Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 18, pp. 1–40.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.18

QUANTITATIVE ESTIMATES OF Lp MAXIMAL REGULARITY

FOR NONAUTONOMOUS OPERATORS AND GLOBAL

EXISTENCE FOR QUASILINEAR EQUATIONS

THÉO BELIN, PAULINE LAFITTE

Abstract. In this work, we obtain quantitative estimates of the continuity

constant for the Lp maximal regularity of relatively continuous nonautonomous
operators A : I → L(D,X), where D ↪→ X densely and compactly. They

allow in particular to establish a new general growth condition for the global
existence of strong solutions of Cauchy problems for nonlocal quasilinear equa-

tions for a certain class of nonlinearities u 7→ A(u). The estimates obtained

rely on the precise asymptotic analysis of the continuity constant with respect
to perturbations of the operator of the form A(·) + λ Id as λ → ±∞. A

complementary work in preparation supplements this abstract inquiry with an

application of these results to nonlocal parabolic equations in noncylindrical
domains depending on the time variable.

Contents

1. Introduction 1
2. Main results 4
2.1. Operators, spaces and measurability 5
2.2. Lp maximal regularity 7
2.3. Relatively continuous operators 9
2.4. Main results and scheme of the proof 11
3. Perturbation and asymptotic behavior 17
3.1. The operator LA := d

dt +A 17
3.2. Diagonal perturbations by constants in mrp(I) and MRp(I) 18
3.3. Nonautonomous perturbations of autonomous operators in mrp 21
4. Gluing maximally regular operators 22
4.1. Behavior of [·]MRp(I) under a change of variable in time 23
4.2. Gluing finitely many MRp operators 24
5. Proofs of the main estimates and examples 30
5.1. Estimation for general relatively continuous operators 30
5.2. Estimation for Hölder-type relatively continuous operators 32
5.3. Examples 33

2020 Mathematics Subject Classification. 35K90, 35K59, 35D35, 47D06.
Key words and phrases. Nonautonomous Cauchy problem; Lp maximal regularity;

relatively continuous operators; quasilinear equations.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted October 8, 2024. Published February 26, 2025.

1



2 T. BELIN, P. LAFITTE EJDE-2025/18

6. Global existence for quasilinear equation 36
6.1. A general existence result 36
6.2. Existence of strong solutions for relatively continuous operators 38
7. Conclusion and perspectives 39
References 39

1. Introduction

The Lp maximal regularity theory deals with the well-posedness in the strong
sense of the abstract nonautonomous Cauchy problem

d

dt
u(t) +A(t)u(t) = f(t), t ∈ I = (a, b),

u(a) = 0.
(1.1)

The problem is set on a time interval I = (a, b) ⊂ R bounded from below, the
source term f lies in Lp(I;X) and A : I → L(D(A);X) is a nonautonomous linear
operator with a uniform-in-time domain D(A) ⊂ X.

For a generic source term f ∈ Lp(I;X), a strong solution of (1.1), i.e. a solution
u such that t 7→ [ ddtu](t), A(t)u(t) are elements of Lp(I;X), lies in the Banach

space MRp(I) := W 1,p(I;X) ∩ Lp(I;D(A)) called the maximal regularity space.
Given the regularity of the source term f ∈ Lp(I;X), this is the best regularity one
can possibly obtain for u, giving a precise semantic meaning to the terminology
of Lp maximal regularity. More precisely if there exists K ≥ 0 such that for any
f ∈ Lp(I;X) there exists a unique solution u ∈ MRp(I) to (1.1) and

∥u∥MRp(I) ≤ K∥f∥Lp(I;X), (1.2)

then we say that A is maximally regular in Lp.
It is of interest to study the following questions which have numerous applica-

tions, e.g. to the study of the well-posedness and regularity of solutions of Cauchy
problems for general parabolic equations:

(Q1) Under which conditions on A is (1.1) well-posed with respect to the source
term f ∈ Lp(I;X)?

(Q2) Can we identify quantities related to the operator A which give fine esti-
mates and precise asymptotic behaviors of the constant K in (1.2)?

(Q3) In which topologies on the set of nonautonomous operators is the set of
maximally regular operators closed? In these topologies, does the continuity
of strong solutions in MRp(I) hold?

(Q4) Can answering (Q2) and (Q3) allow to infer well-posedness criteria of the
Cauchy problem for nonlocal quasilinear equations of the form

d

dt
u(t) + A(u, t)u = F(u, t), u(0) = x ?

Logically, answers to questions (Q2), (Q3) and (Q4) must rely on the already
existing answers to question (Q1), and we hereby give a short summary of the
standard results found for Lp maximal regularity regarding (Q1) for autonomous
and nonautonomous operators.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 3

These results were applied to the study of boundary value parabolic problems,
e.g. for Dirichlet and Neumann conditions (see [24] and references therein) and
more recently for Robin conditions (see [10]).

For autonomous operators A(t) = A, (Q1) has received a lot of attention. Since
de Simon [14] it is known that if X is a Hilbert space, the maximal regularity on
the unbounded interval (0,+∞) is equivalent to the sectoriality of the operator −A
which is also equivalent to the analyticity of the semi-group generated by −A on a
sector of the complex plane. In the setting where X is merely a Banach space, the
sectoriality of −A is necessary (see e.g. [24]) but not sufficient in general. Besides,
it is known that, in this autonomous case, the Lp maximal regularity is independent
of p ∈ (1,+∞) [24, Proposition 2.4].

More recently, in [21], it has been proved that theR-boundedness of the resolvent
operator of A yields maximal regularity in Banach spaces X of the UMD class,
that is, the Banach spaces for which the Hilbert transform is a bounded operator
in Lq(R;X), for some 1 < q < +∞.

For a thorough introduction on maximal regularity for autonomous operators we
refer the reader to the lecture notes of Kunstmann and Weis [21].

The study of the autonomous case relies on a Fourier transform in the time
variable and the study of the regularizing properties of the subsequent kernel. Such
a technique logically leads to difficulties in the nonautonomous case. Sufficient
conditions known in the literature for maximal regularity usually require a certain
degree of regularity in time of A(·) as well as the maximal regularity of each A(t),
t ∈ I. Therefore in general nonautonomous operators t 7→ A(t) satisying such
sufficient conditions will have Lm maximal regularity for any m ∈ (1,+∞). Note
however that in general, it is not straightforward that a nonautonomous operator
with Lp maximal regularity for some p ∈ (1,+∞) is also Lm maximally regular for
a different m ∈ (1,+∞).

The first notable results were obtained by Acquistapace and Terreni [1] who
have uncovered a Hölder-type condition in time of the operator, which may have a
time-dependent domain D(A(t)). In the case of a constant domain D(A), which is
our interest here, Prüss and Schnaubelt [26] showed that the strong continuity in
time along with the maximal regularity of each A(t) is sufficient to guarantee the
maximal regularity of the nonautonomous operator. These inquiries were further
refined and developed by Amann [3], Yagi [28, Chapter 3], Gallarati and Veraar [16].
In this direction, Arendt and coauthors [8] gave a weaker sufficient time-regularity
condition called relative continuity.

Overall, on the subject of autonomous and nonautonomous maximal regularity,
we refer to Monniaux’s work, and especially her review [24], or more recently by
Pyatkov [27].

We rely on relative continuity assumptions to provide new answers to questions
(Q2), (Q3) and (Q4). We introduce the space RC(I; (D,X)) of relatively contin-
uous operators and isolate meaningful quantities through which one can estimate
the constant K.

The Lp maximal regularity has shown its wide uses in various settings for the
study of strong solutions of very wide classes of parabolic-type equations. Again
we refer the reader to the monograph [24] and references therein. The framework
of maximal regularity is a crucial tool to study the strong solutions of nonlinear
equations. Let us mention the recent work of Danchin and coauthors [13] who



4 T. BELIN, P. LAFITTE EJDE-2025/18

explore the critical setting p = 1. It turns out that this L1 setting is natural to
study the equivalence of Lagrangian and Eulerian formulations globally in time.
Their inquiries lead to well-posedness results in free boundary problems of fluid
mechanics.

Answers to (Q4) can rely on answers to (Q2). To fix some ideas, we focus our
interest on a class of nonlocal quasilinear equations of the form

d

dt
u(t) + A(u, t)u(t) = F(u, t), t ∈ I,

u(a) = x
(1.3)

on a bounded time interval I = (a, b) ⊂ R, b < +∞. The nonlinearities u 7→ A(u)
and u 7→ F(u) are assumed to be defined on X (I), a functional space on I for which
the following chain of embeddings hold:

MRp(I) ↪→c X (I) ↪→ Lp(I;X),

where the first embedding is compact. As such these nonlinear operators may in-
clude nonlocal effects in time. They are sometimes assumed to satisfy a certain
causality principle translated by the Volterra property (see Amann [2]): the re-
striction of A(u) and F(u) to any subinterval of the form (a, t), t ≤ b only depends
on u|It .

These equations appear in various models of elasticity and solid mechanics with
memory and nonlocal effects (see [12, 4, 15] and references therein) or population
dynamics [20]. They generally model phenomena which have a memory of their
past states. Their interest is both found in applications and in theoretical studies
since they also appear when regularizing certain singular equations, for instance in
the Perona-Malik model of anisotropic diffusion [5, 18, 19].

The well-posedness of these Cauchy problems in an abstract framework has been
studied by Amann (see [6, 3, 2, 4]). His most important result in [2] shows that if
X (I) = MRp(I) and under the Lipschitz continuity of u 7→ A(u) in L∞(I;L(D,X))
and u 7→ F(u) in Lp(I;X) then the local well-posedness of (1.3) holds. Amann
further shows existence of a Lipschitz semi-flow associated to the equation. These
results rely on a Banach fixed point theorem which inherently requires the Lipschitz
continuity of the operators.

Since the embedding D ↪→ X is compact, we rely here on Schauder’s fixed
point theorem to obtain existence results. No Lipschitz continuity is required,
thereby permitting lower regularity assumptions on the nonlinearities A and a mere
weak continuity condition for F (see e.g. Arendt [9]). The existence is guaranteed
through growth conditions on the regularity constantKA(u) in the regime ∥u∥X (I) →
+∞. We use our answers to (Q2) in the setting of relatively continuous operators
of Hölder-type to obtain quantitative growth conditions on A(u) and F(u). An
interesting interplay between the pointwise maximal regularity constant of each
operator A(u) and their Hölder semi-norm is observed.

In Section 2, we introduce the notion of Lp maximal regularity on bounded and
unbounded intervals as well as the maximal regularity constant. We describe the
maximal regularity space MRp(I) and a special discussion is dedicated to scale-
invariant norms. Then relatively continuous operators and their ranges of relative
continuity are introduced. Related preliminary results pertaining to relatively con-
tinuous operators are further presented. This context allows the exposition of our
main result: we first present the quantitative behavior of the regularity constant



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 5

with respect to perturbations of the form A(·)+λ Id, in particular in the asymptotic
regime λ→ ±∞ (Theorem 2.15), we then show our main quantitative estimates of
the regularity constant for relatively continuous operators (Theorem 2.16, Theorem
2.18). As an application to the estimate in Theorem 2.16, we present various con-
ditions giving positive answers to (Q3) (Proposition 2.20, Theorem 2.21, Theorem
2.22). The growth condition for the global existence for quasilinear equations is
then presented for a Hölder-type class of relatively continuous operators Theorem
2.23, Theorem 2.24.

In Section 3 we describe the operators LIA := d
dt+AI related to the Cauchy prob-

lem on I ⊂ R and recall known properties in the autonomous case. We then prove
precise perturbation results with asymptotic behavior through fine manipulations
of the chosen norm of the maximal regularity space. Section 4 is dedicated to the
proof of intermediate results, in particular Theorem 4.7 which provides the frame
of the quantitative estimate Theorem 2.16 proved in Section 5. Lastly Section 6
treats the global existence of nonlocal quasilinear equations for a class of relatively
continuous operators. Some proofs, which are classical or standard, were omitted
here, but can be found in [11].

2. Main results

In the quantitative study of Lp maximal regularity, the choice of the norm in
the maximal regularity space MRp(I) can have a noticeable impact and should be
chosen with care. Hence in Section 2.1 we set basic notations for operators and
norms and we motivate our choices for the norm of the maximal regularity space
via time-scaling invariance. We further introduce in Section 2.2 basic notations of
Lp maximal regularity and elementary results. Section 2.3 contains an introduction
to the class of relatively continuous operators and we describe topological properties
of this space of operators. Section 2.4 finally showcases our main results.

2.1. Operators, spaces and measurability.
Graph norms and domains of operators. If two norms N1 and N2 are equivalent in
a given Banach space, meaning there exist c, C > 0 such that

cN2 ≤ N1 ≤ CN2,

we write N1
c,C∼ N2 or equivalently N2

1
C ,

1
c∼ N1. If the constants are universal or

not of interest for further development we may simply write N1 ∼ N2.
A linear operator A : D(A) → X on a Banach space X, defined on a linear

subspace D(A) ⊂ X, is said to be closed if its graph Γ(A) := {(x,Ax);x ∈ D(A)}
is closed. In such a case, we can equip D(A) with the p-graph norm induced by A,
p ∈ [1,+∞), and defined by

∥ · ∥A,p := (∥ · ∥pX + ∥A · ∥pX)
1/p

.

The domain D(A) is then a Banach space. It is straightforward to see that for

any two p, q ∈ (1,+∞), ∥ · ∥A,p
1
2 ,2∼ ∥ · ∥A,q. Since the constants of equivalence are

universal, we denote without ambiguity ∥.∥A for any p-graph norm of A.

Let (X, ∥ · ∥X) be a Banach space and let D
d
↪→ X be a Banach space with

norm denoted ∥ · ∥D, which injects densely and compactly in X. We denote by
L(D;X) the space of bounded linear operators from D to X. For the rest of our



6 T. BELIN, P. LAFITTE EJDE-2025/18

developments X and D are fixed, as well as p, q ∈ (1,+∞) two conjugate exponents
satisfying 1

p +
1
q = 1.

Functional spaces for Lp maximal regularity. Let I ⊆ R be any open interval
bounded from below i.e. I = (a, b), with −∞ < a < b ≤ +∞ and Y be a Ba-
nach space.

We define the vector valued Lebesgue space Lp(I;Y ) as the space of Bochner-
measurable functions u : I → Y such that

∥u∥Lp(I;Y ) =
(∫

I

∥u(t)∥pY dt
)1/p

< +∞.

We denote by W 1,p(I;Y ) the Y -valued Sobolev space. It is the space of functions
u ∈ Lp(I;Y ) which admit a weak derivative d

dtu ∈ Lp(I;Y ) which satisfies

−
∫
I

[ d

dt
ϕ
]
(t)u(t)dt =

∫
I

ϕ(t)
[ d

dt
u
]
(t)dt

for all ϕ ∈ C∞
c (I;R). It is endowed with the semi-norm

[u]W 1,p(I;Y ) :=
∥∥∥ d

dt
u
∥∥∥
Lp(I;Y )

=
(∫

I

∥∥[ d
dt
u
]
(t)

∥∥p
Y
dt
)1/p

. (2.1)

As usual in the spaces Lp(I;Y ) andW 1,p(I;Y ) we identify two elements u and v
if they agree L1-almost everywhere on I for the Lebesgue measure L1. Throughout
we endow the equality relation = with this meaning when we identify elements of
such spaces.

Hereafter we define the maximal regularity space on I along with its norm.

Definition 2.1. Define MRp(I), the Lp maximal regularity space as

MRp(I) := Lp(I;D) ∩W 1,p(I;X).

