Special issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 23–89.

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

SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS TO

SEMILINEAR PARABOLIC SYSTEMS WITH VARIABLE

COEFFICIENTS

FALKO BAUSTIAN, PETER TAKÁČ

Dedicated to the memory of Professor Alan C. Lazer

Abstract. We study analytic smooth solutions of a general, strongly para-
bolic semilinear Cauchy problem of 2m-th order in RN × (0, T ) with analytic

coefficients (in space and time variables) and analytic initial data (in space

variables). They are expressed in terms of holomorphic continuation of global
(weak) solutions to the system valued in a suitable Besov interpolation space

of Bs;p,p-type at every time moment t ∈ [0, T ]. Given 0 < T ′ < T ≤ ∞, it

is proved that any Bs;p,p-type solution u : RN × (0, T ) → CM with analytic
initial data possesses a bounded holomorphic continuation u(x + iy, σ + iτ)

into a complex domain in CN × C defined by (x, σ) ∈ RN × (T ′, T ), |y| < A′

and |τ | < B′, where A′, B′ > 0 are constants depending upon T ′. The proof
uses the extension of a weak solution to a Bs;p,p-type solution in a domain in

CN ×C, such that this extension satisfies the Cauchy-Riemann equations. The

holomorphic extension is obtained with a help from holomorphic semigroups
and maximal regularity theory for parabolic problems in Besov interpolation

spaces of Bs;p,p-type imbedded (densely and continuously) into an Lp-type
Lebesgue space. Applications include risk models for European options in

Mathematical Finance.

Contents

1. Introduction 23
2. Notation 29
3. Statement of main results 31
3.1. Hypothesis 32
4. Abstract Cauchy problem in an interpolation space 37
4.1. Hypothesis 41
5. Analyticity in time for the abstract Cauchy problem 45
5.1. Auxiliary linear perturbation results 45
5.2. Proof of analyticity in time 51
5.3. Hypothesis 51

2010 Mathematics Subject Classification. 35B65, 35K10, 32D05, 91G40.
Key words and phrases. Space-time analyticity; parabolic PDE; holomorphic semigroup;

Besov space; maximal regularity; Hardy space; holomorphic continuation to a complex strip;

European option; bilateral counterparty risk.
c©2021 This work is licensed under a CC BY 4.0 license.

Published October 6, 2021.

23



24 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

6. Analyticity in space for the Cauchy problem in RN × (0, T ) 57
6.1. Hypothesis 58
6.2. Hypothesis 60
6.3. Hypothesis 66
7. Space-time analyticity for the Cauchy problem in RN × (0, T ) 69
8. Proofs of main results 70
9. An application to a risk model in mathematical finance 76
10. Historical remarks and comments 84
11. Discussion and possible generalizations 85
Acknowledgments 85
References 86

1. Introduction

In this article we investigate the analyticity (in space and time variables) of strict
Lp-type solutions u = (u1, . . . , uM ) : RN × (0, T ) → CM (or CM ) of the classical
Cauchy problem for a strongly parabolic system of M (coupled) semilinear partial
differential equations of order 2m (m ≥ 1 – an integer) with analytic coefficients and
with analytic initial data u0 belonging to the real interpolation space Bs;p,p(RN ),
such that the function u : [0, T ] → Bs;p,p(RN ) is continuous. Here, Bs;p,p(RN ) =
[Bs;p,p(RN )]M where Bs;p,p(RN ) denotes the Besov space

Bs;p,p(RN ) :=
(
Lp(RN ), W 2m,p(RN )

)
s/(2m),p

=
(
Lp(RN ), W 2m,p(RN )

)
1−(1/p),p

with 1 < p < ∞, p > 2 + N
m , and s = 2m

(
1 − 1

p

)
∈ (0, 2m). This space is defined

by real interpolation, e.g., in Adams and Fournier [1, Chapt. 7], §7.6–§7.23, pp.
208–221, Lunardi [65, Chapt. 1], §1.2.2, pp. 20–25, or in Triebel [84, Chapt. 1],
§1.2–§1.8, pp. 18–55. Since the Besov space Bs;p,p(RN ) is not imbedded into the
Hilbert space L2(RN ) whenever 2 < p < ∞, we find it convenient to consider
strict Lp-type solutions u : RN × (0, T ) → CM having the maximal regularity
property (cf. Ashyralyev and Sobolevskii [9, Chapt. 3, pp. 21–36] and Prüss [74])
rather than weak L2-type solutions treated in Takáč [82] for the corresponding
linear partial differential equation, but with arbitrary nonsmooth initial data u0 ∈
L2(RN ). Consequently, we will be able to apply the classical theory of linear and
semilinear evolutionary problems of parabolic type in a Besov space as presented,
e.g., in Amann [6, Chapt. III, §4, pp. 128–191], Clément and Li [20], Lunardi [65,
Chapt. 7, pp. 257–289], Köhne, Prüss, and Wilke [55], and Tanabe [81, Chapt. 5–6,
pp. 117–229]. Our Cauchy problem has the following general form for a semilinear
2mth-order parabolic problem,

∂u

∂t
+ P

(
x, t,

1

i

∂

∂x

)
u = f

(
x, t;

(∂|β|u
∂xβ

)
|β|≤m

)
for (x, t) ∈ RN × (0, T ) ;

u(x, 0) = u0(x) for x ∈ RN .
(1.1)

Here, ∂/∂x = (∂/∂x1, . . . , ∂/∂xN ) stands for the spatial gradient and ξ 7→ P(x, t, ξ)
is a polynomial of order 2m in the variable ξ = (ξ1, . . . , ξN ) ∈ RN (or CN ); its
coefficients are M ×M matrices (real or complex) which are assumed to be real
analytic (jointly) in both variables x ∈ RN and t ∈ (0, T ). Also the nonlinearity
(x, t;X) 7→ f(x, t;X) (a reaction function valued in RM or CM ) is assumed to



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 25

be analytic in all variables x ∈ RN , t ∈ (0, T ), and X = (Xβ)|β|≤m ∈ RMÑ (or

CMÑ ), where we have substituted Xβ = ∂|β|u
∂xβ

∈ RM (or CM ) for the (mixed)

partial derivative of u with a multi-index β = (β1, . . . , βN ) ∈ (Z+)N of order
|β| = β1 + · · ·+βN , |β| ≤ m. Here, Z+ = {0, 1, 2, . . . } and the Euclidean dimension

of the m-jet X equals to MÑ with

Ñ =

m∑
k=0

∑
|β|=k

(
k

β

)
=

m∑
k=0

Nk where

(
k

β

)
:=

k!

β1!β2! . . . βN !
. (1.2)

As usual, RN and CN , respectively, denote the N -dimensional real and complex
Euclidean spaces, i =

√
−1, and M,N ∈ N where N = {1, 2, 3, . . . }. We have

identified Xβ = ∂|β|u
∂xβ

≡ u(x, t) for β = (0, 0, . . . , 0) of order |β| = 0.
As already indicated, we impose certain standard strong ellipticity and analyt-

icity hypotheses on the coefficients of the partial differential operator P
(
x, t, 1

i
∂
∂x

)
and on the reaction function f(x, t;X) as well. Assuming that u0 ∈ Bs;p,p(RN )
(p > 2 + N

m ) possesses a complex analytic extension to a strip X(κ0) of constant

width in CN = RN + iRN and the first-order partial derivatives

∂

∂t
f(x, t;X) and

∂

∂Xβ
f(x, t;X) , for |β| ≤ m,

are locally uniformly bounded for (x, t;X) ∈ RN × (0, T ) × RMÑ , in this work we
show that the (unique) strict (Lp-type) solution u = u(x, t) of problem (1.1) is real
analytic in (x, t) ∈ RN ×(0, T ). Notice that the latter condition (local boundedness
of all first-order partial derivatives ∂f/∂Xβ) is equivalent with X 7→ f(x, t;X) being
locally uniformly Lipschitz continuous.

This analyticity claim is motivated by the standard formula for the solution of
the Cauchy problem for the heat equation in RN (with the Laplace operator ∆,
i.e., P

(
x, t, 1

i
∂
∂x

)
= −∆, f(x, t;X) = 0, and M = 1); see e.g. John [50], Chapt. 7,

Sect. 1, eq. (1.11), p. 209. The heat equation case has been significantly general-
ized in Takáč et al. [83, Theorem 2.1, p. 429], where only the leading coefficients
of the operator P

(
x, t, 1

i
∂
∂x

)
are assumed to be constant, but it is required that

u0 ∈ L∞(RN ) = [L∞(RN )]M . In our present work, the analyticity hypothesis on
the initial data u0 resembles more to a nonlocal version of the classical Cauchy-
Kowalewski theorem (John [50], Chapt. 3, Sect. 3(d), pp. 73–77). We will show that,
under this analyticity hypothesis on u(·, 0) = u0, if a solution u : RN×[0, T )→ CM
exists, then it must be analytic in RN×(0, T ). We are able to specify also the domain
of analyticity in terms of a complex analytic extension. The restriction on the initial
data u0 ∈ Bs;p,p(RN ), with the conditions p > 2+ N

m and s = 2m
(
1− 1

p

)
∈ (0, 2m),

allows us to take advantage of (the continuity of) the Sobolev(-Besov) imbedding
Bs;p,p(RN ) ↪→ Cm(RN )∩Wm,∞(RN ); see, e.g., Adams and Fournier [1, Chapt. 7],
Theorem 7.34(c), p. 231. This more restrictive condition on the initial data u0

enables us to work with an m-jet X = (Xβ)|β|≤m ∈ CMÑ whose components

Xβ = ∂|β|u
∂xβ

∈ CM are bounded continuous functions of (x, t) ∈ RN × [0, T ); thus,

each Xβ(·, t) (|β| ≤ m) belongs to L∞(RN ) at every time t ∈ [0, T ). Consequently,
we can apply the Banach fixed point theorem to problem (1.1) in a way similar to
[83, Theorem 2.1, p. 429]. For instance, in a typical second-order parabolic problem
(i.e., (1.1) with m = 1) we can allow for a reaction function f

(
x, t; u, ∂u∂x

)
depending



26 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

on u and its gradient ∂u/∂x ( ≡ iDxu), besides the independent variables x ∈ RN
and t ∈ (0, T ).

The main contribution of our present article is that we are able to remove the
hypothesis that the leading coefficients must be constant, in analogy with Takáč [82,
Theorem 3.3, p. 59] where the corresponding linear system is treated. In contrast
to [83, Proposition A.4, p. 446], this means that we cannot calculate the Green
function for the Cauchy problem with the leading coefficients only,

∂u

∂t
+ (−1)m

∑
|α|=2m

P(α)(x, t)
∂|α|u

∂xα
= 0 for (x, t) ∈ RN × (0, T ) ;

u(x, 0) = u0(x) for x ∈ RN ,

(1.3)

and then simply take advantage of the variation-of-constants formula [83, eq. (3.22),
p. 437] to obtain the solution of the original problem (1.1). Fortunately, the methods
from [82], based on a priori L2-type estimates combined with the Cauchy-Riemann
equations, are applicable also to our semilinear system (1.1) provided that already
the initial data u0 are analytic. Here, each P(α)(x, t) is an M ×M matrix and

recall that ∂|α|u/∂xα = ∂|α|u
∂x
α1
1 ... ∂x

αN
N

denotes the (mixed) partial derivative of u :

RN × (0, T ) → CM with a multi-index α = (α1, . . . , αN ) ∈ (Z+)N of order |α| =
α1 + · · · + αN . This means that, for the semilinear parabolic Cauchy problem
(1.1), we do not improve the regularity properties of (in general) nonsmooth initial
data to analytic regularity as time passes by (for t ∈ (0, T )). We show only that
the analytic regularity of the initial data u0 (at t = 0) is preserved for all times
t ∈ (0, T ). In contrast, analytic regularity of the initial data is not assumed in
[82, 83].

As in [82, 83], our method is based on the simple fact that a function u : RN ×
(0, T )→ R (or C) is real analytic if and only if it has a holomorphic (i.e., complex
analytic) extension ũ : Ω→ C to some complex domain Ω such that RN × (0, T ) ⊂
Ω ⊂ CN × C , i.e., u = ũ|RN×(0,T ), the restriction of ũ to RN × (0, T ). If the
domain Ω is fixed then the holomorphic extension ũ of u to Ω is always unique,
see e.g. John [50], Chapt. 3, Sect. 3(c), pp. 70–72. Thus, in order to show that
the weak solution u = u(x, t) of problem (1.1) is real analytic in RN × (0, T ),
it suffices to construct a holomorphic extension ũ of u to some complex domain
Ω (RN × (0, T ) ⊂ Ω ⊂ CN × C). Because of the uniqueness (of a holomorphic

extension), we often drop the tilde “
∼

” in the notation for the (unique) holomorphic
extension. Analogous ideas (holomorphic extension, uniqueness, and Bergman and
Szegő spaces of holomorphic functions) were used earlier in Hayashi [35, 36, 37, 38].

Instead of using the Green function method (cf. [83]), we establish the existence
of solutions to the Cauchy problem (1.1) in a complex parabolic domain X(r)×[0, T )
in CN×C with initial data u0 from a space of holomorphic functions whose domain
X(r) = RN + iQ(r) is a tube in CN with base Q(r) = (−r, r)N , for some 0 < r <∞,
see Takáč [82, (21), p. 58]. The (complex) analyticity in space is then verified
by means of the Cauchy-Riemann equations, whereas the (complex) analyticity in
time is obtained from the properties of holomorphic semigroups in the Besov space
Bs;p,p(RN ) = [Bs;p,p(RN )]M . Our use of the Cauchy-Riemann equations already
at the initial time t = 0 requires that u0 be (complex) analytic in X(r).

To provide a quick, nontechnical hint to our approach, we now give an illustrative
weaker version of our main result, Theorem 3.4 in Section 3, for a single equation



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 27

in one space dimension (M = N = 1),

∂u

∂t
= a(x, t)

∂2u

∂x2
+ b(x, t)

∂u

∂x
+ c(x, t)u+ f

(
x, t;u,

∂u

∂x

)
for (x, t) ∈ R1 × (0, T ) ;

u(x, 0) = u0(x) for x ∈ R1 .

(1.4)
We begin with the complexifications of the spatial and temporal variables, x ∈ R1

and t ∈ (0, T ), respectively: Given any real numbers 0 < r <∞ and 0 < T ′ ≤ T <
∞, we introduce the complex domains

X(r) := {z = x+ iy ∈ C : |y| < r} = R + i(−r, r) ,

∆ϑ := {t = %eiθ ∈ C : % > 0 and θ ∈ (−ϑ, ϑ)} , ϑ = arctan(r/T ′) , (1.5)

∆
(T ′)
ϑ := ∆ϑ ∩ {t ∈ C : 0 < <et < T ′}

= {t = %eiθ ∈ C : |θ| < ϑ and 0 < % < T ′/ cos θ} , (1.6)

∆T ′,T
ϑ := ∆

(T )
ϑ ∩ {t ∈ C : |=mt| < T ′ · tanϑ} = ∪0≤ξ≤T−T ′

(
ξ + ∆

(T ′)
ϑ

)
= ∪0≤ξ≤T−T ′{ξ + t′ ∈ C : t′ ∈ ∆

(T ′)
ϑ } = [0, T − T ′] + ∆

(T ′)
ϑ (1.7)

with the angle ϑ ∈ (0, π/2) given by tanϑ = r/T ′. Of course, if T = T ′ then

∆T ′,T
ϑ = ∆

(T ′)
ϑ is an open triangle. Clearly, we have

∆T ′,T
ϑ = ∪0<r≤T ′T

(r)
r·cotϑ,T = ∪0<r≤T ′ [(r · cotϑ, T ) + i(−r, r)],

where

T
(r)
T ′,T := {t = σ + iτ ∈ C : T ′ < σ < T and |τ | < r} = (T ′, T ) + i(−r, r) . (1.8)

We set T
(r)
0,T = (0, T ) + i(−r, r) if T ′ = 0. The closures in C of X(r), ∆ϑ, ∆

(T ′)
ϑ ,

∆T ′,T
ϑ , and T

(r)
T ′,T are denoted by X̄(r), ∆̄ϑ, ∆̄

(T ′)
ϑ , ∆̄T ′,T

ϑ , and T̄
(r)
T ′,T , respectively.

∆
(T ′)
ϑ ∆T ′,T

ϑ

=mt

0
T ′ T

<et

Figure 1. Triangle ∆ϑ starting at the origin has been defined in

(1.5). Its shifts to the right create the region ∆
(T ′)
ϑ in (1.6)



28 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

The Banach space of all continuous (Bs;p,p(R)-valued) functions u : [0, T ] →
Bs;p,p(R) is denoted by C ([0, T ]→ Bs;p,p(R)); it is endowed with the natural supre-
mum norm

|||u|||L∞(0,T ) := sup
t∈[0,T ]

‖u(·, t)‖Bs;p,p(R) <∞.

Theorem 1.1 (M = N = 1). Let p > 2 + 1
m , s = 2m

(
1 − 1

p

)
, and 0 < T < ∞.

Assume that there are constants A,B > 0 such that all coefficients a, b, and c
and the partial derivative ∂a/∂x are bounded, continuously differentiable functions

in the Cartesian product X̄(A) × T̄
(B)
0,T , with <ea ≥ const > 0, and all a, b, and c

are holomorphic in X(A) × T
(B)
0,T . Furthermore, let us assume that the first-order

time derivatives of all functions a, b, c, and ∂a/∂x are bounded in X̄(A) × T̄
(B)
0,T .

Finally, assume that f is holomorphic in X(A) × T
(B)
0,T × C2, f = f(x, t;u, η) where

η = ∂u/∂x, with all functions f , ∂f/∂t, ∂f/∂u, and ∂f/∂η being locally bounded

in X(A) × T
(B)
0,T × C2, and it satisfies∫ ∞

−∞
|f(x+ iy, t; 0, 0)|p dx ≤ Kp (K = const <∞)

for all y ∈ [−A,A] and t ∈ T̄
(B)
0,T .

(i) Given any u0 ∈ Bs;p,p(R), the Cauchy problem (1.4) possesses a unique weak
solution u ∈ C ([0, T0]→ Bs;p,p(R)) defined on a (possibly shorter) time interval
[0, T0] ⊂ [0, T ] of some positive length T0 ∈ (0, T ].

(ii) Furthermore, if u0 : R1 → C possesses a (unique) holomorphic extension to
a complex strip X(r0) ⊂ C, 0 < r0 ≤ A, denoted by u0 : X(r0) → C again, such that

N(r0)(u0) := sup
y∈[−r0,r0]

‖u0(·+ iy)‖Bs;p,p(R) <∞

holds (cf. (3.10) below), then also any (global) weak solution u ∈ C ([0, T ]→ Bs;p,p(R))

can be (uniquely) extended to a holomorphic function in X(A′) × ∆T ′,T
ϑ , denoted

again by u(x+iy, t), where all numbers A′ ∈ (0, A], T ′ ∈ (0, T ], and ϑ ∈ (0, π/2) are

sufficiently small, T ′ ·tanϑ ≤ B, and u(·+iy, ·) : t 7→ u(·+iy, t) : ∆̄T ′,T
ϑ → Bs;p,p(R)

is continuous for every fixed y ∈ [−r0, r0] together with

sup
t∈∆T ′,T

ϑ

N(A′)(u(·+ iy, t)) = sup
(y,t)∈[−r0,r0]×∆T ′,T

ϑ

‖u(·+ iy, t)‖Bs;p,p(RN ) <∞ .

In particular, the extension u is holomorphic in X(A′)×T(B′)
T ′,T with B′ = T ′ ·tanϑ ≤

B, where T ′ > 0 and ϑ > 0 are small enough.

We remark that T
(B′)
T ′,T ⊂ ∆T ′,T

ϑ ⊂ T
(B)
0,T , by B′ = T ′ · tanϑ ≤ B. If 0 < T0 < T in

Part (i) of this theorem, then we have to replace T by T = T0 in part (ii). Notice
that the condition that both (continuous) partial derivatives ∂f/∂u and ∂f/∂η are

locally bounded in X(A) × T
(B)
0,T × C2 is equivalent with (u, η) 7→ f(x, t;u, η) being

locally Lipschitz continuous.

Remark 1.2. It follows easily from Theorem 1.1 that every weak solution u ∈
C ([0, T ]→ Bs;p,p(R)) to the Cauchy problem (1.4) is classical in the sense that it
is of class C∞ over the open set R×(0, T ) and satisfies (1.4) pointwise and the initial
condition u(·, 0) = u0 ∈ Bs;p,p(R) in theBs;p,p(R)-limit, i.e., ‖u(·, t)−u0‖Bs;p,p(R) →



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 29

0 as t → 0+. For the special case of the reaction function f : (u, η) 7→ f(x, t;u, η)
being linear, a weak solution u ∈ C

(
[0, T ]→ L2(R)

)
to the (linear) Cauchy problem

(1.4) is defined e.g. in Evans [26], Chapt. 7, §1.1, p. 352, or Lions [62], Chapt. IV,
§1, p. 44, or [63], Chapt. III, (1.11), p. 102. In that case the initial condition
u(·, 0) = u0 ∈ L2(R) holds in the sense of the L2(R)-limit, ‖u(·, t)−u0‖L2(R) → 0 as
t→ 0+. The main reason why we prefer to work with the notion of a weak solution
as opposed to a classical solution of the Cauchy problem (1.4) is the fact that
already a weak solution is unique. The uniqueness of a weak solution is an important
technical argument in our proofs of Theorem 1.1 and Theorem 3.4 (Section 3).

In fact, we work sometimes also with the so called mild solutions to the Cauchy
problem (1.4) that make sense in C

(
[0, T ]→ L2(R)

)
; cf. Takáč [82, Sect. 3 and

4], even though we use them in the Besov space Bs;p,p(R) in place of L2(R). Mild
solutions do not require any additional regularity knowledge as they are defined
by the well known variation-of-constants formula (Pazy [72, §5.7, p. 168]). Thus,
they are even “weaker” than the weak solutions, but in our situation one can easily
verify that every mild solution is also a weak solution to problem (1.4) and vice
versa; see e.g. Ball [10] (or [72, Theorem on p. 259]).

The same remarks apply also to the more general Cauchy problem (1.1).

This article is organized as follows. We introduce some basic notation (mostly
complex domains) in Section 2. Our main analyticity result, Theorem 3.4, supple-
mented by an additional explanation in Proposition 3.5, is stated in Section 3. Their
proofs are gradually built up in Sections 4 through 8: First, the Cauchy problem
in RN × (0, T ) is treated as an abstract initial value problem in Section 4. There,
an important abstract a priori Bs;p,p-type estimate is established in Theorem 4.7.
Analyticity in time for this abstract problem is proved in Section 5 (Theorem 5.3).
Then, in Section 6, we treat analyticity in space for the semilinear parabolic Cauchy
problem in RN × (0, T ), see Proposition 6.5, provided the initial data are already
analytic (in space). We show that the analyticity in space is preserved for all times
in [0, T ]. We combine the time and space analyticity results from Sections 5 and 6
into Theorem 7.1 in Section 7. This theorem is still only “local” in time. Our main
analyticity result, Theorem 3.4, is proved in Section 8, together with Proposition 3.5
and, in particular, the “regularity” estimates (3.13) and (3.14). Section 9 treats an
application to a Risk Model in Mathematical Finance. Finally, Section 10 contains
some historical remarks and comments concerning the analyticity of solutions to
linear and semilinear elliptic and parabolic systems and its applications to relevant
classical problems.

2. Notation

We stick to the classical notation N = {1, 2, 3, . . . }, Z+ = {0, 1, 2, 3, . . . } =
N∪ {0}, and Z = {0,±1,±2,±3, . . . } = Z+ ∪ (−Z+), together with R = (−∞,∞),
R+ = [0,∞), and C = R + iR ∼= R2. Typically, we denote by x = (x1, x2, . . . , xN )
and y = (y1, y2, . . . , yN ) points in RN and by z = (z1, z2, . . . , zN ) points in CN .
We often write ζ = ξ + iη for ζ ∈ C and ξ, η ∈ R, i.e., <eζ = ξ and =mζ = η
are the real and imaginary parts of ζ ∈ C, respectively. Similarly, z = x + iy for
z ∈ CN and x, y ∈ RN , or equivalently zi = xi + iyi (i = 1, 2, . . . , N) for zi ∈ C
and xi, yi ∈ R, i.e., <ez = x and =mz = y. Hence, we identify CN = RN ⊕ iRN
(or simply CN = RN + iRN ) as vector spaces over the field R and thus consider
RN to be a (vector) subspace of CN . We use a bar (¯) to denote the complex



30 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

conjugate ζ̄ of a number ζ ∈ C. The complex conjugate of a vector z ∈ CN
is denoted by z̄ = (z̄1, z̄2, . . . , z̄N ). Similarly, the complex conjugate function of
a complex-valued function f(z) (for f : CN → C, for instance) is denoted by

f̄(z) ≡ ¯f(z). Furthermore, we denote by (z, w) =
∑N
i=1 ziw̄i the standard Euclidean

inner product of z, w ∈ CN and by |z| =
(∑N

i=1 |zi|2
)1/2

the induced (Euclidean)

norm of z ∈ CN . We will often use the sum (`1-) and the maximum (`∞-) norms
of z ∈ CN , respectively:

|z|1 =

N∑
i=1

|zi| and |z|∞ = max
1≤i≤N

|zi| .

Finally, we write z · w =
∑N
i=1 ziwi for the bilinear product of z, w ∈ CN , which is

not to be confused with the inner product (z, w) =
∑N
i=1 ziw̄i if w 6∈ RN (which is

sesquilinear). The Euclidean (`2-) norm of z ∈ CN is abbreviated as |z| ≡ |z|2 =√
(z, z) =

(∑N
i=1 |zi|2

)1/2
.

The vector space (over the field R) of all real-valued (square) M ×M matrices
A = (aij)

M
i,j=1 is denoted by RM×M . Similarly, the vector space (over the field C)

of all complex-valued M ×M matrices A is denoted by CM×M .
Given r ∈ (0,∞), we denote by Q(r) = (−r, r)N = {y ∈ RN : |y|∞ < r} the

N -dimensional open cube in RN (centered at the origin) with side lengths 2r, and
by Q̄(r) = [−r, r]N its closure.

To formulate our main hypotheses, given r, T ∈ (0,∞) and T ′ ∈ [0, T ), we
introduce the following complex domains for the complexifications of the spatial
and temporal variables, x ∈ RN and t ∈ (0, T ), respectively:

X(r) := {z = x+ iy ∈ CN : |y|∞ < r} = RN + iQ(r) ,

T
(r)
T ′,T := {t = σ + iτ ∈ C : T ′ < σ < T and |τ | < r} ;

(2.1)

see (1.8).
The former, X(r), is a tube (often called a strip) in CN with base Q(r) and

the latter, T
(r)
T ′,T , is an open rectangle in the complex plane C. Notice that T

(r)
T ′,T

contains the interval (T ′, T ). The remaining temporal domains, ∆ϑ and ∆
(T ′)
ϑ , have

been introduced in (1.5) and (1.6), respectively, with the angle ϑ ∈ (0, π/2) given
by tanϑ = r/T ′.

