Quasilinear parabolic problems with nonlinear boundary conditions

Here f(A, ·) ∈ H0(Σ π
2 +η; B(X)) results from the functional calculus of A:
f(A, λ ′ ) = 1
2πi
�
Γψ
f(λ, λ ′ )(λ − A) −1 dλ, λ ′ ∈ Σ π
2 +η.
Since the resolvent of A is a pointwise operator with respect to t ∈ R+, we deduce from
(2.14) that f(A, B) is causal for all f ∈ H0(Ση × Σ π
2 +η). For an arbitrary function
f ∈ H ∞ (Ση × Σ π
2 +η), this property of f(A, B) can be seen by approximating f with a
sequence (fn) ⊂ H0(Ση × Σ π
2 +η).
2.5 Operator-valued Fourier multipliers
Let 1 < p < ∞ and X be a Banach space. We denote by D(R; X) the space of X-valued
C ∞ -functions with compact support on R. Let D ′ (R; X) := B(D(R), X) be the space
of X-valued distributions on the real line. Further we denote by S(R; X) the Schwartz
space of smooth rapidly decreasing X-valued functions on R, see e.g. Amann [5, p. 129].
By S ′ (R; X) := B(S(R), X) we mean the space of X-valued temperate distributions. Let
Y be another Banach space. Given M ∈ L1, loc(R; B(X, Y )), one may define the operator
TM : F −1 D(R; X) → S ′ (R; Y ) by means of
TMφ := F −1 MFφ, for all Fφ ∈ D(R; X), (2.15)
F denoting the Fourier transform. Note that F −1 D(R; X) is dense in Lp(R; X). Thus
TM is well-defined and linear on a dense subset of Lp(R; X).
One can now ask what conditions have to be imposed on M so that TM is bounded
in Lp-norm, i.e. TM ∈ B(Lp(R; X), Lp(R; Y )). The following theorem, which is due to
Weis [80], contains the operator-valued version of the famous Mikhlin Fourier multiplier
theorem in one variable.
Theorem 2.5.1 Suppose X and Y are Banach spaces of class HT and let 1 < p < ∞.
Let M ∈ C 1 (R \ {0}; B(X, Y )) be such that the following conditions are satisfied.
(i) R({M(ρ) : ρ ∈ R \ {0}}) =: κ0 < ∞;
(ii) R({ρM ′ (ρ) : ρ ∈ R \ {0}}) =: κ1 < ∞.
Then the operator TM defined by (2.15) is bounded from Lp(R; X) into Lp(R; Y ) with
norm |TM| B(Lp(R;X),Lp(R;Y )) ≤ C(κ0 + κ1), where C > 0 depends only on p, X, Y .
A rather short and elegant proof of this theorem is given in [29].
2.6 Kernels
In this section we collect some of the basic definitions and properties concerning scalar
kernels which we need for the treatment of parabolic Volterra equations. Definition 2.6.1
and 2.6.2 as well as Lemma 2.6.1 were taken from the monograph Prüss [63, Sections 3
and 8].
Let a ∈ L1, loc(R+). We say that a is of subexponential growth if for all ε > 0,
� ∞
0 e−εt |a(t)| dt < ∞. If this is the case, then it is readily seen that the Laplace transform
â(λ) exists for Re λ > 0.
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