If b < +∞ we choose the norm

∥u∥MRp(I) :=
(
∥u∥pLp(I;X) + ∥u∥pLp(I;D) + |I|p[u]W 1,p(I;X)

)1/p

. (2.2)

Otherwise if b = +∞ then

∥u∥MRp(I) :=
(
∥u∥pLp(I;X) + ∥u∥pLp(I;D) + [u]W 1,p(I;X)

)1/p

. (2.3)

The norm (2.3) is more classical: it allows for a unified treatment of maximal
regularity on bounded and unbounded intervals. However in our work, we mostly
study maximal regularity in the bounded setting b < +∞. The norm (2.2) is well
behaved under time rescalings thanks to the weight |I|p in front of the W 1,p semi-
norm. In fact, for any λ ̸= 0 a natural isometry from MRp(I) to MRp(λI) can be
defined as ιλ : u 7→ λ−1/pu(λ−1·). This rescaling property will prove useful in the
development of the arguments throughout.

For further use, let us also define the closed subspace MRp0(I) ⊂ MRp(I) defined
as follows

MRp0(I) := {u ∈ MRp(I) : u(a) = 0}. (2.4)

Remark 2.2. It makes sense to evaluate u ∈ MRp(I) at any point of I (e.g.
u(a)) as the following continuous embedding MRp(I) ↪→ C

(
I; Trp

)
holds, where

Trp := (D,X) 1
q ,p

is a real interpolation between D and X of parameters
(

1
q , p

)



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 7

(see e.g. [23, Theorem 3.12.2]). Equivalently Trp can be interpreted as a trace space
of MRp(I) meaning Trp = {u(a) : u ∈ MRp(I)} equipped with the norm

∥x∥Trp,I = inf{∥u∥MRp(I) : u ∈ MRp(I), u(a) = x}.
In the case of a bounded interval b < +∞, under the isometry ιλ : MRp(I) →

MRp(λI) the norm of the trace space is scaled accordingly. More precisely we have
for any x ∈ Trp,

∥x∥Trp,I = inf{∥u∥MRp(I) : u ∈ MRp(I), u(a) = x}
= inf{∥ι−1

λ (v)∥MRp(I) : v ∈ MRp(λI), ι−1
λ (v)(a) = x}

= inf{∥v∥MRp(λI) : v ∈ MRp(λI), λ1/pv(λa) = x}

= |λ|−1/p∥x∥Trp,λI .
Using the time-reversing rescaling ι−1 we see that the trace can either be taken

on the left or on the right of the interval in the definition i.e. Trp = {u(b) : u ∈
MRp(a, b)}.

Strong measurability of nonautonomous operators. Let I = (a, b) ⊂ R be an open
interval, bounded from below. For nonautonomous operators A : I 7→ L(D;X), the
notion of strong measurability is necessary to allow the definition of the associated
multiplication operators A defined in the next paragraph. We want to give meaning
to the quantity

∥Av∥Lp(I;X) =
(∫

I

∥A(t)v(t)∥pXdt
)1/p

,

which requires the measurability of the map t 7→ ∥A(t)v(t)∥X . And if t 7→ A(t) is
strongly measurable as defined below, the measurability of this map will hold for
any v ∈ Lp(I;D).

Definition 2.3. A nonautonomous operator A : I → L(D;X) is said to be strongly
measurable if for any x ∈ D, the map A(·)x : I → X is Bochner-measurable in X.

The following remarks motivate the use of this notion of strong measurability
for applications.

Remark 2.4. (1) The notion of strong measurability for operators is a weaker type
of measurability than the measurability of t 7→ A(t) in L(D;X). Even with D and
X separable, these two notions of measurability are not equivalent in general.

(2) By Pettis’ theorem [25], since X is separable, if t 7→ A(t)x is weakly contin-
uous in X for any x ∈ D, then A is strongly measurable. Pettis’ theorem allows
to show the strong measurability of differential operators with continuous coeffi-
cients. For example fix c ∈ Cb(I × R), let X = L2(R), D = H1(R) and define for
each t ∈ I the operator A(t)u := {x 7→ c(t, x) d

dxu(x)} ∈ L(D,X). We see that

t 7→ [A(t)]u = c(t, x) d
dxu(x) is weakly continuous in L2(R) if u ∈ H1(R). Since

L2(R) is separable, Pettis’ theorem applies and we obtain the strong measurability
of A.

We can then equip this notion of strong measurability to Lebesgue spaces of
nonautonomous operators.

Definition 2.5. We define the Lebesgue space

L∞ (I;L(D;X)) :=
{
A : I → L(D;X) : A is strongly measurable and



8 T. BELIN, P. LAFITTE EJDE-2025/18

for any x ∈ D, ∥A(·)x∥X ∈ L∞(I;R)
}
.

The separability of D and the uniform boundedness principle ensure that the
map t 7→ ∥A(t)∥L(D;X) is essentially bounded in I for any A ∈ L∞ (I;L(D;X)).
As usual, the identification in L∞ (I;L(D;X)) is performed almost everywhere in
I (for the one-dimensional Lebesgue measure) and the norm ∥A∥L∞(I;L(D;X)) :=
∥∥A(·)∥L(D;X)∥L∞(I) gives it the structure of a Banach space. When there is no
ambiguity, we shall often write ∥A∥∞ for this norm.

2.2. Lp maximal regularity. Let I = (a, b) ⊆ R be an open interval bounded
from below, i.e. −∞ < a < b ≤ +∞. We define the maximal regularity property
and the maximal regularity constant for autonomous and nonautonomous opera-
tors on I. We then derive elementary properties of the classes defined below, in
particular the stability of the maximal regularity property under restriction of the
time interval.
Autonomous operators: mrp(I).

Definition 2.6. An operator A ∈ L(D;X) has the Lp maximal regularity property
on I if

(aut.i) ∥ · ∥A ∼ ∥ · ∥D.
(aut.ii) For each f ∈ Lp(I;X) there exists a unique solution vAf ∈ MRp(I) of the

abstract Cauchy problem

d

dt
v(t) +Av(t) = f(t), t ∈ I,

v(a) = 0
(2.5)

and there exists K ≥ 0, independent of f such that

∥vAf ∥MRp(I) ≤ K∥f∥Lp(I;X). (2.6)

We denote mrp(I) this family of maximally regular autonomous operators and for
each A ∈ mrp(I), the infimum K in (2.6) is denoted by [A]mrp(I). It is called the
maximal regularity constant of A on I.

Nonautonomous operators: MRp(I).

Definition 2.7. A nonautonomous operator A ∈ L∞(I;L(D;X)) has the Lp max-
imal regularity property on I if

(nonaut-i) For any t ∈ I, ∥ · ∥A(t) ∼ ∥ · ∥D.
(nonaut-ii) For any f ∈ Lp(I;X) there exists a unique solution vAf ∈ MRp(I) of the

abstract nonautonomous Cauchy problem

d

dt
v(t) +A(t)v(t) = f(t), t ∈ I,

v(a) = 0.
(2.7)

And there exists K ≥ 0, independent of f such that

∥vAf ∥MRp(I) ≤ K∥f∥Lp(I;X). (2.8)

We denote MRp(I) this family of nonautonomous and maximally regular operators
and for each A ∈ MRp(I) the infimum K in (2.8) is denoted by [A]MRp(I). It is
also called the maximal regularity constant of A.

Following these definitions several clarifying remarks are in order.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 9

Remark 2.8. (i) The condition ∥ · ∥A ∼ ∥ · ∥D is equivalent to asking that
A be a closed unbounded operator on X with domain D, provided D ⊊
X. If D = X, then maximal regularity will hold for all A ∈ L(X) by
the abstract Cauchy-Lipschitz theorem. In the case X = Lp(Ω) with ∂Ω

smooth, D =W 2,p(Ω)∩W 1,p
0 (Ω), it is known that autonomous second order

and uniformly elliptic operators with continuous coefficients are maximally
regular, condition (aut.i) being given by the theory of elliptic regularity
(see e.g. [17]).

(ii) See thatmrp(I) (resp. MRp(I)) is not stable by finite sum because require-
ment (aut.i) (resp. (nonaut-i)) fails in general for the sum of two operators.

(iii) When the time interval I is clear by context, we shall write mrp and MRp

instead of mrp(I) or MRp(I) respectively. Moreover we easily see that
mrp(I) ⊂ MRp(I).

(iv) For autonomous operators, mrp(I) = mrp(J) for any two bounded inter-
vals I, J ⊂ R, in other words, in the case of bounded intervals, maximal Lp

regularity is independent of the interval. However we remark that, accord-
ing to Theorem 2.6 mrp(0, 1) ̸= mrp(R+) in general. Fix an open bounded
space interval Ω ⊂ R and consider the diagonally perturbed Dirichlet Lapla-
cian operator A := −∆D − λ Id defined on D = H2(Ω) ∩H1

0 (Ω) ⊂ L2(Ω),
where λ ≥ 0 is some eigenvalue of −∆D. We have A ∈ mrp(0, 1) while

A ̸∈ mrp(R+). Indeed, if uλ ∈ C∞(Ω) is an eigenvector of −∆D associated
to λ, then t ∈ R+ 7→ uλ, is the unique solution of the homogeneous Cauchy
problem with initial condition u(0) = uλ ∈ Trp. Hence u ∈ MRp(0, 1) while
u ̸∈ MRp(R∗

+). Another way to see that A ̸∈ mrp(R∗
+) is to remark that it

is not a sectorial operator.

If we are given a nonautonomous operator A : I 7→ L(D,X), a source term
f ∈ Lp(I;X) and an initial condition x ∈ Trp we can consider the Cauchy problem

d

dt
v(t) +A(t)v(t) = f(t), t ∈ I,

v(a) = x,
(2.9)

where recall that I = (a, b) ⊂ R is bounded from below. We use the notation
(2.9)Ix,f to refer to the above Cauchy problem: we explicitly specify the initial
datum x, the source term f and the time interval I under consideration. It turns
out that condition (nonaut-ii) is equivalent to the well-posedness of the general
Cauchy problem (2.9)Ix,f for any x ∈ Trp, f ∈ Lp(I;X). We omit the proof of this
result which relies on standard lifting techniques.

Proposition 2.9. Let A ∈ L∞(I;L(D;X)) be a nonautonomous operators satis-
fying (nonaut-i). Then A ∈ MRp(I) if and only if for any x ∈ Trp, f ∈ Lp(I;X)
there exists a unique solution to (2.9)Ix,f and there exists K ≥ 0, independent of f
and x such that

∥v∥MRp(I) ≤ K
(
∥f∥Lp(I;X) + ∥x∥Trp,I

)
.

It is quite natural to want to be able to restrict an operator A ∈ MRp(I) to any
subinterval J ⊂ I while keeping the maximal regularity property of the restricted
operator A|J . This property amounts to the existence of a regular evolution oper-
ator for the equation. In our case, the compactness of the embedding D ↪→ X is a
key ingredient that allows to derive this property. The proof of the result can be
found in [11, Appendix D].



10 T. BELIN, P. LAFITTE EJDE-2025/18

Proposition 2.10. Let I ⊂ R be an open interval bounded from below and let
A ∈ MRp(I). Then for any subinterval J ⊂ I, A|J ∈ MRp(J). Moreover, if
either |J | = |I| = ∞ or |J | ≤ |I| < +∞ then

[A|J ]MRp(J) ≤ [A]MRp(I). (2.10)

Otherwise if |J | < |I| = +∞, then

[A|J ]MRp(J) ≤ (|J | ∨ 1) [A]MRp(I). (2.11)

2.3. Relatively continuous operators. Regarding nonautonomous operators,
relative continuity is a weaker type of continuity than strong continuity. Rela-
tive continuity incorporates the idea of perturbing continuous operators and as
such allows to treat with generality various kinds of such perturbations. The def-
inition, which we reformulate here, was first given by Arendt and coauthors in [8,
Definition 2.5]. We state elementary topological properties of RC(I), the space of
relatively continuous operators on I. In particular RC(I) is a closed linear sub-
space of L∞(I;L(D;X)). Then we exhibit a theorem, analogous in shape, to the
Arzelà-Ascoli theorem for continuous maps.

Definition 2.11. Let I = (a, b) be a bounded open interval.

• We say that a strongly measurable nonautonomous operator A : I →
L(D;X) is relatively continuous if for any ϵ > 0, there exists δ > 0 and
η ≥ 0 such that

∀x ∈ D,∀t, s ∈ I, |t− s| ≤ δ =⇒ ∥A(t)x−A(s)x∥X ≤ ϵ∥x∥D + η∥x∥X . (2.12)

We denote by RC(I; (D;X)) the nonautonomous operators of L(D;X)
which are relatively continuous.

• For A ∈ RC(I; (D;X)), we call the ranges of relative continuity the follow-
ing set-valued function, which is nondecreasing with respect to set inclusion

r∗A :

{
(0,+∞) → P ((0,+∞)× [0,+∞))

ϵ 7→ {(δ, η) : (2.12) holds }.

• We say that rA = (δ, η) : (0,+∞) → (0,+∞)× [0,+∞) is a specific range
of relative continuity of A if
(⋆) rA(ϵ) ∈ r∗A(ϵ) for any ϵ ∈ (0,+∞).
(⋆) ϵ 7→ δ(ϵ) is nondecreasing.
(⋆) ϵ 7→ η(ϵ) is nonincreasing.

Here D and X are fixed, so we write RC(I) unambiguously for RC(I; (D;X)).
Furthermore, when we wish to select a specific range of relative continuity rA, with
a slight abuse of notation, we may simply write rA ∈ r∗A. If A ∈ RC(I), then it is
always possible to select a specific range of relative continuity for A.

We now state some topological facts about relatively continuous operators which
are proved in [11, Appendix B].

Proposition 2.12. RC(I) is a closed, proper linear subspace of L∞(I;L(D;X)).

Before stating our Arzelà-Ascoli type result, let us define the notion of relative
equicontinuity.

Definition 2.13. We say that a family F ⊂ RC(I) is relatively equicontinuous, if
for any ϵ > 0, there exists δ > 0 and η ≥ 0 such that (2.12) holds for any A ∈ F .



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 11

Equivalently, this means that

∩A∈Fr
∗
A ̸= ∅.

For the statement of the Arzelà-Ascoli-type theorem, we describe several topolo-
gies one may put on the space L∞(I;L(D;X)), here translated in terms of conver-
gence. Given a family {Aλ}λ∈Λ ⊂ L(D;X)I of nonautonomous operators, where Λ
is some unspecified metric space, we can describe the following convergences:

(t, x)-pointwise convergence in X. In this topology Aλ →
λ→λ

Aλ if for all (t, x) ∈

I ×D, Aλ(t)x →
λ→λ

Aλ(t)x strongly in X. It is generated by the product topology

of XI×D.

t-pointwise convergence in L(D;X). In this topology Aλ →
λ→λ

Aλ if for all t ∈ I,

Aλ(t) →
λ→λ

Aλ(t) for the norm topology in L(D;X). It is generated by the product

topology of L(D;X)I .

Uniform convergence. This is the topology generated by ∥ · ∥∞ as defined in
Theorem 2.5.

There is an obvious hierarchy of these three topologies, the former being the
weakest and the latter being the strongest.

Proposition 2.14. Let F ⊂ RC(I) be a family of relatively continuous operators
on I. The following properties hold:

(i) If F is relatively compact in L∞(I;L(D,X)) then F is relatively equicon-
tinuous.

