Quasilinear parabolic problems with nonlinear boundary conditions

(i) (6.9) has a unique solution w in Z T ;
(ii) for every u ∈ Σ(ρ, T, φ), (6.11) has a unique solution v = Υ(u) in Z T ;
(iii) there exist positive constants M and µ(T ) both not depending on ρ, with M being
also independent of T and µ(T ) → 0 as T → 0, such that for all u, v ∈ Σ(ρ, T, φ) ∪
{φ|J} and K = D, N, the subsequent inequalities are fulfilled:
|(A0 − A(u)) : ∇ 2 u| X T ≤ M(µ(T ) + ρ) 2 , (6.14)
|(A0 − A(u)) : ∇ 2 u − (A0 − A(v)) : ∇ 2 v| X T ≤ M(µ(T ) + ρ)|u − v| Z T , (6.15)
|F (u) − F (v)| X T 1 + |G(u) − G(v)| X T ≤ µ(T )|u − v| Z T , (6.16)
|R φ
K (u) − Rφ
K (v)| Y T K ≤ M(µ(T ) + ρ)|u − v| ZT . (6.17)
Proof. To prove (i) and (ii), we have to consider the linear problem (6.8). Since φ|t=0 =
u0, it follows from (H7) and the compactness of Γ that there exist T3 ∈ (0, T2] and
c > 0 such that |B◦ D (φ)(t, x)| ≥ c as well as |B◦ N (φ)(t, x) · ν(x)| ≥ c for all t ∈ [0, T3] and
x ∈ ΓD resp. x ∈ ΓN. Hereafter, we suppose that T ∈ (0, T3]. We may then normalize
the boundary conditions in (6.8) by dividing by B◦ D (φ)(t, x) resp. B◦ N (φ)(t, x) · ν(x), and
integrate the first equation in (6.8) w.r.t. time. This way we can rewrite (6.8) as a
problem of the form (4.40). One has now to check that Theorem 4.3.1 is applicable to
the reformulations of (6.9) and (6.11).
As to regularity of the data, clearly the initial data u0 and u1 belong to the right
regularity classes, by assumption (H4). Let us next look at the term which involves the
function g. Suppose u ∈ Σ(ρ, T, φ) ∪ {φ|J}. By (H3b) and (H4), we have
|g(·, ·, u, ∇u)| X T ≤ |g(·, ·, u, ∇u) − g(·, ·, u0, ∇u0)| X T + |g(·, ·, u0, ∇u0)| X T
≤ |Cg| X T (|u − u0|∞ + |∇u − ∇u0|∞) + |g(·, ·, u0, ∇u0)| X T .
So G(u) ∈ X T , that is, this term lies in the right regularity class. Furthermore, we have
(A0 − A(u)) : ∇ 2 u ∈ X T , in view of (6.14), which will be shown below. Concerning the
other terms, we refer to Section 6.2, where we shall prove that the regularity assumptions
on f, b D , and b N , together with (H4), ensure that these terms enjoy the regularity
required for the application of Theorem 4.3.1, i.e. that F (u) ∈ X T 1 and BK(φ)−B ◦ K (φ)φ+
R φ
K (u) ∈ Y T K for all u ∈ Σ(ρ, T, φ) ∪ {φ|J}, K = D, N. We shall also demonstrate that
this regularity is also preserved under the above normalization on the boundary.
Observe further that the compatibility conditions are satisfied for (6.9) and (6.11).
This follows from (H5), the definition of Σ(ρ, T, φ), and from the fact that φ|t=0 = u0
and ∂tφ|t=0 = u1 in case α > 1/p. Normality has already been discussed above. Hence,
(i) and (ii) are established for all T ∈ (0, T3].
Turning to (iii), we here only show (6.14), (6.15), and one half of (6.16). The remaining
estimates, which take more effort to be proved, are subject of Section 6.2. In
the subsequent inequalities, M and µ(T ) are constants, which may differ from line to
line, but which are such that both do not depend on ρ, M is independent of T , too, and
µ(T ) → 0 as T → 0.
Let u, v ∈ Σ(ρ, T, φ) ∪ {φ|J}. By means of (H3a) we get
|(A0 − A(u)) : ∇ 2 u| X T ≤ (|A0 − A(w)|∞ + |A(w) − A(u)|∞)
× (|∇ 2 u − ∇ 2 w| (XT ) n2 + |∇2w| (XT ) n2 )
≤ (µ(T ) + Mρ)(ρ + µ(T )),
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