Quasilinear parabolic problems with nonlinear boundary conditions

Following these conventions, BVloc(R+) designates the space of all scalar functions that
are locally of bounded variation on R+.
If not indicated otherwise, by f ∗ g we mean the convolution defined by (f ∗ g)(t) =
� t
0 f(t − τ)g(τ) dτ, t ≥ 0, of two functions f, g supported on the halfline.
For u ∈ L1, loc(R+; X) of exponential growth, i.e. � ∞
0 e−ωt |u(t)| dt < ∞ with some
ω ∈ R, the Laplace transform of u is defined by
� ∞
û(λ) = e −λt u(t) dt, Re λ ≥ ω.
0
For a comprehensive account of the vector-valued Laplace transform we refer to Hille
and Phillips [44], and Arendt, Batty, Hieber, Neubrander [7]; see also Prüss [63]. For
the classical Laplace transform, one of the standard references is Doetsch [30].
We conclude this section by stating a result on the inversion of the vector-valued
Laplace transform. It is due to Prüss, see [64, Corollary 1].
Proposition 2.1.1 Let X be a Banach space. Suppose g : C+ → X is holomorphic and
satisfies
|g(λ)| + |λg ′ (λ)| ≤ c|λ| −β , Re λ > 0, (2.1)
for some β > 0. Then, with n := [β], there is an n-times continuously differentiable
function u : (0, ∞) → X such that û(λ) = g(λ) for all λ ∈ C+. Moreover,
|u (k) (t)| ≤ M t β−k−1 , t > 0, 0 ≤ k ≤ n, (2.2)
where M > 0 is a constant depending only on c and β.
2.2 Sectorial operators
This section contains the definitions and certain known properties of sectorial operators,
operators which admit a bounded H ∞ -calculus, operators with bounded imaginary
powers, R-sectorial operators, and operators with R-bounded functional calculus. A
general reference for the material presented here is the extensive work by Denk, Hieber
and Prüss [29].
We begin with the definition of sectorial operators. Let X be a complex Banach
space, and A be a closed linear operator in X. Then A is called pseudo-sectorial if
(−∞, 0) is contained in the resolvent set of A and the resolvent estimate
|t(t + A) −1 | B(X) ≤ M, t > 0,
holds, for some constant M > 0. If in addition N (A) = {0}, D(A) = X, and R(A) = X,
then A is called sectorial. The class of sectorial operators in X is denoted by S(X).
We recall that in case X is reflexive and A is pseudo-sectorial, the space X decomposes
according to X = N (A) ⊕ R(A). Thus in such a situation A is sectorial on R(A).
Putting
Σθ = {λ ∈ C \ {0} : |arg λ| < θ}
it follows by means of the Neumann series that if A ∈ S(X), then ρ(−A) ⊃ Σθ, for some
θ > 0 and sup{|λ(λ + A) −1 | : | arg λ| < θ} < ∞. Therefore one may define the spectral
angle φA of A ∈ S(X) by
φA = inf{φ : ρ(−A) ⊃ Σπ−φ, sup
λ∈Σπ−φ
|λ(λ + A) −1 | < ∞}.
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