(ii) If F is relatively equicontinuous and for each t ∈ I, {A(t) : A ∈ F} is
bounded in L(D,X) then F is bounded in L∞(I;L(D,X)).

(iii) If F is relatively equicontinuous then F ⊂ RC(I) and is also relatively
equicontinuous, where F is the sequential closure of F for the (t, x)-pointwise
convergence.

Note that, contrary to the standard Arzelà-Ascoli theorem for continuous maps,
if we replace the two occurrences of “bounded” by the expression “relatively com-
pact” in (ii) then the conclusion may fail. A counter-example is provided by the
sequence of nonautonomous operators, n ≥ 2, t ∈ [0, 1], An(t) := A + 1[ 1n ,

2
n ](t)B

where A ∈ L(D,X) and B ∈ L(D,X) satisfies an interpolation inequality i.e. there
exists L > 0 and θ ∈ (0, 1) such that for any x ∈ D,

∥Bx∥X ≤ L∥x∥θX∥x∥1−θD .

By Young’s inequality, it is straightforward to check that the family {An}n∈N ⊂
RC(0, 1) is relatively equicontinuous and that An(t)x → A in X as n → +∞ for
all t ∈ [0, 1], x ∈ D. However

∀n ∈ N, ∥An −A∥∞ = ∥B∥L(D,X) > 0.

2.4. Main results and scheme of the proof. In our main result, we derive
quantitative estimates of the maximal regularity constant for relatively continuous
operators which satisfy pointwise maximal regularity up to a small perturbation.
This result is proved through a combination of intermediate results which have
their own interest. In Section 2.4.1 we describe the asymptotic behavior of the
regularity constant under diagonal perturbations (i.e. taking the form A + λ Id).



12 T. BELIN, P. LAFITTE EJDE-2025/18

In Section 2.4.2 we describe the main quantitative estimate as a result of a gluing-
up of operators defined on subintervals. Section 2.4.3 discusses limiting behavior
of operators in MRp in particular its persistence in the limit. Finally in Section
2.4.4 a general result on the global existence of a solution to the Cauchy problem
for abstract quasilinear equations on bounded intervals and a corollary concerning
relatively continuous operators in this context are presented.

2.4.1. Asymptotic behavior of diagonal perturbations. For our quantitative inquiry
it becomes important to evaluate quite precisely the behavior of the maximal reg-
ularity constant with respect to diagonal perturbations. Indeed, the proof of the
main result relies on a perturbation argument. It is stated in the following.

Theorem 2.15. Let I ⊂ R be a bounded open interval and let A ∈ L∞(I;L(D;X))
be a nonautonomous operator. The following statements are equivalent:

(i) There exists λ0 ∈ R such that A+ λ0 Id ∈ MRp(I).
(ii) A ∈ MRp(I).
(iii) For any λ ∈ R, A+ λ Id ∈ MRp(I).

And if one of these statements holds true, we have for any λ ∈ R,

[A+ λ Id]MRp(I) ≤ cp(λ|I|)[A]MRp(I), (2.13)

where the function cp : R → R+ satisfies

cp(ν) := (|ν|+ κp(ν))
(
1 + epν− − e−pν+

)1/p
(2.14)

with
κp(ν)
|ν| →

|ν|→+∞
0 and κp(0) = 1. In particular the following asymptotic behavior

is observed for cp:

cp(ν) ∼
ν→−∞

−νe−ν

and

cp(ν) ∼
ν→+∞

ν.

It is not surprising to see the exponential factor e−ν appear in the regime ν →
−∞ since this exponential behavior is already sharp for autonomous operators (see
later Lemma 3.3). It is however still not clear to the authors whether one can
discard the extra factor ν appearing. This asymptotic result is shown in Section
3. We can remark (2.13) possesses a particular invariance related to the behavior
of [A] with respect to the rescaling A 7→ µA(µ·) (see Remark 4.2 later). This is a
direct consequence of structure of the time-homogeneous norm (2.2) for MRp(I).
In fact any estimate in the form

[A+ λ Id]MRp(I) ≤ C(λ, |I|)[A]MRp(I)

can be reduced to an estimate with C̃(λ, |I|) := infµ>0 C(µλ,
|I|
µ ) thereby forcing

the invariance of the inequality with respect to these transformations.

2.4.2. Maximal regularity constant of relatively continuous operators. We introduce
here several auxiliary quantities which are involved in the estimate of the maximal
regularity constant of nonautonomous operators. Let I ⊂ R be a bounded open
interval and let A0 : I → L(D,X) satisfy the pointwise autonomous maximal
regularity on the unbounded interval R+:

∀t ∈ I, A0(t) ∈ mrp(R+). (2.15)



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 13

Then it is well-known that for each t ∈ I, −A0(t) is the generator of a bounded
analytic semi-group {e−τA0(t)}τ≥0 on X and from Proposition 2.10, we infer that
[A(t)]mrp(I) < +∞. Hence we can define the following nonnegative maps:

K0 : t 7→ [A(t)]mrp(I), M0 : t 7→ sup
τ≥0

∥e−τA0(t)∥X , (2.16)

ϵ0 : t 7→ 1

2(K0(t) + 1)(M0(t) + 1)
. (2.17)

Now let us describe the main assumption on A : I → L(D,X) which guarantees
nonautonomous maximal regularity of A.

(A1) (a) Relative continuity: A ∈ RC(I; (D;X)).
(b) Decomposition: There exist A0 : I → L(D,X) and B : I × I →

L(D,X) such that
(b.1) For all t, s ∈ I, A(s) = A0(t) +B(t, s).
(b.2) For all t ∈ I, A0(t) ∈ mrp(R+).
(b.3) There exists a range rA = (δA, ηA) ∈ r∗A such that for all x ∈ D,

∀t, s ∈ I, |t− s| ≤ ρA(t)

=⇒ ∥B(t, s)x∥X ≤ ϵ0(t)∥x∥D + µA(t)∥x∥X
(2.18)

where ρA := δA ◦ ϵ0, µA := ηA ◦ ϵ0 and ϵ0 is defined by (2.16)-
(2.17).

Assumption (b) says that we can decompose A into the sum of a pointwise maxi-
mally regular operator A0 and a perturbation B, the meaning of this perturbation
being expressed by (b.3). Note that we may have B : (t, s) 7→ A0(s) − A0(t) in
which case (A1) amounts to the assumptions of relative continuity and pointwise
maximal regularity of A given in [8, Theorem 2.11] of Arendt and coauthors.

Theorem 2.16. Assume that A satisfies (A1) on the bounded open interval I ⊂ R.
Then A ∈ MRp(I) and there exists a subdivision T := {τi}0≤i≤N of I and center
points C := {ti}1≤i≤N such that

a = τ0 < t1 < τ1 < · · · < tN < τN = b (2.19)

ρA(ti) ≤ τi − τi−1 ≤ 2ρA(ti), ∀i ∈ {1, . . . , N} (2.20)

and there exists K := K(p, T , C,K0,M0, ∥A∥∞) ≥ 0 such that

[A]MRp(I) ≤ K. (2.21)

Quantitative description of K. To obtain quantitative growth conditions for the
global existence for quasilinear evolution equations, it is of interest to describe the
form of K found in (2.21). Define for (τ, µ,M0,K0) ∈ R4

+, the function

G(τ, µ,M0,K0) := 4(M0 + 1)cp(−4Mτµ)K0,

where cp is given in Theorem 2.15. Given T , C as in Theorem 2.16, for 1 ≤ i ≤ N
denote

Gi := G(τi − τi−1, µA(ti),M0(ti),K0(ti)).

We show that K in (2.21) is written as

K :=

N∑
i=1

τi − τ0
τi − τi−1

( N∏
j=i+1

Hj(p, T , ∥A∥∞, Gj)
)
Gi, (2.22)



14 T. BELIN, P. LAFITTE EJDE-2025/18

where for fixed p, T , C, the functions Hj(p, T , C, ·) : R+ → R+, j ∈ {2, . . . , N} have
linear growth at most. Observe in particular that K only depends on the evaluation
of the maps K0,M0 : I → R+ at the center points C ⊂ I, and remark that this
dependence is not invariant under permutation of the values (K0(t1), . . . ,K0(tN )):
this is due to the intrinsic direction of time imposed in the Cauchy problem. The
proof of the estimate is found in Section 5.

We obtain an even more explicit estimate in the particular class of relatively
continuous operators which satisfy a Hölder-type regularity and when an additional
integrability condition holds for the maps K0 and M0.

For α ∈ [1,+∞) and β ∈ [0,+∞), let us introduce the subspace RCα,β(I) ⊂
RC(I).

Definition 2.17. We say that A ∈ RCα,β(I) if there exists a range of relative
continuity rA = (δA, ηA) ∈ r∗A for A and mδ ∈ (0, 1], mη ≥ 0 such that

∀ϵ > 0, δA(ϵ) = mδϵ
α,

∀ϵ > 0, ηA(ϵ) ≤ mηϵ
−β .

RCα,β(I) is a closed linear subspace of RC(I) equipped with the topology in-
duced by L∞(I;L(D,X)). In many applications the differential operators encoun-
tered fall in the class RCα,β(I), since it amounts to a Hölder regularity of or-
der 1

α in the higher order coefficients and bounded lower-order coefficients. Let
A : I → L(D,X) satisfy assumption (A1) with the decomposition A0 and B, for
the explicit estimate we ask the additional assumptions:

(A2) (a) (α, β)-Hölder regularity: A ∈ RCα,β(I; (D;X)) for some α ≥ 1,
β ≥ 0.

(b) Lαr(I) Integrability condition: There exists r > 1 such that

Γ = Γ(α, r,M0,K0) :=
(∫

I

(K0(t) + 1)αr(M0(t) + 1)αrdt
)1/r

< +∞.

Theorem 2.18. Assume that A satisfies (A1) and (A2). Then there exists a
constant C = C(|I|, ∥A∥∞, p, r, α, β) such that

log
(
[A]MRp(I)

)
≤ C

(( Γ

mδ

)r∗{
1 +mηm

s2(α,β,r)
δ Γs1(α,β,r)s+ log

( Γ

mδ

)}
+mη + 1

)
,

(2.23)

where

s1(α, β, r) := r∗
(β + 1

α
− 1

)
,

s2(α, β, r) :=
β + 1

α+ 1
−

(α+ 1
r

α+ 1

)
s1(α, β, r),

and r∗ := r
r−1 .

Estimate (2.23) is only suitable to evaluate [A]MRp(I) near +∞ with some pre-
cision. Indeed by definition Γ > 0 therefore, the left-hand side cannot be made
smaller than a given positive constant. However the estimate allows the derivation
of asymptotic behavior of the constant in terms of the three quantities mδ,mη and
Γ. We will see that the asymptotic behaviors have practical uses in existence results
for quasilinear equations.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 15

2.4.3. Topologies for the persistence of maximal regularity in limits. Consider a
map of strongly measurable and nonautonomous operators,

A :

{
Λ → L∞(I;L(D,X))
λ 7→ A(λ) (2.24)

and a corresponding map of source terms in Lp(I;X),

F :

{
Λ → Lp(I;X)
λ 7→ F(λ). (2.25)

where (Λ, dΛ) is a metric space, we exhibit topologies in the set L(D,X)I of nonau-
tonomous operators in which continuity of the maps A,F allows the persistence of
maximal regularity upon taking limits; second to understand in which cases the
solution map,

U :

{
Λ → MRp(I;X)
λ 7→ L−1

A(λ)F(λ)
(2.26)

when well-defined, is continuous in either the strong topology or the weak topology
of MRp(I).

Consider now a subset Σ ⊂ Λ such that

∀λ ∈ Σ, A(λ) ∈ MRp(I). (2.27)

To study (Q2), we rely on the following proposition whose proof is rather ele-
mentary and is given in the long version of this work [11].

Proposition 2.19. Assume that K := supλ∈Σ[A(λ)]MRp(I) < +∞ and consider

the following continuity hypothesis for A at a point λ ∈ Λ:

(CVI) A is continuous at λ for the uniform convergence in L∞(I;L(D,X)).
(CVII) A is continuous at λ for the strong pointwise convergence in L(D,X).

The following conclusions hold:

(i) If (CVI) holds at a point λ ∈ Σ, then A(λ) ∈ MRp(I).

(ii) If (CVI) holds at a point λ ∈ Σ, supλ∈Σ ∥A∥L∞(I;L(D,X)) < +∞ and F is

continuous at λ in the weak topology of Lp(I;X) then

U(λ) ⇀
λ→λ

U(λ) weakly in MRp(I). (2.28)

(iii) If (CVII) holds at a point λ ∈ Σ and F is continuous at λ ∈ Σ in the strong
topology of Lp(I;X) then

U(λ) →
λ→λ

U(λ) strongly in MRp(I). (2.29)

As an almost direct consequence of the Arzelà-Ascoli-type result Proposition
2.14 and Theorem 2.16, we find

(HΣ) (a) For each λ ∈ Σ, t ∈ I, A(λ)(t) ∈ MRp(R+) and

sup
λ∈Σ

K0(λ)(t) < +∞. (2.30)

(b) {A(λ)}λ∈Σ is relatively equicontinuous and {A(λ)(t)}λ∈Σ is bounded
in L(D,X) for each t ∈ I.

These assumptions can be used straightforwardly along with Theorem 2.16 to infer
the following uniform boundedness of the maximal regularity constant on Σ:



16 T. BELIN, P. LAFITTE EJDE-2025/18

Proposition 2.20. If A satisfies (HΣ), then {A(λ)}λ∈Σ ⊂ MRp(I) and

sup
λ∈Σ

[A(λ)]MRp(I) < +∞.

In conjunction with the second part of Proposition 2.19, Proposition 2.20 yields
two interesting corollaries pertaining to (Q2). The rather elementary proofs of
these results rely on Proposition 2.14; for the convenience of the reader we chose
to collect them in [11, Appendix C].

Corollary 2.21. If A satisfies (HΣ)(a) and that there exists λ ∈ Λ such that (CVI)
holds, then A(λ) ∈ MRp(I) and (ii) of Proposition 2.19 holds.

Corollary 2.22. If A satisfies (HΣ) and that there exists λ ∈ Λ such that (CVII)
holds, then A(λ) ∈ MRp(I) and (iii) of Proposition 2.19 holds.

2.4.4. Global existence for a quasilinear equation. We state a global existence result
for a class of abstract quasilinear equations. Assume that X (I) ↪→ Lp(I;X) is a
functional space on I, and we make the further assumption that the injection

MRp(I) ↪→ X (I) is continuous and compact.

Since the injection D ↪→ X is compact, we already know that MRp(I) injects in
Lp(I;X) compactly. Hence there exist such spaces X (I), take for example any
real interpolation between Lp(I;X) and MRp(I). We consider the nonlocal-in-time
Cauchy problem for a quasilinear equation

d

dt
u(t) + A(u)(t)u(t) = F(u)(t) t ∈ I

u(a) = x,
(2.31)

where the initial condition x ∈ Trp.
To ensure existence of global strong solutions to (2.31), i.e. solutions u ∈ MRp(I)

which satisfy the equation for a.e. t ∈ I, we introduce the following assumptions

on the maps A(·) : X (I) → L(D;X)I and F : X (I) → Lp(I;X):

(EI) For all R ≥ 0, define the quantities:

γ(R) := sup
∥u∥X(I)=R

[A(u)]MRp(I), (2.32)

κ(R) := sup
∥u∥X(I)=R

{∥A(u)∥∞ + ∥F(u)∥Lp(I;X) + 1} (2.33)

and assume that for any L ≥ 0 the following sublinear growth condition
holds,

sup
R1,R2∈[R−L,R+L]

γ(R1)κ(R2)

R
→

R→+∞
0. (2.34)

(EII) We ask for the following continuity properties:

(i) F :

{
X (I) → Lp(I;X)

u 7→ F(u) is weakly continuous.