Our techniques will use holomorphic semigroups in an open sector ∆ϑ ⊂ C
defined in (1.5), with a given angle ϑ ∈ (0, π/2), but often locally in time in an open

triangle ∆
(T ′)
ϑ ⊂ C defined in (1.6), where 0 < T ′ < ∞. Their respective closures

in C are denoted by ∆̄ϑ and ∆̄
(T ′)
ϑ ; both contain the origin 0 ∈ C. Finally, for

0 < T ′ ≤ T <∞ we recall the definition of the temporal domain ∆T ′,T
ϑ introduced

in (1.7) with the angle ϑ ∈ (0, π/2) given by tanϑ = r/T ′. Its closure in C is

denoted by ∆̄T ′,T
ϑ . Clearly, ξ + ∆

(T ′)
ϑ = {ξ + t′ ∈ C : t′ ∈ ∆

(T ′)
ϑ }. Of course, if

T = T ′ then ∆T ′,T
ϑ = ∆

(T ′)
ϑ is an open triangle.

Throughout this article we work with complex-valued functions; hence, all Ba-
nach and Hilbert spaces of functions we consider are complex (over the field C). We
work with the standard inner product in L2(RN ) defined by (u, v)L2 :=

∫
RN uv̄ dx

for u, v ∈ L2(RN ). The induced norm is abbreviated by ‖u‖L2 ≡ ‖u‖L2(RN ). We

warn the reader that we identify the dual space H ′ =
(
L2(RN )

)′
of the complex



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 31

Hilbert space H = L2(RN ) with H itself by means of the (complex) Riesz rep-
resentation theorem which yields an anti-linear isomorphism of H onto H ′ (cf.
Adams and Fournier [1, Chapt. 2], Theorem 2.44, p. 47).

The following notation is taken from Krantz [57, Chapt. 0]. Given a domain Ω
in Rr, we denote by Ck(Ω) (k ∈ Z+) the vector space of all k-times continuously
differentiable functions f : Ω→ C and by Ck(Ω̄) the vector space of all f : Ω̄→ C
such that f |Ω ∈ Ck(Ω) and each partial derivative ∂|α|f

∂xα of f (α ∈ (Z+)r) of order

|α| ≤ k can be extended to a continuous function on Ω̄. Of course, f |Ω stands for
the restriction of f to Ω and all partial derivatives are taken in the real variable
sense (x ∈ Rr). In case Ω ⊂ Cr = Rr ⊕ iRr ∼= R2r (r ∈ N) is a complex domain,

the (mixed) partial derivative∂
|α|+|β|f
∂xα ∂yβ

of f (α, β ∈ (Z+)r) of order |α|+ |β| ≤ k is

taken in the real variable sense, where x, y ∈ Rr in (x, y) ∼= z = x + iy ∈ Cr. The
vector spaces Ck(Ω) and Ck(Ω̄) are defined analogously, with |α| + |β| ≤ k. If Ω
is bounded then Ck(Ω̄) is a Banach space endowed with a maximum-type norm.
If Ω is not bounded in general then we denote by C0

unif(Ω̄) the vector space of all
uniformly continuous functions f : Ω̄→ C, and by C0

bdd(Ω) = C0(Ω) ∩ L∞(Ω) the
vector space of all bounded continuous functions f : Ω → C. Alternatively, if Ω
is not bounded (in Rr or Cr endowed with the Euclidean metric d) then we may

consider a compactification Ω̃ of Ω, i.e., a compact metric space Ω̃ (endowed with

a metric d̃) such that there is a homeomorphism j : Ω → Ω̃ of Ω onto a dense

subset j(Ω) of Ω̃, such that for any pair of sequences {xn}∞n=1, {yn}∞n=1 ⊂ Ω we

have d(xn, yn) → 0 if and only if d̃(j(xn), j(yn)) → 0 as n → ∞. Clearly, by

identifying Ω with the subset j(Ω) of Ω̃ we identify Ω with a dense subset of Ω̃.

Hence, we can identify C0(Ω̃) with a vector subspace of C0
bdd(Ω) ∩ C0

unif(Ω̄). As a

simple example, we may take Ω̃ to be the one-point compactification of a domain

Ω ⊂ Rr (or Ω ⊂ Cr). In particular, we have R̃r = Rr ∪ {∞} and C̃r = Cr ∪ {∞}
endowed with the metric d̃ defined in the next section (Section 3, (3.2)).

Finally, if Ω ⊂ Cr is a complex domain, we denote by A(Ω) the Fréchet space
of all holomorphic functions f : Ω → C endowed with the (complete metrizable)
topology of uniform convergence on compact subsets of Ω. As usual, we abbreviate
the Cauchy-Riemann partial differential operators

∂

∂zi
:=

1

2

( ∂

∂xi
− i

∂

∂yi

)
and

∂

∂z̄i
:=

1

2

( ∂

∂xi
+ i

∂

∂yi

)
. (2.2)

Remark 2.1. We will often use the following classical fact; see e.g. John [50,
Theorem, p. 70] or Krantz [57, Definition II, p. 3]: Let Ω ⊂ CN be a complex
domain (N ≥ 1). A continuously differentiable function h : Ω → C (in the real
variable sense, partially with respect to xi, yi ∈ R; i = 1, 2, . . . , N) is holomorphic
(i.e., complex analytic) if and only if it verifies the Cauchy-Riemann equations in
Ω, i.e., ∂h/∂z̄i = 0 in Ω; for i = 1, 2, . . . , N .

3. Statement of main results

Let us abbreviate the differential operators (and derivatives)

Dx :=
1

i

∂

∂x
=
(1

i

∂

∂xi

)N
i=1

, ∂t :=
∂

∂t
,



32 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

Dα
x := i−|α|

∂|α|

∂xα
= i−|α|

∂|α|

∂xα1
1 . . . ∂xαNN

for α = (αi)
N
i=1 ∈ (Z+)N .

We assume that the operator

P(x, t,Dx) =
∑

|α|,|β|≤m

Dα
x

(
Pαβ(x, t)Dβ

x

)
≡

∑
|α|,|β|≤m

i−|α|−|β|
∂|α|

∂xα

(
Pαβ(x, t)

∂|β|

∂xβ

)
,

(3.1)

for (x, t) ∈ RN × (0, T ), is a linear partial differential operator of order 2m in
divergence form with the coefficients i−|α|−|β|Pαβ(x, t) indexed by α, β ∈ (Z+)N

with |α| ≤ m and |β| ≤ m, where each Pαβ(x, t) = (Pαβjk )Mj,k=1 is an M ×M matrix

with real (or complex) entries Pαβjk = Pαβjk (x, t). The reader is referred to Friedman

[32, Part. 1, Sect. 12, pp. 32–37] or John [50, Chapt. 6, Sect. 2, pp. 190–195] for
general facts about such operators.

Let us abbreviate the product domain Ω = X(r0) ×∆T0,T
ϑ0

⊂ CN × C, with some

r0 ∈ (0,∞), 0 < T0 ≤ T < ∞, and ϑ0 ∈ (0, π/2); the closure of Ω in CN × C is

denoted by Ω̄. We introduce also a compactification Ω̃ of Ω,

Ω̃ = X̃(r0) × ∆̄T0,T
ϑ0

=
(
R̃N + iQ̄(r)

)
× ∆̄T0,T

ϑ0

∼= R̃N × Q̄(r) × ∆̄T0,T
ϑ0

,

where R̃N = RN ∪{∞} denotes the one-point compactification of RN endowed with
the standard metric

d̃(x, y) = d̃(y, x) =


|x−y|

1+|x−y| if x, y ∈ RN ;

1 if x ∈ RN , y =∞;

0 if x = y =∞ .

(3.2)

Hence, X̃(r0) = R̃N + iQ̄(r) ∼= R̃N × Q̄(r) is a compactification of X(r0) ⊂ CN .
We assume that the operator P and the function f satisfy the following hypothe-

ses in the product domain Ω = X(r0) ×∆T0,T
ϑ0

defined in (1.7) and (2.1).

3.1. Hypothesis.

(H1) For each pair α, β ∈ (Z+)N with |α| ≤ m and |β| ≤ m, the entries Pαβjk :

Ω̄ → C (j, k = 1, 2, . . . ,M) of the coefficient Pαβ = (Pαβjk )Mj,k=1 belong to

C1(Ω̄)∩L∞(Ω)∩A(Ω). Moreover, we assume that also all partial derivatives
∂|α
′|

∂xα′
Pαβjk (x, t) of order |α′| ≤ |α| (α′ ∈ (Z+)N ) are in C1(Ω̄). The entries

Pαβjk of the leading coefficients (|α| = |β| = m) are assumed to belong also

to C0
unif(Ω̄) besides being in C1(Ω̄) ∩ L∞(Ω) ∩A(Ω).

This is the case if the entries Pαβjk of the leading coefficients

(|α| = |β| = m) belong also to C0(Ω̃) besides being in C1(Ω̄) ∩
L∞(Ω)∩A(Ω). This claim follows directly from C0(Ω̃) ⊂ C0

unif(Ω̄)

which means that any continuous function f : Ω̃→ C is uniformly

continuous on Ω̃ with the restriction f |Ω to Ω being uniformly
continuous and, thus, f |Ω : Ω → C possesses a (unique) contin-

uous extension f̃ : Ω̄ → C to Ω̄ which turns out to be uniformly
continuous, as well.



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 33

(H2) Operator P is strongly elliptic in Ω̄, i.e., there exists a constant c ∈ (0,∞)
such that the inequality

<e
( M∑
j,k=1

∑
|α|=|β|=m

Pαβjk (z, t) ξα+β ηkη̄j

)
≥ c |ξ|2m |η|2 (3.3)

holds for all (z, t) ∈ Ω̄ and for all ξ = (ξ1, . . . , ξN ) ∈ RN and η =

(η1, . . . , ηM ) ∈ CM , where ξα+β = ξα1+β1

1 . . . ξαN+βN
N and α = (α1, . . . , αN )

is in (Z+)N , and β = (β1, . . . , βN ) is in (Z+)N .

(H3) The components fj : Ω̄ × CMÑ → C (j = 1, 2, . . . ,M) of the reaction
function f = (f1, . . . , fM ) belong to

C1(Ω̄× CMÑ ) ∩A(Ω× CMÑ ) .

(Recall that the integer Ñ is defined in (1.2).) Moreover, we assume that,

for every bounded subset Σ ⊂ CMÑ , their first-order time derivatives
∂
∂tfj(x, t;X) together with their first-order partial derivatives

∂

∂Xβ,k
fj(x, t;X) , for |β| ≤ m and j, k = 1, 2, . . . ,M ,

with respect to the componentsXβ,k of the vectorXβ = (Xβ,1, . . . , Xβ,M ) =
∂|β|u
∂xβ

∈ CM (or CM ) are uniformly bounded on the set Ω × Σ. Finally, we

assume that the function f : Ω̄× CMÑ → CM satisfies∫
RN
|f(x+ iy, t;~0)|p dx ≤ Kp for all y ∈ Q̄(r0) and t ∈ ∆̄T0,T

ϑ0
, (3.4)

where K ∈ (0,∞) is a constant and ~0 := (0)|β|≤m ≡ (0, . . . , 0) ∈ CMÑ .

Remark 3.1. The local boundedness condition in (H3) on the first-order partial

derivatives ∂
∂tfj(x, t;X) and

∂fj
∂Xβ,k

(x, t;X) on the set Ω× Σ will be used later (cf.

(6.8) in Section 6) in the following equivalent form:

For a bounded subset Σ ⊂ CMÑ and indices j = 1, 2, . . . ,M , there is a constant
Cj ≡ Cj(Σ) ∈ (0,∞) such that the following inequalities,∣∣∂fj

∂t
(x, t;X)

∣∣ ≤ Cj(Σ) and
∣∣ ∂fj
∂Xβ,k

(x, t;X)
∣∣ ≤ Cj(Σ) (3.5)

for all (x, t) ∈ Ω and for all X = (Xβ)|β|≤m ∈ Σ, hold for all β = (β1, . . . , βN ) ∈
(Z+)N with |β| ≤ m and for all k = 1, 2, . . . ,M .

A simple, but more restrictive alternative to formulate (H1)–(H3) is to replace

Ω = X(r0) × ∆T0,T
ϑ0

by a larger, but simpler product domain Ω0 = X(r0) × T
(τ0)
0,T

(defined in (1.8) and (2.1)) with τ0 = T0 · tanϑ0; hence, Ω ⊂ Ω0, thanks to ∆T0,T
ϑ0

⊂
T

(τ0)
0,T .

Let us recall our abbreviation X = (Xβ)|β|≤m ∈ RMÑ (or CMÑ ) with Ñ =∑m
k=0N

k from (1.2) and make it more precise as follows: When dealing with com-
plex (partial) derivatives of the function fj with respect to the variableXβ,k, we pre-
fer to replaceXβ,k by the complex variable Zβ,k = Xβ,k+iYβ,k ∈ C (Xβ,k, Yβ,k ∈ R);

j, k = 1, 2, . . . ,M , and write
∂fj
∂Zβ,k

(x, t;Z) in place of
∂fj
∂Xβ,k

(x, t;X).



34 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

The strong ellipticity inequality (3.3) can be improved as follows; cf. Takáč [82,
Remark 3.1, p. 57]:

Remark 3.2. In a smaller domain Ω′ = X(r0) × ∆T0,T
ϑ′0

⊂ Ω, with some number

ϑ′0 ∈ (0, ϑ0] (small enough), inequality (3.3) holds in the following qualitatively
stronger form, cf. Agmon [2], Theorem 7.12, ineq. (7.21) on p. 87:

<e
(

eiθ ·
M∑

j,k=1

∑
|α|=|β|=m

Pαβjk (z, t) ξα+β ηkη̄j

)
≥ c′ |ξ|2m |η|2 (3.3′)

for all θ ∈ [−ϑ′0, ϑ′0], for all (z, t) ∈ Ω̄′, and for all ξ = (ξ1, . . . , ξN ) ∈ RN and
η = (η1, . . . , ηM ) ∈ CM , where c′ ∈ (0, c] is a constant. Recall that 0 < ϑ′0 ≤ ϑ0

and Ω′ ⊂ Ω. Consequently, without loss of generality, we may remove the prime (′)
from both ϑ′0 and c′ in (3.3′) and assume that

<e
(

eiθ ·
M∑

j,k=1

∑
|α|=|β|=m

Pαβjk (z, t) ξα+β ηkη̄j

)
≥ c |ξ|2m |η|2 (3.6)

for all θ ∈ [−ϑ0, ϑ0] and for all (z, t) ∈ Ω̄, ξ = (ξ1, . . . , ξN ) ∈ RN , and η =
(η1, . . . , ηM ) ∈ CM , where c > 0 is a constant. We prefer to use inequality (3.6) in
place of (3.3).

The G̊arding inequality (in the whole space RN ) below is an important conse-
quence of inequality (3.6); see e.g. Agmon [2, Theorem 7.6, p. 78]:

Corollary 3.3 (G̊arding’s inequality). Under Hypotheses (H1), (H2), there exist
constants c1 and c2, c1 > 0 and 0 ≤ c2 <∞, such that

<e
[
eiθ ·

∑
|α|=|β|=m

∫
RN

¯Dα
xw ·Pαβ(x+ iy, t)Dβ

xw dx
]

≥ c1
∑
|α|=m

‖Dα
xw‖2L2(RN ) − c2‖w‖

2
L2(RN )

(3.7)

holds for all w ∈Wm,2(RN ) and for all θ ∈ [−ϑ0, ϑ0], y ∈ Q̄(r0), and t ∈ ∆̄T0,T
ϑ0

.

Proof. The reader is referred to Agmon [2, Theorem 7.6, pp. 78–86] for a proof.
We remark that the proof of G̊arding’s inequality ([2, Lemma 7.9, p. 81]) requires
the uniform equicontinuity of the leading coefficients Pαβ(x + iy, t) as functions

of x ∈ RN parametrized by y ∈ Q̄(r0) and t ∈ ∆̄T0,T
ϑ0

, where Pαβ(x + iy, t) =

(Pαβjk )Mj,k=1 for |α| = |β| = m. This is guaranteed by our Hypothesis (H1) that all

Pαβjk (|α| = |β| = m) belong to C0
unif(Ω̄) as functions of (z, t) = (x+ iy, t) ∈ Ω̄. �

To give a natural lower estimate on the domain of holomorphy (i.e., the domain
of complex analyticity) of a weak solution u to the Cauchy problem (1.1), we
introduce a few more subsets of CN × C (cf. [82, p. 58] or [83, p. 428]).

We use the subdomain Γ
(T ′)
T (r′, ϑ′) = X(r′) ×∆T ′,T

ϑ′ of the (larger) domain

Ω = Γ
(T0)
T (r0, ϑ0) := X(r0) ×∆T0,T

ϑ0
⊂ CN × C (3.8)

defined above (H1)–(H3). The three constants T ′ ∈ (0, T0], r′ ∈ (0, r0], and ϑ′ ∈
(0, ϑ0] used below will be specified later (in Theorem 3.4).



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 35

We recall the tube X(r0) = RN + iQ(r0) (called often a strip) in CN with base

Q(r0) ⊂ RN defined in (2.1) and recall from (1.6) also the definition of the set ∆
(T ′)
ϑ

in C. In formula (1.7) we employ the time translation of ∆
(T ′)
ϑ by r units, i.e., the

set r+∆
(T ′)
ϑ , to define the union ∆T ′,T

ϑ of such translations for 0 ≤ r ≤ T−T ′ <∞.
It is easy to see that, for 0 < s ≤ T <∞ and 0 < ϑ0 < π/2, we have

∆s,T
ϑ0

= ∆
(s)
ϑ0
∪ ([s, T ) + i(−s · tanϑ0, s · tanϑ0))

=
{
t = σ + iτ ∈ C : 0 < σ < T and |τ | < r · tanϑ0 where r = min{σ, s}

}
.

(3.9)

Recall that the closure of ∆s,T
ϑ0

(and ∆
(s)
ϑ0

, respectively) in C is denoted by ∆̄s,T
ϑ0

(and ∆̄
(s)
ϑ0

). Given any r ∈ [0, T ), we observe that the (real) time r section of ∆s,T
ϑ0

is given by

{t ∈ ∆s,T
ϑ0

: <et = r} = r + i(−r′ · tanϑ0, r
′ · tanϑ0) ⊂ C

where r′ = min{r, s}. These sets in the complex plane C have already been sketched
in Figure 1 above and will be sketched more precisely in Figure 2 below.

The Cartesian product Γ
(s)
T (r0, ϑ0) = X(r0) × ∆s,T

ϑ0
defined in (3.8) is our most

important complex analyticity domain in CN × C. Recall that Bs;p,p(RN ) =(
Lp(RN ), W 2m,p(RN )

)
1−(1/p),p

↪→ Lp(RN ) with p > 2 + N
m . Our main result reads

as follows.

Theorem 3.4. Let m,M,N ≥ 1, p > 2 + N
m , 0 < T <∞, and assume that (H1)–

(H3) are satisfied in the product domain Ω = Γ
(T0)
T (r0, ϑ0) = X(r0)×∆T0,T

ϑ0
⊂ CN×C

with some constants 0 < r0 < ∞, 0 < T0 ≤ T , and 0 < ϑ0 < π/2; cf. (1.7) and
(2.1).

(i) For t0 ∈ [0, T ) and any initial value u0 ∈ Bs;p,p(RN ) at time t = t0, there is a
number T1 ∈ (t0, T ], depending on t0 and u0, such that the Cauchy problem (1.1)
on the (local) time interval [t0, T1] ⊂ [0, T ] with the initial condition u(·, t0) = u0

in Bs;p,p(RN ) possesses a unique weak solution u ∈ C
(
[t0, T1]→ Bs;p,p(RN )

)
.

(ii) Furthermore, for any initial data u0 ∈ Bs;p,p(RN ) at time t = 0, any (global)
weak solution u ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
of the Cauchy problem (1.1), if it exists,

is always unique and it possesses a unique (temporal) extension to the space-time

domain RN × ∆̄T ′,T
ϑ′ , denoted again by u, such that the Bs;p,p(RN )-valued function

u : ∆̄T ′,T
ϑ′ 7→ Bs;p,p(RN ) is continuous in ∆̄T ′,T

ϑ′ and its restriction to ∆T ′,T
ϑ′ is

holomorphic, provided the numbers T ′ ∈ (0, T0] and ϑ′ ∈ (0, ϑ0] are small enough.
This temporal extension is a unique weak solution to the Cauchy problem (1.1) in

RN × ∆̄T ′,T
ϑ′ .

(iii) If, in addition to u0 ∈ Bs;p,p(RN ), u0 possesses a (unique) holomorphic ex-
tension ũ0 : X(κ0) → CM from RN to the complex domain X(κ0) ⊂ CN , for some
κ0 ∈ (0, r0], such that the function

ũ0(·+ iy) : x 7→ ũ0(x+ iy) : RN → CM

belongs to Bs;p,p(RN ) for each y ∈ Q(κ0) and

N(κ0)(ũ0) := sup
y∈Q(κ0)

‖ũ0(·+ iy)‖Bs;p,p(RN ) <∞ , (3.10)



36 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

then also any (global) weak solution u ∈ C
(
[0, T ]→ Bs;p,p(RN )

)
of the Cauchy

problem (1.1), if it exists, possesses a unique extension to the space-time domain

X(r′) × ∆̄T ′,T
ϑ′ , denoted by ũ, with the following properties, provided the numbers

T ′ ∈ (0, T0], r′ ∈ (0, κ0], and ϑ′ ∈ (0, ϑ0] are small enough:

(iii.1) ũ(·+ iy, t) ∈ Bs;p,p(RN ) holds for all (y, t) ∈ Q(r′) × ∆̄T ′,T
ϑ′ , together with

sup
t∈∆̄T ′,T

ϑ′

N(r′) (ũ(·, t)) = sup
t∈∆̄T ′,T

ϑ′

sup
y∈Q(κ0)

‖ũ(·+ iy, t)‖Bs;p,p(RN ) <∞ , (3.11)

(iii.2) the function

ũ : (y, t) 7→ ũ(·+ iy, t) : Q(r′) × ∆̄T ′,T
ϑ′ → Bs;p,p(RN )

is continuous in the space-time variable (y, t) ∈ Q(r′) × ∆̄T ′,T
ϑ′ ,

(iii.3) ũ is holomorphic in the complex domain

Γ
(T ′)
T (r′, ϑ′) = X(r′) ×∆T ′,T

ϑ′ ⊂ X(κ0) ×∆T0,T
ϑ0

⊂ Ω = Γ
(T0)
T (r0, ϑ0) = X(r0) ×∆T0,T

ϑ0
.

Finally, the extension ũ verifies the partial differential equation in the Cauchy

problem (1.1) pointwise in Γ
(T ′)
T (r′, ϑ′), i.e., in the classical sense, and obeys the

initial data as follows,

‖ũ(·+ iy, t)− ũ0(·+ iy)‖Bs;p,p(RN ) → 0 as t→ 0 , t ∈ ∆
(T ′)
ϑ′ , (3.12)

for every y ∈ Q(r′).

In Part (iii), properties (iii.1) and (iii.2) combined with the Sobolev(-Besov)
imbedding Bs;p,p(RN ) ↪→ Cm(RN ) ∩ Wm,∞(RN ) guarantee the continuity and

boundedness of the function ũ : X(r′) × ∆̄T ′,T
ϑ′ → CM .

Our condition p > 2 + N
m is natural (and sharp) to guarantee the continuity of

the Sobolev imbedding

Bs;p,p(RN ) = [Bs;p,p(RN )]M

↪→ Cm(RN ) ∩Wm,∞(RN ) = [Cm(RN ) ∩Wm,∞(RN )]M

which follows from the Sobolev inequalities and the Sobolev imbedding

Bs−m;p,p(RN ) ↪→ C0
bdd(RN ) := C(RN ) ∩ L∞(RN )

for N/p < s−m <∞; see, e.g., Adams and Fournier [1, Chapt. 7], Theorem 7.34(c),
p. 231. In our proposition below we abbreviate ς1(s) = min{s, 1} for s ∈ R+.

The above proposition follows directly from our proof of Part (iii) of Theorem 3.4.
The temporal integration path in the double space-time integrals (3.13) and (3.14)
is sketched in Figure 2.

Proposition 3.5. In the situation of Theorem 3.4 above, there exist constants
c0 > 0 and C0 > 0 depending solely on κ0, ϑ0, K, T ′, r′, ϑ′ and the supremum
norm

|||u|||L∞(0,T ) := ‖u‖C([0,T ]→Bs;p,p(RN )) = sup
0≤t≤T

‖u(·, t)‖Bs;p,p(RN ) (<∞)



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 37

t = (1 + iτ
T ′ )s t = s+ iτ

τ = =mt

τ0

τ

0
s T ′ s T

σ = <et

−τ0

Figure 2. Here is a more detailed version of Figure 1 used in Proposition 3.5.

of the (global) weak solution u ∈ C
(
[0, T ]→ Bs;p,p(RN )

)
, such that the following

estimate holds for all pairs (y, t) ∈ Q(r′) × ∆̄T ′,T
ϑ′ with t = σ + iτ (σ, τ ∈ R):∫ σ

0

∫
RN
|∂tũ (x+ iy, s+ iς1(s/T ′)τ) |p dxds

+ c0
∑
|α|≤2m

∫ σ

0

∫
RN
|Dα

x ũ (x+ iy, s+ iς1(s/T ′)τ)|p dxds ≤ C0 .
(3.13)

Similarly, there are constants c′0 > 0 and C ′0 > 0 depending solely on the constants

c0 and C0, such that the following estimate holds for all pairs (y, t) ∈ Q(r′)× ∆̄T ′,T
ϑ′

with t = σ + iτ (σ, τ ∈ R):

‖ũ(·+ iy, σ + iς1(σ/T ′)τ)‖p
Bs;p,p(RN )

+ c′0
∑
|α|≤2m

∫ σ

0

∫
RN
|Dα

x ũ (x+ iy, s+ iς1(s/T ′)τ)|p dxds ≤ C ′0 .
(3.14)

Remark 3.6. (See Figure 2). Notice that, in (3.13) and (3.14), the temporal
argument in the function Dα

xu (x+ iy, s+ iς1(s/T ′)τ) reads s+ iς1(s/T ′)τ = (1 +
i(τ/T ′))s whenever 0 ≤ s < T ′ <∞, whereas s+iς1(s/T ′)τ = s+iτ holds whenever
0 < T ′ ≤ s (≤ σ ≤ T <∞).

