Quasilinear parabolic problems with nonlinear boundary conditions

is a Banach space, cp. Corollary 2.8.2. Recall that for g ∈ 0Ξl := 0H α p ([0, Tl]; Lp(R n )),
one has |g| (k,1)
H α p ([0,Tl];Lp(R n )) = |Bkg|Xl , Bk denoting the inverse convolution operator in Xl
associated with the kernel k. For the spaces Zl, l = 1, . . . , M, we choose the norm
|w|Zl = |w|(k,1)
Hα p ([0,Tl];Lp(Rn )) + |∇2xw| Xn2 l
Claim 1: Let i ∈ {0, . . . , M − 1}, j ∈ {0, . . . , N}, and l ∈ {1, . . . , M}. Then
w + k ∗ A ij w = g, t ∈ [0, Tl], x ∈ R n , (4.6)
possesses a unique solution w =: L ij
l g in the space Zl if and only if g ∈ Ξl. Further,
there exists a constant C > 0 not depending on i, j, l such that
|L ij
l g|Zl ≤ C|g|Ξl , ∀g ∈ 0Ξl. (4.7)
Claim 1 is an immediate consequence of Theorem 3.1.4. Note that after a rotation
and a stretch of the spatial coordinates, the elliptic operator Aij becomes the negative
Laplacian. The constant C in (4.7) can be selected to be independent of i and j, due to
the uniform ellipticity assumption (H3); the independence on l is clear, because in (4.7),
functions g in the subspace 0Ξl are considered, only.
By applying the solution operator L ij
i+1 to (4.5) we get
where
and
(I − S ij )ϕjv = L ij
i+1 (ϕjf) + L ij
i+1 k ∗ Cj(·, x, Dx)v =: h ij (f, v), (4.8)
S ij w = L ij
i+1 k ∗ (Aj # (Ti, xj, Dx) − A j
# (·, x, Dx))w, (4.9)
Cj(t, x, Dx) = [A#(t, x, Dx), ϕj] − ϕjAR(t, x, Dx). (4.10)
One immediately verifies that S ij ∈ B(Zi+1). Furthermore, S ij enjoys the subsequent
properties, which will be shown at the end of this proof.
Claim 2: There exists η0 > 0 such that whenever η ≤ η0, i ∈ {0, . . . , M − 1}, and
j ∈ {0, . . . , N},
(a) |S ij w|Zi+1
≤ 1
2 |w|Zi+1 for all w ∈ Zi+1(0) (Z1(0) := 0Z1);
(b) if w ∈ Zi+1(0) and w0 = (I − Sij )w, then |w|Zi+1 ≤ 2|w0|Zi+1 ;
(c) for each g ∈ Ξi+1 and v ∈ Zi+1(V −1
i g), the equation (I − Sij )w = h ij (g, v) ad-
mits a unique solution w =: (I − S ij )| −1
Zi+1(ϕjV −1
.
i g)hij (g, v) in Zi+1(ϕjV −1
i g). Here
Z1(V −1
0 g) := {v ∈ Z1 : ∂ m t v|t=0 = ∂ m t g|t=0, if α > m + 1/p, m = 0, 1}.
Let η ≤ η0. By employing the operators (I − Sij )| −1
Zi+1(ϕjv i desribed in Claim 2 we infer
)
from (4.8) that
ϕjv = (I − S ij )| −1
Zi+1(ϕjv i ) hij (f, v).
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