(ii) A :

{
X (I) → L∞(I;L(D,X))

u 7→ A(u) is continuous.

The sublinear growth condition stated in (EI) does not allow local singularities
or degeneracies to develop in the operator A and imposes a quite restrictive growth
condition on the source term F. Indeed, in the case of a constant operator A(u) = A,
it is required that F have a strictly sublinear growth.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 17

The statement of the growth condition involves the supremum over the two
variables R1 and R2 in the interval [R− L,R+ L]. This allows to treat any initial
condition x ∈ Trp. One can restrict the range of L > 0 if one has a smallness
assumption on the initial data x. In the case of a homogeneous initial data x = 0, we
can in fact reduce the assumption to L = 0 and take κ(R) := sup∥u∥=R{∥A(u)∥∞+

∥F(u)∥Lp(I;X)} instead.
Compared to Amann’s results [2], the existence of strong solutions does not

require the Lipschitz continuity of the nonlinearities A and F. The method of
proof uses Schauder’s fixed-point theorem contrary to the proof in [2] which uses
Banach’s fixed point. The uniqueness of the solution is thus not guaranteed with
this method.

Theorem 2.23. Assume that A and F satisfy conditions (EI)–(EII). Then for
any x ∈ Trp there exists a global solution u ∈ MRp(I) of (2.31).

This framework is then used to infer quantitative growth conditions on the regu-
larity constant for relatively continuous operators in RCα,β(I) which satisfy (A1)-
(A2). Denote by A0 : u 7→ A0(u) the operator given by the decomposition (A1)(b)
and the subsequent u 7→ K0(u),M0(u) given in (2.16). Naturally this gives rise to
the following maps:

(i) mδ : X (I) → (0, 1],
(ii) mη : X (I) → R+,

(iii) Γ :

{
X (I) → R+

u 7→ ∥(K0(u) + 1)(M0(u) + 1)∥αLαr(I).

Our existence criterion reads as follows.

(E′
I) For any R ≥ 0, define the following quantities:

γlog(R) := sup
∥u∥X(I)=R

[( Γ(u)

mδ(u)

)r∗(
1 +mη(u)mδ(u)

s2Γ(u)s1

+ log
( Γ(u)

mδ(u)

))
+mη(u)

]
,

κlog(R) := sup
∥u∥X(I)=R

{log
(
∥A(u)∥∞ + ∥F(u) + 1∥Lp(I;X)

)
}

and assume that there exists h0 > 0, such that for any L > 0, there exists
R0 ≥ 0 such that for all R ≥ R0 we have

sup
R1,R2∈[R−L,R+L]

γlog(R1) + κlog(R2)

log (R)
≤ 1− h0, (2.35)

where s1(α, β, r) = r∗
(
β+1
α − 1

)
and s2(α, β, r) :=

β+1
α+1 −

(α+ 1
r

α+1

)
s1(α, β, r).

Theorem 2.24. Assume that for each u ∈ X (I), A(u) satisfies (A1) and (A2).
Moreover assume that A,F satisfy (E′

I) and (EII). Then for any x ∈ Trp, there
exists a global solution u to (2.31).

3. Perturbation and asymptotic behavior

We here study the influence of well-chosen perturbations on the maximal regular-
ity constant. First we recall useful notions of Lp maximal regularity for autonomous
operators on unbounded intervals. For such operators, we describe the asymp-
totic behavior of their maximal regularity constant under diagonal perturbations



18 T. BELIN, P. LAFITTE EJDE-2025/18

on bounded intervals. The same prospect is then carried out for nonautonomous
operators, even though much less is known about. Lastly we exhibit the influence
of suitable nonautonomous perturbations on the regularity constant.

3.1. The operator LA := d
dt +A. We assume throughout that A : I → L(D;X)

is a strongly measurable operator. Before our asymptotic study, let us describe the
operators of interest in the Cauchy problem (2.9):

• AI is the multiplication operator associated to A on Lp(I;X); it is defined
on the domain

D(AI) := {u ∈ Lp(I;X) : t 7→ A(t)u(t) ∈ Lp(I;X)}.
By definition,

(
AIu

)
: t 7→ A(t)u(t) for any u ∈ D(AI);

• d
dt is the time-derivative operator defined on the domain of Lp(I;X)

D

(
d

dt

)
=W 1,p

0 (I;X) := {u ∈W 1,p(I;X) : v(a) = 0}.

Their sum is an unbounded operator on Lp(I;X) denoted by

LIA :=
d

dt
+AI

and defined on the domain D(LIA) ⊃ D(AI) ∩ D( d
dt ) = MRp0(I) where MRp0(I) is

defined in (2.4).

Lemma 3.1. A ∈ MRp(I) if and only if LIA is closed on D(LIA) = MRp0(I) and
LIA is invertible in L(MRp0(I);L

p(I;X)). In such a case we have

[A]MRp(I) = ∥(LIA)−1∥L(Lp(I;X);MRp
0(I))

. (3.1)

The proof of the above lemma can be found in [11].
We can concisely describe

MRp(I) := {A : I → L(D;X), ∥ · ∥A(t) ∼ ∥ · ∥D for all t ∈ I, [A]MRp(I) < +∞}.

As before, we shall often write A and LA instead of AI and LIA when the time
interval is clear by context.

3.2. Diagonal perturbations by constants in mrp(I) and MRp(I). Let us
show that the maximal regularity property of the operator on a bounded interval
I is not affected by a constant diagonal perturbation of the form λ Id. We first
show it for autonomous operators in mrp(R+) (Lemma 3.3) with associated decays
and then for nonautonomous operators (Proposition 3.4); in each case we give
asymptotic bounds of [A+ λ Id]mrp(I),MRp(I) when |λ| → +∞.
The autonomous case. Let A ∈ mrp(R+). Recall (see e.g. [24, Proposition 2.2])
that −A is the generator of an analytic semi-group

(
e−tA

)
t≥0

. It means that the

semi-group (e−tA)t≥0 can be analytically extended to a sector

Σθ := {z ∈ C : | arg(z)| < θ} ∪ {0},
θ ∈ (0, π/2] of the complex plane and supz∈Σθ

∥e−zA∥L(X) < +∞.
These facts will prove useful in the study of the resolvent operator Rλ :=

(λ Id+LA)
−1 of −LA for real values of λ, since this operator can be represented

by a Laplace transformation of the semi-group generated by −LA, as stated in the
following result.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 19

Lemma 3.2. If A ∈ mrp(R+), then for any bounded open interval I ⊂ R and
λ ∈ R, λ Id+LIA is invertible in L (Lp(I;X)) and we can express

Rλ =

∫ ∞

0

e−λτe−τLAdτ. (3.2)

The proof of the above lemma can be found in [11]. The main argument that
allows to consider any value of λ is that the semi-group generated by LA is nilpotent.

Using representation (3.2), we further refine our study of Rλ by obtaining bounds
and asymptotic behaviors when λ→ ±∞.

Lemma 3.3. Assume that A ∈ mrp(R+), then for any bounded interval I ⊂ R and
any λ ∈ R,

∥Rλ∥L(Lp(I;X)) ≤M
1− e−λ|I|

λ
(3.3)

and

[A+ λ Id]mrp(I) = ∥Rλ∥L(Lp(I;X);MRp(I)) ≤
(
1 +M |1− e−λ|I||

)
[A]mrp(I), (3.4)

where M := supt≥0 ∥e−tA∥X .

Proof. From the uniform boundedness of ∥e−zA∥L(X) on Σθ, we obtain the following
uniform boundedness (see e.g. [24, Proposition 2.2]),

sup
τ≥0

∥e−τA∥L(Lp(I;X)) ≤ sup
τ≥0

∥e−τA∥L(X) =:M < +∞. (3.5)

From (3.2), since Sτ is a contraction semi-group and vanishes for τ ≥ |I|, we have
for any λ ̸= 0 and v ∈ Lp(I;X), using the triangle inequality and (3.5),

∥Rλv∥Lp(I;X) ≤
∫ |I|

0

e−λτ∥e−τAv∥Lp(I;X))dτ

≤M
(∫ |I|

0

e−λτdτ
)
∥v∥Lp(I;X)

≤M
1− e−λ|I|

λ
∥v∥Lp(I;X)

and the above inequality also holds for λ = 0 if we extend λ 7→ 1−e−λ|I|

λ by conti-

nuity. This yields estimate (3.3). Observe that Rµ = (LA + µ Id)
−1

= L−1
A+µ Id for

any µ ∈ R. Now use the resolvent identity Rλ = R0 − λR0Rλ = R0(Id−λRλ) and
(3.3) to find (3.4). □

The nonautonomous case. Similarly we show that a constant diagonal perturbation
λ Id does not change the maximal regularity property of a nonautonomous opera-
tor on a bounded interval. Moreover we describe the asymptotic behavior of the
maximal regularity constant as the diagonal perturbation goes to ±∞. Parts of the
proof techniques are inspired by the proof of [22, Proposition 2.2].

Theorem 3.4. Let A : I → L(D;X) be a nonautonomous operator on the bounded
open interval I. The following statements are equivalent:

(i) There exists λ0 ∈ R such that A+ λ0 Id ∈ MRp(I).
(ii) A ∈ MRp(I).
(iii) For any λ ∈ R, A+ λ Id ∈ MRp(I).



20 T. BELIN, P. LAFITTE EJDE-2025/18

And if one of these statements holds true, we have for any λ ∈ R,

[A+ λ Id]MRp(I) ≤ cp(λ|I|)[A]MRp(I), (3.6)

where the function cp : R → R+ satisfies

cp(ν) := (|ν|+ κp(ν))
(
1 + epν− − e−pν+

)1/p
(3.7)

with κp(ν) = o±∞(|ν|) and κp(0) = 1. In particular the following asymptotic
behavior is observed for cp:

cp(ν) ∼
ν→−∞

−νe−ν ,

cp(ν) ∼
ν→+∞

ν.

Remark 3.5. Note that the bounds in the autonomous case (3.4) and nonau-
tonomous case (3.6) do not have the same asymptotic behaviors in ν → ±∞ as
they differ by a factor ν. However note that cp(ν) is sharp around 0 since cp(0) = 1.
It is not yet known to the authors whether or not this asymptotic at ±∞ for the
nonautonomous case is sharp or not.

Proof of Proposition 3.4. First see that (iii) implies (ii) implies (i) obviously. Also
for t ∈ [a, b], since ∥·∥A(t)+λ Id ∼ ∥·∥A(t)+µ for any λ, µ ∈ R, we obtain ∥·∥A(t)+λ Id ∼
∥ · ∥D if and only if ∥ · ∥A(t)+µ ∼ ∥ · ∥D.

There only remains to show that (i) implies (iii).

Existence and estimate. Possibly renaming A+λ0 Id as A, let us assume without
loss of generality that λ0 = 0. Also upon performing a time translation, we can
further assume that I = (0, T ), with T = |I|. Let λ ∈ R, f ∈ Lp(0, T ;X). Define
g : t 7→ eλtf(t) ∈ Lp(0, T ;X), because A ∈ MRp, there exists v ∈ MRp(0, T ) such
that

d

dt
v(t) +A(t)v(t) = g(t),

v(0) = 0.
(3.8)

And further for any t ∈ (0, T ], using Proposition 2.10 and (2.10), we find that

∥v∥MRp(0,t) ≤ [A]MRp(0,T )∥g∥Lp((0,t);X). (3.9)

Now let u : t 7→ e−λtv(t) ∈ Lp((0, T );X) and for all t ∈ [0, T ] the integrand of the
norm of MRp(a, b) defined in (2.2) is

[u]p(t) := ∥u(t)∥pX + ∥u(t)∥pD + T p
∥∥∥ d

dt
u(t)

∥∥∥p
X

∈ L1((0, T );R+).

Note that by convexity of r 7→ rp, for any µ ∈ (0, 1),∥∥∥ d

dt
u(t)

∥∥∥p
X

≤
(
∥λe−λtv(t)∥X + ∥e−λt d

dt
v(t)∥X

)p
=

(
µ
|λ|∥v(t)∥X

µ
+ (1− µ)

∥ d
dtv(t)∥X
1− µ

)p
e−pλt

≤
(
µ1−p|λ|p∥v(t)∥pX + (1− µ)1−p

∥∥∥ d

dt
v(t)

∥∥∥p
X

)
e−pλt.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 21

Therefore we have the bound

[u]p(t)

≤
( (

1 + µ1−pT p|λ|p
)
∥v(t)∥pX + ∥v(t)∥pD + (1− µ)1−pT p

∥∥∥ d

dt
v(t)

∥∥∥p
X

)
e−pλt

≤ Rp(µ, λT )e
−pλt[v]p(t),

(3.10)

where Rp(µ, ν) :=
(
1 + µ1−p|ν|p

)
∨ (1−µ)1−p. Performing an integration by parts,

we obtain the following if λ ̸= 0,∫ T

0

e−pλt[v]p(t)dt

= e−pλT
∫ T

0

[v]p(t)dt+ pλ

∫ T

0

e−pλt
∫ t

0

[v]p(t)drdt︸ ︷︷ ︸
≤0 if λ<0

≤ [A]pMRp

(
e−pλT ∥g∥pLp((0,T );X) + pλ+

∫ T

0

e−pλt
∫ t

0

∥g(r)∥pXdrdt
)

≤ [A]pMRp

(
e−pλT ∥g∥pLp((0,T );X) + pλ+

∫ T

0

∥f(r)∥pX
∫ T

r

e−pλ(t−r)dtdr
)

≤ [A]pMRp

(
epλ−T ∥f∥pLp((0,T );X) +

pλ+
pλ

∫ T

0

∥f(r)∥pX(1− e−pλ(T−r))dr
)

≤ [A]pMRp

(
epλ−T ∥f∥pLp((0,T );X) + (1− e−λ+pT )∥f∥pLp((0,T );X)

)
≤ [A]pMRp

(1 + eλ−pT − e−λ+pT )∥f∥pLp((0,T );X),

where λ+ := λ∨0 and λ− := (−λ)∨0. Note that this final inequality also obviously
holds if λ = 0. Now combining with (3.10) we find that for any µ ∈ (0, 1),

∥u∥MRp(0,T ) ≤ Rp(µ, λT )
1/p

(
1 + eλ−pT − e−λ+pT

)1/p
[A]MRp

∥f∥Lp(I;X). (3.11)

From (3.11) we infer that u ∈ MRp(I) and solves

d

dt
u(t) +A(t)u(t) + λu(t) = f(t), t ∈ I,

u(0) = 0.
(3.12)

Uniqueness. If ũ ∈ MRp(I) also solves (3.12) then ṽ : t 7→ eλtũ(t) ∈ MRp(I) must
also solve (3.8) and by unique solvability, v = ṽ hence u = ũ.

Optimization of Rp(µ, ν) For any ν ̸= 0, there exists a unique minimum µν :=
argmin
µ∈(0,1)

Rp(µ, ν). Indeed any minimizer µν of Rp(·, ν) is the unique zero of the

continuous and strictly decreasing function µ 7→ 1 + µ1−p|ν|p − (1 − µ)1−p which
diverges to +∞ and −∞ at µ = 0 and µ = 1 respectively. Moreover ν 7→ µν is
increasing on ν ∈ R+.