4. Abstract Cauchy problem in an interpolation space

We assume that E = (E0, E1) is a Banach couple, that is, E0, E1 are Banach
spaces such that E1 is densely and continuously imbedded into E0, i.e., E1 ↪→ E0.
We consider only complex Banach spaces over the field C. Given a number 1 < p <
∞, we denote by

E1− 1
p ,p
≡ (E0, E1)1− 1

p ,p

the real interpolation space between E1 and E0 obtained by the trace method as
follows, with the paremeter θ = 1 − 1

p ∈ (0, 1). We define such an interpolation

space for any θ ∈ (0, 1) below, cf. Lunardi [65, Chapt. 1], §1.2.2, pp. 20–25. The



38 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

reader is referred to Adams and Fournier [1, Chapt. 7], §7.6–§7.23, pp. 208–221, or
Triebel [84, Chapt. 1], §1.8, pp. 41–55, for further details. The trace spaces were
originally introduced in Lions [59, 60, 61].

Let Xp
θ denote the Banach space of all Bochner-measurable functions u : R+ →

E0 endowed with the weighted Lebesgue norm

‖u‖Xpθ :=
(∫ ∞

0

‖t1−θ u(t)‖pE0

dt

t

)1/p

≡
(∫ ∞

0

‖u(t)‖pE0

dt

t1−(1−θ)p

)1/p

<∞ . (4.1)

Notice that Xp

1− 1
p

= Lp(R+ → E0). Analogously, we define the Banach space Y pθ of

all functions u ∈ Xp
θ with the following properties: u can be identified (by equality

a.e. in R+) with a Bochner-measurable function u : R+ → E1 satisfying

[u]Y pθ :=
(∫ ∞

0

‖t1−θu(t)‖pE1

dt

t

)1/p

≡
(∫ ∞

0

‖u(t)‖pE1

dt

t1−(1−θ)p

)1/p

<∞ , (4.2)

and there is a function v ∈ Xp
θ , denoted by v = u′ in the sequel, such that the

equality

u(t2)− u(t1) =

∫ t2

t1

v(s) ds holds in E0 for all 0 < t1 ≤ t2 <∞ . (4.3)

Applying Hölder’s inequality to this equation, it is easy to show that every u ∈ Y pθ
is θ-Hölder continuous on any compact interval [0, T ] as a function valued in E0,
see, e.g., Lunardi [65, Chapt. 1], §1.2.2, p. 20. The Banach space Y pθ is endowed
with the norm

‖u‖Y pθ := [u]Y pθ + ‖u′‖Xpθ <∞ . (4.4)

For the special case θ = 1− 1
p , a useful equivalent norm on Y p

1− 1
p

is defined by

‖u‖]
Y p
1− 1

p

:=
(∫ ∞

0

‖u(t)‖pE1
dt+

∫ ∞
0

‖u′(t)‖pE0
dt
)1/p

. (4.5)

Thus,

Y p
1− 1

p

= Lp(R+ → E1) ∩W 1,p(R+ → E0)

is an abstract Sobolev space Amann [6, Chapt. III, §1.1], pp. 88–89).
Finally, the interpolation trace space is the vector space

Eθ,p ≡ (E0, E1)θ,p := {u(0) ∈ E0 : u ∈ Y pθ }
of the initial values x = u(0) ∈ E0 of all functions u ∈ Y pθ endowed with the trace
norm

‖x‖Eθ,p := inf
{
‖u‖Y pθ : x = u(0) for some u ∈ Y pθ

}
(4.6)

which makes the (linear) trace mapping τ : u 7→ u(0) : Y pθ → Eθ,p bounded
(i.e., continuous), with the operator norm ≤ 1. Equivalently to (4.6), we have
‖x‖Eθ,p ≤ ‖u‖Y pθ for every u ∈ Y pθ with u(0) = x and there exists a sequence

{un}∞n=1 ⊂ Y pθ such that un(0) = x and ‖un‖Y pθ → ‖x‖Eθ,p as n → ∞. It can be

shown that there is a constant c = c(θ, p) > 0, depending only on E = (E0, E1),
θ ∈ (0, 1), and p ∈ (1,∞), such that

‖x‖Eθ,p ≤ c‖x‖
1−θ
E0
‖x‖θE1

holds for all x ∈ E1 ; (4.7)

see, e.g., Triebel [84, Chapt. 1], Theorem 1.3.3(g), p. 25, combined with Theorem
1.8.2, pp. 44–45. As an easy consequence of the definition of Eθ,p for θ = 1− 1

p , i.e.,



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 39

(1 − θ)p = 1, one can show that the abstract Sobolev space Y p
1− 1

p

is continuously

imbedded into the Fréchet space C
(
R+ → E1− 1

p ,p

)
of all continuous functions u :

R+ → E1− 1
p ,p

endowed with the (locally convex) topology of uniform convergence

on every compact time interval [t1, t2] ⊂ R+,

Y p
1− 1

p

= Lp(R+ → E1) ∩W 1,p(R+ → E0) ↪→ C
(
R+ → E1− 1

p ,p

)
.

We complete our definition by setting Eθ,p := Eθ if θ ∈ {0, 1}.
In what follows we deal with applications of the interpolation trace space Eθ,p

(with θ = 1− 1
p ) to abstract linear and nonlinear evolutionary problems of type

du

dt
−A(t, u(t))u(t) = f(t, u(t)) + g(t) for a.e. t ∈ (0, T ) ;

u(0) = u0 ∈ E1− 1
p ,p

.
(4.8)

Here, u : (0, T )→ E0 is the unknown function valued in the Banach space E0 and
0 < T ≤ ∞. A rigorous definition of a strict solution u of the initial value problem
(4.8) will be given below, in Definition 4.4. Essentially, we follow Clément and Li
[20], Section 1, pp. 17–18. A closely related approach is carried out also in Köhne,
Prüss, and Wilke [55].

We denote by L(E1 → E0) the Banach space of all bounded (i.e., continu-
ous) linear operators B : E1 → E0 endowed with the standard operator norm
‖B‖L(E1→E0). Let us denote by I : E1 ↪→ E0 the continuous imbedding of E1 into
E0; hence, I ∈ L(E1 → E0). We identify I with the identity mapping in the whole
of E0 and abbreviate L(E0) ≡ L(E0 → E0).

If, for some complex number λ ∈ C, the operator λI − B ∈ L(E1 → E0) is
invertible with an inverse denoted by (λI − B)−1 : E0 → E1 ↪→ E0 such that
this inverse is bounded from E0 into itself, i.e., (λI − B)−1 ∈ L(E0 → E0), then
we alternatively (equivalently) view B as a densely defined, closed linear operator
B : E0 → E0 with the domain D(B) = E1, by the closed graph theorem, cf.
Amann [6, Chapt. I, Lemma 1.1.2], p. 10. Indeed, if the graph G(B) of B is closed
in E0 × E0, it is closed also in E1 × E0. In this case, the norm ‖ · ‖E1 on E1 is
equivalent with the graph norm

‖x‖D(B) := ‖Bx‖E0
+ ‖x‖E0

, x ∈ D(B) ,

on D(B) = E1. An important class of such operators, denoted by Gen(E) ≡
Gen(E1 → E0), is formed by all closed linear operators B : E0 → E0 with the
domain D(B) = E1 that generate a strongly continuous semigroup {etB : t ≥ 0}
on E0. We will consider only generators B with domain D(B) = E1. Finally,
we denote by Hol(E) ≡ Hol(E1 → E0) the subset of all (infinitesimal) generators
B ∈ Gen(E) that generate a holomorphic (i.e., analytic) semigroup on E0. We
refer to Amann [6, Chapt. I, §1], pp. 9–24, Pazy [72, Chapt. 1–2], pp. 1–75, or
Tanabe [81, Chapt. 3, §3.1–§3.4], pp. 51–72, for details about strongly continuous
(and holomorphic) semigroups.

Next, given an operator B ∈ Hol(E), let us consider the following special (linear)
case of problem (4.8), namely,

du

dt
−Bu(t) = g(t) for a.e. t ∈ (0, T ) ;

u(0) = u0 ∈ E1− 1
p ,p

.
(4.9)



40 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

Here, u0 ∈ E1− 1
p ,p

is a given initial value, g ∈ Lp((0, T )→ E0) is a given function,

1 < p < ∞, and 0 < T < ∞. In analogy with our definition of the Banach spaces
Xp
θ and Y pθ of functions u : R+ → E0 on the entire half line R+, endowed with the

norms given by eqs. (4.1) and (4.4), respectively, we introduce the corresponding
Banach spaces Xp

θ (0, T ) and Y pθ (0, T ) of functions u : [0, T ) → E0 on a bounded

interval [0, T ), 0 < T < ∞. Of course, in (4.1), the integral
∫∞

0
. . . dt

t has to

be replaced by
∫ T

0
. . . dt

t . It is not difficult to show that if one replaces the pair
of spaces Xp

θ and Y pθ by Xp
θ (0, T ) and Y pθ (0, T ), respectively, in the definition of

the trace space Eθ,p and its norm in (4.6), the same interpolation trace space is
obtained. These facts can be inferred easily from the treatment of trace spaces in
the monographs [1, 6, 65, 84] or from the original works by Lions [59, 60, 61]. In
particular, we have the continuous imbedding

Y p
1− 1

p

(0, T ) = Lp((0, T )→ E1) ∩W 1,p((0, T )→ E0) ↪→ C
(

[0, T ]→ E1− 1
p ,p

)
,

(4.10)
see, e.g., [1, Chapt. 7], §7.67, p. 255. Thus, the (linear) trace mapping

τ : u 7→ u(0) : Y p
1− 1

p

(0, T )→ E1− 1
p ,p

is continuous.
We say that a function u : [0, T ) → E0 is a strict solution of the initial value

problem (4.9) if

u ∈ Y p
1− 1

p

(0, T ) , τu ≡ u(0) = u0 ,

and the differential equation in (4.9) is satisfied with all terms in Xp

1− 1
p

(0, T ) =

Lp((0, T )→ E0).

Definition 4.1. An infinitesimal generator B ∈ Hol(E) of a holomorphic semi-
group on E0 with domain D(B) = E1 is said to possess the maximal Lp-regularity
property, symbolically B ∈ MRp(E) ≡ MRp(E1 → E0), if for any given initial
condition u0 ∈ E1− 1

p ,p
and any given function g ∈ Lp((0, T ) → E0), problem

(4.9) possesses a unique strict solution u ∈ Y p
1− 1

p

(0, T ) that satisfies the following

estimate:

There exists a constant M ≡M(p,E,B, T ) > 0, independent of u0

and g, such that∫ T

0

∥∥du

dt

∥∥∥∥du

dt

∥∥p
E0

dt+

∫ T

0

‖Bu(t)‖pE0
dt

≤M
(
‖u0‖pE

1− 1
p
,p

+

∫ T

0

‖g(t)‖pE0
dt
)
.

(4.11)

We have adopted this definition of class MRp(E) from Clément and Li [20, p. 18]
and from the monograph by Ashyralyev and Sobolevskii [9, Chapt. 1], §3.5, p. 28.
It may be viewed as some kind of ellipticity hypothesis for the linear operator
B ∈ L(E1 → E0) or a stability hypothesis for the linear parabolic problem (4.9).
Equivalently, the abstract (linear evolutionary) partial differential operator

(∂t −B, τ) : Y p
1− 1

p

(0, T )→ Lp((0, T )→ E0)× E1− 1
p ,p

,



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 41

defined for every u ∈ Y p
1− 1

p

(0, T ) = Lp((0, T )→ E1) ∩W 1,p((0, T )→ E0) by

(∂t −B, τ) : u 7→
(du

dt
−Bu(t) , u(0)

)
, (4.12)

possesses a bounded inverse furnished by the strict solution

u = (∂t −B, τ)
−1

(g, u0) ∈ Y p
1− 1

p

(0, T )

of problem (4.9); cf. Angenent [7, Lemma 2.2, p. 95] for the parallel interpolation
case p =∞ introduced in Da Prato and Grisvard [33].

Remark 4.2. (a) It is not difficult to show that the maximal Lp-regularity class
MRp(E) is independent from a particular choice of T ∈ (0,∞); see Dore [24, Sect. 5,
p. 310], Corollary 5.4, or Prüss [74, p. 4], remarks after Corollary 1.3. More impor-
tantly, this class is independent from p ∈ (1,∞) as well, i.e., MRp(E) = MRp0(E)
holds for all p, p0 ∈ (1,∞), by a classical result due to Sobolevskii [78, 79]; see,
e.g., [78, §3.1, pp. 343–345]. Further details on the independence of MRp(E) from
p ∈ (1,∞) can be found in Ashyralyev and Sobolevskii [9, Chapt. 1], §3.5, Theorem
3.6 on p. 35, Dore [24, Sect. 7, p. 313], Theorem 7.1, or Hieber [42, Corollary 4.4,
p. 371], where in (4.11) one may take M ≡ M(p) = p2(p − 1)−1M(p0) < ∞ if the
constant M(p0) ∈ (0,∞) is known, by [9].

(b) We are allowed to specify the constant M ≡ M(p,E,B, T ) > 0 in (4.11)
to be the smallest nonnegative number M ∈ R+ for which (4.11) is valid; cf.
Clément and Li [20, Proposition 2.2, p. 19]. Then, clearly, T 7→ M(p,E,B, T ) is
a nondecreasing (nonnegative) function of time T ∈ (0,∞). Indeed, if T ′ ∈ (0, T )
and g ∈ Lp((0, T ′)→ E0) is arbitrary, it suffices to apply (4.11) with the function

g̃(t) =

{
g(t) if 0 ≤ t ≤ T ′ ;
0 if T ′ < t ≤ T ,

in place of g in order to derive (4.11) for T ′ in place of T with the same constant M .
Hence, M(p,E,B, T ′) ≤M(p,E,B, T ) holds for 0 < T ′ < T . It is easy to see that
M(p,E,B, T ) > 0. (The case M = 0 would easily lead to a contradiction.) In what
follows we always use this optimal value of M , i.e., M ≡M(p,E,B, T ) > 0.

(c) Simple perturbation theory for linear operators shows that the set Hol(E) ≡
Hol(E1 → E0) is open in the Banach space L(E1 → E0). Even a more precise,
relative perturbation result is valid; see Kato [52, Chapt. IX], §2.2, Theorem 2.4
on p. 499. A similar result can be derived for the class MRp(E) ≡ MRp(E1 →
E0) applying the perturbation technique from either Amann [6, Chapt. III, §1.6],
Proposition 1.6.3 on p. 97, or from Clément and Li [20, Proof of Theorem 2.1], pp.
19–23: The set MRp(E) is open in L(E1 → E0); see Lemma 5.1 below. Indeed,
this follows from the fact that the set of all bounded linear operators from

L
(
Y p

1− 1
p

(0, T )→ Lp((0, T )→ E0)× E1− 1
p ,p

)
that possess a bounded inverse is open in this Banach space, and the inverse
(∂t −B, τ)

−1
is a locally Lipschitz-continuous function of B ∈ L(E1 → E0), by

Lemma 5.1 below and formula (5.4) thereafter, with B ∈ MRp(E) being fixed and
A ∈ L(E1 → E0) having a sufficiently small operator norm ‖A‖L(E1→E0) depending
on B.



42 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

Now we are ready to define a strict solution u to our abstract nonlinear evolution-
ary problem (4.8). We assume that 1 < p <∞, 0 < T <∞, g ∈ Lp((0, T )→ E0),
u0 ∈ U where U is an open set in E1− 1

p ,p
, and the mappings

A : (t, v) 7→ A(t, v) : [0, T ]× U ⊂ [0, T ]× E1− 1
p ,p
→ L(E1 → E0) ,

f : (t, v) 7→ f(t, v) : [0, T ]× U ⊂ [0, T ]× E1− 1
p ,p
→ E0

satisfy the following “natural” hypotheses (cf. Clément and Li [20, p. 19], (H1)–
(H3)):

4.1. Hypothesis.

(H4) A : [0, T ]× U → L(E1 → E0) is a Lipschitz continuous mapping such that
A(t, v) ∈ MRp(E) for all (t, v) ∈ [0, T ]× U .

(H5) f : [0, T ]× U → E0 is a Lipschitz continuous mapping.

Of course, the metric on [0, T ]×U is induced by the canonical norm on R×E1− 1
p ,p

.

It is a matter of a straight forward calculation to verify that both substitution
mappings,

(v, u) 7→ [t 7→ A(t, v(t))u(t)] : C([0, T ]→ U)× Lp((0, T )→ E1)→ Lp((0, T )→ E0) ,

v 7→ [t 7→ f(t, v(t))] : C ([0, T ]→ U)→ Lp((0, T )→ E0) ,

are locally Lipschitz continuous with values in Lp((0, T )→ E0); see, e.g., Clément
and Li [20, Proof of Theorem 2.1], pp. 19–23.

Remark 4.3. In (H4) we did not have to assume that A(t, v) ∈ MRp(E) holds for
all (t, v) ∈ [0, T ]×U . We could assume only A(0, u0) ∈ MRp(E); cf. results to follow
below (e.g., Theorems 4.5 and 4.7 and Remark 4.6). However, the set MRp(E) being
open in L(E1 → E0), A(0, u0) ∈ MRp(E) would imply that there are a number
T0 ∈ (0, T ] and an open neighborhood U0 of u0 in E1− 1

p ,p
, u0 ∈ U0 ⊂ U , such

that A(t, v) ∈ MRp(E) holds for all (t, v) ∈ [0, T0]×U0, by the Lipschitz continuity
of A. But this statement is qualitatively the same as A(t, v) ∈ MRp(E) for all
(t, v) ∈ [0, T ]× U in our (H4).

Definition 4.4 (Clément and Li [20, p. 18]). Recall that U is an open set in E1− 1
p ,p

and u0 ∈ U . We say that a function u : [0, T )→ E0 is a strict solution of the initial
value problem (4.8) if u ∈ Y p

1− 1
p

(0, T ), u(t) ∈ U for every t ∈ [0, T ], u(0) = u0,

and the differential equation in (4.8) is satisfied with all terms (summands) in
Lp((0, T )→ E0).

We recall that the Banach space Y p
1− 1

p

(0, T ) has been introduced in (4.10).

The main result in [20, Theorem 2.1, p. 19] is local in time and reads as follows,
with (H4) being somewhat weakened in the sense of our Remark 4.3 above.

Theorem 4.5. Let 1 < p < ∞ and 0 < T < ∞. Let U be a nonempty open set
in E1− 1

p ,p
and u0 ∈ U . Assume that both mappings A : [0, T ] × U → L(E1 → E0)

and f : [0, T ]×U → E0 are Lipschitz continuous. If A(0, u0) ∈ MRp(E) then there
exists some time T1 ≡ T1(u0) ∈ (0, T ], depending on u0, such that the abstract
initial value problem (4.8) possesses a unique strict solution

u ∈ Y p
1− 1

p

(0, T1)(
= Lp((0, T1)→ E1) ∩W 1,p((0, T1)→ E0) ↪→ C

(
[0, T1]→ E1− 1

p ,p

)) (4.13)



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 43

on the time interval [0, T1]. Consequently, one has u(t) ∈ U for every t ∈ [0, T1].

This theorem is proved in [20], Section 2, pp. 20–23, using the Banach contraction
principle in the closed ball

Σ
(u0)
ρ1,T1

=
{
v ∈ Y T1 : v(0) = u0 and ‖v − w‖Y T1 ≤ ρ1

}
of radius ρ1 ∈ (0,∞) centered at the point w ∈ Y T1 in the Banach space

Y T1 = Y p
1− 1

p

(0, T1) = Lp((0, T1)→ E1) ∩W 1,p((0, T1)→ E0) .

Here, the “center” function w ∈ Y T1 is defined to be the restriction to [0, T1] of the
unique strict solution w̃ ∈ Y T = Y p

1− 1
p

(0, T ) to the abstract initial value problem

(4.9) in the time interval [0, T ] with the linear operator B = A(0, u0) ∈ MRp(E)
and the right-hand side g(t) replaced by the sum f(t, u0) + g(t),

dw̃

dt
−A(0, u0)w̃(t) = f(t, u0) + g(t) for a.e. t ∈ (0, T ) ;

w̃(0) = u0 ∈ E1− 1
p ,p

.
(4.14)

Although the proof in [20] has been carried out only for A(t, u) = A(u) independent
from time t ∈ [0, T ], it is a matter of straight forward calculations to adapt this
proof to the case of A(t, u) depending on time t, cf. [20, p. 23], Remark at the end
of Section 2. A detailed treatment of the latter case is presented in Prüss [74, pp.
9–13], Chapt. 3, under slightly different assumptions (see also Köhne, Prüss, and
Wilke [55]).

Remark 4.6. Furthermore, one can easily conclude from the proof of Theorem 2.1
in [20, pp. 20–23] that if B̄R0

(w0) is any closed ball in the Banach space E1− 1
p ,p

of radius R0 ∈ (0,∞) centered at a point w0 ∈ E1− 1
p ,p

, such that B̄R0
(w0) ⊂ U

and R0 > 0 is small enough, then the constants ρ1 ∈ (0,∞) and T1 ∈ (0, T ]
can be chosen small enough to depend solely on R0, but not on w0, provided
u0 ∈ B̄R0(w0) ⊂ U . The estimates in [20, pp. 20–23], based on the Lipschitz
constants for A and f in [0, T ]×U and the estimate in (4.11), remain valid for any
u0 ∈ B̄R0

(w0). Thus, we have T1 ≡ T1(R0) ∈ (0, T ] and ρ1 ≡ ρ1(R0) ∈ (0,∞).
Finally, using similar estimates, cf. [20, p. 22], (2.14)–(2.17), one can show that the
(strict) solution mapping

u0 7→ u : B̄R0
(w0) ⊂ U ⊂ E1− 1

p ,p
→ Y T1 = Y p

1− 1
p

(0, T1)

is Lipschitz continuous with a Lipschitz constant L ≡ L(R0) ∈ (0,∞) independent
from w0 ∈ E1− 1

p ,p
, such that B̄R0

(w0) ⊂ U and R0 > 0 is small enough. This

means that if u1, u2 : [0, T1] → E1− 1
p ,p

are two strict solutions to problem (4.8)

on the time interval [0, T1], with (possibly different) initial values u1(0) = u0,1 and
u2(0) = u0,2 in B̄R0

(w0) ⊂ U , then one has u1(t), u2(t) ∈ U for all t ∈ [0, T1] and

‖u1 − u2‖Y T1 ≤ L‖u0,1 − u0,2‖E
1− 1

p
,p
. (4.15)

Combining this with the continuous imbedding Y T1 ↪→ C
(
[0, T1] → E1− 1

p ,p

)
in

(4.10), we obtain

‖u1(t)− u2(t)‖E
1− 1

p
,p
≤ L1‖u0,1 − u0,2‖E

1− 1
p
,p

for all t ∈ [0, T1] , (4.16)

with another Lipschitz constant L1 ≡ L1(R0) ∈ (0,∞).



44 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

A number of sufficient conditions that guarantee the existence of a global weak
solution u : RN × (0, T ) → RM (CM ) for all times t ∈ (0, T ) to the parabolic
Cauchy problem (1.1) can be found in Amann [4, 5] for systems similar to ours.
As we do not wish to impose those kinds of restrictive growth conditions on the
reaction function f on the right-hand side of (1.1), we prefer to assume the existence
of a fixed global strict solution (cf. (4.13))

w ∈ Y p
1− 1

p

(0, T )(
= Lp((0, T )→ E1) ∩W 1,p((0, T )→ E0) ↪→ C

(
[0, T ]→ E1− 1

p ,p

)) (4.17)

to problem (4.8) on the whole time interval [0, T ], for some T ∈ (0,∞), with a
prescribed initial value w(0) = w0 ∈ U ⊂ E1− 1

p ,p
and such that w(t) ∈ U and

A(t, w(t)) ∈ MRp(E) for all t ∈ [0, T ]. Then the local Theorem 4.5 and Remark 4.6
from above may be applied on any time interval [t0, t0 + T1] ⊂ [0, T ] of sufficiently
short length T1 > 0 in order to obtain unique strict solutions u “along” the known
solution w to the following abstract initial value problem:

du

dt
−A(t, u(t))u(t) = f(t, u(t)) + g(t) for a.e. t ∈ (t0, t0 + T1) ;

u(t0) = u0 ∈ E1− 1
p ,p

.
(4.18)

Here, u0 ∈ B̄R0(w(t0)) is arbitrary, where the radius R0 > 0 is small enough, as
described in Remark 4.6, such that B̄R0

(w(t0)) ⊂ U . By Theorem 4.5, the strict
solution u : [t0, t0 + T1]→ E1− 1

p ,p
satisfies u(t) ∈ U for every t ∈ [t0, t0 + T1]. The

image w([0, T ]) = {w(t) : t ∈ [0, T ]} of the solution w being compact in the open
set U ⊂ E1− 1

p ,p
, we may choose R0 > 0 even smaller, such that B̄R0

(w(t)) ⊂ U

holds for all t ∈ [0, T ].
In addition to these claims that follow immediately from the proof of [20, The-

orem 2.1, pp. 20–23], one can deduce from inequalities analogous to those in [20,
p. 22], (2.14)–(2.17), cf. Remark 4.6 above, (4.16), that there exists a Lipschitz
constant L1 ∈ [1,∞), such that if u1, u2 : [t0, t0 + T1] → E1− 1

p ,p
are two strict

solutions to problem (4.18) with initial values u1(t0) = u0,1 and u2(t0) = u0,2 in
B̄R0

(w(t0)) ⊂ U , then one has u1(t), u2(t) ∈ U and

‖u1(t)− u2(t)‖E
1− 1

p
,p
≤ L1‖u1(t0)− u2(t0)‖E

1− 1
p
,p

(4.19)

for all t ∈ [t0, t0 + T1]. Consequently, fixing the smallest integer m ∈ N such that
m ≥ T/T1 ( ≥ 1), we obtain, by “induction” on k = 1, 2, 3, . . . ,m, first

‖u(t)− w(t)‖E
1− 1

p
,p
≤ Lk1‖u0 − w0‖E

1− 1
p
,p
≤ Lk1 ·R0/L

k
1 = R0 (4.20)

for all t ∈ [0, min{kT1, T}], whenever ‖u0−w0‖E
1− 1

p
,p
≤ R0/L

k
1 ; also u1(t), u2(t) ∈

B̄R0
(w(t)) ⊂ U and

‖u1(t)− u2(t)‖E
1− 1

p
,p
≤ Lk1‖u0,1 − u0,2‖E

1− 1
p
,p

(4.21)

for all t ∈ [0, min{kT1, T}], whenever

‖u0,j − w0‖E
1− 1

p
,p
≤ Rk := R0/L

k
1 (> 0) ; j = 1, 2 ,

for k = 1, 2, 3, . . . ,m.



