Quasilinear parabolic problems with nonlinear boundary conditions

Turning to (c), let g ∈ Ξi+1 and v ∈ Zi+1(V −1
i g). Evidently hij (g, v) ∈ Zi+1. If i > 0,
we further have
h ij (g, v)| [0,Ti] = (L ij
i+1 (ϕjg + k ∗ Cj(·, x, Dx)v))| [0,Ti]
= L ij
i (ϕjg| [0,Ti] + k ∗ Cj(·, x, Dx)V −1
i
= ϕjV −1
i
g − Lij
i
g) (here ∗ is meant on [0, Ti])
k ∗ (Aj
# (·, x, Dx) − A j
# (Ti, xj, Dx))ϕjV −1
i g.
Thus, w ∈ Zi+1(ϕjV −1
i g) implies F (w) := Sijw + hij (g, v) ∈ Zi+1(ϕjV −1
i g). In fact,
(F (w))| [0,Ti] = L ij
i k ∗ (Aj # (·, x, Dx) − A j
# (Ti, xj, Dx))ϕjV −1
i g + hij (g, v)| [0,Ti]
= ϕjV −1
i g.
So F is a self-mapping of Zi+1(ϕjV −1
i g). This is true for i = 0, too. Compare the
initial values of all terms occurring above to see this. In view of (4.17), F is also a strict
contraction. Hence the assertion follows by the contraction principle. �
4.2 Half space problems
This section is devoted to parabolic problems of second order in a half space subject
to general boundary conditions. In the first subsection we study the case in which the
coefficients are constant and the differential operators consist only of their principal
parts. Then we shall prove pointwise multiplication properties for the function spaces
arising as the natural regularity classes on the boundary. These results allow us to treat
also the case of variable coefficients by means of perturbation arguments. This will be
done in the last part of this paragraph.
4.2.1 Constant coefficients
Let J = [0, T ] and R n+1
+ = {x := (x′ , y) ∈ R n+1 : x ′ ∈ R n , y > 0}. We separately
consider the problems
� u − k ∗ a : ∇ 2 xu = f, t ∈ J, x ∈ R n+1
+ ,
u = g, t ∈ J, x ′ ∈ R n , y = 0,
� u − k ∗ a : ∇ 2 xu = f, t ∈ J, x ∈ R n+1
+ ,
−∂yu + b · ∇x ′u = h, t ∈ J, x′ ∈ R n , y = 0,
(4.18)
(4.19)
where k is as in Section 4.1, a is an (n + 1)-dimensional real matrix, and b ∈ R n . We
look for unique solutions u in the maximal regularity space
As to (4.18), we have the following result.
Z := H α p (J; Lp(R n+1
+ )) ∩ Lp(J; H 2 p(R n+1
+ )).
Theorem 4.2.1 Let 1 < p < ∞, n ∈ N, and a ∈ Sym{n + 1}. Let further k ∈ K 1 (α, θ),
where θ < π and α ∈ (0, 2) \
definite.
� 1
p , 2
2p−1
1 3
, 1 + p , 1 + 2p−1
63
�
. Assume that a is positive