Since (µν)ν>0 is bounded from below, by monotonicity, there exists µ0 ∈ [0, 1)
such that µν →

ν→0
µ0. Assume by contradiction that µ0 > 0 then we see that upon

taking the limit ν → 0, in the identity 0 = 1 + µ1−p
ν νp − (1− µν)

1−p, we obtain

0 = 1− (1− µ0)
1−p,

1 = (1− µ0)
1−p,



22 T. BELIN, P. LAFITTE EJDE-2025/18

µ0 = 0.

We find a contradiction, meaning µ0 = 0. Hence αp(ν) := Rp(µν , ν)
1/p = (1 −

µν)
1−p
p →

ν→0
1. Since (µν)ν>0 is bounded from above, again by monotonicity, there

exists µ∞ ∈ (0, 1] such that µν →
ν→+∞

µ∞. And observe that

(1− µν)
1−p = 1 + µ1−p

ν νp,

(1− µν)
1−p →

ν→+∞
+∞.

Meaning µ∞ = 1. We then infer that αp(ν) =
(
1 + µ1−p

ν νp
)1/p ∼

ν→+∞
µ

1−p
p

ν ν ∼
ν. □

We will use this result later to estimate the effect of well-suited nonautonomous
perturbations on the maximal regularity constant.

3.3. Nonautonomous perturbations of autonomous operators in mrp. Let
us state the following perturbation result, also found in [8, Proposition 2.7]. The
novelty being that a quantitative estimate of the maximal regularity constant is
given.

Theorem 3.6. Let A0 ∈ mrp(R+) and B : I → L(D;X) be strongly measurable on
the bounded interval I. Assume there exists η ≥ 0, such that for any t ∈ I, x ∈ D,

∥B(t)x∥X ≤ ϵ0∥x∥D + η∥x∥X ,

where ϵ0 := 1
2([A0]mrp(I)+1)(M0+1) and M0 = supτ≥0 ∥e−τA0∥X . Then A0 + B ∈

MRp(I) and we have the bound

[A0 +B]MRp(I) ≤ G(|I|, η,M0, [A0]mrp(I)), (3.13)

where G(τ, η,M,K) := 4(1 +M)cp(−4Mητ)K for (τ, η,M,K) ∈ (R+)
4 and cp is

given by (3.7).
In such a case we call B a regular perturbation of A0.

Proof. First see that for any t ∈ I, ∥ · ∥A0+B(t) ∼ ∥ · ∥A0
∼ ∥ · ∥D, since ϵ0 ≤ 1

2 .

As before, denote for λ ∈ R, Rλ := (λ Id+LA)
−1 and observe that λ Id+LA0+B =

(Id+BRλ)(λ Id+LA) where B is the multiplication operator on Lp(I;X) associated
with B, this operator is well-defined by strong measurability of t 7→ B(t). Now for
λ > 0 and using the bounds (3.3) and (3.4) of Lemma 3.3 compute the following,

∥BRλf∥Lp(I;X) ≤ ϵ0∥Rλf∥Lp(I;D) + η∥Rλf∥Lp(I;X)

≤ 1

2
(
[A0]mrp(I) + 1

)
(M0 + 1)

∥Rλf∥MRp(I) + η∥Rλf∥Lp(I;X)

≤ 1

2
∥f∥Lp(I;X) + ηM

1− e−λ|I|

λ
∥f∥Lp(I;X)

≤
(1
2
+
ηM0

λ

)
∥f∥Lp(I;X).

Fix λ0 := 4ηM for which we then have

∥BRλ0
f∥Lp(I;X) ≤

3

4
∥f∥Lp(I;X).



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 23

Hence Id+BRλ0
is an invertible operator of Lp(I;X), and so

LA0+B+λ0 Id = (Id+BRλ0
)(λ0 Id+LA0

)

is invertible as an operator of L(MRp(I), Lp(I;X)). By (3.4) in Lemma 3.1 we infer
that A0 +B + λ0 Id ∈ MRp(I) with

[A0 +B + λ0 Id]MRp(I) ≤
(
1− 3

4

)−1

(1 +M)[A0]mrp(I) = 4(1 +M)[A0]mrp(I).

Apply Proposition 3.4 with −λ0 to obtain that A0+B = (A0+B+λ0 Id)−λ0 Id ∈
MRp(I) with the desired bound (3.13). □

This perturbation result is key to the analysis of the maximal regularity property
of nonautonomous operators which are relatively continuous.

4. Gluing maximally regular operators

As a tool for the quantitative study of regularity constant of relatively continuous
operators, let us derive an estimate concerning the gluing of finite family (Ai)1≤i≤N
of maximally regular operators defined on a subdivision of the bounded time interval
I = (a, b).

4.1. Behavior of [·]MRp(I) under a change of variable in time. To refine the
bounds on the maximal regularity constants obtained in the next section, it is useful
to study the behavior of [·]MRp(I) with respect to a change of variables in time. An
invariance stemming from the choice of the norm of MRp(I) (2.2) is observed when
the change of variables is linear (see Remark 4.2 later).

Lemma 4.1. Let I, J ⊆ R be two bounded open intervals and ϕ : I → J be a C1-
diffeomorphism and denote ψ := ϕ−1. Let A : J → L(D,X) ∈ MRp(J), then the
nonautonomous operator Aϕ defined by

Aϕ = ϕ′(·) (A ◦ ϕ) (·) : I → L(D,X)

belongs to MRp(I) and we have,

[Aϕ]MRp(I) ≤ |ψ′|∞ max
(
1,

|I|
|J |

|ϕ′|∞
)
[A]MRp(J). (4.1)

Proof. Before proving inequality (4.1) we first estimate the continuity constant of
the isomorphism ιϕ : u 7→ u ◦ ϕ from MRp(J) to MRp(I) induced by ϕ.

Let u ∈ MRp(J) and let us estimate ∥ιϕ(u)∥MRp(I).

Estimation of the Lp(I;X) and Lp(I;D) norms. We compute

∥u ◦ ϕ∥pLp(I;X) =

∫
I

∥u(ϕ(t))∥pXdt

=

∫
J

∥u(s)∥pX |ψ′(s)|ds

≤ |ψ′|∞∥u∥pLp(J;X).

(4.2)

Applying the previous computations with the Banach spaceD, we obtain the similar
result

∥u ◦ ϕ∥pLp(I;D) ≤ |ψ′|∞∥u∥pLp(J;D). (4.3)



24 T. BELIN, P. LAFITTE EJDE-2025/18

Estimation of the W 1,p(I;X) semi-norm.

|I|p∥ d

dt
(u ◦ ϕ) ∥pLp(I;X) = |I|p

∫
I

∥ϕ′(t)
( d

ds
u
)
(ϕ(t))∥pXdt

= |I|p
∫
J

|ϕ′(ψ(s))|p
∥∥∥ d

ds
u(s)

∥∥∥p
X
|ψ′(s)|ds

≤ |ψ′|∞
(
|I|
|J |

|ϕ′|∞
)p

|J |p∥ d

ds
u∥pLp(J;X).

(4.4)

perator norm of ιϕ. By combining inequalities (4.2), (4.3) and (4.4) we obtain a
bound on the operator norm ∥ιϕ∥L(MRp(J),MRp(I)) of ιϕ:

∥ιϕ∥ ≤ |ψ′|1/p∞ max
(
1,

|I|
|J |

|ϕ′|∞
)
. (4.5)

Inequality (4.5) becomes an identity in the case of affine homomorphisms (i.e. ϕ of

the form ϕ(t) = λt+ t0), since inequalities (4.2) - (4.4) saturate and |I|
|J| |ϕ

′|∞ = 1.

Remark that if ϕ′ ≥ 0, then ιϕ also induces an isomorphism of MRp0(J) onto MRp0(I)
with the same operator norm.

Maximal regularity of Aϕ. Denote aI := inf I, aJ := inf J and let f ∈ Lp(I;X).
By the isomorphism ιϕ, the unique solvability in MRp0(I) of the nonautonomous
Cauchy problem

d

dt
v +Aϕv = f, in I,

v(aI) = 0
(4.6)

is equivalent to the unique solvability in MRp0(J) of

d

ds
u+Au =

1

ϕ′ ◦ ψ
f ◦ ψ, in J,

u(aJ) = 0.
(4.7)

Since

∥ 1

ϕ′ ◦ ψ
f ◦ ψ∥Lp(J;X) ≤ |ψ′|1−

1
p

∞ ∥f∥Lp(I;X) < +∞,

the assumption A ∈ MRp(J) shows existence and uniqueness of v ∈ MRp0(I)
solution of (4.6). Let us then estimate [Aϕ]MRp(J):

∥v∥MRp(I) ≤ ∥ιϕ∥∥u∥MRp(J)

≤ ∥ιϕ∥[A]MRp(J)∥
1

ϕ′ ◦ ψ
f ◦ ψ∥Lp(J,X)

≤ ∥ιϕ∥[A]MRp(J)|ψ
′|1−

1
p

∞ ∥f∥Lp(I;X)

≤ |ψ′|∞ max
(
1,

|I|
|J |

|ϕ′|∞
)
[A]MRp(J)∥f∥Lp(I;X).

From which we infer (4.1). □

As a direct application of Lemma 4.1 we derive the two following important
remarks for any A ∈ MRp(J):



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 25

Remark 4.2. (1) Note that (Aϕ)ψ = A, hence using (4.1) with ψ we obtain a

lower bound for [Aϕ]MRp(I) as

1

|ϕ′|∞
min

(
1,

|I|
|J |

1

|ψ′|∞

)
[A]MRp(J) ≤ [Aϕ]MRp(I). (4.8)

(2) Let ϕ : I 7→ J be a linear homomorphism, denote ϕ′ = λ := |J|
|I| and

Aλ := Aϕ. Using both (4.1) and (4.8) we find

1

λ
[A]MRp(J) ≤ [Aλ]MRp(I) ≤

1

λ
[A]MRp(J),

yielding

[Aλ]MRp(I) =
1

λ
[A]MRp(J). (4.9)

4.2. Gluing finitely many MRp operators. We here show that finitely many
adjacent maximally regular operators can be glued-up together to form a maximally
regular operator on the joined interval. The results are all stated as quantitatively
as possible with the tools developed before. The estimations here described are
new in the theory of Lp maximal regularity.

A first lemma is derived on the gluing of functions in MRp. Working with the
weighted W 1,p semi-norm in (2.2) yields nonstandard norm behavior with respect
to restriction and extension of functions in MRp.

Lemma 4.3. Let v1 ∈ MRp(a, b), v2 ∈ MRp(b, c), with a < b < c ∈ R and assume
that v1(b) = v2(b). Then v := v11[a,b) + v21[b,c] ∈ MRp(a, c) and we have

∥v∥MRp(a,c) ≤ κ1∥v1∥MRp(a,b) + κ2∥v2∥MRp(b,c), (4.10)

where κ1 = c−a
b−a , κ2 := c−a

c−b .

Proof. Let ξ ∈ C1
c ((a, c);R), note that∫ c

a

( d

dt
ξ(t)

)
v(t)dt

=

∫ b

a

( d

dt
ξ(t)

)
v1(t)dt+

∫ c

b

( d

dt
ξ(t)

)
v2(t)dt

= [ξ(t)v1(t)]
b
a −

∫ b

a

ξ(t)
d

dt
v1(t)dt+ [ξ(t)v2(t)]

c
b −

∫ c

b

ξ(t)
d

dt
v2(t)dt

= ξ(b)v1(b)− ξ(b)v2(b)︸ ︷︷ ︸
=0

−
∫ c

a

ξ(t)
(( d

dt
v1(t)

)
1[a,b)(t) +

( d

dt
v2(t)

)
1[b,c](t)

)
dt

= −
∫ c

a

ξ(t)
(( d

dt
v1(t)

)
1[a,b)(t) +

( d

dt
v2(t)

)
1[b,c](t)

)
dt.

Therefore
d

dt
v =

( d

dt
v1

)
1[a,b] +

( d

dt
v2

)
1[b,c] ∈ Lp((a, c);X).

Since also v ∈ Lp((a, c);D) we infer that v ∈ MRp(a, c). Furthermore observe that

∥v∥pMRp(a,c) = ∥v∥pLp((a,c);X) + ∥v∥pLp((a,c);D) + (c− a)p∥ d

dt
v∥pLp((a,c);X)

= ∥v1∥pLp((a,b);X) + ∥v2∥pLp((b,c);X) + ∥v1∥pLp((a,b);D) + ∥v2∥pLp((b,c);D)



26 T. BELIN, P. LAFITTE EJDE-2025/18

+ (c− a)p
(
∥ d

dt
v1∥pLp((a,b);X) + ∥ d

dt
v2∥pLp((b,c);X)

)
≤

(
c− a

b− a

)p
∥v1∥pMRp(a,b) +

(
c− a

c− b

)p
∥v2∥pMRp(b,c).

Now see that (αp + βp)
1/p ≤ α+ β for any α, β ≥ 0 to conclude (4.10). □

As a first step, we glue two operators together and then proceed by induction
to show maximal regularity and bound on the regularity constant for finitely many
operators. Some asymptotic behaviors are discussed as well as the sharpness of
obtained estimate.

Lemma 4.4. Assume that A1 ∈ MRp(a, b) and A2 ∈ MRp(b, c) for some a < b <
c ∈ R. Then the glued-up operator A defined by

A(t) :=

{
A1(t), a ≤ t < b,

A2(t), b ≤ t ≤ c

belongs to MRp(a, c) and we have

[A]MRp(a,c) ≤ κ1[A1] + κ2[A2] + κ
1/p
1 κ

1/q
2 Qp (c− b, ∥A2∥∞, [A2]) [A1], (4.11)

where κ1 := c−a
b−a and κ2 := c−a

c−b . Moreover denoting wp := 21/q(p− 1) we have

Qp(T,C,G) :=


1 + 21/q

(
CG ∨ G

T

)
if wp

G
T ≤ 1,

2
1
pq p

(p−1)1/q

(
G
T

)1/p
if 1 ≤ wp

G
T ≤ 1

CT ,(
1
CT ∨ 1

)1/p (
1 + 21/qCG

)
if 1

CT ≤ wp
G
T .

(4.12)

Proof. * We remark that A obviously satisfies (nonaut-i) by maximal regularity of
A1 and A2.

* Let f ∈ Lp((a, c);X), let us show that there exists a unique v ∈ MRp(a, c)

satisfying (2.9)
(a,c)
0,f and that (2.8) holds.

Uniqueness. Assume that v is a solution to (2.9)
(a,c)
0,0 . It implies that v|[a,b] is a

solution of the Cauchy problem with operator A1 and source term 0, hence v = 0
on [a, b]. Now this implies that v|[b,c] starts at 0 and is a solution of the Cauchy
problem with operator A2 and source term 0, again v = 0 on [b, c].

Existence and initial estimate. Denote f1 := f |[a,b] ∈ Lp((a, b);X), f2 =
f |[b,c] ∈ Lp((b, c);X). For convenience of notations let us set [A1] := [A1]MRp(a,b)

and [A2] := [A2]MRp(b,c).
Since A1 ∈ MRp(a, b), there exists a (unique) solution v1 ∈ MRp(a, b), solution

of the Cauchy problem starting at 0 with source term f1 and such that

∥v1∥MRp(a,b) ≤ [A1]∥f1∥Lp((a,b);X). (4.13)

Denote x = v1(b) ∈ Trp. Fix µ ≥ 1 and let z ∈ {w ∈ MRp(b, b+µ(c−b)) : w(b) = x},
which is nonempty since x ∈ Trp. By definition, LA2

z|(b,c) ∈ Lp((b, c);X), let us
then denote gz := f2−LA2z|[b,c] ∈ Lp((b, c);X). Since A2 ∈ MRp(b, c) there exists
vz ∈ MRp(b, c) such that

d

dt
vz(t) +A2(t)vz(t) = gz(t), a.e. t ∈ (b, c),

vz(b) = 0.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 27

And by linearity v2 := vz + z|[b,c] must be the (unique) solution to

d

dt
v2(t) +A2(t)v2(t) = f2(t), a.e. t ∈ (b, c),

v2(b) = x.