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 45

We have thus obtained the following result, global in time on an arbitrary time
interval (t0, T ), 0 ≤ t0 < T , with the constants R0 ∈ (0,∞), T1 ≡ T1(R0) ∈ (0, T ],
and L1 ∈ [1,∞) specified above in (4.19)–(4.21):

Theorem 4.7. Let 1 < p < ∞, 0 < T < ∞, and g ∈ Lp((0, T ) → E0). Assume
that U is a nonempty open subset of E1− 1

p ,p
and A and f satisfy (H4) and (H5),

respectively. Finally, assume that w : [0, T ] → U ⊂ E1− 1
p ,p

is a fixed global

strict solution to problem (4.8) satisfying (4.17), with a prescribed initial value
w(0) = w0 ∈ U and such that w(t) ∈ U and A(t, w(t)) ∈ MRp(E) for all t ∈ [0, T ].
Then there exist some constant R0 ∈ (0,∞), sufficiently small, with the following
two properties, where Rm = R0/L

m
1 ∈ (0, R0] is the constant defined in (4.21):

(i) If t0 ∈ [0, T ) and u0 ∈ B̄Rm(w(t0)) ⊂ U , then the abstract initial value problem
(4.8) on the time-interval (t0, T ) with u(t0) = u0 possesses a unique strict solution
u ∈ Y p

1− 1
p

(t0, T ) (cf. (4.13))

u ∈ Y p
1− 1

p

(t0, T )(
= Lp((t0, T )→ E1) ∩W 1,p((t0, T )→ E0) ↪→ C

(
[t0, T ]→ E1− 1

p ,p

))
such that u(t) ∈ B̄R0

(w(t)) ⊂ U for every t ∈ [t0, T ].
(ii) If t0 ∈ [0, T ) and u1, u2 : [t0, T ] → E1− 1

p ,p
are two strict solutions to problem

(4.8) on the time-interval (t0, T ) with initial values u1(t0) = u0,1 and u2(t0) = u0,2

in B̄Rm(w(t0)) ⊂ U , then one has u1(t), u2(t) ∈ B̄R0
(w(t)) ⊂ U and

‖u1(t)− u2(t)‖E
1− 1

p
,p
≤ Lm1 ‖u0,1 − u0,2‖E

1− 1
p
,p

for all t ∈ [t0, T ] . (4.22)

5. Analyticity in time for the abstract Cauchy problem

In this section we establish a few temporal analyticity results, Theorem 5.3 being
the most important among them, that will be used later (in Section 8) in order to
prove Part (ii) of Theorem 3.4.

5.1. Auxiliary linear perturbation results. We begin by quoting a well-known
result: If B ∈ Gen(E) is the generator of a holomorphic semigroup on E0 with the
domain D(B) = E1, i.e., B ∈ Hol(E), then so is every operator Bν = (1 + iν)B :
E1 ⊂ E0 → E0, ν ∈ R, provided |ν| is small enough, |ν| ≤ δ1 < 1; see, e.g.,
Amann [6, Chapt. I, §1], pp. 9–24, Pazy [72, §3.2, pp. 80–81], or Tanabe [81,
Chapt. 3, §3.1–§3.4], pp. 51–72. A more general perturbation theorem for generators
of holomorphic semigroups is proved in Pazy [72, §3.2], Theorem 2.1 on p. 80. An
analogous perturbation result for the smaller class MRp(E)

(
MRp(E) ⊂ Hol(E) ⊂

Gen(E)
)

is proved in Amann [6, Chapt. III, §1.6], Proposition 1.6.3 on p. 97. Since
we take advantage of the latter in an essential manner, we now give its precise
formulation.

Let 1 < p < ∞ and 0 < T < ∞. Given any generator B ∈ Gen(E), let us

consider the bounded linear operator K̃B : L1((0, T ) → E0) → L∞((0, T ) → E0)
defined by

(K̃Bg)(t) :=

∫ t

0

e(t−s)Bg(s) ds ∈ E0 (5.1)

for all t ∈ [0, T ] and all g ∈ L1((0, T ) → E0). It is proved in [6, Chapt. III,
§1.5], Theorem 1.5.2 on p. 95, that if B ∈ Hol(E) and B possesses the maximal



46 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

Lp-regularity property, i.e., B ∈ MRp(E) ≡ MRp(E1 → E0), then the restriction

KB = K̃B |XT of K̃B to XT := Xp

1− 1
p

(0, T ) = Lp((0, T )→ E0)

is a bounded linear operator from the Banach space XT into another Banach space

Y T := Y p
1− 1

p

(0, T ) = Lp((0, T )→ E1) ∩W 1,p((0, T )→ E0)

with the operator norm ‖KB‖L(XT→Y T ) < ∞. For the perturbed initial value
problem

du

dt
− (B +A)u(t) = g(t) for a.e. t ∈ (0, T ) ;

u(0) = u0 ∈ E1− 1
p ,p

,
(5.2)

the following result is established in [6, Chapt. III, §1.6], Proposition 1.6.3 on p. 97:

Lemma 5.1. Assume that B ∈ MRp(E) and let A ∈ L(E1 → E0) be arbitrary with
the norm

‖A‖L(E1→E0) ≤ γ
/
‖KB‖L(XT→Y T ) for some γ ∈ (0, 1) .

Then also the operator BA = B + A ∈ L(E1 → E0) belongs to the class MRp(E)
and the operator norms of the inverses of the abstract (linear) partial differential
operators

(∂t −B, τ) , (∂t −B −A, τ) : Y p
1− 1

p

(0, T )→ Lp((0, T )→ E0)× E1− 1
p ,p

defined in (4.12) satisfy

‖ (∂t −B −A, τ)
−1 ‖ ≤ C · (1− γ)−1‖ (∂t −B, τ)

−1 ‖ , (5.3)

where C ≡ C(p,E, T ) > 0 is a constant independent of A, B, and γ.

More precisely, we have

(∂t −B −A, τ)
−1

= (I −KBA)−1 (∂t −B, τ)
−1

(5.4)

with the operator norm of the product

KBA : Y T → Y T = Y p
1− 1

p

(0, T ) = Lp((0, T )→ E1)

bounded above by

‖KBA‖L(Y T→Y T ) ≤ ‖KB‖L(XT→Y T ) · ‖A‖L(E1→E0) ≤ γ < 1 .

Here, I stands for the identity mapping in L(Y T → Y T ). Hence, the Neumann
series (I − KBA)−1 =

∑∞
k=0(KBA)k converges absolutely in L(Y T → Y T ) and

‖(I −KBA)−1‖L(Y T→Y T ) ≤ (1− γ)−1 <∞.
The following claims are trivial applications of this lemma: MRp(E) ≡ MRp(E1 →

E0) is an open subset of the Banach space L(E1 → E0). Furthermore, if B ∈
MRp(E) and A ∈ L(E1 → E0) then also BνA = B + νA ∈ MRp(E) holds for every
ν ∈ C provided |ν| is small enough,

|ν| ≤ δ1 = γ‖A‖−1
L(E1→E0)‖KB‖−1

L(XT→Y T )
<∞ .

Naturally, the special case A = iB is of interest.
The following perturbation lemma for problem (5.2) is related to Angenent [7,

Lemma 2.5, p. 97]; see also Denk, Hieber, and Prüss [23], Proposition 4.3 on p. 44
and Theorem 4.4 on p. 45.



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 47

Lemma 5.2. Assume that B ∈ MRp(E). Then there exists a number δ ∈ (0, 1) and
a constant Cδ ∈ R+ with the following property: If A ∈ L(E1 → E0) is arbitrary
with the norm

‖Au‖E0
≤ δ‖Bu‖E0

+ Cδ‖u‖E0
for all u ∈ E1 , (5.5)

then also the operator BA = B + A ∈ L(E1 → E0) belongs to the class MRp(E).

Furthermore, there exists a constant M̃ ≡ M̃(p,E,B, δ, Cδ, T ) > 0, independent
of (g, u0) ∈ Lp((0, T ) → E0) × E1− 1

p ,p
, such that the unique strict solution v =

(∂t −B −A, τ)
−1

(g, u0) to the perturbed initial value problem (5.2) satisfies the
inequality ∫ T

0

∥∥dv

dt

∥∥p
E0

dt+

∫ T

0

‖(B +A)v(t)‖pE0
dt

≤ M̃
(
‖u0‖pE

1− 1
p
,p

+

∫ T

0

‖g(t)‖pE0
dt
)
,

(5.6)

whenever u0 ∈ E1− 1
p ,p

and g ∈ Lp((0, T )→ E0).

Proof. Step 1. First, we prove the lemma for [0, T ] ⊂ R+ replaced by a sufficiently
short time interval [t0, t0 + T1] ⊂ [0, T ], i.e., 0 ≤ t0 < t0 + T1 ≤ T with T1 ∈ (0,∞)
small enough. Without loss of generality, we may assume t0 = 0 and 0 < T1 ≤ T .

Let us recall our notation and the continuous imbedding (cf. (4.13))

Y T1 = Y p
1− 1

p

(0, T1) = Lp((0, T1)→ E1) ∩W 1,p((0, T1)→ E0)

↪→ C
(

[0, T1]→ E1− 1
p ,p

)
.

(5.7)

It is easy to see that a function v ∈ Y T1 is a strict solution of the perturbed initial
value problem (5.2) on (0, T1) if and only if it satisfies

dv

dt
−Bv(t) = Av(t) + g(t) for a.e. t ∈ (0, T1) ;

v(0) = u0 ∈ E1− 1
p ,p

,
(5.8)

in the strict sense, again. Notice that g̃ = Av + g ∈ Lp((0, T1)→ E0). We observe
that problem (5.8) has a unique strict solution v ∈ Y T1 as soon as we have shown
that the affine self mapping F : v 7→ v̂ : Y T1 → Y T1 , defined by

dv̂

dt
−Bv̂(t) = Av(t) + g(t) for a.e. t ∈ (0, T1) ;

v̂(0) = u0 ∈ E1− 1
p ,p

,
(5.9)

possesses a unique fixed point v ∈ Y T1 . Obviously, such a fixed point must belong
to the (closed) affine subspace

Y T1

(u0) =
{
v ∈ Y T1 : v(0) = u0

}
of the Banach space Y T1 ;

hence, Y T1

(u0) = u0 + Y T1

(0) . Clearly, Y T1

(0) =
{
v ∈ Y T1 : v(0) = 0

}
is a closed vector

subspace of Y T1 . The former one inherits the norm from the latter.
Next, we prove that F : v 7→ v̂ is a contraction on Y T1

(u0). To this end, let

vi ∈ Y T1

(u0) be arbitrary and set v̂i = F (vi); i = 1, 2. The differences z = v1− v2 and



48 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

ẑ = v̂1 − v̂2 are in Y T1

(0) and, by (5.9), they satisfy

dẑ

dt
−Bẑ(t) = Az(t) for a.e. t ∈ (0, T1) ;

ẑ(0) = 0 ∈ E1− 1
p ,p

.
(5.10)

By Remark 4.2, Part (a), the operator B ∈ MRp(E) satisfies (4.11) with a constant
M(p,E,B, T1) ≤M(p,E,B, T ) ≡MT <∞. Hence, we have∫ T1

0

∥∥dẑ

dt

∥∥p
E0

dt+

∫ T1

0

‖Bẑ(t)‖pE0
dt ≤MT

∫ T1

0

‖Az(t)‖pE0
dt .

Now we estimate the integrand on the right-hand side of (5.5),∫ T1

0

∥∥dẑ

dt

∥∥p
E0

dt+

∫ T1

0

‖Bẑ(t)‖pE0
dt

≤MT

∫ T1

0

(
δ‖Bz(t)‖E0

+ Cδ‖z(t)‖E0

)p
dt

≤ 2p−1MT

(
δp
∫ T1

0

‖Bz(t)‖pE0
dt+ Cpδ

∫ T1

0

‖z(t)‖pE0
dt
)
.

(5.11)

The integrand in the second integral on the right-hand side is estimated by Hölder’s
inequality:

‖z(t)‖E0
=
∥∥∥∫ t

0

dz

dt
(s) ds

∥∥∥
E0

≤
∫ t

0

‖z′(s)‖E0
ds

≤
(∫ t

0

∥∥dz

dt
(s)
∥∥p
E0

ds
)1/p(∫ t

0

ds
)1/p′

for all t ∈ [0, T1] ,

where p′ = p/(p− 1) ∈ (1,∞). Here, we have used z(0) = 0 ∈ E0. Hence,

‖z(t)‖pE0
≤ tp/p

′
(∫ t

0

∥∥dz

dt
(s)
∥∥p
E0

ds
)
.

After integration we thus obtain, thanks to p/p′ = p− 1,∫ T1

0

‖z(t)‖pE0
dt ≤ 1

p
T p1

∫ T1

0

∥∥dz

dt
(s)
∥∥p
E0

ds . (5.12)

Of course, the same inequality is valid for ẑ ∈ Y T1

(0) in place of the function z. We

apply the last inequality to the right-hand side of (5.11),∫ T1

0

∥∥dẑ

dt

∥∥p
E0

dt+

∫ T1

0

‖Bẑ(t)‖pE0
dt

≤ 2p−1MT

(
δp
∫ T1

0

‖Bz(t)‖pE0
dt+

1

p
CpδT

p
1

∫ T1

0

∥∥dz

dt

∥∥p
E0

dt
)
.

(5.13)

The integrals on both sides containing the generator B ∈ MRp(E) are estimated
as follows. First, there are constants c1, C1 ∈ (0,∞) and c2, C2 ∈ R+ such that the
inequalities

c1‖u‖E1
− c2‖u‖E0

≤ ‖Bu‖E0
≤ C1‖u‖E1

+ C2‖u‖E0
hold for all u ∈ E1 .

Consequently, we have

cp1‖u‖
p
E1
≤ 2p−1

(
‖Bu‖pE0

+ cp2‖u‖
p
E0

)
and



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 49

‖Bu‖pE0
≤ 2p−1

(
Cp1‖u‖

p
E1

+ Cp2‖u‖
p
E0

)
for all u ∈ E1 .

Applying these inequalities to (5.13), we arrive at∫ T1

0

∥∥dẑ

dt

∥∥p
E0

dt+ 2−(p−1)cp1

∫ T1

0

‖ẑ(t)‖pE1
dt− cp2

∫ T1

0

‖ẑ(t)‖pE0
dt

≤ 2p−1MT · 2p−1Cp1 δ
p

∫ T1

0

‖z(t)‖pE1
dt

+ 2p−1MT

(
2p−1Cp2 δ

p

∫ T1

0

‖z(t)‖pE0
dt+

1

p
CpδT

p
1

∫ T1

0

∥∥dz

dt

∥∥p
E0

dt
)
.

Finally, we estimate the integrals
∫ T1

0
‖ẑ(t)‖pE0

dt and
∫ T1

0
‖z(t)‖pE0

dt above by
(5.12), thus obtaining(

1− 1

p
T p1 c

p
2

)∫ T1

0

∥∥dẑ

dt

∥∥p
E0

dt+ 2−(p−1)cp1

∫ T1

0

‖ẑ(t)‖pE1
dt

≤ 22(p−1)Cp1 δ
pMT

∫ T1

0

‖z(t)‖pE1
dt

+
2p−1

p
T p1MT

(
2p−1Cp2 δ

p + Cpδ
) ∫ T1

0

∥∥dz

dt

∥∥p
E0

dt .

(5.14)

We finish this step by choosing first δ ∈ (0, 1) then T1 ∈ (0, T ] small enough, such
that

22(p−1)Cp1 δ
pMT ≤

1

2
· 2−(p−1)cp1 and

2p−1

p
T p1MT

(
2p−1Cp2 δ

p + Cpδ
)
≤ 1

2

(
1− 1

p
T p1 c

p
2

)
,

respectively, or, equivalently,

0 < δ ≤ 2−(3p−2)/p (c1/C1)M
−1/p
T and (5.15)

T p1 [2pMT (2p−1Cp2 δ
p + Cpδ ) + cp2] ≤ p . (5.16)

With these choices of δ and T1, we obtain

‖ẑ‖[Y T1 ≤
1

2
‖z‖[Y T1 (5.17)

in the new, equivalent norm

‖u‖[Y T1 :=
[(

1− 1

p
T p1 c

p
2

) ∫ T1

0

∥∥du

dt

∥∥p
E0

dt+ 2−(p−1)cp1

∫ T1

0

‖u(t)‖pE1
dt
]1/p

(5.18)

on the abstract Sobolev space Y T1 = Y p
1− 1

p

(0, T1) = Lp((0, T1)→ E1)∩W 1,p((0, T1)→
E0); see (4.5) and below in (5.19). Inequality (5.17) shows that F : v 7→ v̂ is a con-

traction on Y T1

(u0) (⊂ Y T1) with the Lipschitz constant 1
2 with respect to the new

norm ‖ · ‖[
Y T1

. Consequently, problem (5.8) has a unique strict solution v ∈ Y T1 ;

in fact, we have v ∈ Y T1

(u0).

The following estimate for v can be proved by the same arguments as those
used in our proof of contraction above: There is a constant Γ ≡ Γ(T1) ∈ (0,∞),



50 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

independent from u0 ∈ E1− 1
p ,p

and g ∈ Lp((0, T )→ E0), such that

(
‖v‖]

Y T1

)p
=

∫ T1

0

∥∥dv

dt

∥∥p
E0

dt+

∫ T1

0

‖v(t)‖pE1
dt

≤ Γ
(
‖u0‖pE

1− 1
p
,p

+

∫ T1

0

‖g(t)‖pE0
dt
)
.

(5.19)

Recall that ‖ · ‖]
Y T1

is an equivalent norm on the Banach space Y T1 ; cf. (4.5). In
analogy with Remark 4.2, Part (b), we may take the constant Γ ≡ Γ(T1) > 0 in
(5.19) above to be the smallest nonnegative number Γ ∈ R+ for which (5.19) is
valid. It is easy to see that Γ ≡ Γ(T1) ∈ R+ is a nondecreasing function of time
T1 ∈ (0, T ] and Γ > 0. The last estimate, (5.19), easily implies (5.6) with T1 in place
of T . The imbedding (5.7) being continuous, by (5.19), there is another constant

Γ̂ ≡ Γ̂(T1) ∈ [1,∞), independent from u0 ∈ E1− 1
p ,p

and g ∈ Lp((0, T )→ E0), such

that

‖v(T1)‖pE
1− 1

p
,p
≤ Γ̂

(
‖u0‖pE

1− 1
p
,p

+

∫ T1

0

‖g(t)‖pE0
dt
)
. (5.20)

Again, similarly to Γ ≡ Γ(T1) > 0 in (5.19), we may take the constant Γ̂ ≡ Γ̂(T1) >

0 in (5.20) above to be the smallest number Γ̂ ∈ [1,∞) for which (5.20) is valid. It

is now easy to see that also the constant Γ̂ ≡ Γ̂(T1) ≥ 1 is a nondecreasing function
of time T1 ∈ (0, T ].

Step 2. We may take T1 = T/m sufficiently small in Step 1 above, where m ∈ N
is a sufficiently large positive integer. Next, we replace the interval [0, T1] from
Step 1 by any subinterval [t0, t0 + T1] = Jk = [(k − 1)T1, kT1] of [0, T ] of length
T1 for k = 1, 2, . . . ,m; hence, ∪mk=1Jk = [0, T ]. We make use of the existence and
uniqueness of a strict solution

v ∈ Y p
1− 1

p

(t0, t0 + T1) = Lp((t0, t0 + T1)→ E1) ∩W 1,p((t0, t0 + T1)→ E0)

of the perturbed initial value problem (5.2) in every subinterval Jk; k = 1, 2, . . . ,m,
together with the estimates (5.19) and (5.20) on Jk, by Step 1. Thus, from (5.19)
and (5.20) we obtain, respectively,∫ kT1

(k−1)T1

∥∥dv

dt

∥∥p
E0

dt+

∫ kT1

(k−1)T1

‖v(t)‖pE1
dt

≤ Γ
(
‖v((k − 1)T1)‖pE

1− 1
p
,p

+

∫ kT1

(k−1)T1

‖g(t)‖pE0
dt
)
,

(5.21)

‖v(kT1)‖pE
1− 1

p
,p
≤ Γ̂

(
‖v((k − 1)T1)‖pE

1− 1
p
,p

+

∫ kT1

(k−1)T1

‖g(t)‖pE0
dt
)
. (5.22)

We recall that Γ̂ ≥ 1. Consequently, iterating inequalities (5.22) for k = 1, 2, . . . , `,
1 ≤ ` ≤ m, we arrive at

‖v(`T1)‖pE
1− 1

p
,p
≤ Γ̂`‖u0‖pE

1− 1
p
,p

+
∑̀
k=1

Γ̂`−k+1

∫ kT1

(k−1)T1

‖g(t)‖pE0
dt

≤ Γ̂`
(
‖u0‖pE

1− 1
p
,p

+

∫ `T1

0

‖g(t)‖pE0
dt
)
.

(5.23)



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 51

Next, we sum inequalities (5.21) for k = 1, 2, . . . ,m, thus obtaining(
‖v‖]

Y T

)p
=

∫ T

0

∥∥dv

dt

∥∥p
E0

dt+

∫ T

0

‖v(t)‖pE1
dt

≤ Γ
( m∑
k=1

‖v((k − 1)T1)‖pE
1− 1

p
,p

)
+ Γ

∫ T

0

‖g(t)‖pE0
dt .

(5.24)

To estimate the first summand on the right-hand side from above, we apply (5.23)
with ` = k − 1 for k = 1, 2, . . . ,m, thus arriving at

m∑
k=1

‖v((k − 1)T1)‖pE
1− 1

p
,p

≤
(m−1∑
`=0

Γ̂`
)
‖u0‖pE

1− 1
p
,p

+

m−1∑
`=0

Γ̂`
∫ `T1

0

‖g(t)‖pE0
dt

≤ mΓ̂m−1‖u0‖pE
1− 1

p
,p

+ (m− 1)Γ̂m−1

∫ (m−1)T1

0

‖g(t)‖pE0
dt

≤ M̂Γ−1
(
‖u0‖pE

1− 1
p
,p

+

∫ T

0

‖g(t)‖pE0
dt
)
,

where M̂ = mΓ̂m−1Γ ∈ [1,∞) is a constant independent from v. We apply this
estimate to the right-hand side of (5.24) to obtain(

‖v‖]
Y T

)p ≤ M̂‖u0‖pE
1− 1

p
,p

+ (M̂ + Γ)

∫ T

0

‖g(t)‖pE0
dt . (5.25)

We conclude the proof by applying (5.5) with B,A ∈ L(E1 → E0) to the left-
hand side of (5.6),∫ T

0

∥∥dv

dt

∥∥p
E0

dt+

∫ T

0

‖(B +A)v(t)‖pE0
dt

≤
∫ T

0

∥∥dv

dt

∥∥p
E0

+ (1 + δ)

∫ T

0

‖Bv(t)‖pE0
dt+ Cδ

∫ T

0

‖v(t)‖pE0
dt

≤
∫ T

0

∥∥dv

dt

∥∥p
E0

+ (1 + δ)‖B‖L(E1→E0)

∫ T

0

‖v(t)‖pE1
dt+ Cδ

∫ T

0

‖v(t)‖pE0
dt

≤Mδ

(
‖v‖]

Y T

)p
,

by (5.24), with a constant Mδ ∈ (0,∞) independent from v. Now we apply (5.25)
to the last estimate to arrive at the desired inequality (5.6) with the constant

M̃ = Mδ(M̂+Γ) > 0. We have proved that the operator BA = B+A ∈ L(E1 → E0)
belongs to the class MRp(E). �

5.2. Proof of analyticity in time. Now we are ready to prove that any global
strict solution w : [0, T ]→ U ⊂ E1− 1

p ,p
to problem (4.8) that satisfies the hypothe-

ses of Theorem 4.7 above must be analytic in time t ∈ (0, T ). Let us recall that a
strict solution to problem (4.8) has been introduced in Definition 4.4. Indeed, below
we will prove a more detailed result on a complex analytic (i.e., holomorphic) exten-
sion of u(t) from the real time interval (0, T ) ⊂ R ⊂ C to the open complex domain

∆T ′,T
ϑ which is the intersection of the (open) triangle ∆

(T ′)
ϑ with the (open) complex

strip T(r) defined in (1.6), (1.7), and (1.8), respectively, where ϑ ∈ (0, π/2) is a given



52 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

angle and 0 < T ′ ≤ T < ∞. Here, the constants ϑ ∈ (0, π/2) and T ′ ∈ (0, T ] will

be chosen sufficiently small, but positive; hence, we have (0, T ) ⊂ ∆T ′,T
ϑ . Finally,

we denote by ∆̄T ′,T
ϑ the closure of ∆T ′,T

ϑ in C.
In addition to (H4) and (H5), we assume that A and f satisfy the following

analyticity hypotheses (cf. Lunardi [65, Chapt. 8], §8.3.3, p. 308):

5.3. Hypothesis. Recall that both spaces, E0 and E1, in the Banach couple E =
(E0, E1) are assumed to be complex Banach spaces (over the field C) with E1 ↪→ E0

densely and continuously. Furthermore, we assume that there are positive constants
ϑ0 ∈ (0, π/2) and T0 ∈ (0, T ], and open sets U ⊂ C and Ũ ⊂ E1− 1

p ,p
containing

the compact set ∆̄T0,T
ϑ0

and the open set U , respectively, i.e., ∆̄T0,T
ϑ0

⊂ U ⊂ C and

U ⊂ Ũ ⊂ E1− 1
p ,p

, such that

(H4’) A : [0, T ]×U → L(E1 → E0) possesses a holomorphic extension à : U×Ũ →
L(E1 → E0) to the complex domain U×Ũ which satisfies Ã(t, v) ∈ MRp(E)

for all (t, v) ∈ U × Ũ .

(H5’) f : [0, T ] × U → E0 possesses a holomorphic extension f̃ : U × Ũ → E0 to

the complex domain U × Ũ .

Again, the metric on U × Ũ is induced by the canonical norm on C × E1− 1
p ,p

.

A precise definition of a holomorphic (i.e., complex analytic) mapping F : O ⊂
X → Y from an open subset O of a complex Banach space X into another complex
Banach space Y is given in Deimling [22, Definition 15.1, p. 150] (see also [22,
Proposition 15.2, p. 150]).

Without assuming (H4) and (H5), we observe that (H4’) and (H5’) still guar-
antee the following claims, respectively: Given any compact set K ⊂ U and any
continuous function z : [0, T ] → K, one can easily verify that both substitution
mappings,

v 7→ [t 7→ A (z(t), v(z(t)))] :

C(K → Ũ)→ L (Lp((0, T )→ E1)→ Lp((0, T )→ E0)) and

v 7→ [t 7→ f (z(t), v(z(t)))] : C(K → Ũ)→ Lp((0, T )→ E0) ,

the former one meaning that

(v, u) 7→ [t 7→ A (z(t), v(z(t))) u(z(t))] :

C(K → Ũ)× Lp((0, T )→ E1)→ Lp((0, T )→ E0) ,

are locally Lipschitz continuous, the former one with values in L(Lp((0, T )→ E1)→
Lp((0, T )→ E0)) and the latter one with values in Lp((0, T )→ E0). We will take
advantage of this local Lipschitz continuity in our proof of Theorem 5.3 below. We
remark that the operator norm in L (Lp((0, T )→ E1)→ Lp((0, T )→ E0)) of the
linear substitution operator

u 7→ [t 7→ A (z(t), v(z(t)))u(t)] : Lp((0, T )→ E1)→ Lp((0, T )→ E0) ,

with z ∈ C([0, T ] → K) and v ∈ C(K → Ũ) being fixed, is bounded above by the
supremum norm

|||A (z(·), v(z(·))) |||L∞(0,T ) := ‖[t 7→ A(z(t), v(z(t)))]‖C([0,T ]→L(E1→E0))

= sup
0≤t≤T

‖A (z(t), v(z(t))) ‖L(E1→E0) (<∞) .



