Quasilinear parabolic problems with nonlinear boundary conditions

We conclude this section by illustrating the usefulness of the operators T in connection
with transformations of Volterra equations.
Let X be a Banach space of class HT , p ∈ (1, ∞), and J = [0, T ] be a compact
interval. We consider in X the Volterra equation
u + a ∗ Au = f, t ∈ J, (2.33)
where A is a closed linear operator in X with domain D(A), and the kernel a is assumed
to belong to the class a ∈ K r (α, θa) for some r ∈ N, α ≥ 0, and 0 < θa < θ. Given λ > 0,
our aim is to transform (2.33) in such a way that the operator A is shifted to λ + A.
To this end, we choose an ω ≥ 0 such that aω defined by aω(t) = a(t)e −ωt , t ≥ 0,
is an L1(R+)-function and ν := λ|aω| L1(R+) < 1, as well as θν := θa + arcsin(ν) < θ.
According to Lemma 2.6.2, we have aω ∈ K r (α, θa), and there is a unique kernel b ∈
K r (α, θν) ∩ L1(R+) such that b − λb ∗ aω = aω. Further, the kernel λaω fulfills the
assumptions of Corollary 2.8.3, hence the operator T := Tλaω := (I − λaω∗) is welldefined
and is an isomorphism of Lp(J; X), of (0)H α p (J; X), and also of Lp(J; D(A)).
Observe that aω ∗ g = b ∗ T g for all g ∈ Lp(J; X). Multiply now (2.33) by e −ωt , put
uω(t) = u(t)e −ωt as well as fω(t) = f(t)e −ωt and add a zero-term to obtain
uω − λaω ∗ uω + aω ∗ (λ + A)uω = fω, t ∈ J. (2.34)
If we set v = T uω, then b ∗ v = aω ∗ uω and so (2.34) transforms to
v + b ∗ (λ + A)v = fω, t ∈ J. (2.35)
This equation is equivalent to (2.33). The kernel b enjoys the same properties as a, and
instead of A we have now the shifted operator λ + A.
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