Quasilinear parabolic problems with nonlinear boundary conditions

the data, and R(v) contains only terms of lower order, see equation (4.52) below. By
means of the contraction principle, this fixed point equation can be solved first on a
small interval [0, T1] (denote the (unique) solution by v 1 ), then on [0, T2], where T2 > T1
and the unknown v equaling v 1 on [0, T1], and proceeding in this way, finally, after
finitely many steps, it can be solved on the entire interval [0, T ]. Here it is essential that
maxi |Ti+1 − Ti| is sufficiently small to ensure that, in each step of this procedure, the
mapping G is a strict contraction.
Let us start with the spatial localization. By boundedness of Γ, there exists r0 > 0
such that Γ is entirely contained in the open ball Br0 (0). If Ω is unbounded we set
U0 = {x ∈ Rn+1 : |x| > r0}, otherwise we may assume that Ω ⊂ Br0 (0) and put
U0 = ∅. The other assumptions on Ω allow us to cover Br0 (0) by finitely many open sets
Uj, j = 1, . . . , N, which are subject to the following conditions.
(L1) Uj ∩ Γ = ∅ and Uj = Brj (xj) for all j = 1, . . . , N1;
(L2) Uj ∩ ΓD �= ∅ for N1 + 1 ≤ j ≤ N2, Uj ∩ ΓN �= ∅ for N2 + 1 ≤ j ≤ N, and
for each j in either index set, there exist xj ∈ Uj ∩ ΓD resp. Uj ∩ ΓN and ˜ ζj ∈ C 2 (R n )
with compact support such that - using coordinates corresponding to xj (i.e. xj = 0
and n(xj) = (0, . . . , 0, −1)) - Γ ∩ Uj = {x = (x ′ , y) ∈ Uj : y = ˜ ζj(x ′ )} as well as
Ω ∩ Uj = {x = (x ′ , y) ∈ Uj : y > ˜ ζj(x ′ )}, and Uj = ˜ ϑ −1
j (Brj (xj)), where ˜ ϑj is related to
˜ζj as described above.
(L3) Ui ∩ Uj = ∅ for all N1 + 1 ≤ i ≤ N2 and N2 + 1 ≤ j ≤ N.
It is then not difficult to construct local operators Aj = Aj(t, x, Dx), j = 0, . . . , N,
and Bj = Bj(t, x, Dx), j = N2 + 1, . . . , N, of second resp. first order which enjoy the
subsequent properties.
(LO1) Aj is defined on J ×R n+1 if 0 ≤ j ≤ N1, and on J ×Ωj otherwise; here the set
Ωj is given in coordinates corresponding to xj by means of Ωj = {x = (x ′ , y) ∈ R n+1 :
y ≥ ˜ ζj(x ′ )}; Bj is defined on J × Γj for all j = N2 + 1, . . . , N, where Γj = {x = (x ′ , y) ∈
R n+1 : y = ˜ ζj(x ′ )};
(LO2) the coefficients of Aj coincide with the corresponding coefficients of A(t, x, Dx)
on Ω∩Uj, for all j = 0, . . . , N, and the coefficients of Bj coincide with those of B(t, x, Dx)
on Γ ∩ Uj, for all j = N2 + 1, . . . , N;
(LO3) Aj satisfies the assumptions of Theorem 4.1.1 for all j = 0, . . . , N1; A ˜ ϑj
j =
Θ −1
j AjΘj defined on J × R n+1
+
fulfills the assumptions of Theorem 4.2.3 for all j =
N1 + 1, . . . , N; finally, B ˜ ϑj
j = Θ−1
j BjΘj defined on J × Rn satisfies the assumptions of
Theorem 4.2.4 for all j = N2 + 1, . . . , N.
Here we use the fact that ellipticity and normality, as well as smoothness of the
coefficients of A(t, x, Dx) resp. B(t, x, Dx) are preserved in Ω ∩ Uj resp. Γ ∩ Uj under
the coordinate transformations ¯x = ˜ ϑj(x) for all j = N1 + 1, . . . , N. We refer to [29,
Section 8.2], where appropriate extensions of the coefficients are constructed by means
of reflection and cut-off techniques.
We choose next a partition of unity {ϕj} N j=0 ⊂ C∞ (R n+1 ) such that � N
j=0 ϕj(x) ≡ 1
on Ω, 0 ≤ ϕj(x) ≤ 1, and supp ϕj ⊂ Uj; we fix also a family {ψj} N j=0 ⊂ C∞ (R n+1 ) that
satisfies ψj ≡ 1 on an open set Vj containing supp ϕj, as well as supp ψj ⊂ Uj. As to
localization w.r.t. time, we subdivide the interval [0, T ] according to 0 =: T0 < T1 <
. . . < TM−1 < TM := T and put δ := maxi |Ti+1 − Ti|. Then, owing to (LO1) and (LO2),
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