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 53

Theorem 5.3. Let 1 < p < ∞, ϑ0 ∈ (0, π/2), 0 < T0 ≤ T < ∞, and assume that
g ∈ Lp((0, T ) → E0) possesses a holomorphic extension g̃ : U → E0 to an open

set U ⊂ C containing ∆̄T0,T
ϑ0

, i.e., ∆̄T0,T
ϑ0

⊂ U ⊂ C. Assume that Ũ is a nonempty

open subset of E1− 1
p ,p

and à and f̃ satisfy (H4’) and (H5’), respectively, and their

respective restrictions A = Ã|[0,T ]×U and f = f̃ |[0,T ]×U to [0, T ]× U ⊂ R×E1− 1
p ,p

satisfy (H4) and (H5) with an open set U ⊂ Ũ ⊂ E1− 1
p ,p

. Finally, assume that

w : [0, T ] → U ⊂ E1− 1
p ,p

is a fixed global strict solution to problem (4.8) (hence,

satisfying (4.17)) with a prescribed initial value w(0) = w0 ∈ U and such that

w(t) ∈ U and Ã(t, w(t)) ∈ MRp(E) for all t ∈ [0, T ].
Then there exist constants ϑ′ ∈ (0, ϑ0] and T ′ ∈ (0, T0], small enough, and a

holomorphic function w̃ : ∆T ′,T
ϑ′ → E1− 1

p ,p
with the following two properties:

(a) w̃(t) ∈ Ũ for every t ∈ ∆T ′,T
ϑ′ and w̃ verifies the abstract nonlinear evolu-

tionary problem (4.8) in the complex domain ∆T ′,T
ϑ′ , i.e.,

du

dt
− Ã(t, u(t))u(t) = f̃(t, u(t)) + g̃(t) for every t ∈ ∆T ′,T

ϑ′ ;

lim
t→0,t∈∆T ′,T

ϑ′

u(t) = w0 ∈ E1− 1
p ,p

.
(5.26)

(b) w̃(t) = w(t) holds for a.e. t ∈ (0, T ).

Such a holomorphic extension w̃ : ∆T ′,T
ϑ′ → Ũ ⊂ E1− 1

p ,p
of w : (0, T )→ U ⊂ E1− 1

p ,p

from (0, T ) to ∆T ′,T
ϑ′ is unique.

Before proceeding to prove this theorem, we clarify our notation with the open
sets U and Ũ in E1− 1

p ,p
as follows.

Remark 5.4. We need to take advantage of our (H4) and (H5) (with an open

set U ⊂ E1− 1
p ,p

) and (H4’) and (H5’) (with another open set Ũ ⊂ E1− 1
p ,p

) only

for the values of v = w(t) ∈ U ⊂ Ũ (t ∈ ∆̄T ′,T
ϑ′ ) near the (compact) image K =

{w(t) ∈ E1− 1
p ,p

: t ∈ [0, T ]} of the (continuous) curve w : [0, T ]→ E1− 1
p ,p

. Indeed,

(H4’) and (H5’) imply that both holomorphic extensions à : U × Ũ → L(E1 → E0)

and f̃ : U × Ũ → E0 of A : [0, T ] × U → L(E1 → E0) and f : [0, T ] × U → E0,
respectively, are locally Lipschitz continuous. Consequently, the Cartesian product
[0, T ] × K being compact in the complex Banach space C × E1− 1

p ,p
, we use a

finite open subcover by open balls to find two bounded open sets U ⊂ C and
U = Ũ ⊂ E1− 1

p ,p
, such that both mappings à and f̃ are Lipschitz continuous in

U × Ũ . We conclude that, in our proof of Theorem 5.3 below, we may assume that
[0, T ] ⊂ U ⊂ C and K ⊂ U = Ũ ⊂ E1− 1

p ,p
with both U and U being open and

bounded. In particular, if the numbers ϑ0 ∈ (0, π/2) and T0 ∈ (0, T ] are taken

sufficiently small, then we have also ∆̄T0,T
ϑ0

⊂ U together with w̃(t) ∈ U for all

t ∈ ∆̄T ′,T
ϑ′ , provided ϑ′ ∈ (0, ϑ0] and T ′ ∈ (0, T0] are small enough. Consequently,

∆̄T ′,T
ϑ′ ⊂ ∆̄T0,T

ϑ0
⊂ U . To simplify our notation, we work only with the holomorphic

extensions g̃ : U → E0, Ã : U × U → L(E1 → E0), and f̃ : U × U → E0



54 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

of the mappings g, A, and f , respectively. Hence, we may remove the “tilde”
from these symbols and write simply g = g̃, A = Ã, and f = f̃ . We also may and
will assume that both mappings A and f are Lipschitz continuous in all of U × Ũ .

In our construction of the continuous extension w̃ : ∆̄T0,T
ϑ0

→ C of the strict

solution w : [0, T ]→ C, holomorphic in ∆T0,T
ϑ0

, we take advantage of a factorization

approach for the complex time variable t = %µ where % ∈ (0, τ0) and µ ∈ C with
|µ − 1| < sinϑ. The numbers τ0 ∈ (0, T ) and ϑ ∈ (0, ϑ0) are suitable constants.
Fixing such a constant µ, we obtain a mild solution, ω ≡ ωµ : [0, τ0]→ U ⊂ E1− 1

p ,p
,

of the corresponding initial value problem with the real time variable t = % ∈ [0, τ0].
Of course, this solution depends on the complex parameter µ from the open disc

Dr(1) := {µ ∈ C : |µ− 1| < r}
centered at the point 1 ∈ C with radius r = sinϑ. We will complete the proof
by showing that the mild solution, ω, is holomorphic with respect to µ. This
factorization approach has been used earlier in Henry [40, Chapt. 3, §3.4] and
Lunardi [65, Chapt. 8, §8.3.3].

Proof of Theorem 5.3. Given any two numbers ϑ ∈ (0, π/2) and τ0 ∈ (0,∞), we
define a bounded open sector in the complex plane C by

A
(τ0)
ϑ := {t = %µ ∈ C : 0 < % < τ0 and µ ∈ C with |µ− 1| < sinϑ} (5.27)

with vertex at the origin 0 ∈ C and angle 2ϑ. Its closure in C is denoted by Ā
(τ0)
ϑ .

Recalling our definition of the triangle ∆
(T )
ϑ by (1.6), and setting r = sinϑ (hence,

0 < r < 1), we deduce that

∆
(T1)
ϑ′ ⊂ A

(τ0)
ϑ ⊂ ∆

(T2)
ϑ

holds whenever

0 < T1 ≤ τ0 , 0 < ϑ′ < arctan r , (1 + r)τ0 ≤ T2 <∞ .

Following this factorization of the complex time t ∈ C in t = %µ with % ∈ (0, τ0)
and µ ∈ Dr(1) = {µ ∈ C : |µ − 1| < r}, so that 0 < 1 − r < <eµ < 1 + r with

r = sinϑ (< 1), we replace the complex time t ∈ ∆T ′,T
ϑ in the initial value problem

(5.26) by the product µt ∈ C with t ∈ (0, τ0) and µ ∈ Dr(1), where we will choose

both ϑ ∈ (0, π/2) and τ0 ∈ (0,∞) sufficiently small, so that A
(τ0)
ϑ ⊂ ∆T0,T

ϑ0
holds,

i.e., µt ∈ ∆T0,T
ϑ0

for every pair (t, µ) ∈ (0, τ0)×Dr(1). Hence, we must have

0 < ϑ ≤ ϑ0 , rτ0 ≤ T0 · tanϑ0 , and (1 + r)τ0 ≤ T .
Given a fixed number µ ∈ Dr(1) ⊂ C, we look for an unknown continuous

mapping ω ≡ ωµ : [0, τ0] → U ⊂ E1− 1
p ,p

, ω(t) ≡ ωµ(t) = w̃(µt), that according

to (5.26) must be a strict solution to the following evolutionary problem (with the

tilde “
∼

”, marking holomorphic extensions, having been removed),

dω

dt
− µA(µt, ω(t))ω(t) = µ [f(µt, ω(t)) + g(µt)] for every t ∈ (0, τ0) ;

ω(0) = w0 ∈ E1− 1
p ,p

.
(5.28)

Of course, for µ = 1 we will have ω(t) ≡ ω1(t) = w(t) for a.e. t ∈ (0, T ), by
uniqueness. We remark that, thanks to our hypothesis w(t) ∈ U and A(t, w(t)) ∈
MRp(E) for all t ∈ [0, T ], we have also µA(t, w(t)) ∈ MRp(E) for all t ∈ [0, T ] and



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 55

all µ ∈ C satisfying |µ−1| < sinϑ with ϑ ∈ (0, π/2) small enough, say, 0 < ϑ < π/6
in which case |µ−1| < 1/2. This claim follows easily from the perturbation lemma,
Lemma 5.2, thanks to µ = 1 +ν with ν ∈ C satisfying |ν| < sinϑ. Even Lemma 5.1
would do if ϑ ∈ (0, π/2) were chosen sufficiently small. Clearly, it suffices to prove
that, for each fixed t ∈ (0, τ0), the function µ 7→ ωµ(t) : Dr(1) → E1− 1

p ,p
is

holomorphic. This approach to the analyticity in time of solutions to semilinear
parabolic problems can be found, e.g., in the monographs by Henry [40, Chapt. 3],
§3.4, Theorem 3.4.4 and Corollary 3.4.6 on pp. 63–66, and by Lunardi [65, Chapt. 8],
§8.3.3, p. 308.

Choosing ϑ ∈ (0, π/2) and τ0 ∈ (0, T ] small enough, such that Ā
(τ0)
ϑ ⊂ U , and

recalling r = sinϑ ∈ (0, 1), we abbreviate

F (t, v, µ) := µ[f(µt, v) + g(µt)] for all (t, v, µ) ∈ [0, τ0]× U ×Dr(1) . (5.29)

By (H5’), the mapping (v, µ) 7→ F (t, v, µ) : U×Dr(1)→ E0 is holomorphic for each
t ∈ [0, τ0], with all partial derivatives of F with respect to v and µ being continous
in [0, τ0]×U ×Dr(1). According to Amann [6, Chapt. III, §4.10], pp. 180–191, and
Clément and Li [20, Sect. 2], p. 18, given a fixed parameter value µ ∈ Dr(1), every
strict solution ω ≡ ωµ ∈ Y p1− 1

p

(0, τ0) of the initial value problem (5.28) satisfies the

following integral equation for the unknown function ω ≡ ωµ ∈ Y p1− 1
p

(0, τ0),

ω(t) = F(t, ω, µ) for every t ∈ [0, τ0] , (5.30)

with the right-hand side equal to

F(t, v, µ)

:= eµtA(0,w0) w0 + µ

∫ t

0

eµ(t−s)A(0,w0) [A(µs, v(s))−A(0, w0)] v(s) ds

+

∫ t

0

eµ(t−s)A(0,w0) F (s, v(s), µ) ds for every t ∈ [0, τ0]

(5.31)

and for all v ∈ Y p
1− 1

p

(0, τ0) satisfying v(t) ∈ U for every t ∈ [0, τ0]. In contrast

to defining a contraction mapping using the (unique) strict solution to prove local
existence in Theorem 4.5, in the case of problem (5.28) we prefer to use the (unique)
mild solution defined by an integral representation (variation-of-constants formula);
cf. (5.31). The equivalence between strict and mild solutions is treated in Ball [10],
Henry [40, Chapt. 3], and Pazy [72, Theorem on p. 259].

Clearly, (5.30) is a fixed point equation for the unknown function ω ∈ Y p
1− 1

p

(0, τ0).

Here, one can choose τ0 ∈ (0, T1], where T1 ∈ (0, T ] and ϑ ∈ (0, π/2) are sufficiently

small, such that Ā
(T1)
ϑ ⊂ U and the mapping

Φ ≡ Φµ : v 7→ [t 7→ F(t, v, µ)]

is a contraction in a closed ball

Σ
(w0)
ρ1,T1

=
{
v ∈ Y T1 : v(0) = w0 and ‖v − w|[0,T1]‖Y T1 ≤ ρ1

}
in the Banach space

Y T1 = Y p
1− 1

p

(0, T1) = Lp((0, T1)→ E1) ∩W 1,p((0, T1)→ E0)

of radius ρ1 ∈ (0,∞) centered at the point w ∈ Y T1 . As usual, the function
w|[0,T1] ∈ Y T1 denotes the restriction to [0, T1] of the strict solution w ∈ Y T =



56 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

Y p
1− 1

p

(0, T ) from the hypotheses of our theorem. The proof of this contraction

property follows the same ideas and steps as the proof of Theorem 4.5 taken from
Clément and Li [20, Theorem 2.1, p. 19]. The reader is referred to Prüss [74,
pp. 9–13], Chapt. 3, for further details. Notice that the numbers ρ1 ∈ (0,∞),
T1 ∈ (0, T ], and ϑ ∈ (0, π/2), if chosen small enough, such that the contraction
holds with the Lipschitz constant 1

2 , are independent from the particular choice

of the parameter µ ∈ Dr(1) since Ā
(T1)
ϑ is a compact subset of U ; cf. our remarks

before this proof (Remark 5.4) that remain valid also for the compact set Ā
(T1)
ϑ ×K

in the complex Banach space C×E1− 1
p ,p

. Of course, r = sinϑ ∈ (0, 1) is sufficiently

small, and both ρ1 ∈ (0,∞) and T1 ∈ (0, T ] must be also so small that v(t) ∈ U
holds for all t ∈ [0, T1], whenever v ∈ Σ

(w0)
ρ1,T1

. Finally, the constants ρ1, T1, and ϑ
can be chosen independent from w0 ∈ K, so that one may use them in any time
interval [t0, t0 +T1] ⊂ [0, T ] of sufficiently short length T1 > 0; the initial condition
w(0) = w0 ∈ K at t = 0 is replaced by w(t0) ∈ K at arbitrary time t0 ∈ [0, T −T1].

Next, we analyze the holomorphy properties of the fix point mapping

F : [0, T1]× Σ
(w0)
ρ1,T1

×Dr(1)→ Σ
(w0)
ρ1,T1

defined in (5.31) where we may take τ0 = T1; more precisely, those of the mapping

(v, µ) 7→ F(t, v, µ) : Σ
(w0)
ρ1,T1

×Dr(1)→ Σ
(w0)
ρ1,T1

,

for each fixed t ∈ [0, T1]. To begin with, for 0 ≤ s < t ≤ T1, µ ∈ Dr(1), and

v ∈ Σ
(w0)
ρ1,T1

, we rewrite

A(µs, v(s))−A(0, w0)

= {I − [λI −A(µs, v(s))][λI −A(0, w0)]−1}[λI −A(0, w0)] ,

where λ ∈ (0,∞) is large enough in order to guarantee that the (bounded) linear

operator λI −A(0, w0) : E1 → E0 has a bounded inverse [λI −A(0, w0)]
−1

: E0 →
E1, and observe that the function (integrand)

µ 7→ eµ(t−s)A(0,w0) [A(µs, v(s))−A(0, w0)] v(s) : Dr(1)→ E0

is holomorphic and so is the integral

µ 7→
∫ t

0

eµ(t−s)A(0,w0) [A(µs, v(s))−A(0, w0)] v(s) ds : Dr(1)→ E0 .

We have used here the fact that the operator-valued function

µ 7→ eµ(t−s)A(0,w0) : Dr(1)→ L(E0 → E0)

is holomorphic for any fixed numbers s, t ∈ R satisfying 0 ≤ s < t ≤ T1. Similarly,
the function

µ 7→ eµ(t−s)A(0,w0) F (s, v(s), µ) : Dr(1)→ E0

being holomorphic, so is the integral

µ 7→
∫ t

0

eµ(t−s)A(0,w0) F (s, v(s), µ) ds : Dr(1)→ E0 .

We conclude that the sum

µ 7→ F(t, v, µ) : Dr(1)→ E0

defined by (5.31) with τ0 = T1 is holomorphic for every t ∈ [0, T1].



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 57

Finally, from the fixed point equation (5.30) we deduce that the function ω ≡
ωµ : [0, τ0] → U ⊂ E1− 1

p ,p
, which is continuous thanks to ω ∈ Σ

(w0)
ρ1,T1

⊂ Y T1 =

Y p
1− 1

p

(0, T1), is holomorphic in the variable µ ∈ Dr(1). Although this holomorphy

claim follows directly from a well-known result in Deimling [22, Theorem 15.3,
Chapt. 4, §15, p. 151], cf. also Krantz and Parks [58, Theorem 6.1.2, §6.1, p. 118],
we sketch a constructive proof below for the sake of completeness.

Indeed, any standard proof of the Banach fixed point theorem for the (contrac-
tive) self mapping

Φ ≡ Φµ : v 7→ [t 7→ F(t, v, µ)] : Σ
(w0)
ρ1,T1

→ Σ
(w0)
ρ1,T1

shows that, given an arbitrary “initial” function ϕ0 ∈ Σ
(w0)
ρ1,T1

, the iterates

ϕn = Φ(ϕn−1) = Φ2(ϕn−2) = . . . = Φn−1(ϕ1) = Φn(ϕ0) ; for n = 1, 2, 3, . . . ,

form a Cauchy sequence in Σ
(w0)
ρ1,T1

which converges to the unique fixed point ω ≡ ωµ
of Φ, namely, ϕn → ω in Σ

(w0)
ρ1,T1

⊂ Y T1 as n→∞. The convergence is uniform for

µ ∈ Dr(1). Recalling the continuous imbedding (5.7), we have also ϕn(t)→ ω(t) in
E1− 1

p ,p
as n→∞, uniformly for t ∈ [0, T1] and µ ∈ Dr(1). Choosing ϕ0 = w|[0,T1],

a function of time t ∈ [0, T1] which does not depend on the parameter µ ∈ Dr(1),
we observe that each iterate

ϕn(t) = F(t, ϕn−1, µ); n = 1, 2, 3, . . . , t ∈ [0, T1] ,

is a holomorphic function in the variable (parameter) µ ∈ Dr(1). Applying Os-
good’s theorem and the Cauchy integral formula for discs to each iterate ϕn(t)
(see e.g. Krantz [57], Theorem 1.2.2 (p. 24), or John [50], Chapt. 3, Sect. 3(c), eq.
(3.22c), p. 71), we conclude that also the limit function ω ≡ ωµ is holomorphic in
the variable µ ∈ Dr(1) and satisfies Φ(ω) = ω.

We have thus verified that the strict solution w : [0, T ]→ U ⊂ E1− 1
p ,p

of problem

(4.8) possesses a holomorphic extension to the bounded open sector A
(T1)
ϑ . In fact,

we have proved that this claim is valid in any time shift of this sector by a number
t0 ∈ [0, T − T1], that is, in any sector

t0 + A
(T1)
ϑ := {t = t0 + %µ ∈ C : 0 < % < T1 and |µ− 1| < sinϑ}

with vertex at the point t0 ∈ C and angle 2ϑ. We apply the last result with t0
ranging from 0 to T − T1 over the interval [0, T − T1] to conclude that the function
w : [0, T ] → U ⊂ E1− 1

p ,p
possesses a holomorphic extension to the bounded open

set

∪t0∈[0,T−T1]

(
t0 + A

(T1)
ϑ

)
⊂ C

which contains the open complex domain ∆T ′,T
ϑ′ defined in (1.7), whenever T ′ = T1

and 0 < ϑ′ ≤ arctan(sinϑ), owing to ∆
(T1)
ϑ′ ⊂ A

(T1)
ϑ .

Hence, we have proved that there are constants ϑ′ ∈ (0, ϑ0] and T ′ ∈ (0, T0],

small enough, and a holomorphic function w̃ : ∆T ′,T
ϑ′ → E1− 1

p ,p
with the desired

properties (a) and (b) in the conclusion of our theorem. Since (0, T ) ⊂ ∆T ′,T
ϑ′ , such

a holomorphic function w̃ must be unique. The proof is complete. �



58 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

6. Analyticity in space for the Cauchy problem in RN × (0, T )

In the previous two sections, Sections 4 and 5, we have treated the initial value
problem (4.8) for a strict solution u : (0, T ) → E0 with the initial condition u0 in
the real interpolation space E1− 1

p ,p
≡ (E0, E1)1− 1

p ,p
between E1 and E0, E1 ↪→

E1− 1
p ,p

↪→ E0.

By Theorem 4.7, such a strict solution belongs to the abstract Sobolev space
Y T = Y p

1− 1
p

(0, T ) introduced in (4.10). Hence, we have u(t) ∈ E1− 1
p ,p

for every

t ∈ [0, T ].
In this section we replace the triplet of abstract (complex) Banach spaces E1 ↪→

E1− 1
p ,p

↪→ E0 by the following complex Sobolev, Besov, and Lebesgue spaces,

respectively,

W 2m,p(RN ) ↪→ Bs;p,p(RN ) ↪→ Lp(RN ) , s = 2m
(
1− 1

p

)
∈ (0, 2m) ,

where

Bs;p,p(RN ) :=
(
Lp(RN ), W 2m,p(RN )

)
s/(2m),p

=
(
Lp(RN ), W 2m,p(RN )

)
1−(1/p),p

is the Besov space obtained by real interpolation (see, e.g., [1, 65, 84]). We recall
that 2+ N

m < p <∞ which guarantees (s−m)p > N and, thus, the Sobolev(-Besov)

imbeddings Bs−m;p,p(RN ) ↪→ C0(RN ) ∩ L∞(RN ) and Bs;p,p(RN ) ↪→ Cm(RN ) ∩
Wm,∞(RN ) are continuous.

Throughout this section we restrict ourselves to the easiest case of analytic initial
conditions that we are able to treat in our present work.

6.1. Hypothesis. We use the following assumptions:

(H6) The initial data u0 : RN → CM , u0 = (u0,1, u0,2, . . . , u0,M ), can be ex-

tended to a holomorphic function ũ0 = (ũ0,1, ũ0,2, . . . , ũ0,M ) : X(r) → CM
in a complex strip X(r) ⊂ CN defined in (2.1), for some r ∈ (0,∞), such
that every component ũ0,j : X(r) → C; j = 1, 2, . . . ,M , has the following
properties:

(H6.1) the function x 7→ ũ0,j(x + iy) : RN → C is in the (complex) Besov
space Bs;p,p(RN ),

(H6.2) the Besov norm ‖ũ0,j(· + iy)‖Bs;p,p(RN ) is uniformly bounded for all

y ∈ Q(r), and
(H6.3) y 7→ ũ0,j(·+ iy) : Q(r) → Bs;p,p(RN ) is continuously (partially) differ-

entiable with respect to the parameter y = (y1, . . . , yN ) ∈ Q(r) = {y ∈
RN : |y|∞ < r}; j = 1, 2, . . . ,M .

Equivalently, the function x 7→ ũ0(x+iy) : RN → CM belongs to the Carte-
sian product Bs;p,p(RN ) = [Bs;p,p(RN )]M , its norm ‖ũ0(· + iy)‖Bs;p,p(RN )

satisfies (cf. (3.10))

N(r)(ũ0) = sup
y∈Q(r)

‖ũ0(·+ iy)‖Bs;p,p(RN ) <∞ ,

and it is continuously differentiable with respect to the parameter y ∈ Q(r).

The “shift” isometry ‖ũ0(·+x0 +iy0)‖Bs;p,p(RN ) = ‖ũ0(·+iy0)‖Bs;p,p(RN ) is obvious

for all pairs (x0, y0) ∈ RN ×Q(r), i.e., for all complex numbers z0 = x0 +iy0 ∈ X(r).



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 59

The restriction in (H6) is motivated by the following approximation property of
the Sobolev and Besov spaces, see e.g. Triebel [84, Chapt. 2].

Remark 6.1. The Fréchet space S(RN ) of all complex-valued, rapidly decreasing
infinitely differentiable functions ϕ : RN → C being dense in all of the spaces
Lp(RN ), W 2m,p(RN ), Bs;p,p(RN ), and L2(RN ), by [84, Chapt. 2], §2.3, Theorem
2.3.2 on p. 172, we take u0 : RN → CM so smooth and rapidly decreasing near
infinity that its holomorphic extension ũ0 : X(r) → CM satisfies even the following
stronger regularity condition: The family of functions x 7→ ũ0(x+ iy) : RN → CM ,
parametrized by y ∈ Q(r), belongs to a bounded subset of

L2(RN ) ∩W2m,p(RN ) for some p ∈ R, 2 +
N

m
< p <∞ .

For instance, all complex linear combinations of Hermite functions form a dense
vector subspace V of the Fréchet space S(RN ), by Reed and B. Simon [75, Chapt. V,
§3], Theorem V.13 on p. 143. Hermite functions are entire complex functions
h : CN → C of the form

h(z) = P (z1, z2, . . . , zN ) · exp
(
− 1

2

N∑
i=1

z2
i

)
for z = (zi)

N
i=1 = x+ iy ∈ CN ,

where P (z) is a complex polynomial in N complex variables zi ∈ C; i = 1, 2, . . . , N ,
see [75, p. 142]. One may take functions from V as components ũ0,j of ũ0; j =
1, 2, . . . ,M . Indeed, notice that∣∣∣ exp

(
− 1

2

N∑
i=1

z2
i

)∣∣∣ = exp
(
− 1

2

N∑
i=1

x2
i +

1

2

N∑
i=1

y2
i

)
≤ exp

(1

2
Nr2

)
· exp

(
− 1

2
|x|22
)

holds for all z = x+ iy ∈ X(r) ⊂ CN , where

|x|2 =
( N∑
i=1

|xi|2
)1/2

, |y|∞ = max
i=1,2,...,N

|yi| < r .

It is well known that all three vector spaces V ⊂ S(RN ) ⊂ L2(RN ) are invariant
under the (unitary) Fourier transformation F : L2(RN ) → L2(RN ). (We always
consider the unitary Fourier transformation F as described in Stein and Weiss
[80, Chapt. I].) Consequently, if the Fourier transform Fu0,j : RN → C of each
component of u0 : RN → CM decays at least exponentially fast at infinity, then the
holomorphic extension of the function u0,j : RN → C to a complex strip X(r) ⊂ CN ,
for some r ∈ (0,∞), is easily obtained in the form of the inverse Fourier-Laplace
transform F−1(Fu0,j) : X(r) → C of Fu0,j , by the classical Paley-Wiener-Schwartz
theory, see e.g. Hörmander [44, Theorem 7.4.2, p. 192] or Stein and Weiss [80,
Chapt. III], §2, pp. 91–101, and §6.12, pp. 127–128. An interested reader is referred
to Takáč [82, Chapt. 5] for a brief review of the (inverse) Fourier-Laplace transform
that applies to our current setting.

