Quasilinear parabolic problems with nonlinear boundary conditions

where c2 does not depend on f.
To see the equivalence of the norms for α < 1 + 1/p, replace the non-existing traces
in the above estimates with zero and use H α p (J; X) ↩→ C(J; X) instead of (2.31), if
α ∈ (1/p, 1 + 1/p). We thus have proved
Corollary 2.8.2 Let the assumptions of Corollary 2.8.1 hold. Let J = [0, T ], µ > 0,
and assume that α ∈ (0, 2) \ {1/p, 1 + 1/p}. Then (2.30) defines an equivalent norm for
H α p (J; X).
We continue to consider the setting of Corollary 2.8.1, where J = [0, T ] and α ∈ (0, 2) \
{1/p, 1 + 1/p}.
Let us additionally assume that a ∈ L1(R+) with ν := |a| L1(R+) < 1, and define the
operator Ta in Lp(J; X) by
By Young’s inequality,
Taf = f − a ∗ f, f ∈ Lp(J; X). (2.32)
|a ∗ f| Lp(J;X) ≤ |a| L1(J)|f| Lp(J;X) ≤ ν|f| Lp(J;X), f ∈ Lp(J; X),
i.e. Ta ∈ B(Lp(J; X)) and |Ta| B(Lp(J;X)) ≤ 1 + ν. Since ν < 1, we also see that Ta is
invertible with |T −1
a | B(Lp(J;X)) ≤ 1/(1 − ν).
Suppose now that f ∈ Y := H α p (J; X). Then trivially f ∈ Lp(J; X), and thus
Corollary 2.8.1 yields Taf ∈ Y . Assuming µ ≥ ν −1 T 1/p max{1, T/(1 + p) 1/p } in (2.30),
we obtain in the case α > 1 + 1/p,
|a ∗ f| (a,µ)
Y = |B(a ∗ f)| Lp(J;X) = |f| Lp(J;X)
≤ |f − f(0) − t ˙
f(0)| Lp(J;X) + |f(0)| Lp(J;X) + |t ˙
f(0)| Lp(J;X)
T p+1
p + 1
≤ |a| L1(J)|B(f − f(0) − t ˙ f(0))| Lp(J;X) + µν(|f(0)|X + | ˙ f(0)|X)
≤ |a ∗ B(f − f(0) − t ˙ f(0))| Lp(J;X) + T 1
p |f(0)|X + (
≤ ν |f| (a,µ)
Y .
In the same way, we see that |a ∗ f| (a,µ)
Y
) 1
p | ˙
f(0)|X
≤ ν|f| (a,µ)
Y is valid for α < 1 + 1/p. This
shows Ta ∈ B(Y ) and |Ta| B(Y ) ≤ 1 + ν, where the operator norm is induced by | · | (a,µ)
Y ,
which is a norm for Y , due to Corollary 2.8.2. Moreover, in view of ν < 1, it follows
that Ta is invertible in Y and |T −1
a | B(Y ) ≤ 1/(1 − ν). This is also true in the subspace
0Y := 0H α p (J; X). Here (2.30) reduces to |f| (a,µ)
0Y = |Bf| Lp(J;X), f ∈ 0Y .
We record these properties of Ta in
Corollary 2.8.3 Let the assumptions of Corollary 2.8.1 hold. Let J = [0, T ], and assume
α ∈ (0, 2) \ {1/p, 1 + 1/p}. Suppose further a ∈ L1(R+) and ν := |a| L1(R+) < 1.
Then the operator Ta defined by (2.32) is an isomorphism of the spaces Y = Lp(J; X),
H α p (J; X), and 0H α p (J; X), satisfying
|Ta| B(Y ) ≤ 1 + ν, |T −1
a | B(Y ) ≤ 1
1 − ν ,
where in case Y = H α p (J; X) or 0H α p (J; X), the operator norm is induced by (2.30) with
µ ≥ ν −1 T 1/p max{1, T/(1 + p) 1/p }.
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