Now by maximal regularity of A2 we infer that

∥vz∥MRp(b,c) ≤ [A2]∥g∥Lp((b,c);X),

≤ [A2]∥f2∥Lp((b,c);X) + [A2]∥LA2
z∥Lp((b,c);X).

Observe that

∥LA2z∥
p
Lp((b,c);X)

=

∫ c

b

∥∥∥ d

dt
z(t) +A2(t)z(t)

∥∥∥p
X
dt

≤ 2p−1
(∫ c

b

∥∥∥ d

dt
z(t)

∥∥∥pdt+ Cp
∫ c

b

∥z(t)∥pDdt
)

≤ 2p−1
(∫ b+µ(c−b)

b

∥∥∥ d

dt
z(t)

∥∥∥pdt+ Cp
∫ b+µ(c−b)

b

∥z(t)∥pDdt
)

≤ 2p−1
(
Cp ∨ 1

µp(c− b)p

)
∥z∥pMRp(b,b+µ(c−b)),

(4.14)

according to (2.2) of Theorem 2.1 and where C := ∥A2∥L∞((b,c);L(D,X)) < +∞.
Therefore we can infer the following bound using (4.14):

∥v2∥MRp(b,c) ≤ ∥vz∥MRp(b,c) + ∥z∥MRp(b,c)

≤ [A2]∥f2∥Lp((b,c);X) + 21/q
(
C ∨ 1

µ(c− b)

)
[A2]∥z∥MRp(b,b+µ(c−b))

+ ∥z∥MRp(b,b+µ(c−b))

≤ [A2]∥f2∥Lp((b,c);X)

+
(
1 + 21/q

(
C ∨ 1

µ(c− b)

)
[A2]

)
∥z∥MRp(b,b+µ(c−b)).

Passing to the infimum over z ∈ {w ∈ MRp(b, b+ µ(c− b)) : w(b) = x}, we find

∥v2∥MRp(b,c)

≤
(
1 + 21/q

(
C ∨ 1

µ(c− b)

)
[A2]

)
∥x∥Trp,(b,b+µ(c−b)) + [A2]∥f2∥Lp(b,c;X).

Using the rescaling of the norms on the trace space described in Remark 2.2 together
with (4.13), we find

∥x∥Trp,(b,b+µ(c−b)) = (µκ∗)
1/p ∥x∥Trp,(a,b)

≤ (µκ∗)
1/p ∥v1∥MRp(a,b)

≤ (µκ∗)
1/p

[A1]∥f1∥Lp((a,b);X),

where κ∗ := c−b
b−a , therefore we finally obtain

∥v2∥MRp(b,c) ≤ (µκ∗)
1/p

(
1 + 21/q

(
C ∨ 1

µ(c− b)

)
[A2]

)
[A1]∥f1∥Lp((a,b);X)

+ [A2]∥f2∥Lp((b,c);X).
(4.15)



28 T. BELIN, P. LAFITTE EJDE-2025/18

Define the function v := v11[a,b) + v21[b,c]. Since v1(b) = v2(b), we infer from
Lemma 4.3 that v ∈ MRp(a, c). Further we observe that v is in fact a solution to
(2.9) for a.e. t ∈ [a, c]. Moreover recall that

∥v∥MRp(a,c) ≤ κ1∥v1∥MRp(a,b) + κ2∥v2∥MRp(b,c), (4.16)

where κ1 = c−a
b−a and κ2 = c−a

c−b . We remark that 1
κ1

+ 1
κ2

= 1 and κ∗ = κ1

κ2
. Hence

assembling (4.15) and (4.16), and the fact that ∥f1∥Lp((a,b);X), ∥f2∥Lp((b,c);X) ≤
∥f∥Lp((a,c);X):

∥v∥MRp(a,c) ≤
(
κ1[A1] + κ2

(
(µκ∗)

1/p
(
1

+ 21/q
(
C ∨ 1

µ(c− b)

)
[A2]

)
[A1] + [A2]

))
∥f∥Lp((a,c);X).

By remarking that κ2(κ∗)
1/p = κ

1/p
1 κ

1/q
2 we deduce the following estimate of

[A]MRp(a,c):

[A]MRp(a,c)

≤ κ1[A1] + κ2[A2] + κ
1/p
1 κ

1/q
2 µ1/p

(
1 + 21/q

(
C ∨ 1

µ(c− b)

)
[A2]

)
[A1].

(4.17)

Optimization with µ. We denote Q̃p(T,C,G, µ) := µ1/p
(
1 + 21/q

(
C ∨ 1

µT

)
G
)
.

Let G,C, T be given and let us minimize µ 7→ P (µ) := Q̃p(T,C,G, µ) on [1,+∞).
Remark that P (µ) = α(µ) ∨ β(µ), where

α(µ) := µ1/p
(
1 + 21/qCG

)
is increasing and

β(µ) := µ1/p
(
1 +

21/qG

µT

)
.

We note that µ 7→ β(µ) admits a unique global minimum on R∗
+ attained at µ1 :=

(p−1)21/q GT =: wp
G
T , because β is decreasing on (0, µ1] and increasing on [µ1,+∞).

Moreover observe that α and β both meet at a unique point µ2 = 1
CT . Therefore

the global minimum of P (µ) on R∗
+ is attained at µ∗ := µ1 ∧ µ2. We can therefore

infer that the minimum of P on [1,+∞) is attained at 1∨
(

1
CT ∧ wp GT

)
which yields

our result. □

Remark 4.5. Note that in (4.11), the roles of A1 and A2 are not symmetric,
this accounts for the fact that the maximal regularity constant is defined with the
initial condition x = 0. It is worth noting that it is the multiplicative interaction
between [A1] and [A2] of the term qκ1κ2Qp(c − b, C, [A2])[A1] which gives rise to
the asymmetry.

It might seem possible to further refine (4.11) by exploiting a piecewise linear
change of variable ϕ such that |ϕ((a, b))| = |ϕ((b, c))| so as to obtain optimality on
the factors κ1 and κ2. Unfortunately the factor in (4.1) cancels out the optimality.

We remark that for any λ > 0, C, T,G ∈ R+, we have the invarianceQp(C, T,G) =

Qp(λC,
T
λ ,

G
λ ). This invariance is compatible with a linear change of variables in

time (see Remark 4.2 (4.9)). This then implies that (4.11) is invariant with respect
to a linear change of variables in time.



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 29

Now we derive the result for the glued-up operator of a finite family of nonau-
tonomous operators. We first define the following geometrical parameters related
to the subdivision.

Definition 4.6. Let N ≥ 2 and TN = {τi}0≤i≤N be such that a = τ0 < τ1 < · · · <
τN = b, define for each j ∈ {2, . . . , N} the function Hj(p, T , ·, ·) : R+ × R+ → R+

as

Hj(p, T ,C,G) := κ1,j + κ
1/p
1,j κ

1/q
2,j Qp (τj − τj−1,C,G) , (4.18)

where κ1,j :=
τj−τ0
τj−1−τ0 , κ2,j :=

τj−τ0
τj−τj−1

.

We now give an estimate of the MRp constant for the glueing of finitely many
operators in terms of these geometrical parameters.

Theorem 4.7. Let T = {τi}0≤i≤N be any finite subdivision of I. Let us be given
a family {Ai}1≤i≤N of nonautonomous operators such that Ai ∈ MRp(τi−1, τi).
Then the operator A defined by

A(t) :=


A1(t), if τ0 ≤ t < τ1,

A2(t), if τ1 ≤ t < τ2,

. . .

AN (t), if τN−1 ≤ t ≤ τN

belongs to MRp(a, b) and we have

[A]MRp(a,b)

≤
N∑
i=1

τj − τ0
τj − τj−1

( N∏
j=i+1

Hj(p, T , ∥Aj∥∞, [Aj ])
)
[Ai]MRp(τi−1,τi).

(4.19)

Proof. We show the estimate by induction on N . * For N = 1, the result is trivial.
* Fix some N ≥ 2 and assume that the result holds for any subdivision of size

N − 1, let us show that the result holds for any subdivision of size N .
Let a, b ∈ R, a < b, T := {τi}0≤i≤N be a subdivision of size N and let {Ai}1≤i≤N

be a family of maximally regular nonautonomous operators on each [τi−1, τi] respec-
tively. Denote by A the resulting glued-up operator on [a, b].

We denote B1 := A|[τ0,τN−1) and B2 := AN . By our induction hypothesis, we
are able to apply Lemma 4.4 on B1 and B2. We obtain

[A]MRp(a,b) ≤
(
κ1,N + κ

1/p
1,Nκ

1/q
2,NQp (τN−1 − τN , ∥B2∥∞, [B2])

)
[B1]MRp(τ0,τN−1)

+ κ2,N [B2]MRp(τN−1,τN )

= HN (p, T , ∥B2∥∞, [B2])[B1]MRp(τ0,τN−1) + κ2,N [B2]MRp(τN−1,τN ).

Let us then apply our induction hypothesis on the subdivisions T ′ = {τi}0≤i≤N−1

and the corresponding family of operators {Ai}1≤i≤N−1 glueing-up to B1 and each
of size N − 1. For simplicity denote Hj := Hj(p, T , ∥Aj∥∞, [Aj ]MRp(τj−τj−1)). We
infer that

[A]MRp(a,b) ≤ HN

N−1∑
i=1

( N−1∏
j=i+1

Hj

)
κ2,i[Ai]MRp(τi−1,τi) + κ2,N [AN ]MRp(τN−1,τN )



30 T. BELIN, P. LAFITTE EJDE-2025/18

=

N−1∑
i=1

( N∏
j=i+1

Hj

)
κ2,i[Ai]MRp(τi−1,τi) + κ2,N [AN ]MRp(τN−1,τN )

=

N∑
i=1

( N∏
j=i+1

Hj

)
κ2,i[Ai]MRp(τi−1,τi).

This completes our proof. □

5. Proofs of the main estimates and examples

In this section we prove our main estimation results. The general estimate
Theorem 2.16 is derived in Section 5.1. It is then used in Section 5.2 to show
the quantitative version Theorem 2.18 for operators in RCα,β(I). Finally Section
5.3 contains some examples.

5.1. Estimation for general relatively continuous operators. The idea for
the study of the maximal regularity of relatively continuous operators, is to subdi-
vide the time interval [a, b] and consider the operator on each subinterval Ii ⊂ [a, b]
as the sum of an autonomous maximal operator Ai and a regular perturbation B
arising from the relative continuity.

We start with a somewhat precise version of the Besicovitch covering theorem
in dimension 1 (found e.g. in [7, Theorem 2.18]).

Lemma 5.1. Let ρ : [a, b] → (0,+∞) be a positive function, then there exists
N = N(ρ) ∈ N, a set of center points C := C(ρ) = {ti}1≤i≤N ⊂ [a, b] and a
subdivision of [a, b], T := T (ρ) = {τi}0≤i≤N such that

a = τ0 ≤ t1 < τ1 < t2 < . . . τN−1 < tN ≤ τN = b, (5.1)

such that for all i ∈ {1, . . . , N},

|ti − τi|, |ti − τi−1| ≤ ρ(ti), (5.2)

ρ(ti) ≤ |τi − τi−1| ≤ 2ρ(ti). (5.3)

Hence we can write

[a, b] = ∪Ni=1[τi−1, τi].

Proof. We have [a, b] ⊂ ∪t∈[a,b]It, It := (t − ρ(t), t + ρ(t)). By compactness there
exists a finite family of intervals F ⊂ {It}t∈[a,b] covering [a, b].

Claim: there exists a subcover F̃ ⊂ F which satisfies both of the following nonover-
lapping properties:

(P1) For any I, J ∈ F̃ , I ⊂ J ⇒ I = J .

(P2) For any I, J,K ∈ F̃ , I ∩ J ∩K ̸= ∅ ⇒ I = J or J = K or K = I.

We show the claim by induction on the size N of F .

Base. If N = 1, the result is trivial.

Induction. Assume the claim holds for some N ≥ 1. Let F be a covering family
of size N + 1. If F already satisfies (P1) and (P2) we are done. Otherwise, either
(P1) or (P2) fail to be satisfied.

Case 1: Assume that (P1) fails for F . There exist I, J ∈ F such that I ⊂ J and
I ̸= J . Observe then that F \ {I} still covers [a, b] and is of size N .



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 31

Case 2: Assume now that (P2) fails for F . There exist I, J,K ∈ F , all distinct such
that I ∩ J ∩K ̸= ∅. Write I = (aI , bI), J = (aJ , bJ) and K = (aK , bK) and
assume without loss of generality that aI ≤ aJ ≤ aK .
⋆ If bK ≤ bI or bK ≤ bJ then we conclude that K ⊂ I or K ⊂ J hence
F \ {K} covers [a, b] and is of size N .

⋆ Otherwise if bK > bI and bK > bJ , then since I ∩ K ̸= ∅, aK < bI .
Thus we have I ∪K = (aI , bK) and so J ⊂ I ∪K. Therefore F \ {J}
covers [a, b] and is of size N .

In any case we can cover [a, b] with a subfamily F̃ ⊂ F of size N . The induction
hypothesis allows to conclude.
⋆ We still denote by F = {Iti}1≤i≤N some finite subcover of [a, b] satisfying (P1)
and (P2). Let us assume furthermore that (ti)1≤i≤N are ordered in an increasing
fashion, i.e. a ≤ t1 < t2 < · · · < tN ≤ b and define τ0 = a, τN = b and τi :=
ti+1+ti

2 + ρ(ti)−ρ(ti+1)
2 for i ∈ {0, . . . , N − 1}. From the properties (P1) and (P2),

we infer for any i ∈ {1, . . . , N − 1} that

ρ(ti)− ρ(ti+1) < ti+1 − ti < ρ(ti) + ρ(ti+1),

ti+1 − ρ(ti+1) > ti−1 + ρ(ti−1).

From these we straightforwardly infer (5.1) and (5.3). □

Let us now state and prove an explicit version of the result of Arendt (found in
[8, Theorem 2.7, Theorem 2.11]). First let us recall the quantities of interest and
the main assumption of the theorem.

(A1) (a) Relative continuity: A ∈ RC(I; (D;X)).
(b) Decomposition: There exist A0 : I → L(D,X) and B : I × I →

L(D,X) such that
(b.1) For all t, s ∈ I, A(s) = A0(t) +B(t, s).
(b.2) For all t ∈ I, A0(t) ∈ mrp(R+).
(b.3) There exists a range rA = (δA, ηA) such that for all x ∈ D,

∀t, s ∈ I, |t− s| ≤ ρA(t) =⇒ ∥B(t, s)x∥X ≤ ϵ0(t)∥x∥D + µA(t)∥x∥X , (5.4)

where ρA := δA ◦ ϵ0, µA := ηA ◦ ϵ0 and ϵ0 is defined through
(2.16)-(2.17).

Recall also the auxiliary quantities K0 and M0 for A0 defined in (2.16).