In regard to later applications (cf. Proposition 6.5 and Theorem 7.1), in our
(H6) above we have not specified the number r ∈ (0,∞) corresponding to the
half width of the complex strip X(r) = RN + iQ(r), a tube in CN with the base
Q(r) = (−r, r)N . Hypotheses (H1)–(H3) in Section 3 show that only the case



60 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

0 < r ≤ r0 is useful. We will comment on the choice of r ∈ (0, r0] in Remark 7.2
right after Theorem 7.1 below. Concerning this question of choosing (finding) a
suitable half-width r ∈ (0, r0], we begin with the following observation.

Remark 6.2. The Hermite functions h : CN → C described in Remark 6.1 are not
the only way for approximating the initial values u(·, t0) = u0 ∈ Bs;p,p(RN ) at time
t0 = 0 in the Besov space within the Besov norm ‖·‖Bs;p,p(RN ). In our approximation
procedure we need to guarantee the following “uniformity” of the half width of the
complex strip X(r) = RN + iQ(r), i.e., the same half-width r ∈ (0, r0] for each
approximating function ũ0,n : X(r) → CM ; n = 1, 2, 3, . . . . In precise analytic
terms, this means that, for any given radius R1 ∈ (0,∞) of the ball BR1

(0) in
Bs;p,p(RN ), there exists a number r1 ∈ (0, r0] small enough, such that, whenever
r ∈ (0, r1], the approximating sequence of functions {ũ0,n}∞n=1 has the following
properties (cf. (H6)):

(H7.1) each function x 7→ ũ0,n(x+iy) : RN → CM is in the Besov space Bs;p,p(RN ),

for every y ∈ Q(r),
(H7.2) the “proximity to u0” estimate ‖ũ0,n(·+ iy)−u0‖Bs;p,p(RN ) < R1 holds for

all y ∈ Q(r) and n = 1, 2, 3, . . . ,
(H7.3) ũ0,n : X(r) → CM is holomorphic for every n = 1, 2, 3, . . . , and finally
(H7.4) the restrictions u0,n = ũ0,n|R of ũ0,n to the real line R satisfy ‖u0,n −

u0‖Bs;p,p(RN ) → 0 as n→∞.

We keep the natural notation L2(RN ) = [L2(RN )]M etc. introduced for spaces
of vector-valued functions in the Introduction (Section 1). We recall the continuous
Sobolev(-Besov) imbeddings

S(RN ) ↪→ L2(RN ) ∩W 2m,p(RN ) ↪→ Bs;p,p(RN ) ↪→ Cm(RN ) ∩Wm,∞(RN ),

W 2m,p(RN ) ↪→ Bs;p,p(RN ) ↪→ Lp(RN ) ∩ L∞(RN ) , 2 +
N

m
< p <∞ .

We remark that W 2m,p(RN ) 6⊂ L2(RN ) if 2 < p < ∞. From now on we identify
u0 ≡ ũ0 and drop the tilde “˜” in the (unique) holomorphic extension.

By (H1)–(H3) (cf. Theorem 3.4), let us set r = r0 ∈ (0,∞) above. In the Cauchy
problem (1.1) we may replace the real space variable x ∈ RN by its complex shift
z = x + x0 + iy0 by a fixed complex vector z0 = x0 + iy0 ∈ X(r) ⊂ CN with any
x0 ∈ RN and any y0 ∈ Q(r). In the sequel we consider z0 ∈ X(r) to be a parameter
and x ∈ RN an independent variable in the Cauchy problem (1.1) spatially “shifted”
by z0,

∂u

∂t
+ P

(
x+ z0, t,

1

i

∂

∂x

)
u = f

(
x+ z0, t;

(∂|β|u
∂xβ

)
|β|≤m

)
for (x, t) ∈ RN × (0, T ) ;

u(x, 0) = u0(x+ z0) for x ∈ RN .

(6.1)

By our hypothesis on the initial data u0 : RN → CM and its holomorphic extension
u0 ≡ ũ0 : X(r) → CM stated above, for each z0 ∈ X(r), the “shifted” function

x 7→ u
(z0)
0 (x) := u0(x + z0) : RN → CM belongs to L2(RN ) ∩W2m,p(RN ), where

2+N
m < p <∞. Consequently, we have u

(z0)
0 ∈ Bs;p,p(RN ), and thus, we may apply

the local (in time) existence and uniqueness result (Theorem 4.5) on a short time

interval [t0, T1] ⊂ [0, T ] with the initial condition u(·, t0) = u
(z0)
0 in Bs;p,p(RN ) at

time t = t0 ∈ [0, T ) to conclude that the spatially “shifted” Cauchy problem (6.1)



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 61

possesses a unique weak solution u(z0) ∈ C
(
[t0, T1]→ Bs;p,p(RN )

)
, local in time.

Of course, the length of the time interval [t0, T1] depends on the shift z0 ∈ X(r);
more precisely, on its imaginary part y0 = =mz0 ∈ Q(r). However, when making use
of the abstract reformulation (4.8) of the Cauchy problem (6.1), we must guarantee
that the values of the (unique) strict solution u(z0) : [t0, T1] → Bs;p,p(RN ) to
the Cauchy problem (6.1), the (continuous) “shifted” function t 7→ u(· + z0, t) =

u(z0)(·, t) : [t0, T1] → Bs;p,p(RN ), stay in the bounded open set U = Ũ ⊂ E1− 1
p ,p

for all times t ∈ [t0, T1] (cf. Remark 5.4). To avoid this technical problem, we make
the following global existence hypothesis, cf. Theorem 3.4:

6.2. Hypothesis.

(H8) The original Cauchy problem (1.1), i.e., problem (6.1) with z0 = 0 and the
initial data u0 = û0 ∈ Bs;p,p(RN ) at t = 0, possesses a global weak solution
û ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
.

Now define the set U ⊂ E1− 1
p ,p

= Bs;p,p(RN ) that appears in (H4), (H5), (H4’),

(H5’), as follows: First, let

U0 = conv
(
BR0

(0) ∪ ∪t∈[0,T ]BR0
(û(·, t))

)
⊂ Bs;p,p(RN ) (6.2)

be the convex hull of the union of open balls

BR0(0) :=
{
w ∈ Bs;p,p(RN ) : ‖w‖Bs;p,p(RN ) < R0

}
⊂ Bs;p,p(RN ),

BR0(v) := v +BR0(0) with v = û(·, t) for t ∈ [0, T ] ,

where their radius R0 ∈ (0,∞) is an arbitrary positive number. Of course, the
symbol “0” stands for the zero function in Bs;p,p(RN ). Alternatively, we may
take U0 = BR0

(û0) to be any open ball in Bs;p,p(RN ) centered at û0 = û(·, 0) with
(sufficiently large) radius R0 ∈ (0,∞), such that 0 ∈ BR0(û0) and û(·, t) ∈ BR0(û0)
holds for every t ∈ [0, T ]. However, this choice of R0 would not fit in Example 9.2
in Section 9 below. Clearly, U0 is a bounded open set in Bs;p,p(RN ). From now

on we take the initial values u(z0)(·, t0) = u
(z0)
0 = u(z0)(· + z0, t0) ∈ Bs;p,p(RN )

at time t = t0 ∈ [0, T ) in the Cauchy problem (6.1) (and similar related initial

value problems) from the set U0 only, i.e., u
(z0)
0 ∈ U0. This choice will guarantee

that the values of the (unique) strict solution u(z0) : [t0, T1] → Bs;p,p(RN ) to the
“shifted” Cauchy problem (6.1) stay for all times t ∈ [t0, T1] in the bounded open

set U = Ũ ⊂ E1− 1
p ,p

= Bs;p,p(RN ) defined next, U ⊃ U0. We put

U = ∪{BR0
(v) : v ∈ U0} ⊂ Bs;p,p(RN ) ; (6.3)

hence,

U =
{
w ∈ Bs;p,p(RN ) : ‖w − v‖Bs;p,p(RN ) < R0 for some v ∈ U0

}
.

One may call U the open R0-neighborhood of U0 in Bs;p,p(RN ). Also U = Ũ
is a bounded, open, and convex set in the complex Besov space Bs;p,p(RN ) and,
consequently, in Wm,p(RN ) and in Cm

bdd(RN ) = Cm(RN ) ∩Wm,∞(RN ), as well,
owing to the continuous Sobolev(-Besov) imbeddings

Bs;p,p(RN ) ↪→Wm,p(RN ) and Bs;p,p(RN ) ↪→ Cm(RN ) ∩Wm,∞(RN ) , (6.4)



62 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

respectively, where (
1 +

N

2m+N

)
m < s = 2m

(
1− 1

p

)
< 2m

thanks to the inequalities 2 + N
m < p < ∞; see Adams and Fournier [1, Chapt. 7],

Theorem 7.34(a,c), p. 231. This shows that, for any function w ∈ U , the partial

derivatives ∂|β|w
∂xβ

, β = (β1, . . . , βN ) ∈ (Z+)N , of order |β| = β1 + · · ·+ βN ≤ m, are

uniformly bounded on RN ,∣∣∂|β|w(x)

∂xβ
∣∣ ≤ C ≡ C(U) = const <∞ for all x ∈ RN , (6.5)

for a constant C ∈ R+ depending solely on U . These partial derivatives are argu-

ments in the reaction function f
(
x, t;

(
∂|β|u
∂xβ

)
|β|≤m

)
on the right-hand side of (1.1)

and (6.1). In (H3) on f we take Σ ⊂ CMÑ to be the closed polydisc Σ = [D̄C(0)]MÑ

where D̄C(0) := {z ∈ C : |z| ≤ C} is a closed disc. This restriction on the values

of the (unique) strict solution to the bounded open set U = Ũ ⊂ E1− 1
p ,p

will be

used in applications to semilinear Heston-type models in “Mathematical Finance”
treated in Section 9.

From (H3) we deduce immediately that each component fj : Ω̄×[D̄C(0)]MÑ → C
of the reaction function f = (f1, . . . , fM ) is continuously differentiable (i.e., of
class C1) with the time derivative ∂

∂tfj(x, t;X) and all argument first-order partial
derivatives

∂fj
∂Xβ,k

(x, t;X) , for |β| ≤ m and j, k = 1, 2, . . . ,M ,

being uniformly bounded on Ω× Σ. Consequently, each fj is Lipschitz continuous
with respect to the variables t and Xβ,k, uniformly on Ω× Σ.

Recalling the continuous Sobolev(-Besov) imbeddings (6.4), i.e.,

Bs;p,p(RN ) ↪→Wm,p(RN ) ∩ Cm(RN ) ∩Wm,∞(RN ) ,

and the Lp-integrability condition in (3.4), we have just proved the following lemma
(cf. (H4), (H5), (H4’), (H5’)):

Lemma 6.3. Assume that f : Ω̄ × CMÑ → CM satisfies (H3), and (H8) is also

satisfied. Let U = Ũ ⊂ E1− 1
p ,p

= Bs;p,p(RN ) be as in (6.3). Then the Nemytskii

operator F : [0, T ]× U → E0 = Lp(RN ) defined by

F(t,v)(x) := f
(
x, t;

(∂|β|v
∂xβ

)
|β|≤m

)
, x ∈ RN , (6.6)

for all t ∈ [0, T ] and all v ∈ U , satisfies the following properties:

(a) F : [0, T ] × U → E0 is a Lipschitz continuous mapping, i.e., F satisfies
(H5).

(b) The substitution mapping F : C([0, T ]→ U)→ Lp((0, T )→ E0) defined by

F(v)(t) := F(t,v(t)), for all t ∈ [0, T ], v ∈ C([0, T ]→ U) ,

is Lipschitz continuous with values in Lp((0, T )→ E0).

Proof. The only claims in Parts (a) and (b) that remain to be verified are that
F maps [0, T ] × U into E0 and F maps C([0, T ] → U) into Lp((0, T ) → E0),
respectively.



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 63

(a) For each component Fj of F = (F1, . . . , FM ) from (6.6) we derive

Fj(t,v1)(x)− Fj(t,v2)(x)

=
∑
|β|≤m

M∑
k=1

[ ∫ 1

0

∂fj
∂Xβ,k

(
x, t;

(
(1− θ)∂

|β|v1

∂xβ
+ θ

∂|β|v2

∂xβ

)
|β|≤m

)
dθ
]

× ∂|β|

∂xβ
(v1,k(x)− v2,k(x)) , x ∈ RN , j = 1, 2, . . . ,M ,

(6.7)

for all t ∈ [0, T ] and for all v1,v2 ∈ U . Notice that the partial derivatives with
respect to xβ emerge from the chain rule applied to the right-hand side of (6.6)
using the partial derivatives ∂

∂Xβ,k
fj(x, t;X) with respect to the argument Xβ,k and

the convex combination w = (1− θ)v1 + θv2 ∈ U for 0 ≤ θ ≤ 1, thanks to U being
convex. Consequently, with this abbreviation for w and our choice of the constant

C ≡ C(U) in (6.5), all partial derivatives ∂|β|w
∂xβ

, β = (β1, . . . , βN ) ∈ (Z+)N , of order

|β| = β1 + · · · + βN ≤ m, are uniformly bounded on RN , by (6.5). By (H3), cf.
Remark 3.1, all partial derivatives ∂

∂Xβ,k
fj(x, t; ·) : Σ→ C are uniformly bounded,∣∣∂fj(x, t;X)

∂Xβ,k

∣∣ ≤ C1 ≡ C1(C(U)) = const <∞ (6.8)

(cf. (3.5)) for all (x, t) ∈ Ω and all X =
(
(Xβ,k)|β|≤m

)M
k=1
∈ Σ, by a constant

C1 ∈ R+ depending solely on C(U). We apply these estimates to the integrands
in (6.7) to conclude that there is a Lipschitz constant L ≡ L(U) ∈ R+ depending
solely on U (through the constant C1(C(U)) ≥ 0 in (6.8) above), such that

|F(t,v1)(x)− F(t,v2)(x)| ≤ L
∑
|β|≤m

M∑
k=1

∣∣∣ ∂|β|
∂xβ

(v1,k(x)− v2,k(x))
∣∣∣ (6.9)

for all x ∈ RN , t ∈ [0, T ] and v1,v2 ∈ U .
We recall U ⊂ Bs;p,p(RN ) and the imbeddings in (6.4) to deduce from (6.9) that

the mappings F(t, ·) : U → E0 = Lp(RN ) are uniformly Lipschitz continuous (with
the same Lipschitz constant) for all t ∈ [0, T ]. Here, we single out the special case
of v1 = v ∈ U being arbitrary and v2 = 0 ∈ U , i.e., v2(x) ≡ 0 ∈ CM for all
x ∈ RN . Then (6.9) yields

|F(t,v)(x)| ≤ |F(t,0)(x)|+ L
∑
|β|≤m

M∑
k=1

∣∣ ∂|β|
∂xβ

vk(x)
∣∣ , (6.10)

for all x ∈ RN , t ∈ [0, T ] and v ∈ U , where

F(t,0)(x) = f(x, t;~0), x ∈ RN , ~0 = (0)|β|≤m ≡ (0, . . . , 0) ∈ CMÑ .

satisfies F(t,0) ∈ E0 = Lp(RN ), by the Lp-integrability condition in (3.4), i.e.,

‖F(t,0)‖E0
=
(∫

RN
|f(x, t;~0)|p dx

)1/p

≤ K for all t ∈ [0, T ],

where K ∈ (0,∞) is a constant. Now it follows from (6.10) above that also F(t,v) ∈
E0 holds for all t ∈ [0, T ] and for all v ∈ U , as claimed.

(b) Analogous results for the mapping F : C([0, T ] → U) → Lp((0, T ) → E0)
follow from those we have just proved for F(t, ·) : U → E0, t ∈ [0, T ]. Namely, the



64 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

“supremum” (or “maximum”) norm on the Banach space C
(
[0, T ] → E1− 1

p ,p

)
of

all continuous functions u : [0, T ]→ E1− 1
p ,p

is defined by

|||u|||L∞(0,T ) := ‖u‖
C
(

[0,T ]→E
1− 1

p
,p

) = sup
t∈[0,T ]

‖u(t)‖E
1− 1

p
,p
. �

Remark 6.4. Analogous result (to Lemma 6.3) hold for the “shifted” Nemytskii
operator F(z0) : [0, T ]×U → E0 = Lp(RN ), by a complex vector z0 ∈ X(r), defined
by

F(z0)(t,v)(x) := f
(
x+ z0, t;

(∂|β|v
∂xβ

)
|β|≤m

)
, x ∈ RN , (6.11)

for all t ∈ [0, T ] and for all v ∈ U . Both constants, C1 ≡ C1(C(U)) and L ≡ L(U) in
inequalities (6.8) and (6.9), respectively, are independent from the shift by z0 ∈ X(r)

in case x ∈ RN is replaced by x + z0, thanks to (x, t) ∈ Ω where the domain

Ω = Γ
(T0)
T (r0, ϑ0) = X(r0) × ∆T0,T

ϑ0
⊂ CN × C has been introduced in Section 3

before (H1)–(H3) (cf. (3.8) and Theorem 3.4).

In what follows, the initial data u0 in the Cauchy problem (6.1) at time t0 ∈
[0, T ) have nothing to do with the initial value û(·, 0) = û0 ∈ Bs;p,p(RN ) of the
global weak solution û ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
in (H8), except for the restriction

u0 ∈ U0, where U0 is determined by the values of the solution û(·, t) in Bs;p,p(RN )
for 0 ≤ t ≤ T , see (6.2).

We take advantage of Remark 6.4 to recall that, given a holomorphic function
u0 ≡ ũ0 : X(r) → CM as described before (H8), the spatially “shifted” Cauchy prob-
lem (6.1) possesses a unique weak solution u(z0) ≡ u(z0,t0) ∈ C

(
[t0, T1]→ Bs;p,p(RN )

)
,

local in time, for every fixed shift z0 ∈ X(r), locally uniformly in the complex do-
main, provided its real and imaginary parts, x0 = <ez0, y0 = =mz0 ∈ Q̄(r1) ⊂ Q(r),
are small enough, i.e., max{|x0|∞, |y0|∞} ≤ r1 (< r = r0). Indeed, it suf-

fices to choose r1 ∈ (0, r) so small that each “shifted” function x 7→ u
(z0)
0 (x) =

u0(x + z0) : RN → CM (serving as the initial data at time t = t0 ∈ [0, T )), with
z0 = x0 + iy0 satisfying max{|x0|∞, |y0|∞} ≤ r1, lies in the open set U0 specified

in (6.2) after (H8), i.e., u
(z0)
0 = u0(· + z0) ∈ U0 ⊂ E1− 1

p ,p
. Recall that Q̄(r1) and

X̄(r1) = RN + iQ̄(r1) stand for the respective closures of the cube Q(r1) ⊂ RN and
the strip X(r1) ∈ CN . Indeed, our choice of r1 ∈ (0, r) small enough to guaran-

tee u
(z0)
0 = u0(· + x0 + iy0) ∈ U0 for every y0 ∈ Q(r1) (and for all x0 ∈ RN ), is

possible thanks to the closed cube Q̄(r1) ⊂ RN being compact. For “small” shifts
z0 = x0 + iy0 ∈ CN we introduce the complex cube

Q
(r1)
C := Q(r1) + iQ(r1) = {z = x+ iy ∈ C : max{|x|∞, |y|∞} < r1}

⊂ X(r1) = RN + iQ(r1)

and denote by Q̄
(r1)
C its closure in CN ; hence,

Q̄
(r1)
C = Q̄(r1) + iQ̄(r1)

= {z = x+ iy ∈ C : max{|x|∞, |y|∞} ≤ r1}

⊂ X̄(r1) = RN + iQ̄(r1) ⊂ X(r) ,

thanks to 0 < r1 < r. We will call Q̄
(r1)
C the small shift cube.

To obtain a spatially holomorphic extension u ≡ ũ : X(r1) × [0, T ] → CM of
the function û ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
, from (H8), to the spatio-temporal strip



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 65

X(r1) × [0, T ], having a (unique) continuous extension u : X̄(r1) × [0, T ] → CM to
the closed strip X̄(r1) × [0, T ], such that (y0, t) 7→ u(· + iy0, t) : Q̄(r1) × [0, T ] →
Bs;p,p(RN ) is continuous and satisfies u(· + iy0, t) ∈ U for every y0 ∈ Q̄(r1) and
for every t ∈ [0, T ], we construct u first locally in time on a short time interval
[t0, t0 + T1] ⊂ [0, T ] as follows, where T1 > 0 is small enough.

Let u(z0) : t 7→ u(z0)(·, t) ≡ u(z0,t0)(·, t) from C
(
[t0, t0 + T1]→ Bs;p,p(RN )

)
be

as above, such that z0 = iy0 and u(iy0)(·, t) ∈ U for every pair (y0, t) ∈ Q̄(r1) ×
[t0, t0 + T1]. Given any y0 ∈ Q̄(r1), let us define

u(x+ iy0, t) := u(iy0)(x, t) for all (x, t) ∈ RN × [t0, t0 + T1] . (6.12)

Clearly, u : X̄(r1) × [t0, t0 + T1] → CM is a well-defined mapping, and it has the
following properties.

Proposition 6.5. Let M,N ≥ 1, 0 < T < ∞, and assume that (H1)–(H3) are
satisfied with constants 0 < r0 <∞, 0 < T0 ≤ T , and 0 < ϑ0 < π/2. Furthermore,
assume that û ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
is a globally defined weak solution to

the original Cauchy problem (1.1), i.e., (H8) is valid. Let the sets U0 ⊂ U ⊂
E1− 1

p ,p
= Bs;p,p(RN ) be specified as in (6.2) and (6.3). Given any t0 ∈ [0, T ),

let u0 ∈ U0 ⊂ Bs;p,p(RN ) be any initial data at time t = t0, such that u0 has a
holomorphic extension ũ0 : X(r) → CM as described in (H6) and Remark 6.2 (we

identify u0 ≡ ũ0), with u
(z0)
0 = u0(·+ z0) ∈ U0 ⊂ E1− 1

p ,p
whenever z0 = x0 + iy0 ∈

Q̄
(r1)
C ⊂ X̄(r1) for some r1 ∈ (0, r). (We have set r = r0; r1 may depend on u0.)
Then there exists a number T1 ∈ (0, T − t0], depending on r1 and U , but not on

t0, such that the Cauchy problem (6.1) on the (local) time interval [t0, t0 + T1] ⊂
[0, T ] with the initial condition u(·, t0) = u

(z0)
0 possesses a unique weak solution

u(z0) ≡ u(z0,t0) ∈ C
(
[t0, t0 + T1]→ Bs;p,p(RN )

)
, such that u(z0)(·, t) ∈ U holds for

every t ∈ [t0, t0 + T1]. The family u(z0), parametrized by z0 = x0 + iy0 ∈ Q̄(r1)
C , has

the following properties:

(a) u(z0)(x, t) = u(iy0)(x+ x0, t) holds for all (x, t) ∈ RN × [t0, t0 + T1] and for

all z0 ∈ Q̄(r1)
C ; consequently, even for all z0 ∈ X̄(r1).

(b) The mapping u : X̄(r1) × [t0, t0 + T1] → CM defined in (6.12) satisfies
u(x+ x0 + iy0, t) = u(x0+iy0)(x, t) for all (x, t) ∈ RN × [t0, t0 + T1] and for
all z0 = x0 + iy0 ∈ CN with |y0|∞ ≤ r1.

(c) The mapping u : X̄(r1)×[t0, t0 +T1]→ CM : (z, t) = (x+iy, t) 7→ u(x+iy, t)
is continuously (partially) differentiable with respect to all the real variables
xi and yi (i = 1, 2, . . . , N) in x = (x1, . . . , xN ) and y = (y1, . . . , yN ) in RN
with |y|∞ ≤ r1.

(d) For each fixed t ∈ [t0, t0 + T1], the mapping u(·, t) : X(r1) → CM : z =
x+ iy 7→ u(x+ iy, t) is holomorphic, i.e., (partially) complex differentiable
with respect to all the complex variables zi = xi + iyi (i = 1, 2, . . . , N) in
z = (z1, . . . , zN ) ∈ X(r1) ⊂ CN .

As for Part (d) in this proposition, there are several equivalent definitions of a
holomorphic function of several complex variables used in the literature, cf. Krantz
[57, Definitions I–IV, pp. 3–4]. We adopt the most widely used definition in [57],
Definition II (p. 3) and Definition 1.2.1 (p. 24). From this definition it is easy to de-
rive the existence of an absolutely convergent power series ([57, Definition III, p. 3])
and the Cauchy formula in a polydisc ([57, Definition IV, pp. 3–4]). Nevertheless,



66 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

the equivalence of [57, Definition I, p. 3] verified in part (d) in our proposition above
with [57, Definitions II–IV, pp. 3–4] is a deep classical result due to Hartogs; see
[57, Theorem 1.2.5, p. 25]. However, taking also part (c) into account, we observe
that also [57, Definition II, p. 3] is verified in our proposition.

Proof of Proposition 6.5. Recalling our remarks on the local (in time) existence
and uniqueness before this proposition, we observe that it suffices to verify only
our claims in Parts (a)–(d).

(a) Clearly, given any fixed z0 = x0 + iy0 ∈ Q̄(r1)
C , both functions

t 7→ u(z0)(·, t) and t 7→ u(iy0)(·+ x0, t) : [t0, t0 + T1]→ Bs;p,p(RN )

are weak solutions to our Cauchy problem (6.1) on the (sufficiently short) time

interval [t0, t0 + T1] ⊂ [0, T ] with the same initial data u
(z0)
0 (·) = u

(iy0)
0 (· + x0) at

time t = t0 ∈ [0, T ), for some T1 ∈ (0, T − t0]. The uniqueness for problem (6.1)
now forces u(z0)(·, t) ≡ u(iy0)(·+ x0, t) for every t ∈ [t0, t0 + T1] as claimed.

