Quasilinear parabolic problems with nonlinear boundary conditions

v is a solution of (4.40) on Ji+1 := [0, Ti+1] if and only if
⎧
⎨
⎩
ϕjv+k ∗ Aj(·, x, Dx)ϕjv = ϕjf +k ∗ [Aj(·, x, Dx), ϕj]v (Ji+1 × Ω) , 0≤j ≤N
ϕjv = ϕjg (Ji+1 × ΓD), N1+1≤j ≤N2
Bj(t, x, Dx)ϕjv = ϕjh+[Bj(t, x, Dx), ϕj]v (Ji+1 × ΓN), N2+1≤j ≤N.
(4.46)
In case j = 0, . . . , N1, we have to consider full space problems for the functions ϕjv. In
view of (LO3), we can apply Theorem 4.1.1, which ensures existence of corresponding
solution operators LF ij , thereby obtaining
ϕjv = L F ij(ϕjf + k ∗ [Aj, ϕj]v) =: h F ij(f, v), j = 0, . . . , N1. (4.47)
For j = N1 + 1, . . . , N2, we get problems on crooked half spaces with inhomogeneous
Dirichlet boundary condition. Using affine mappings that transform xj to the origin and
n(xj) to (0, . . . , 0, −1) combined with the variable transformations ¯x = ˜ ϑj(x) (denote
these compositions again by ˜ ϑj) leads to
�
Θ −1
j (ϕjv) + k ∗ A ˜ ϑj
j Θ−1
j (ϕjv) = Θ −1
j (ϕjf) + k ∗ Θ −1
j [Aj, ϕj]v (Ji+1 × R n+1
+ )
Θ −1
j (ϕjv) = Θ −1
j (ϕjg) (Ji+1 × R n ),
(4.48)
that is, to half space problems for Θ −1
j (ϕjv), for which Theorem 4.2.3 is applicable, in
virtue of (LO3). Employing the corresponding solution operators denoted by L D ij thus
yields
ϕjv = ΘjL D ij
� −1
Θj (ϕjf) + k ∗ Θ −1
j [Aj, ϕj]v
Θ −1
j (ϕjg)
�
=: h D ij (f, g, v), j = N1 + 1, . . . , N2.
(4.49)
The situation is similar for j = N2 + 1, . . . , N. In this case we have to consider problems
on crooked half spaces with inhomogeneous boundary condition of first order. Using
again the variable substitutions ¯x = ˜ ϑj(x) gives
⎧
⎨
⎩
Θ −1
j (ϕjv) + k ∗ A ˜ ϑj
j Θ−1
j (ϕjv) = Θ −1
j (ϕjf) + k ∗ Θ −1
j [Aj, ϕj]v (Ji+1 × R n+1
+ )
B ˜ ϑj
j Θ−1
j (ϕjv) = Θ −1
j (ϕjh) + [Bj, ϕj]v (Ji+1 × R n ),
(4.50)
which are problems on a half space for Θ −1
j (ϕjv). Without loss of generality, we may
assume that b(t, x) · ν(x) = 1 for all t ∈ J, x ∈ ΓN; in fact, we can always divide
the boundary condition in (4.40) by (b(t, x) · ν(x)) to achieve this without affecting the
smoothness of the inhomogeneity and that of the coefficients of the boundary operator,
see Section 6.2. As a consequence of this normalization, the operators B ˜ ϑj
j take the
form (4.28). By (LO3), we can apply Theorem 4.2.4, which asserts existence of solution
operators L N ij
ϕjv = ΘjL N ij
for the above problems. So we immediately get
�
� Θ −1
j (ϕjf) + k ∗ Θ −1
j [Aj, ϕj]v
Θ −1
j (ϕjh) + [Bj, ϕj]v
=: h N ij (f, h, v), j = N2 + 1, . . . , N.
(4.51)
Multiplying now (4.47), (4.49), and (4.51) by ψj and summing over all j yields the
formula
v =
N1 �
j=0
ψj h F ij(f, v) +
N2 �
j=N1+1
ψj h D ij (f, g, v) +
77
N�
j=N2+1
ψj h N ij (f, h, v) =: G(v), (4.52)