Quasilinear parabolic problems with nonlinear boundary conditions

Since ψj ≡ 1 on supp ϕj, we may multiply this equation by ψj resulting in
Summing over j then yields
v =
N�
j=0
ϕjv = ψj(I − S ij )| −1
Zi+1(ϕjv i ) hij (f, v).
ψj(I − S ij )| −1
Zi+1(ϕjV −1
i f)Lij i+1 (ϕjf + k ∗ Cj(·, x, Dx)v) =: G(v), (4.11)
which is a fixed point equation for v ∈ Zi+1(v i ).
Due to Claim 2(c), G is a self-mapping of Zi+1(v i ). Thus the contraction principle is
applicable to (4.11), if we can verify that G is a strict contraction. It turns out that this
can be achieved by choosing a yet finer partition 0 = T0 < T1 < . . . < TM−1 < TM =
T , more precisely, by making δ := maxi |Ti+1 − Ti| sufficiently small. The following
observation is crucial in this connection.
Suppose u ∈ Zi+1(0). By causality, it is clear that Bku| [0,Ti] = 0. So we can write
Thus, by Young’s inequality,
u = k ∗ Bku = (kχ [0,Ti+1−Ti]) ∗ Bku.
|u|Xi+1 ≤ |k| L1(0,Ti+1−Ti)|Bku|Xi+1 ≤ |k| L1(0,δ)|u|Zi+1 ∀u ∈ Zi+1(0). (4.12)
Another observation concerns the operators Cj(t, x, Dx). Since those are at most of
first order and have bounded coefficients, for each ε > 0, there exists Cε > 0 such that
|Cj(t, x, Dx)w|Xi+1 ≤ ε|∇2xw| Xn2 + Cε|w|Xi+1
i+1
(4.13)
for all i = 0, . . . , M − 1, j = 0, . . . , N, and w ∈ Lp([0, Ti+1]; H 2 p(R n )).
We are now ready to prove the contractivity of G. Let v, ¯v ∈ Zi+1(v i ). In view of
(4.7), Claim 2(b), (4.12), and (4.13), we may estimate
N�
|G(v) − G(¯v)|Zi+1 = | ψj(I − S ij )| −1
Zi+1(0) Lij
i+1k ∗ Cj(·, x, Dx)(v − ¯v)|Zi+1
N�
≤ C0
j=0
j=0
≤ C0C1
j=0
|L ij
i+1 k ∗ Cj(·, x, Dx)(v − ¯v)|Zi+1
N�
|Cj(·, x, Dx)(v − ¯v)|Xi+1
N�
≤ C0C1 (ε|∇
j=0
2 x(v − ¯v)|
Xn2 i+1
+ Cε|(v − ¯v)|Xi+1 )
�
�
≤ C0C1(N + 1) ε + Cε|k| L1(0,δ) |v − ¯v|Zi+1 =: κ(ε, δ)|v − ¯v|Zi+1 , (4.14)
with C0, C1 and N being independent of δ. This shows existence of a left inverse Qi+1
for the operator
Vi+1 = I + k ∗ A(·, x, Dx) : Zi+1(v i ) → Ξi+1(Viv i ), (4.15)
61