Part (b) is an immediate consequence of Part (a) applied to (6.12).
(c) At the initial time t = t0, the continuous (partial) differentiability is valid by

our hypotheses on the initial data u0 : X(r) → CN viewed as a function

z0 = x0 + iy0 = (x0, y0) 7→ u0(·+ z0) = u
(z0)
0 : X(r) = RN ×Q(r) → Bs;p,p(RN )

valued in the Besov space E1− 1
p ,p

= Bs;p,p(RN ); in particular, u0(·+ z0) = u
(z0)
0 ∈

U0 ⊂ E1− 1
p ,p

provided z0 = x0 + iy0 ∈ Q̄(r1)
C ⊂ X̄(r1) (⊂ X(r)). We recall that U0

is an open subset of E1− 1
p ,p

defined in (6.2). We may view this C1 differentiability

as (partial) differentiability with respect to the real parameters x0,i and y0,i in

the complex shift z0 = x0 + iy0 ∈ Q̄
(r1)
C , where x0 = (x0,1, . . . , x0,N ) and y0 =

(y0,1, . . . , y0,N ) are in RN with max{|x0|∞, |y0|∞} ≤ r1.
We now briefly interrupt our proof of Proposition 6.5 to make the following

remarks:

Remarks. The kind of theory on continuous and differentiable dependence of the
solution

z0 7→ u(·+ z0, t) = u(z0)(·, t) : X̄(r1) = RN × Q̄(r1) → Bs;p,p(RN )

for t ∈ [t0, t0 +T1], on the real parameters x0,i and y0,i in z0 ∈ X̄(r1), that has been
developed in Henry [40, Chapt. 3], §3.4, pp. 62–70, or, alternatively, in Lunardi [65,
Chapt. 8], §8.3.1, pp. 302–306, can be adapted also to our setting for the spatially
“shifted” Cauchy problem (6.1), with only minor changes. We should remark that,
in this approach, the following hypotheses on A and f will do; they follow from
(H1)–(H3) (cf. Lemma 6.3 and its proof):

6.3. Hypothesis. In analogy to, (H4’) and (H5’) let us assume that there are

positive constants ϑ0 ∈ (0, π/2) and T0 ∈ (0, T ], and open sets U ⊂ C and Ũ ⊂
E1− 1

p ,p
containing the compact set ∆̄T0,T

ϑ0
and the open set U , respectively, i.e.,

∆̄T0,T
ϑ0

⊂ U ⊂ C and U ⊂ Ũ ⊂ E1− 1
p ,p

, such that

(H4”) A : [0, T ]× U → L(E1 → E0) possesses a continuously (Fréchet-) differen-

tiable extension (i.e., of class C1) Ã : U × Ũ → L(E1 → E0) to the complex

domain U × Ũ which satisfies Ã(t, v) ∈ MRp(E) for all (t, v) ∈ U × Ũ .



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 67

(H5”) f : [0, T ] × U → E0 possesses a continuously (Fréchet-) differentiable ex-

tension f̃ : U × Ũ → E0 to the complex domain U × Ũ .

Clearly, in both these hypotheses, the mappings A and f , respectively, are ex-

tended from the domain [0, T ] × U ⊂ ∆̄T0,T
ϑ0
× U to the complex domain U × Ũ ⊂

C× E1− 1
p ,p

.

Now we continue the proof of Proposition 6.5. Proof of part (c). Recall that the

metric on U × Ũ ( ⊂ C×E1− 1
p ,p

) is induced by the canonical norm on C×E1− 1
p ,p

.

It is evident that (H4’) and (H5’) imply (H4”) and (H5”), respectively.
Applying the results from [40, Chapt. 3, §3.4] or [65, Chapt. 8, §8.3.1], we now

conclude that the mapping

z0 7→ u(·+ z0, t) : X̄(r1) = RN × Q̄(r1) → Bs;p,p(RN ) , t ∈ [t0, t0 + T1] ,

is continuously differentiable with respect to the real parameters x0,i and y0,i in

z0 ∈ X̄(r1). The partial derivatives,

∂u

∂x0,i
≡ ∂u

∂xi
=
∂u

∂zi
,

∂u

∂y0,i
≡ ∂u

∂yi
= i · ∂u

∂zi
: X̄(r1) × [t0, t0 + T1]→ CM

valued in C
(
[t0, t0 + T1]→ Bs;p,p(RN )

)
, are the unique weak solutions of the follow-

ing Cauchy problems derived from (6.1) by the corresponding partial differentiation,
respectively:

∂

∂t

( ∂u

∂xi

)
+
∂P

∂xi

(
x+ z0, t,

1

i

∂

∂x

)
u(x+ z0, t)

+ P
(
x+ z0, t,

1

i

∂

∂x

)( ∂u

∂xi

)
(x+ z0, t)

=
∂f

∂xi

(
x+ z0, t;

(∂|β|u
∂xβ

(x+ z0, t)
)
|β|≤m

)
+
∑
|β|≤m

M∑
k=1

∂f

∂Zβ,k

(
x+ z0, t;

(∂|β|u
∂xβ

)
|β|≤m

) ∂|β|
∂xβ

(∂uk
∂xi

)
(x+ z0, t)

for (x, t) ∈ RN × (t0, t0 + T1) ;

∂u

∂xi
(x+ z0, 0) =

∂u0

∂xi
(x+ z0) for x ∈ RN ,

(6.13)

and

∂

∂t

( ∂u

∂yi

)
+
∂P

∂yi

(
x+ z0, t,

1

i

∂

∂x

)
u(x+ z0, t)

+ P
(
x+ z0, t,

1

i

∂

∂x

)( ∂u

∂yi

)
(x+ z0, t)

=
∂f

∂yi

(
x+ z0, t;

(∂|β|u
∂xβ

(x+ z0, t)
)
|β|≤m

)
+
∑
|β|≤m

M∑
k=1

∂f

∂Zβ,k

(
x+ z0, t;

(∂|β|u
∂xβ

)
|β|≤m

) ∂|β|
∂xβ

(∂uk
∂yi

)
(x+ z0, t)

for (x, t) ∈ RN × (t0, t0 + T1) ;

∂u

∂yi
(x+ z0, 0) =

∂u0

∂yi
(x+ z0) for x ∈ RN ,

(6.14)



68 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

where the complex variable Zβ,k stands for Zβ,k = ∂|β|

∂xβ
uk = i|β|Dβ

xuk ∈ C. This
proves Part (c).

Proof of part (d). We take advantage of the two equations (6.13) and (6.14), to
apply the Cauchy-Riemann operator ∂/∂z̄i from (2.2) to problem (6.1) to conclude
that the Cauchy-Riemann derivative

∂̄z0,iu ≡
∂u

∂z̄0,i
≡ ∂u

∂z̄i
=

1

2

( ∂

∂xi
+ i

∂

∂yi

)
u : X̄(r1) × [t0, t0 + T1]→ CM

is in C
(
[t0, t0 + T1]→ Bs;p,p(RN )

)
and obeys the following homogeneous linear

Cauchy problem, which is a simple linear combination 1
2 ·(6.13) + i

2 · (6.14) = (6.15):

∂

∂t

(
∂̄z0,iu

)
+ P

(
x+ z0, t,

1

i

∂

∂x

)
(∂̄z0,iu)(x+ z0, t)

=
∑
|β|≤m

M∑
k=1

∂f

∂Zβ,k

(
x+ z0, t;

(∂|β|u
∂xβ

)
|β|≤m

) ∂|β|
∂xβ

(
∂̄z0,iuk

)
(x+ z0, t)

for (x, t) ∈ RN × (t0, t0 + T1) ;(
∂̄z0,iu

)
(x+ z0, 0) = 0 for x ∈ RN .

(6.15)

Here, we have used that both operators

z 7→ P
(
z, t,

1

i

∂

∂x

)
and z 7→ f

(
z, t; (Zβ)|β|≤m

)
: X(r) → CM (r = r0)

are holomorphic, i.e., ∂̄ziP
(
z, t, 1

i
∂
∂x

)
= 0 and ∂̄zif

(
z, t; (Zβ)|β|≤m

)
= 0, by (H1)

and (H3), respectively. By our choice of u0 ≡ ũ0 : X(r) → CM being holomorphic,
we have also ∂̄ziu0(z) = 0; i = 1, 2, . . . , N . Notice that (6.15) is valid only for every
z0 ∈ X̄(r1) (⊂ X(r)).

By (H1) and (H2), the linear differential operator on the left-hand side of (6.15),
i.e.,

∂

∂t
+ P

(
x+ z0, t,

1

i

∂

∂x

)
,

is uniformly parabolic of order 2m with smooth coefficients. It is proved in Denk,
Hieber and Prüss [23, p. 67], Theorem 5.7 (cf. also [74], Theorem 2.1 (p. 8) and
remarks thereafter (p. 9)) that, for every z0 ∈ X(r) and for every t ∈ [0, T ],

A(z0)(t) := −P
(
x+ z0, t,

1

i

∂

∂x

)
: W2m,p(RN )→ Lp(RN ) (6.16)

is a bounded linear operator, i.e., A(z0)(t) ∈ L(E1 → E0), and it possesses the
maximal Lp-regularity property, i.e., A(z0)(t) ∈ MRp(E) ≡ MRp(E1 → E0). Let us
recall that E1 = W2m,p(RN ) ↪→ E0 = Lp(RN ).

Furthermore, in view of (H3), the pointwise multiplication and differentiation
operators on the right-hand side of (6.15) are of order |β| (|β| ≤ m < 2m) and
all have bounded continuous coefficients, by u(·, t) ∈ U ⊂ Bs;p,p(RN ) for every
t ∈ [t0, t0 + T1] combined with the Sobolev imbedding Bs;p,p(RN ) ↪→ Cm(RN ) ∩
Wm,∞(RN ), where 2+ N

m < p <∞ and m < s = 2m
(
1− 1

p

)
< 2m. We denote their

sum, which appears in (6.15), by f (z0)(t) : Bs;p,p(RN ) → Lp(RN ), i.e., f (z0)(t) ∈
L
(
E1− 1

p ,p
→ E0

)
. Here, we allow any z0 ∈ X̄(r1). Consequently, the mappings

(t, v) 7→ A(z0)(t) : [0, T ]×U → L(E1 → E0) and (t, v) 7→ f (z0)(t)v : [0, T ]×U → E0



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 69

satisfy (H4) and (H5) for A and f , respectively, with U = E1− 1
p ,p

, A(z0)(t) ∈
L(E1 → E0) being independent from v ∈ U , and v 7→ f (z0)(t)v linear in v ∈ U .

We observe that the homogeneous linear Cauchy problem (6.15) for the Cauchy-
-Riemann derivative ∂̄z0,iu of u takes the following abstract linear form, whenever

z0 ∈ X̄(r1):

d

dt
(∂̄z0,iu)−A(z0)(t)(∂̄z0,iu) = f (z0)(t)(∂̄z0,iu) for a.e. t ∈ (t0, t0 + T1) ;

(∂̄z0,iu)(0) = 0 ∈ E1− 1
p ,p

.
(6.17)

This abstract linear problem corresponds to the nonlinear initial value problem
(4.8) treated in Section 4. We apply the uniqueness part of Theorem 4.5 to deduce
(∂̄z0,iu)(x, t) ≡ 0 for all (x, t) ∈ RN × [t0, t0 + T1]. This implies that the mapping

z 7→ uk(z, t) : X(r1) → C is holomorphic in each complex variable zi ∈ C, for every
fixed time t ∈ [t0, t0+T1]; k = 1, 2, . . . , N . Moreover, by part (c), all complex partial
derivatives ∂ziuk(·, t) are continuous in X(r1). Finally, we take advantage of the
classical fact that such a function uk(·, t) : X(r1) → C is holomorphic (Remark 2.1);
see e.g. John [50, Theorem, p. 70] or Krantz [57, Definition II, p. 3]. Also Part (d)
and, thus, the entire proposition is proved. �

7. Space-time analyticity for the Cauchy problem in RN × (0, T )

We summarize the time and space analyticity results from the last two sections
(Sections 5 and 6), for the mapping u : X̄(r1)× [t0, t0 +T1]→ CM defined in (6.12),
in the following theorem.

Theorem 7.1. Let M,N ≥ 1, 0 < T <∞, and assume that (H1)–(H3)are satisfied
with some constants 0 < r0 < ∞, 0 < T0 ≤ T , and 0 < ϑ0 < π/2. Furthermore,
assume that û ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
is a globally defined weak solution to

the original Cauchy problem (1.1), i.e., (H8) is valid. Let the sets U0 ⊂ U ⊂
E1− 1

p ,p
= Bs;p,p(RN ) be specified as in (6.2) and (6.3). Given any t0 ∈ [0, T ), let

u0 ∈ Bs;p,p(RN ) be any initial data at time t = t0, such that u0 ∈ U0 and u0 has a
holomorphic extension ũ0 : X(r) → CM as described before Lemma 6.3 (we identify

u0 ≡ ũ0), with u
(z0)
0 = u0(· + z0) ∈ U0 ⊂ E1− 1

p ,p
whenever z0 ∈ Q̄(r1)

C ⊂ X̄(r1) for

some r1 ∈ (0, r), cf. (H6). (We have set r = r0; r1 may depend on u0.)
Finally, let u : X̄(r1)× [t0, t0 +T1]→ CM be the continuous mapping obtained in

Proposition 6.5, with T1 ∈ (0, T −t0] depending on r1 and U , but not on t0. Replace
T0 ∈ (0, T ] by min{T0, T1} if necessary, so that 0 < T0 ≤ T1 ≤ T holds. Then
there exist constants ϑ′ ∈ (0, ϑ0] and T ′ ∈ (0, T0], small enough, and a continuous

mapping ũ : X̄(r1) ×
(
t0 + ∆T ′,T1

ϑ′

)
→ CM with the following properties:

(i) For each z0 ∈ X̄(r1), the (unique) weak solution

u(z0) ∈ C
(
[t0, t0 + T1]→ Bs;p,p(RN )

)
to the Cauchy problem (6.1) on the time interval [t0, t0 + T1] ⊂ [0, T ] with

the initial condition u(·, t0) = u
(z0)
0 at time t = t0 satisfying z0 ∈ Q̄

(r1)
C

possesses a unique holomorphic extension from (t0, t0 + T1) to t0 + ∆T ′,T1

ϑ′ ,

such that u(z0)(·, t0 + s) = ũ(·+ z0, t0 + s) ∈ U holds for every s ∈ ∆T ′,T1

ϑ′ .



70 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

(ii) The complex function ũ : X(r1) ×
(
t0 + ∆T ′,T1

ϑ′

)
→ CM is holomorphic

(jointly) in all its variables, z = (z1, z2, . . . , zN ) ∈ X(r1) ⊂ CN and t ∈
t0 + ∆T ′,T1

ϑ′ ⊂ C.

Let us recall that, by our notation in (1.7), for ϑ ∈ (0, π/2), 0 < t0 < T < ∞,
and 0 < T ′ ≤ T − t0, we have

t0 + ∆T ′,T−t0
ϑ =

(
t0 + ∆

(T−t0)
ϑ

)
∩ {t ∈ C : |=mt| < T ′ tanϑ}

= ∪t0≤ξ≤T−T ′{ξ + t′ ∈ C : t′ ∈ ∆
(T ′)
ϑ }

= ∪t0≤ξ≤T−T ′
(
ξ + ∆

(T ′)
ϑ

) (7.1)

with the closure t0 + ∆̄T ′,T−t0
ϑ in C.

Remark 7.2. (a) The main difference between our main result, Theorem 3.4 (Sec-
tion 3), and Theorem 7.1 above is the temporally local character of the latter stated
for the time interval [t0, t0 + T1] ⊂ [0, T ] with the additional analyticity hypothesis
on the initial data u0 (as in part (iii) of Theorem 3.4).

(b) Recalling our choice of the number r ∈ (0,∞) in (H6) (before Remark 6.1)
on the complex analyticity of the initial data u0 = ũ0 : X(r) → CM extended to
the complex strip X(r) ⊂ CN , we observe that the number r1 ∈ (0, r), originally
introduced in the spatially “shifted” Cauchy problem (6.1), is needed for sufficiently
small perturbations (“shifts”) z0 ∈ CN of the space variable z ∈ X(r1) in order to
keep z + z0 ∈ X(r). As we have already mentioned after Remark 6.1, (H1)–(H3)
show that only the case 0 < r ≤ r0 is useful. We now recall from Proposition 6.5
and Theorem 7.1 that, to avoid excessive notation, r1 ∈ (0, r) must be chosen

small enough, such that u
(z0)
0 = u0(· + z0) ∈ U0 ⊂ U ⊂ E1− 1

p ,p
for every z0 ∈

Q̄
(r1)
C ⊂ X̄(r1). We recall that the sets U0 and U are defined in (6.2) and (6.3),

respectively. We stress that both, U0 and U , are open in E1− 1
p ,p

, while being

determined solely by the restriction to the real line R, u0 = ũ0|R ∈ U0 ⊂ E1− 1
p ,p

,

of the initial data ũ0 : X(r) → CM , i.e., by u0 : RN → CM as an element of the
Besov space E1− 1

p ,p
= Bs;p,p(RN ). Consequently, the number r1 ≡ r1(u0) ∈ (0,∞)

is determined by these initial data u0 ∈ U0; we have 0 < r1 < r ≤ r0 where we may
choose r = r0, by Remark 6.1. Such a choice of r1 ∈ (0, r) is possible thanks to the
closed cube Q̄(r1) being compact in RN .

Proof of Theorem 7.1. (i) Let z0 = x0 + iy0 ∈ X̄(r1) be arbitrary, but fixed, with

max{|x0|∞, |y0|∞} ≤ r1, i.e., z0 ∈ Q̄(r1)
C . We apply our time analyticity result in

Theorem 5.3 to the Cauchy problem (6.1) on the time interval [t0, t0 + T1] ⊂ [0, T ]

with the initial condition u(·, t0) = u
(z0)
0 ∈ U0 at time t = t0 to derive the conclusion

of Part (i).
(ii) The second part is obtained by combining Part (i) with Proposition 6.5,

particularly Part (d). Finally, the joint time and space analyticity of the complex

function ũ : X(r1) ×
(
t0 + ∆T ′,T1

ϑ′

)
→ CM is obtained by applying the classical

characterization of holomorphic functions by the Cauchy-Riemann equations (Re-
mark 2.1); see e.g. John [50, Theorem, p. 70] or Krantz [57, Definition II, p. 3]. �



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 71

8. Proofs of main results

Now we are ready to prove Theorem 3.4. Then Proposition 3.5 is a consequence
of Theorem 3.4 and Lemma 5.2, inequality (5.6). We give its proof right after that
of Theorem 3.4.

Proof of Theorem 3.4. (i) The existence and uniqueness of a weak solution u ∈
C
(
[0, T1]→ Bs;p,p(RN )

)
to the Cauchy problem (1.1), local in time for t ∈ [0, T1]

with some T1 ∈ (0, T ], is obtained directly from the abstract result in Theorem 4.5
where E1− 1

p ,p
= Bs;p,p(RN ) = [Bs;p,p(RN )]M . The technical details in applying

Theorem 4.5 (an abstract result) to problem (1.1) have been given in Section 6,
right after problem (6.1). The linear parabolic operator on the left-hand side in
(6.1) is treated by the maximal Lp-regularity described in Remark 4.2(a). The
special case of p in this remark, p0 = 2, is taken care of by standard parabolic
regularity making use of G̊arding’s inequality in Corollary 3.3; see, e.g., Friedman
[31, Chapt. 10]. If a weak solution u ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
exists globally in

time t ∈ [0, T ], then it is unique, by Theorem 4.7, Part (i).
(ii) The (unique) temporal extension of the function u : RN × (0, T )→ CM to a

holomorphic function u] : ∆T ′,T
ϑ′ → Bs;p,p(RN ) that possesses another extension to

a continuous function on the closure ∆̄T ′,T
ϑ′ , denoted again by u], is derived from

Theorem 7.1, Part (i). More precisely, Part (i) of Theorem 7.1 is applied to the
(global) weak solution u ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
of the Cauchy problem (1.1),

which is assumed to exist, in the temporal complex domain t0 + ∆
(T ′)
ϑ′ with the

initial value u(·, t0) ∈ Bs;p,p(RN ) at every initial time t0 ∈ [0, T − T ′]. Here, we
have used that

∆T ′,T
ϑ′ = ∪0≤t0≤T−T ′

(
t0 + ∆

(T ′)
ϑ′

)
(8.1)

(cf. (7.1)).
(iii) We remark that (H8) is satisfied with the function

û = u ∈ C
(
[0, T ]→ Bs;p,p(RN )

)
,

a globally defined weak solution to the original Cauchy problem (1.1), which exists
by our hypothesis. Let us recall the definitions of the bounded, open, and convex
sets U0 and U in Bs;p,p(RN ), U0 ⊂ U = Ũ ⊂ E1− 1

p ,p
= Bs;p,p(RN ), in (6.2) and

(6.3), respectively, where the radius R0 ∈ (0,∞) is an arbitrary positive number.
We recall also our hypothesis that the initial condition u(·, 0) = u0 ∈ Bs;p,p(RN )
possesses a (unique) holomorphic extension ũ0 : X(κ0) → CM from RN to the
complex domain X(κ0) = RN + iQ(κ0) ⊂ CN (a tube), for some κ0 ∈ (0, r0], that
satisfies (3.10).

We begin with a construction of the (unique) spatial extension of the continuous
function u : RN × [0, T ] → CM to a continuous function u[ : X̄(ρ) × [0, T ] → CM
that is holomorphic in the space variable z = x+ iy ∈ X(ρ) = RN + iQ(ρ) with some
ρ ∈ (0, κ0] small enough. Let us recall our notation with the “shifted” function

x 7→ u
(z0)
0 (x) := u0(x + z0) : RN → CM introduced in the Cauchy problem (6.1)

spatially “shifted” by z0 = x0 + iy0 ∈ X(r) ⊂ CN . The constant r ∈ (0,∞) has
been introduced in (H6); only the case 0 < r < κ0 ≤ r0 ( <∞) is useful. We wish
to apply Proposition 6.5 with the constant r1 ∈ (0, r) specified there. We choose



72 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

ρ ∈ (0, r1) small enough, such that also

‖u(iy)
0 − u0‖Bs;p,p(RN ) = ‖u0(·+ iy)− u0‖Bs;p,p(RN ) < R0 holds for all y ∈ Q̄(ρ) .

Here, we have used the (Lipschitz) continuity of the mapping y 7→ u
(iy)
0 := u0(· +

iy) : Q̄(r1) → Bs;p,p(RN ), thanks to 0 < r1 < κ0 supplemented by the Cauchy
formula in a polydisc centered in X̄(r1) (with radius < κ0−r1) and contained in the

complex strip X(κ0) ⊂ CN . From (6.2) we deduce that u
(iy)
0 ∈ U0 for all y ∈ Q̄(ρ).

In analogy with our proof of Part (i) above, we apply Theorem 4.5 to conclude
that the spatially shifted Cauchy problem (6.1), with the shift z0 = iy (y ∈ Q̄(ρ)),
possesses a unique weak solution u(iy) ∈ C

(
[0, T1]→ Bs;p,p(RN )

)
, local in time for

t ∈ [0, T1] with some T1 ∈ (0, T ], that satisfies u(iy)(·, t) ∈ U for every t ∈ [0, T1].
We apply part (c) or part (d) of Proposition 6.5 with t0 = 0 to conclude that there
is a number R1 ∈ (0, R0) small enough, such that even u(iy)(·, t) ∈ U0 ⊂ U holds
for every t ∈ [0, T1], provided ρ ∈ (0, r1) is chosen so small that also

‖u(iy)
0 − u0‖Bs;p,p(RN ) < R1 (< R0) holds for all y ∈ Q̄(ρ) .

Here, besides the (Lipschitz) continuity of the mapping y 7→ u
(iy)
0 := u0(· + iy) :

Q̄(r1) → Bs;p,p(RN ) mentioned above, we have used also the continuous depen-

dence of the solution u(iy) upon the initial data u
(iy)
0 ∈ Bs;p,p(RN ) obtained in

Theorem 4.7, part (ii); see also Remark 4.6. According to (6.12), we define the
function u[ : X̄(ρ) × [0, T1]→ CM by the formula

u[(x+ iy, t) := u(iy)(x, t) for all (x, y, t) ∈ RN × Q̄(ρ) × [0, T1] . (8.2)

Clearly, by Proposition 6.5, part (c), the function u[ : X̄(ρ) × [0, T1] → CM is
continuous and, by Proposition 6.5, part (d), also holomorphic with respect to the
complex variable z = x+ iy ∈ X(ρ) = RN + iQ(ρ) at every time t ∈ [0, T1].

Next, we set u
(iy)
1 := u(iy)(·, T1) ∈ U0 and repeat the procedure from above on

the interval [T1, 2T1] with the initial data u
(iy)
1 ∈ U0 at t0 = T1 in place of u

(iy)
0 ∈ U0

at t0 = 0. We stress that the interval length T1 ∈ (0, T − t0] in Proposition 6.5 is
independent from the choice of the initial time t0 ∈ (0, T ) whenever [t0, t0 + T1] ⊂
[0, T ]. Again, we apply part (c) or part (d) of Proposition 6.5 with t0 = T1 (in place
of t0 = 0) to conclude that there is a number R2 ∈ (0, R1) small enough, such that
even u(iy)(·, t) ∈ U0 ⊂ U holds for every t ∈ [0, 2T1], provided ρ ∈ (0, r1) is chosen
so small that also

‖u(iy)
0 − u0‖Bs;p,p(RN ) < R2 (< R1 < R0) holds for all y ∈ Q̄(ρ) .

The desired function u[ is naturally extended from the domain X̄(ρ) × [0, T1] to
X̄(ρ) × [0, 2T1] by setting (cf. (8.2))

u[(x+ iy, t) := u(iy)(x, t) for all (x, y, t) ∈ RN × Q̄(ρ) × [T1, 2T1] . (8.3)

We keep repeating this procedure (by “induction” on k) with the initial data

u
(iy)
k := u(iy)(·, kT1) ∈ U0 successively for every k = 0, 1, 2, . . . ,m until reaching the

inequalities
(m− 1)T1 ≤ T < mT1 at k = m− 1 .

