Quasilinear parabolic problems with nonlinear boundary conditions

further remark that property (E2) already entails r-regularity of k1 for all r ∈ N as well
as Re ˆ k1(λ) ≥ 0 for all Re λ ≥ 0, λ �= 0, i.e. k1 is of positive type. For a proof of this fact
see Prüss [63, Proposition 3.3]. In comparison to the last inequality, (E3) requires that
|arg ˆ k1(λ)| ≤ θk1 < π/2 for all Re λ > 0.
We recall that the Laplace transform of any function which is locally integrable on
R+ and completely monotonic has an analytic extension to the region C\R−, see e.g. [39,
Thm. 2.6, p. 144]. Thus if k is of type (E), both ˆ k and � dk, given by ˆ k(λ) = (k0+ ˆ k1(λ))/λ
resp. � dk(λ) = k0 + ˆ k1(λ), may be assumed to be analytic in C \ R.
In the sequel we will assume that both kernels a and b are of type (E) and that
a �= 0. Define the parameters α, θa1 and β, θb1 by a1 ∈ K ∞ (α, θa1 ) and b1 ∈ K ∞ (β, θb1 ),
if a1 �= 0 respectively b1 �= 0.
We will study (5.11) in Lp(J; Lp(Ω, R 3 )), where 1 < p < ∞, and J = [0, T ] is a
compact time-interval. We are looking for a unique solution v of (5.11) in the regularity
class
Z := (H δa
p (J; Lp(Ω)) ∩ Lp(J; H 2 p(Ω))) 3 ,
where the exponent δa is defined by δa := 1, if a0 �= 0, and δa := 1 + α, otherwise. In
other words, δa gives the regularization order of a in the sense of Corollary 2.8.1. Here,
the regularity properties of b are not taken into account, since, in some sense, b only
plays a subordinate role in solving problem (5.11) as we shall see below. Nevertheless,
if b �= 0, we have to distinguish two principal cases. Letting δb be the regularization
order of b �= 0, defined in the same way as for a, we have to distinguish the cases δa ≤ δb
and δa > δb. The second case is more difficult, for here the terms involving b have
less regularity than those involving a. In order to cope with this defect, supplementary
regularity conditions have to be introduced.
It is convenient to define δb also in the case b = 0. So we put δb = ∞ in that case.
The strategy in solving (5.11) is the same as in Chapter 4. (5.11) is studied first
in the cases Ω = R 3 and Ω = R 3 +, where in the latter situation one has to consider
both boundary conditions separately. Having solved those cases, a solution of (5.11) can
be constructed by the aid of localization and perturbation arguments. We will restrict
our investigation to the half space case with prescribed normal stress. It will become
apparent that the techniques used here also apply to the much simpler full space case
and the half space case with prescribed velocity.
5.3 A homogeneous and isotropic material in a half space
In this section we consider a homogeneous and isotropic material in a half space with
prescribed normal stress. We do not only look at the three-dimensional situation but
study the general (n + 1)-dimensional case, n ∈ N.
Let R n+1
+ = {(x, y) ∈ Rn+1 : x ∈ Rn , y > 0} and denote the velocity vector by (v, w),
where v is Rn-valued and w is a scalar function. From (5.11) we are then led to the
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