Theorem 5.2. Assume that A satisfies (A1). Then A ∈ MRp(I) and there exists

a subdivision T := {τi}0≤i≤N of I and center points C := {ti}1≤i≤N such that

a = τ0 < t1 < τ1 < · · · < tN < τN = b, (5.5)

ρA(ti) ≤ τi − τi−1 ≤ 2ρA(ti), ∀i ∈ {1, . . . , N} (5.6)

and there exists K := K(p, T , C,K0,M0, ∥A∥∞) ≥ 0 such that

[A]MRp(I) ≤ K. (5.7)

Proof. We apply Lemma 5.1 with ρA = δA ◦ϵ0 on [a, b]. This yields the correspond-
ing N ∈ N, C := (ti)1≤i≤N and T := (τi)0≤i≤N .

We define Ai := A|τi−1,τi for each i ∈ {1, . . . , N}. By the decomposition as-
sumption (b), we have Ai(s) = A0(ti) + B(ti, s) and by (5.2), s 7→ B(ti, s) is a



32 T. BELIN, P. LAFITTE EJDE-2025/18

regular perturbation of A0(ti) on the interval [τi−1, τi]. Hence by Theorem 3.6 we
have Ai ∈ MRp(τi−1, τi) and

[Ai]MRp(τi−1,τi) ≤ G(τi − τi−1, µA(ti),M0(ti),K0(ti)).

Finally we can apply Theorem 4.7 to A with the subdivision T to obtain (5.7). □

5.2. Estimation for Hölder-type relatively continuous operators. To esti-
mate the Lp regularity constant more explicitly, we use an integrability condition
on the mapsK0 andM0 stated in (A2) (b): (K0+1)(M0+1) ∈ Lαr(I). This implies
in fact that 1

ρA
∈ Lr(I). Under this condition, the Besicovitch covering theorem

can be explicited with concrete constants with the use of Chebyshev inequalities.

Proposition 5.3. Let ρ : I → R+ be measurable and such that 1
ρ ∈ Lr(I) for some

r ∈ (1,+∞]. Then there exist (N, {ti}1≤i≤N , {τi}0≤i≤N ) satisfying (5.1) and such
that ⌈1

2
|I|

∥∥∥1
ρ

∥∥∥r∗
Lr(a,b)

⌉
≤ N ≤

⌊
|I|

∥∥∥1
ρ

∥∥∥r∗
Lr(I)

⌋
∨ 1, (5.8)

where r∗ is such that 1
r + 1

r∗ = 1.

Proof. Denote for ϵ > 0, Eϵ := {t ∈ I : ρ(t) > ϵ}. From Markov’s inequality we
have for any ϵ > 0,

|I \ Eϵ| ≤ ϵr
∣∣∣1
ρ

∣∣∣r
Lr(I)

. (5.9)

Hence there exists ϵ∗ := |I|
1
r∥∥ 1

ρ

∥∥
Lr

> 0 such that Eϵ ̸= ∅ for any 0 < ϵ < ϵ∗. If ϵ∗ > |I|,

then we may take N = 1, hence (5.8) obviously holds. From now on we assume
that ϵ∗ ≤ |I|.

Fix some 0 < α < 1, and note that if there exists some 0 < ϵ0 < ϵ∗ such that

|Eϵ0 | ≥ |I| − αϵ0, (5.10)

then for any t ∈ I, we have dist(t, Eϵ0) ≤ αϵ0 otherwise we could fit a ball of diam-
eter strictly bigger than αϵ0 outside Eϵ0 which would contradict (5.10). Therefore

I ⊂ ∪t∈Eϵ0
B(t, ϵ0).

Compactness of I allows to extract a finite subcover, which we may assume satisfies
(P1)-(P2), yielding (5.1) with ϵ0 ≤ |τi − τi−1| ≤ 2ϵ0 for all 1 ≤ i ≤ N . Therefore
we can bound N from above and below:⌈ |I|

2ϵ0

⌉
≤ N ≤

⌊ |I|
ϵ0

⌋
. (5.11)

Now from (5.9), condition (5.10) is satisfied for ϵ0 satisfying

ϵr0

∣∣∣1
ρ

∣∣∣r
Lr(I)

= αϵ0.

After some rearranging we find that ϵ0 := αr∗−1∥∥ 1
ρ

∥∥r∗

Lr(I)

< ϵ∗ because we assumed

ϵ∗ ≥ |I|. Denoting β =
(
1
α

) 1
r∗−1 ∈ (1,+∞), we find from (5.11)⌈β

2
|I|

∥∥1
ρ

∥∥r∗
Lr(I)

⌉
≤ N ≤

⌊
β|I|

∥∥1
ρ

∥∥r∗
Lr(I)

⌋
. (5.12)

Letting β be close to 1 in (5.12), we obtain (5.8). □



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 33

Assume that A satisfies (A1) and (A2). Recall that then A ∈ RCα,β(I): it has
a specific range of relative continuity rA = (δA, ηA) such that there exists mδ ≥ 0,
mδ ∈ (0, 1] and

∀ϵ > 0, δA(ϵ) = mδϵ
α,

∀ϵ > 0, ηA(ϵ) ≤ mηϵ
−β .

The integrability condition (A2) (b), lets us define Γ := ∥(K0 + 1)(M0 + 1)∥αLαr(I).

Theorem 5.4. Assume that A satisfies (A1) and (A2). Then there exists a con-
stant C = C(|I|, ∥A∥∞, p, r, α, β) such that

log
(
[A]MRp(I)

)
≤ C

(( Γ

mδ

)r∗{
1 +mηm

s2(α,β,r)
δ Γs1(α,β,r) + log

( Γ

mδ

)}
+mη + 1

)
,

(5.13)

where

s1(α, β, r) := r∗
(β + 1

α
− 1

)
, s2(α, β, r) :=

β + 1

α+ 1
−
(α+ 1

r

α+ 1

)
s1(α, β, r).

The proof of the above theorem is quite technical, can be found in [11]. This re-
sult is useful for applications, it allows to express an explicit bound of the regularity
constant in terms of the parameters of relative continuity of the nonautonomous op-
erator A. This estimate is then used to infer a global existence result for quasilinear
equations, stated in the next section.

5.3. Examples. Here we give two simple but useful examples of operators in the
class RCα,β(I) for which we can apply our explicit estimate in Theorem 2.18.

Throughout the development of these examples we fix A ∈ mrp(R+) a given
maximally regular operator on R+ and an additional operator B ∈ L(D′;X) where
D ↪→ D′ ↪→ X is an intermediate space between D and X, that is we assume
there exists L ≥ 0, θ ∈ (0, 1) such that for any x ∈ D the following interpolation
inequality holds:

∥Bx∥X ≤ L∥x∥θD∥x∥1−θX .

The reader may think of A as a well-known, autonomous and maximally regu-
lar operator (e.g. A = −∆ on L2(Rn), A = − ÷x (a(x)∇x·) with a, a−1 ∈
L∞(Rn;S++(R))) and B as representing an operator with “low-order coefficients”
(e.g. Gagliardo-Nirenberg interpolation inequalities).

Example 5.5. Fix some λ ∈ C0, 1α (I;R+) such that 1
λ2 ∈ Lαr(I) for some r > 1

and define for t ∈ I,

A(t) := λ(t)A.

Then A ∈ C0, 1α (I;L(D;X)) ⊂ RCα,β(I) with mδ =
(
1 + ∥A∥L(D;X)[λ] 1

α

)−α
,

where [λ] 1
α
:= supt ̸=s∈I

|λ(t)−λ(s)|
|t−s|1/α is the 1

α -Hölder semi-norm of λ.

The operator A satisfies assumption (A1) with A0 = A and B : (t, s) 7→ A0(s)−
A0(t). Now observe that

K0(t) := [A(t)]mrp(I)

=
1

λ(t)
[A]mrp( I

λ(t) )



34 T. BELIN, P. LAFITTE EJDE-2025/18

≤ |I|+ λ(t)

λ(t)2
[A]mrp(R+)

≤ C(A, |I|, ∥λ∥∞)
1

λ(t)2
,

first by time rescaling invariance (see (4.9)) then by Proposition 2.10. AlsoM0(t) :=

supτ≥0 ∥e−τA∥X , therefore

Γ ≤ C(A, |I|, ∥λ∥∞)
∣∣∣ 1
λ2

∣∣∣α
Lαr(I)

.

From Theorem 2.18 (2.23) we find that there exists C(A, |I|, ∥λ∥∞, p, α, r) ≥ 0 such
that

log
(
[A]MRp(I)

)
≤ C

(
1 + [λ] 1

α

)αr∗ ∥∥ 1

λ2
∥∥αr∗
Lαr

(
1+log

((
1+[λ] 1

α

)∥∥ 1

λ2
∥∥
Lαr

))
. (5.14)

We observe from estimate (5.14) that the behavior of the regularity constant of A
can be expressed in terms of the strength of the oscillations of λ represented by [λ] 1

α

and of the term
∣∣ 1
λ2α

∣∣
Lr representing a measure of the degeneracy of the operator.

Example 5.6. We now consider a more general operator of the form

A(t) = λ(t)A+ γ(t)B,

where as before λ ∈ C0, 1α (I;R+) and γ ∈ L∞(I;R). As above we show how to
obtain a log-estimate of [A]MRp(I) using Theorem 2.18.

Relative continuity. A ∈ RCα,
θ

1−θ (I). Let us compute a range of relative conti-
nuity for A(·). Let t, s ∈ I, t ̸= s and x ∈ D, we have

∥A(t)x−A(s)x∥X ≤ |λ(t)− λ(s)|∥A∥L(D;X)∥x∥D + |γ(t)− γ(s)|∥Bx∥X
≤ [λ] 1

α
∥A∥L(D;X)|t− s|1/α∥x∥D + 2L∥γ∥∞∥x∥θD∥x∥1−θX .

Now for any h > 0, from Young’s inequality with p = 1
θ , q = 1

1−θ , we have

aθb1−θ ≤ θh1/θa+ (1− θ)h−
1

1−θ b. Therefore,

∥A(t)x−A(s)x∥X ≤
(
[λ] 1

α
∥A∥L(D;X)|t− s|1/α + 2θL∥γ∥∞h1/θ

)
∥x∥D

+ 2(1− θ)L∥γ∥∞h−
1

1−θ ∥x∥X .

Fix r > 0 and take h = (r|t− s|)
θ
α to find that

∥A(t)x−A(s)x∥X ≤
(
[λ] 1

α
∥A∥L(D;X) + 2θL∥γ∥∞r1/α

)
|t− s|1/α∥x∥D

+ 2(1− θ)L∥γ∥∞ (r|t− s|)−
θ

α(1−θ) ∥x∥X .

Now choose r :=
( [λ] 1

α
∥A∥

2θL∥γ∥∞

)α
to find that

∥A(t)x−A(s)x∥X
≤ 2[λ] 1

α
∥A∥L(D;X)|t− s|1/α∥x∥D

+ 2(1− θ)L∥γ∥∞
(2θL∥γ∥∞
[λ] 1

α
∥A∥

) θ
(1−θ) |t− s|−

θ
α(1−θ) ∥x∥X

= Cλ|t− s|1/α∥x∥D + (2θ)
θ

1−θ (1− θ)Cγ

(
Cγ
Cλ

) θ
(1−θ)

|t− s|−
θ

α(1−θ) ∥x∥X ,



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 35

where Cλ := 2[λ] 1
α
∥A∥L(D;X) and Cγ := 2L∥γ∥∞. Observe then that for any ϵ > 0,

if |t− s| ≤ C−α
λ ϵα, then we find that

∥A(t)x−A(s)x∥X ≤ ϵ∥x∥D + CθCγ

(
Cγ
Cλ

) θ
1−θ

C
θ

1−θ

λ ϵ−
θ

1−θ ∥x∥X

≤ ϵ∥x∥D + CθC
1

1−θ
γ ϵ−

θ
1−θ ∥x∥X .

A range of relative continuity of A can be

δ(ϵ) := (Cλ + 1)
−α

ϵα,

η(ϵ) := CθC
1

1−θ
γ ϵ−

θ
1−θ

and therefore A ∈ RCα,
θ

1−θ (I).

Maximal regularity. A ∈ MRp(I). Here A satisfies assumption (A1) with

A0 : t 7→ λ(t)A and B : (t, s) 7→ (λ(s)− λ(t))A + γ(s)B. Now we have for x ∈ D,
ϵ > 0,

∥B(t, s)x∥X ≤ ∥A(s)x−A(t)x∥X + ∥γ∥∞∥Bx∥X

≤ ∥A(s)x−A(t)x∥X +
ϵ

2
∥x∥D + CθCγϵ

− θ
1−θ ∥x∥X .

Hence upon choosing δA(ϵ) := δ( ϵ2 ) and ηA(ϵ) := η( ϵ2 ) + CθCγϵ
− θ

1−θ we find that

(δA, ηA) is still a range of relative continuity for A, compatible in RCα,
θ

1−θ (I) and
that B satisfies (A1)(b.3).

Logarithmic estimate. log
(
[A]MRp(I)

)
. As in the continuous case, we find that

Γ ≤ C(A, |I|, ∥λ∥∞)
∥∥ 1

λ2
∥∥α
Lαr(I)

< +∞.

Hence A satisfies (A2) for α and β = θ
1−θ . We also observe from the definitions of

δA and ηA that

mδ ∝ (1 + [λ] 1
α
)−α,

mη ∝ ∥γ∥
1

1−θ
∞ ,

where the proportionality factors only depend on θ, L and ∥A∥L(D,X). Also see

that s1(α,
θ

1−θ ) = r∗
(

1
α(1−θ) − 1

)
and

s2

(
α,

θ

1− θ

)
=

1

(1− θ)(α+ 1)
− r∗

(α+ 1
r

α+ 1

)( 1

(1− θ)α
− 1

)
.

Hence we can apply Theorem 2.18, to find that

log
(
[A(·)]MRp(I)

)
≤ C

((
1 + [λ] 1

α

)αr∗ ∥∥ 1

λ2
∥∥αr∗
Lαr

{
1 + ∥γ∥

1
1−θ
∞

(
1 + [λ] 1

α

)αs2 ∥∥ 1

λ2
∥∥αs1
Lαr

+ log

((
1 + [λ] 1

α

)∥∥ 1

λ2
∥∥
Lαr

)}
+ ∥γ∥

1
1−θ
∞ + 1

)
,

(5.15)
where C := C(|I|, ∥λ∥∞, p, α, θ, r).

As for the continuous case described in Theorem 5.5, the behavior of the regular-
ity constant of A in (5.15) is expressed in terms of [λ] 1

α
representing the oscillations

of the diffusion coefficient λ and in terms of ∥ 1
λ2α ∥Lr representing a measure of



36 T. BELIN, P. LAFITTE EJDE-2025/18

degeneracy of λ. Extra terms in ∥γ∥∞ appear which account for the perturba-
tion γ(t)B. Remark that if θ is close to 0, then the log-estimate becomes linear
with respect to ∥γ∥∞ which matches the asymptotic behavior of [A+ γ]MRp(I) as
γ → +∞, γ ∈ R (see Proposition 3.4). If γ ≡ 0 we exactly find (5.14) back.

A regime of interest happens when α(1− θ) < 1, in this case we have s1, s2 < 0.
Therefore the above bound can be simplified as follows:

log
(
[A(·)]MRp(I)

)
≤ C

((
1 + [λ] 1

α

)αr∗∥∥ 1

λ(·)2
∥∥αr∗
Lαr

{
1 + ∥γ∥

1
1−θ
∞

+ log
((

1 + [λ] 1
α

)∥∥ 1

λ(·)2
∥∥
Lαr

)}
+ 1

)
.

