Quasilinear parabolic problems with nonlinear boundary conditions

which is necessary for v to be a solution of (4.40) on [0, Ti+1].
Let Zi+1 = H α p ([0, Ti+1]; Lp(Ω)) ∩ Lp([0, Ti+1]; H 2 p(Ω)) for i = 0, . . . , M − 1, and set
as usual Z1(v 0 ) = {w ∈ Z1 : ∂ m t w|t=0 = ∂ m t f|t=0, if α > m + 1/p, m = 0, 1}, as well as
Zi+1(v i ) := {w ∈ Zi+1 : w| [0,Ti] = v i } for i > 0, where v i denotes the unique solution of
(4.40) on [0, Ti] determined in the ith time step. In similar fashion as in the proofs of
Theorem 4.1.1 and Theorem 4.2.3, one can now show that for each i = 0, . . . , M − 1, the
mapping G leaves Zi+1(v i ) invariant and is a strict contraction, provided that δ is small
enough. In fact, one uses the estimates derived in the above proofs together with the
diffeomorphism property of the coordinate transformations ¯x = ˜ ϑj(x). Observe that we
have uniform bounds for |Θj| and |Θ −1
j | w.r.t. i and j, as only the spatial variables are
transformed and by compactness of Γ. So for sufficiently small δ, equation (4.52) admits
a unique solution v =: Qi+1(f, g, h) in the space Zi+1(v i ). To see that this function
indeed solves (4.40) on [0, Ti+1], one can again argue as in the proofs of Theorem 4.1.1
and Theorem 4.2.3. Here one has to consider the linear operators Ki+1 defined by
+
=
N�
� N1
Ki+1(f1, f2, f3) = (K (1)
i+1 (f1, f2, f3), 0 , K (3)
i+1 (f1, f2, f3))
�
[A, ψj] h F ij(f1, Qi+1(f1, f2, f3)) +
j=0
N2 �
j=N1+1
[A, ψj] h
j=N2+1
N ij (f1, f3, Qi+1(f1, f2, f3)), 0 , γΓN
j=N2+1
[A, ψj] h D ij (f1, f2, Qi+1(f1, f2, f3))
N�
[B, ψj] h N �
ij (f1, f3, Qi+1(f1, f2, f3)) ,
for triples (f1, f2, f3) in the product space of the regularity classes of f, g, h whose temporal
traces at t = 0 coincide with those of f, g, h (including the first temporal derivative)
whenever these traces exist. Since Ki+1 contains only terms of lower order, one can prove
existence of a triple (f1, f2, f3) satisfying (f1, f2, f3)+Ki+1(f1, f2, f3) = (f, g, h), provided
that δ is sufficiently small. By simple computations, one shows then that Qi+1(f1, f2, f3)
solves (4.40) on [0, Ti+1]. Finally, uniqueness implies v = Qi+1(f, g, h) = Qi+1(f1, f2, f3).
This establishes the sufficiency part.
We turn now to necessity. Suppose that v ∈ Z solves (4.40). Clearly, ϕjv ∈ Z for
all j = 0, . . . , N, which in turn implies Θ −1
j (ϕjv) ∈ Hα p (J; Lp(R n+1
+ )) ∩ Lp(J; H2 p(R n+1
+ ))
for all j = N1 + 1, . . . , N. Observe further that all those terms on the right-hand
sides of (4.48), (4.50), and the first equation of (4.46) (i = M − 1) which involve the
function v have the regularity desired for the corresponding inhomogeneity (f, g resp.
h) on J × Rn+1 resp. J × R n+1
+ . In view of the Theorems 4.1.1, 4.2.3, 4.2.4, and the
diffeomorphism property of the variable transformations ¯x = ˜ ϑj(x), it thus follows that
the functions ϕjf, ϕjg, and ϕjh enjoy the desired regularity on J ×Ω, J ×ΓD, and J ×ΓN,
respectively, for each j. On account of �N j=0 ϕj(x) ≡ 1 on Ω, we eventually obtain the
desired regularity for the data f, g, h themselves. The compatibility conditions can be
seen in the same way. �
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