose as natural regularity classes for the
inhomogeneities in the treatment of the linear problem (6.8), and turned out to be the
spaces XT 1 , Y T D , and Y T N , see (6.2), (6.3), (6.4) for their definitions. These are anisotropic
Bessel-potential and Sobolev-Slobodeckij spaces in domains, respectively, compact manifolds
in the euclidean space, which means that the subject of this section is not altogether
trivial.
We remark that in order to get the desired estimates for Y T D and Y T N , which are spaces
on J × ΓD resp. J × ΓN, one considers first the corresponding spaces in domains (w.r.t.
the spatial variable). The results obtained for the latter can then be transferred to Y T D
and Y T N by means of the standard method of local coordinates. In what is to follow, we
shall focus on the first step. As to the second step, we merely point out that the fact that
Ω (in Section 6.1) has a (compact) C2 boundary ensures that the smoothness of bD resp.
bN w.r.t. the spatial variable x is preserved under the local coordinate transformations
studied in Section 4.3, which flatten the boundary.
The reader is further reminded of the embedding ZT ↩→ C(J; C1 (Ω)), which is valid
in view of the assumption (H2), and which considerably simplifies the investigation of the
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