6. Global existence for quasilinear equation

6.1. A general existence result. Let us recall here the criterion for the existence
of global solutions on the bounded interval I of the quasilinear problem

d

dt
u+ A(u)u = F(u),

u(a) = x,
(6.1)

where x ∈ Trp.

(EI) For all R ≥ 0, we define the quantities:

γ(R) := sup
∥u∥X(I)=R

[A(u)]MRp(I), (6.2)

κ(R) := sup
∥u∥X(I)=R

{∥A(u)∥∞ + ∥F(u)∥Lp(I;X) + 1} (6.3)

and assume that for any L > 0 the following sublinear growth condition
holds,

sup
R1,R2∈[R−L,R+L]

γ(R1)κ(R2)

R
→

R→+∞
0. (6.4)

(EII) We ask for the following continuity properties:

(i) F :

{
X (I) → Lp(I;X)

u 7→ F(u) is weakly continuous.

(ii) A(·) :
{

X (I) → L∞(I;L(D,X))
u 7→ A(u) is continuous.

Let us now prove the existence theorem.

Theorem 6.1. Assume that A and F satisfy conditions (EI) and (EII). Then for
any x ∈ Trp there exists a unique global strong solution u ∈ MRp(I) of (6.1).

Proof. Reduction to x = 0. Fix a function w ∈ MRp(I) such that w(a) = x, and
consider the quasilinear problem

d

dt
v + A(v + w)v = F(v + w)− d

dt
w − A(v + w)w,

v(a) = 0.
(6.5)

If v ∈ MRp(I) is a solution of (6.5), then u := v + w is a solution to the original
problem (6.1) starting from x. Let us show that

Ax(·) := A(·+ w),



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 37

Fx(·) := F(·+ w)− d

dt
w − Ax(·)w

satisfy assumptions and (EI) - (EII). It is straightforward to see that they satisfy
(EII). There only needs to show that they satisfy (EI).

We define as required

γx(R) := sup
∥u∥X (I)=R

[Ax(u)]MRp(I),

κx(R) := sup
∥u∥X(I)=R

{∥Ax(u)∥∞ + ∥Fx(u)∥Lp(I;X)}.

For a given u ∈ X (I), from the triangle inequality we have ∥u+w∥X (I)−∥w∥X (I) ≤
∥u∥X (I) ≤ ∥u+w∥X (I) + ∥w∥X (I). We infer that if Lw := ∥w∥X (I), we have for any
R ≥ Lw,

γx(R) ≤ sup
R1∈[R−Lw,R+Lw]

γ(R1).

In a similar fashion, since there exists Cw ≥ 0 such that

∥Fx(u)∥Lp(I;X) ≤ Cw
(
∥F(u+ w)∥Lp(I;X) + ∥A(u+ w)∥∞ + 1

)
,

we find that

κx(R) ≤ sup
R2∈[R−Lw,R+Lw]

Cwκ(R2).

Let L ≥ 0 and let R ≥ 0 and R1, R2 ∈ [R− L,R+ L] we have

γx(R1)κx(R2) ≤ sup
R′

1∈[R1−Lw,R1+Lw]

γ(R′
1) sup
R′

2∈[R2−Lw,R2+Lw]

Cwκ(R
′
2)

≤ Cw sup
R′

1,R
′
2∈[R−L−Lw,R+L+Lw]

γ(R′
1)κ(R

′
2).

Now taking the supremum over R1 and R2, we find from (E′
I) with L+ Lw that

sup
R1,R2∈[R−L,R+L]

γx(R1)κx(R2) →
R→+∞

0.

Existence. The existence of the solution is shown through Schauder’s fixed point
theorem. Define the map

T :

{
X (I) → X (I),

u 7→ L−1
A(u)F(u)

and note that any fixed point of T must be solution of (6.1).
⋆]The map T is continuous from X (I) equipped with the strong topology to

MRp(I) equipped with the weak topology as a consequence of (ii) of Proposition
2.19 and the continuity properties of F and A in (EII). Now since the embed-
ding MRp(I) ↪→ X (I) is compact, this implies the continuity of T onto the strong
topology of X (I).
⋆ T is compact, again by the compact embedding MRp(I) ↪→ X (I).
⋆ Let u ∈ X (I), and denote R := ∥u∥X (I). We have

∥Tu∥MRp(I) ≤ [A(u)]MRp(I)∥F(u)∥Lp(I;X)

≤ γ(R)κ(R)

≤ R
(γ(R)κ(R)

R

)
.



38 T. BELIN, P. LAFITTE EJDE-2025/18

Denote by C > 0 the continuity constant of the embedding MRp(I) ↪→ X (I), to
find that

∥Tu∥X (I) ≤ CR

(
γ(R)κ(R)

R

)
. (6.6)

For any r > 0, we denote Br = BX (I)(0, r) the closed ball of X (I) of radius
r ∈ R+ and centered at 0.

By (EI) there exists R0 > 0 large enough such that for any R ≥ R0 we have

C γ(R)κ(R)
R ≤ 1

2 . By continuity of the map T there exists R1 > 0 large enough such
that T (BR0

2
) ⊂ BR1

. We denote R∗ := R0 ∨R1 and define the quantity

α := sup
R0
2 ≤R≤R∗

C

(
γ(R)κ(R)

R

)
< +∞.

⋆ We claim that T (BαR∗) ⊂ BαR∗ . Indeed, if α ≤ 1 then it is obvious because
from (6.6) that T (BαR∗) ⊂ T (BR∗) ⊂ BαR∗ . Otherwise if α > 1 let us writeBαR∗ =
BR0

2
∪ S1 ∪ S2 where S1 := BR∗ \ BR0

2
, S2 := BαR∗ \ BR∗ . We already know that

T (BR0
2
) ⊂ BR1 ⊂ BR∗ ⊂ BαR∗ . Now from (6.6) we find that T (S1) ⊂ BαR∗ , and

from the definition of R0 and again from (6.6) we find that T (S2) ⊂ Bα
2R

∗ ⊂ BαR∗ .

Finally since BαR∗ is a closed and convex subset of the Banach space X (I), we
can apply Schauder’s fixed point theorem which yields the existence of a solution
to (6.1). □

6.2. Existence of strong solutions for relatively continuous operators. We
assume throughout this subsection that the map A : X (I) → RC(I) maps to
the space of relatively continuous operators and that A(u) is globally bounded in
L∞(I;L(D,X)).

Let α ≥ 1, β ≥ 0, r > 1 be fixed and assume that for each u ∈ X (I), A(u) satisfies
(A1)- (A2). Denote by A0 : u 7→ A0(u) the operator given by the decomposition
(A1)(b) and the subsequent u 7→ K0(u),M0(u) given in (2.16).

Naturally this gives rise to the following maps:

(i) mδ : X (I) → (0, 1],
(ii) mη : X (I) → R+,

(iii) Γ :

{
X (I) → R+,
u 7→ ∥(K0 + 1)α(M0 + 1)α∥αLαr(I).

(E′
I) For any R ≥ 0, define the following quantities:

γlog(R) := sup
∥u∥X(I)=R

{( Γ(u)

mδ(u)

)r∗(
1 +mη(u)mδ(u)

s2Γ(u)s1

+ log
( Γ(u)

mδ(u)

))
+mη(u)

}
,

κlog(R) := sup
∥u∥X(I)=R

{log
(
∥A(u)∥∞ + ∥F(u)∥Lp(I;X) + 1

)
}

and assume that there exists h > 0, such that for any L > 0, there exists
R0 ≥ 0 such that for all R ≥ R0 we have

sup
R1,R2∈[R−L,R+L]

γlog(R1) + κlog(R2)

log (R)
≤ 1− h, (6.7)



EJDE-2025/18 QUANTITATIVE ESTIMATES IN Lp MAXIMAL REGULARITY 39

where s1(α, β, r) := r∗
(
β+1
α − 1

)
and s2(α, β, r) :=

β+1
α+1−

(
α+ 1

r

α+1

)
s1(α, β, r).

Theorem 6.2. Assume that for each u ∈ X (I), A(u) satisfies (A1) and (A2).
Moreover assume that A,F satisfy (E′

I) and (EII). Then for any x ∈ Trp, there
exists a unique solution u to (6.1).

Proof. The growth condition (6.7) straightforwardly implies the growth condition
(EI). (EII) is fulfilled by assumption. We simply apply Theorem 6.1. □

7. Conclusion and perspectives

This work gives new estimates of the maximal regularity constants of nonau-
tonomous relatively continuous operators and describes the weak topologies of op-
erators in L∞(I;L(D,X)) for which the maximal regularity is persistent.

These inquiries are used to derive a new well-posedness criterion for a general
class of Cauchy problems for nonlocal in time and quasilinear equations. This
criterion allows to bypass the standard Lipschitz continuity of the nonlinearities and
instead requires growth conditions on the pointwise constant of maximal regularity.

We hope that these inquiries will help the study of nonlinear parabolic equations
arising in various models involving time-delays or memory effects with moderately
low-regularity coefficients in time. A first inquiry pursued by the authors is to find
concrete applications of the result.

Remaining inquiries would involve the existence and regularity of the semi-flow
associated to (2.31) and better asymptotic estimates of the maximal regularity
constant to possibly improve the criterion (2.35).

References

[1] P. Acquistapace, B. Terreni; A unified approach to abstract linear nonautonomous parabolic
equations, Rendiconti del seminario matematico della Università di Padova 78 (1987), 47–107.

[2] H Amann; Quasilinear parabolic problems via maximal regularity, Advances in Differential

Equations 10 (2005), no. 10, 1081–1110.
[3] Herbert Amann; Maximal regularity for nonautonomous evolution equations, Advanced Non-

linear Studies 4 (2004), no. 4, 417–430.

[4] Herbert Amann; Quasilinear parabolic functional evolution equations, Recent advances on
elliptic and parabolic issues, World Scientific, 2006, pp. 19–44.

[5] Herbert Amann; A time-delayed Perona-Malik type problems, Proceedings of Equadiff 11

(2007), 15–38.
[6] Herbert Amann et al.; Linear and quasilinear parabolic problems, vol. 1, Springer, 1995.
[7] L. Ambrosio, N. Fusco, D. Pallara; Functions of bounded variation and free discontinuity

problems, Oxford Science Publications, Clarendon Press, 2000.
[8] W. Arendt, R. Chill, S. Fornaro, C. Poupaud; emphLp-maximal regularity for non-

autonomous evolution equations, Journal of Differential Equations 237 (2007), no. 1, 1–26.
[9] Wolfgang Arendt, Ralph Chill; Global existence for quasilinear diffusion equations in

isotropic nondivergence form, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze
9 (2010), no. 3, 523–539.

[10] Wolfgang Arendt, Sylvie Monniaux; Maximal regularity for non-autonomous Robin boundary
conditions, Mathematische Nachrichten 289 (2016), no. 11-12, 1325–1340.

[11] Théo Belin, Pauline Lafitte; Quantitative estimates of Lp maximal regularity for nonau-
tonomous operators and global existence for quasilinear equations, working paper, version 1,

February 2024.
[12] Philippe Clément, Jan Prüss; Global existence for a semilinear parabolic Volterra equation,

Mathematische Zeitschrift 209 (1992), 17–26.
[13] Raphaël Danchin, Matthias Hieber, Piotr Boguslaw Mucha, Patrick Tolksdorf; Free Boundary

Problems via Da Prato–Grisvard Theory, Memoirs of the American Mathematical Society (to
appear).



40 T. BELIN, P. LAFITTE EJDE-2025/18

[14] L. De Simon; Un’applicazione della teoria degli integrali singolari allo studio delle equazioni

differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della

Università di Padova 34 (1964), 205–223.
[15] Qiang Du, Jiang Yang, Zhi Zhou; Analysis of a nonlocal-in-time parabolic equation, Discrete

Contin. Dyn. Syst. Ser. B 22 (2017), no. 2, 339–368.

[16] Chiara Gallarati, Mark Veraar; Maximal regularity for non-autonomous equations with mea-
surable dependence on time, Potential Analysis 46 (2017), 527–567.

[17] D. Gilbarg,N.S. Trudinger; Elliptic partial differential equations of second order, vol. 224(2),

Springer, 1977.
[18] Patrick Guidotti; A new nonlocal nonlinear diffusion of image processing, Journal of Differ-

ential Equations 246 (2009), no. 12, 4731–4742.

[19] Patrick Guidotti; Anisotropic diffusions of image processing from Perona-Malik on, Varia-
tional Methods for Evolving Objects, vol. 67, Mathematical Society of Japan, 2015, pp. 131–

157.
[20] N Kosovalić, FMG Magpantay, Y Chen, J Wu; Abstract algebraic-delay differential systems

and age structured population dynamics, Journal of Differential Equations 255 (2013), no. 3,

593–609.
[21] Peer C. Kunstmann, Lutz Weis; Maximal lp-regularity for parabolic equations, fourier mul-

tiplier theorems and h∞-functional calculus, pp. 65–311, Springer Berlin Heidelberg, Berlin,

Heidelberg, 2004.
[22] H. Laasri, O. El-Mennaoui; Stability for non-autonomous linear evolution equations with

lp-maximal regularity, Czechoslovak Mathematical Journal 63 (2013), no. 4, 887–908 (eng).

[23] J. Lofstrom, J. Bergh; Interpolation spaces, Springer Verlag, Newe (1976).
[24] S. Monniaux; Maximal regularity and applications to PDEs, Analytical and numerical aspects

of partial differential equations (2009), 247.

[25] B.J. Pettis; On integration in vector spaces, Transactions of the American Mathematical
Society 44 (1938), no. 2, 277–304.

[26] Jan Prüss, Roland Schnaubelt; Solvability and maximal regularity of parabolic evolution equa-
tions with coefficients continuous in time, Journal of mathematical analysis and applications

256 (2001), no. 2, 405–430.

[27] SG Pyatkov; On the maximal regularity property for evolution equations., Azerbaijan Journal
of Mathematics 9 (2019), no. 1.

[28] Atsushi Yagi; Abstract parabolic evolution equations and their applications, Springer Science

& Business Media, 2009.

Théo Belin
Université Paris-Saclay, Laboratoire MICS, Centrale Supélec, France

Email address: theo.belin@centralesupelec.fr

Pauline Lafitte

Université Paris-Saclay, Fédération de Mathématiques de Centrale Supélec (FR CNRS

3487), France
Email address: pauline.lafitte@centralesupelec.fr


	1. Introduction
	2. Main results
	2.1. Operators, spaces and measurability
	2.2. Lp maximal regularity
	2.3. Relatively continuous operators
	2.4. Main results and scheme of the proof

	3. Perturbation and asymptotic behavior
	3.1. The operator LA:= ddt + A
	3.2. Diagonal perturbations by constants in mrp(I) and MRp(I)
	3.3. Nonautonomous perturbations of autonomous operators in mrp

	4. Gluing maximally regular operators
	4.1. Behavior of []MRp(I) under a change of variable in time
	4.2. Gluing finitely many MRp operators

	5. Proofs of the main estimates and examples
	5.1. Estimation for general relatively continuous operators
	5.2. Estimation for Hölder-type relatively continuous operators
	5.3. Examples

	6. Global existence for quasilinear equation
	6.1. A general existence result
	6.2. Existence of strong solutions for relatively continuous operators

	7. Conclusion and perspectives
	References