In fact, setting u
(iy)
m−1 := u(iy)(·, (m− 1)T1) ∈ U0 and repeating the procedure from

above on the time interval [(m − 1)T1,mT1] with the initial data u
(iy)
m−1 ∈ U0 at

t0 = (m−1)T1 in place of u
(iy)
0 ∈ U0 at t0 = 0, we can apply Theorem 4.5 to conclude



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 73

that the spatially shifted Cauchy problem (6.1), with the shift z0 = iy (y ∈ Q̄(ρ)),
possesses a unique weak solution u(iy) ∈ C

(
[(m− 1)T1,mT1]→ Bs;p,p(RN )

)
, local

in time for t ∈ [(m − 1)T1,mT1], that satisfies u(iy)(·, t) ∈ U for every t ∈ [(m −
1)T1,mT1]. Consequently, we may assume T = mT1 instead of (m − 1)T1 ≤ T <
mT1. In this way we have constructed a finite set of numbers R1, R2, . . . , Rm−1, Rm
such that 0 < Rm < Rm−1 < · · · < R1 < R0 and, provided ρ ∈ (0, r1) is chosen
small enough, also

‖u(iy)
k − u0‖Bs;p,p(RN ) < Rk holds for all y ∈ Q̄(ρ) , k = 0, 1, 2, . . . ,m− 1 ,

together with u(iy)(·, t) ∈ U0 ⊂ U for every t ∈ [0, (m− 1)T1] and u(iy)(·, t) ∈ U for
every t ∈ [(m− 1)T1,mT1]. Finally, the desired function u[ is defined successively
on the domains X̄(ρ) × [(k − 1)T1, kT1] for each k = 1, 2, 3, . . . ,m by the formula

u[(x+ iy, t) := u(iy)(x, t) for all (x, y, t) ∈ RN × Q̄(ρ) × [0, T ] . (8.4)

To summarize the result of the procedure described above, we have determined
a constant ρ ∈ (0, r1), small enough, such that for each shift y ∈ Q̄(ρ) there is
a unique weak solution u(iy) ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
to the spatially shifted

Cauchy problem (6.1) with the shift z0 = iy, that satisfies u(iy)(·, t) ∈ U0 ⊂ U for
every t ∈ [0, (m− 1)T1] and u(iy)(·, t) ∈ U for every t ∈ [0, T ], where T = mT1. We
apply Proposition 6.5, Parts (c) and (d), once again to conclude that the function
u[ : X̄(ρ) × [0, T ]→ CM constructed above in (8.4) has the desired properties: it is
continuous and holomorphic in the space variable z = x + iy ∈ X(ρ) = RN + iQ(ρ)

with some ρ ∈ (0, r1) small enough, where 0 < r1 < κ0 ≤ r0.
Now we are ready to finish our proof of part (iii) by further extending the (global)

weak solution u ∈ C
(
[0, T ]→ Bs;p,p(RN )

)
of the Cauchy problem (1.1) from the

domain X̄(ρ) × [0, T ] of the (unique) spatial extension u[ : X̄(ρ) × [0, T ] → CM

to another continuous function ũ : X̄(ρ) × ∆̄T ′,T
ϑ′ → CM which is holomorphic in

X(ρ) ×∆T ′,T
ϑ′ . We recall that the solution u ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
is assumed

to exist by hypothesis (in part (iii)) with the initial data u0 ∈ Bs;p,p(RN ) having a
(unique) holomorphic extension ũ0 : X(κ0) → CM from RN to the complex domain
X(κ0) ⊂ CN , for some κ0 ∈ (0, r0].

We apply Theorem 7.1 to the function u[ : X̄(ρ) × [0, T ] → CM on every time
interval [t0, t0 + T1] ⊂ [0, T ]. We remark that the number T1 ∈ (0, T − t0] depends
on r1 and U , but not on t0 ∈ [0, T ), provided [t0, t0 + T1] ⊂ [0, T ]. In fact, making
use of the same argument as above, where we have extended the function u[ from
the domain X̄(ρ) × [0, T ] to X̄(ρ) × [0,mT1] in case (m − 1)T1 ≤ T < mT1, we
can extend u[ from the domain X̄(ρ) × [t0, T ] to X̄(ρ) × [t0, t0 + T1] in case 0 ≤
t0 < T < t0 + T1. Thus, if T < t0 + T1 then we may replace T by T = t0 + T1

and, hence, assume that [t0, t0 + T1] ⊂ [0, T ]. Consequently, by Theorem 7.1,
the (unique) weak solution u ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
to the Cauchy problem

(1.1) possesses a unique holomorphic extension from the time interval (t0, t0 + T1)

to the complex temporal domain t0 + ∆T ′,T1

ϑ′ , such that the continuous mapping

ũ : X̄(r1) ×
(
t0 + ∆T ′,T1

ϑ′

)
→ CM constructed in Theorem 7.1 is holomorphic in the

space-time domain X(r1) ×
(
t0 + ∆T ′,T1

ϑ′

)
. Since the interval [t0, t0 + T1] ⊂ [0, T ] is

arbitrary, both statements (iii1) and (iii2) and the complex analyticity statement
(iii3) in part (iii) of Theorem 3.4 follow from (8.1). More precisely, the desired

holomorphic extension ũ : X̄(r1)×∆T ′,T
ϑ′ → CM is obtained by shifting the temporal



74 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

domain t0 + ∆T ′,T1

ϑ′ with the vertex t0 ranging from the left to the right over the
time interval [0, T −T1]. In this process, the uniqueness result in Theorem 7.1, Part
(i), guarantees that the function ũ(·+iy, t) = u(iy)(t) = ũ(·+iy, t0 +s) = u(iy)(t) ∈
Bs;p,p(RN ), with 0 ≤ s = t − t0 ≤ T1, is well defined for all (y, t) ∈ Q(r1) ×∆T ′,T

ϑ′

independently from the particular choice of the vertex t0 ∈ [0, T − T1] of the the

complex temporal domain t0 + ∆T ′,T1

ϑ′ . Furthermore, in part (iii), (iii1), (3.11)
holds with ρ in place of κ0, whereas r′ ∈ (0, κ0] has to be replaced by ρ, as well.

This concludes our proof of Theorem 3.4. �

We conclude this section with the proof of the estimate in (3.14).

Proof of Proposition 3.5. We recall from Section 6 that the shifted continuous func-
tion u(iy) : t 7→ u(iy)(t) = ũ(·+iy, t) : [0, T ]→ Bs;p,p(RN ) is a unique weak solution
of the spatially shifted Cauchy problem (6.1) with the shift z0 = iy (y ∈ Q(r1)).
Consequently, u(iy) ∈ C

(
[0, T ]→ Bs;p,p(RN )

)
is also a strict solution (cf. Defini-

tion 4.4) to the following abstract initial value problem, for every y ∈ Q(r1):

d

dt
u(iy) −A(iy)(t)u(iy) = F(iy)

(
t, u(iy)(t)

)
for a.e. t ∈ (0, T ) ;

u(iy)(0) = ũ0(·+ iy) ∈ E1− 1
p ,p

= Bs;p,p(RN ) ,
(8.5)

cf. (6.1) and (6.17). Here, A(z0)(t) ∈ L(E1 → E0) is the bounded linear (partial
differential) operator introduced in (6.16), satisfying A(z0)(t) ∈ MRp(E1 → E0) for

every t ∈ [0, T ], and F(z0) : [0, T ] × U → E0 = Lp(RN ) stands for the “shifted”
Nemytskii operator defined in Remark 6.4, (6.11). We recall that both constants,
C1 ≡ C1(C(U)) and L ≡ L(U) in inequalities (6.8) and (6.9), respectively, are
independent from the shift by z0 ∈ X(r1) in case x ∈ RN is replaced by x+ z0; with
z0 = iy (y ∈ Q(r1)) in our case.

We now derive an estimate analogous to (4.11) for our shifted Cauchy prob-
lem (8.5) in place of the (original) abstract problem (4.9). Inspecting the proof of
Theorem 2.1 in Clément and Li [20, pp. 20–23] and combining it with our linear
perturbation result in Lemma 5.2 and the estimate in (5.6), we arrive at the follow-
ing estimate for our shifted Cauchy problem (8.5) in place of the abstract problem
(4.9), ∫ T

0

∥∥du(iy)

dt

∥∥p
E0

dt+

∫ T

0

∥∥∥A(iy)(0)u(iy)(t)
∥∥∥p
E0

dt

≤Mp,T

(∥∥u(iy)(0)
∥∥p
E

1− 1
p
,p

+

∫ T

0

∥∥F(iy)
(
t, u(iy)(t)

)∥∥p
E0

dt
)
,

(8.6)

where Mp,T ∈ (0,∞) is a constant independent from the initial data u(iy)(0) =

ũ0(· + iy) ∈ E1− 1
p ,p

= Bs;p,p(RN ) and the right-hand side F(iy)
(
t, u(iy)(t)

)
of

(8.5), as well. Since A(iy)(0) ∈ MRp(E1 → E0) ⊂ Hol(E1 → E0) holds by the proof
of Proposition 6.5, part (d), there is a number λ0 ∈ R+ = [0,∞), sufficiently large,
such that the bounded linear operator λ0I −A(iy)(0) : E1 → E0 is an isomorphism
of E1 onto E0. Hence, its inverse satisfies (λ0I − A(iy)(0))−1 ∈ L(E0 → E1). We
conclude that there are constants c1, C1 ∈ (0,∞) and c2, C2 ∈ R+ (both sufficiently
large, depending on λ0 ≥ 0) such that both inequalities

c1‖u‖E1
− c2‖u‖E0

≤ ‖A(iy)(0)u‖E0
≤ C1‖u‖E1

+ C2‖u‖E0



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 75

hold for all u ∈ E1. Consequently, we have (respectively)

2−(p−1)c1‖u‖pE1
≤ ‖A(iy)(0)u‖pE0

+ cp2‖u‖
p
E0

and (8.7)

‖A(iy)(0)u‖pE0
≤ 2p−1

(
Cp1‖u‖

p
E1

+ Cp2‖u‖
p
E0

)
for all u ∈ E1 . (8.8)

Furthermore, for every t ∈ [0, T ], we split the expression

F(iy)
(
t, u(iy)(t)

)
= F(iy) (t, 0) +

[
F(iy)

(
t, u(iy)(0)

)
− F(iy)

(
t, 0
)]

and apply Lemma 6.3 and Remark 6.4 to derive the following analogue of (6.10)
(where we insert v = u(iy)(t)):

∣∣F(iy)
(
t,u(iy)(t)

)
(x)
∣∣ ≤ ∣∣F(iy)(t,0)(x)

∣∣+ L
∑
|β|≤m

M∑
k=1

∣∣ ∂|β|
∂xβ

u
(iy)
k (x, t)

∣∣ , (8.9)

for all x ∈ RN , t ∈ [0, T ]. Here,

F(iy)(t,0)(x) = f(x+ iy, t;~0) , x ∈ RN , ~0 = (0)|β|≤m ≡ (0, . . . , 0) ∈ CMÑ ,

satisfies F(iy)(t,0) ∈ E0 = Lp(RN ), by the Lp-integrability condition in (3.4), i.e.,

‖F(iy)(t,0)‖E0
=
(∫

RN
|f(x+ iy, t;~0)|p dx

)1/p

≤ K for all t ∈ [0, T ],

where K ∈ (0,∞) is a constant. Each term ∂|β|

∂xβ
u

(iy)
k (·, t) ∈ Lp(RN ) on the right-

hand side of (8.9) above belongs to the Besov space Bs−|β|;p,p(RN ) which, thanks

to |β| ≤ m < s = 2m
(

1− 1
p

)
< 2m, is continuously imbedded into another Besov

space, Bs−|β|;p,p(RN ) ↪→ Bs−m;p,p(RN ) ↪→ Lp(RN ). Applying these estimates to
the right-hand side of (8.9), we thus obtain∥∥F(iy)

(
t, u(iy)(t)

)∥∥p
E0
≤ 2p−1

(
Kp + γN,m,M Lp · ‖u(iy)(t)‖p

Bs;p,p(RN )

)
(8.10)

for all t ∈ [0, T ] and every y ∈ Q(r1), where γN,m,M ∈ (0,∞) is a numerical constant

depending only on N ,m, and M . Recalling u(iy)(t) ∈ U for all (y, t) ∈ Q(r1)× [0, T ]
and the definition of the set U ⊂ E1− 1

p ,p
= Bs;p,p(RN ) in (6.3), we conclude that

the right-hand side of (8.10) can be estimated from above by a constant C ≡
C(K,U) ∈ (0,∞) independent from (y, t) ∈ Q(r1) × [0, T ]:∥∥F(iy)

(
t, u(iy)(t)

)∥∥p
E0
≤ C(K,U) for all (y, t) ∈ Q(r1) × [0, T ] . (8.11)

Finally, we combine this estimate with Theorem 7.1 to arrive at∥∥F(iy)
(
t, ũ(·+ iy, t)

)∥∥p
E0
≤ C̃(K,U) for all (y, t) ∈ Q(r1) ×∆T ′,T

ϑ′ , (8.12)

where C̃ ≡ C̃(K,U) ∈ (0,∞) is a constant independent from (y, t) ∈ Q(r1)×∆T ′,T
ϑ′ .

Recall from our proof of Theorem 3.4, part (iii), that the function ũ : X̄(r1) ×
∆T ′,T
ϑ′ → CM stands for the unique holomorphic temporal extension of the function

u[ : X̄(r1) × (0, T ) → CM defined in formula (8.4). This extension, ũ, satisfies
ũ(x, y, t) = u(iy)(x, t) for all (x, y, t) ∈ RN × Q̄(r1)× [0, T ] and ũ(·+iy, t) ∈ U for all



76 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

(y, t) ∈ Q̄(r1)×∆T ′,T
ϑ′ . We employ the norm defined in (3.10) and (8.12) to estimate

the right-hand side of (8.6):∫ T

0

∥∥du(iy)

dt

∥∥p
E0

dt+

∫ T

0

∥∥A(iy)(0)u(iy)(t)
∥∥p
E0

dt

≤Mp,T

(
sup

y∈Q(r1)

‖ũ0(·+ iy)‖Bs;p,p(RN ) + C̃(K,U)T
) (8.13)

for all (y, t) ∈ Q(r1) × [0, T ]. However, in the norm

N(r1)(ũ0) := sup
y∈Q(r1)

‖ũ0(·+ iy)‖Bs;p,p(RN ) <∞

we have u(iy)(0) = ũ0(· + iy) ∈ U ⊂ E1− 1
p ,p

= Bs;p,p(RN ) for every y ∈ Q(r1)

owing to our choice of the number r1 ∈ (0, r) being sufficiently small in (and
before) Proposition 6.5. Consequently, we can estimate the right-hand side of (8.13)

above by another constant C̃ ′ ≡ C̃ ′(p, T,K,U) ∈ (0,∞) independent from (y, t) ∈
Q(r1) × [0, T ]:∫ T

0

∥∥du(iy)

dt

∥∥p
E0

dt+

∫ T

0

∥∥A(iy)(0)u(iy)(t)
∥∥p
E0

dt ≤ C̃ ′(p, T,K,U) .

We estimate the left-hand side of this inequality by (8.7) combined with u(iy)(t) ∈ U
for all (y, t) ∈ Q(r1) × [0, T ], thus arriving at∫ T

0

∥∥du(iy)

dt

∥∥p
E0

dt+ 2−(p−1)c1

∫ T

0

∥∥u(iy)(t)
∥∥p
E1

dt

≤ C̃ ′(p, T,K,U) + cp2

∫ T

0

∥∥u(iy)(t)
∥∥p
E0

dt ≤ Ĉ(p, T,K,U) ,

(8.14)

where Ĉ ≡ Ĉ(p, T,K,U) ∈ (0,∞) is a constant independent from (y, t) ∈ Q(r1) ×
[0, T ].

Within the restriction to the real time t ∈ [0, T ], the desired estimate in (3.13) is

derived directly from (8.14) above for all pairs (y, t) ∈ Q(r′)× [0, T ] ⊂ Q(r′)×∆̄T ′,T
ϑ′ .

To extend (3.13) to the complex time t ∈ ∆̄T ′,T
ϑ′ with t = σ + iτ (σ, τ ∈ R),

we take advantage of Theorem 7.1 once again. We will consider the function ũ :

X̄(r1) × ∆T ′,T
ϑ′ → CM constructed in our proof of Theorem 3.4, part (iii), along

the complex temporal path θ̃ : [0, T ] → ∆T ′,T
ϑ′ : s 7→ θ̃(s) := s + iς1(s/T ′)τ which

consists of two straight line segments,

θ̃1 : [0, T ′]→ ∆T ′,T
ϑ′ : s 7→ θ̃1(s) :=

(
1 + i

τ

T ′

)
s with 0 ≤ s ≤ T ′ ,

θ̃2 : [T ′, T ]→ ∆T ′,T
ϑ′ : s 7→ θ̃2(s) := s+ iτ with T ′ ≤ s ≤ T .

Notice that θ̃1(0) = 0, θ̃1(T ′) = θ̃2(T ′) = T ′ + iτ , and θ̃2(T ) = T + iτ . We replace
the (complex) time variable t in the original Cauchy problem (1.1) by the new (real)
time variable s ∈ [0, T ], thus obtaining two new abstract differential equations with
the time derivatives

∂u

∂s
=
(
1 + i

τ

T ′
)∂u

∂t
(x, t)

∣∣∣
t=θ̃1(s)

for 0 ≤ s ≤ T ′, (8.15)

∂u

∂s
=
∂u

∂t
(x, t)

∣∣∣
t=θ̃2(s)

for T ′ ≤ s ≤ T , (8.16)



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 77

respectively.
Finally, we apply Lemma 5.2 and the estimate in (5.6) to the new problem for

0 ≤ s ≤ T ′, thus arriving at the desired estimate in (3.13) for 0 ≤ σ ≤ T ′. For
T ′ ≤ s ≤ T we can use the definition of a strict solution (Definition 4.4) directly
and combine it with the estimate in (4.11) to obtain the estimate in (3.13) for

T ′ ≤ σ ≤ T . We conclude that (3.13) is valid also for all pairs (y, t) ∈ Q(r′)× ∆̄T ′,T
ϑ′

with t = σ + iτ (σ, τ ∈ R).
The desired estimate in (3.14) now follows from (3.13) by applying (4.5) and

(4.6). Proposition 3.5 is proved. �

9. An application to a risk model in mathematical finance

Standard models in derivative pricing, including the Black-Scholes model (see
Black and Scholes [13] and Merton [70]) and the Heston model (see Heston [41])
take advantage of risk neutral valuation methods for the arbitrage-free (“fair”)
price of the derivative. The methods are economically justified by riskless hedging
arguments introduced in [13, 70]; see also Fouque, Papanicolaou, and Sircar [29] and
Hull [45] for detailed explanations of these arguments. An important assumption
of these models, which is used in most of the hedging arguments, is the possibility
to borrow and lend any amount of money at a risk-free interest rate. This crucial
conjecture has been questioned as a consequence of the financial crisis that started
in 2007 and resulted in the bankruptcy of major financial entities like Lehman
Brothers. Enron’s bankruptcy in 2001 is briefly described in [45, p. 537], Business
Snapshot 23.1. Namely, traders have to take into consideration the increased chance
of a default. For this reason many trades contain a collateral against default and also
the pricing of non-collateralized derivatives has to be adjusted. A standard book
on risk management has been written by Hull [46]. Piterbarg [73] discusses the
differences or convexity adjustments between the price processes of collateralized
and non-collaterlized contracts which could result in funding value adjustments of
the price processes. It is only natural that traders have different funding costs for
transactions and try to include them in the price of the contract. Hull and White
reasoned in [47] that there exists no theoretical basis for such a funding value
adjustment (FVA). Also Burgard and Kjaer [16, 18] came to a similar conclusion,
by using different arguments. However, since these theoretical arguments are not
convincing from a practitioner’s point of view, but traders make the adjustments
anyway, Hull and White studied the consequences of funding value adjustment in
a more practice-oriented way in [48, 49]. Further common adjustments of the no-
-default value of a derivative are credit value adjustments (CVA) and the related
debit value adjustments (DVA); see, e.g., [16, 17, 18]. In a particular trade both
parties have to take the possibility of default of the counterparty into account which
is the bilateral counterparty risk. Price-reducing credit value adjustments are made
by the trader to have a collateral against default of the counterparty (e.g., a bank),
whereas debit value adjustments are the corresponding adjustments made by the
counterparty. The sum of all adjustments to the value of the derivative evaluated
in the absence of default is often refered to as XVA with

XVA = FVA− CVA + DVA .

A general partial differential equation for the adjusted value under the bilateral
counterparty risk and funding value adjustments has been derived in Burgard and



78 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

Kjaer [17, Section 3] using hedging arguments. This partial differential equation is
nonlinear if the mark-to-market value at default is considered to be the total value
of the derivative including all value adjustments (see [17, Section 4]) and linear
if the mark-to-market value is given by the no-default value of the derivative (see
[17, Section 5]). In both cases, the partial differential equations are well suited for
numerical calculations of the adjusted value of the derivative; see, e.g., Arregui,
Salvador, and Vázquez [8] for recent results. Following notation introduced in [17],
let us consider a derivative contract with payoff H between a trader, B, and a
counterparty, C, on an asset S, e.g., a stock, that is not affected in case of a default
by one of the two counterparties and follows the stochastic dynamics,

dSt = µ(t)St dt+ σ(t)St dWt , (9.1)

where the drift µ : [0,∞) → R and the volatility σ : [0,∞) → R are positive de-
terministic (Borel measurable) functions and (Wt)t>0 is a one-dimensional Brow-
nian motion. Let V denote the fair price (the “risk-less” value) of the derivative

in the setting without default and let V̂ denote the adjusted price (the “risky”
value) including funding value adjustments (= FVA) and bilateral counterparty risk
(= −CVA + DVA),

V̂ = V + XVA = V + FVA− CVA + DVA .

By Itô’s formula, the generator At of the Markov process (9.1) is the partial differ-
ential operator

At =
1

2
σ2S2 ∂2

∂S2
+ (qS − γS)S

∂

∂S
, (9.2)

where γS is the dividend income rate of S and qS represents the financing costs
that depend on the risk-free rate r and repo-rate of the asset (e.g., under the Fed
Repurchase Agreement (Repo)). The decisive variable in the bilateral counterparty
risk models studied in [8, 16, 17, 18] is the mark-to-market value (cf. “close-out”),
M , introduced in [17, Sect. 3, Eq. (24)]. Only two different values of M seem to be

of significant interest, namely, M = V̂ and M = V as described below:
If we set the mark-to-market value at default M = V̂ , then the total value V̂

satisfies the nonlinear partial differential equation

∂

∂t
V̂ +AtV̂ − rV̂ = −(1−RB)λBV̂

− + (1−RC)λC V̂
+ + sF V̂

+ , (9.3)

with the final value V̂ (S, T ) = H(S) at maturity time t = T , by [17, Sect. 4,
Eq. (26)]. Here, we have abbreviated x+ := max{x, 0} and x− := max{−x, 0} for
x ∈ R; hence, x = x+ − x−. We remark that the definition of the negative part
x− of x ∈ R often differs in the literature ([8, 16, 17]); it may be used with the
negative sign, i.e., x− = min{x, 0} (≤ 0), whence x = x+ + x−. We will respect
this convention below only when approximating the function x 7→ x− within the
sum x = x+ + x−. Otherwise we use x− = max{−x, 0} (≥ 0). The parameters λB
and λC are given by λB = rB − r and λC = rC − r, where rB and rC are the yields
on recovery-less bonds for B and C, respectively. RB and RC , respectively, are
recovery rates on the derivatives’ mark-to-market value at default and sF = rF − r
is the funding spread between the sellers funding rate rF for borrowed cash and the
risk-free rate r. We refer the interested reader to Burgard and Kjaer [17, Sect. 2,
pp. 2–4] for further details concerning recovery-less bonds.



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 79

In contrast, if we assume the mark-to-market value M = V , then the resulting
partial differential equation for V̂ is linear, albeit inhomogeneous with source terms
on the right-hand side,

∂

∂t
V̂ +AtV̂ −(r+λB+λC)V̂ = (RBλB+λC)V −−(λB+RCλC)V + +sFV

+ , (9.4)

with the final value V̂ (S, T ) = H(S), by [17, Sect. 5, Eq. (46)]. Of course, it
is assumed that the fair price of the derivative, V , is known. It is claimed in
[17, Sect. 3] that the vast majority of papers on valuation of conterparty risk
uses this choice (M = V ) for contracts that follow the well known “2002 ISDA
Master Agreement” initiated by the International Swaps and Derivatives Associa-
tion (ISDA). From the mathematical point of view, also any convex combination

M = (1 − θ) · V + θ · V̂ = V + θ · (XVA) of V and V̂ , with a constant θ ∈ [0, 1],
might be of economic interest, as well.

We would like to investigate the question of market completeness for the nonlin-
ear model (9.3) raised for related financial market models in Davis and Ob lój [21].
There, the authors have shown that the problem of market completeness in Mathe-
matical Finance is closely connected to (in fact, follows from) the analyticity of the
derivative price. We refer to Takáč [82, Section 8, pp. 74–83] for a survey of results
regarding the correlation between market completeness and the analyticity of the
solution and an application of analyticity results to the stochastic volatility model
in Fouque, Papanicolaou, and Sircar [29, p, 47]. The Heston stochastic volatility
model (Heston [41], which is more popular) is treated in Alziary and Takáč [3].
Market completeness for other stochastic volatility models is discussed in [82, Re-
mark 8.7, pp. 82–83]. In our present work, we are primarily interested in analyticity
of the solution for the nonlinear partial differential equation (9.3) since the linear
case (9.4) can be studied by applying the results from [82].

The nonlinearities in (9.3) are uniformly Lipschitz continuous which enables us
to apply standard existence and uniqueness results for regular, strongly parabolic
semilinear Cauchy problems from, e.g., Eidel′man [25], Friedman [30, 31, 32], Pazy
[72, Chapt. 6, §6.1, pp. 183–191], or Tanabe [81]. Due to the fact that the nonlin-

earities V̂ 7→ V̂ ± : R → R are not real-analytic, we cannot expect any analyticity
of the solution (S, t) 7→ V̂ (S, t) : (0,∞) × (0,∞) → R, neither in space nor in

time. In our approach we therefore modify the functions V̂ 7→ V̂ ± : R → R as
follows: We approximate them by real-analytic functions with complex-analytic
extensions to a domain (⊃ R) in the complex plane C. We attempt to justify this
rather “nonrigorous” step by arguing that we deal with a model in Social Sciences
(Economics) where a precise nonlinear response (i.e., the reaction function of type

V̂ 7→ V̂ ± : R → R) is hard to determine, while facing the dominant influence of
stochastic (and possibly also random) phenomena. In our example with a single
equation in one space dimension (M = N = 1), see (1.4), we thus replace the
nonlinearity f(u) = f+(u)+f−(u) by a suitable linear combination of real-analytic
approximations of the functions u 7→ u+ : R → R and u 7→ −u− having the same
asymptotic behavior at ±∞ (as u → ±∞) and denoted by f (+)(u) and f (−)(u),
respectively, with a complex variable u ∈ C. We postpone explaining the details of
this modification until Example 9.2 below.

It is a common approach to replace V̂ as a function of (S, t) by the function

v̂(X, τ) = V̂ (S, t) that depends on the logarithm of the asset price X = lnS and



80 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

the time to maturity τ = T − t. Hence, by (9.3), this function satisfies the initial
value problem

∂

∂τ
v̂ − Ãtv̂ + rv̂ = −(1−RB)λBf

(−)(v̂)− (1−RC)λCf
(+)(v̂)− sF f (+)(v̂) (9.5)

with the initial value v̂(X, 0) = Ĥ(X) := H(exp(X)) and the partial differential
operator

Ãt =
1

2
σ2 ∂2

∂X2
+
(
qS − γS +

1

2
σ2
) ∂

∂X
. (9.6)

As an alternative to the previous variable substitution it is also possible to directly
alter the stochastic processes by

S̃t = er(T−t)St and X̃t = ln S̃t = Xt + r(T − t) ,

which yields the same partial differential equation (9.5) and allows for a financial
interpretation of the new variables.

Since the coefficients of the operator Ãt defined in (9.6) are independent of the
variables X and τ , the analyticity of the solution can be studied by means of the
Green function; see Takáč et al. [83]. But if we replace the stochastic process (9.1)
that drives the value process of the asset by a stochastic volatility process, e.g. the
mean-reverting process from the classical paper of Heston [41],

dSt = µSt dt+
√
Vt St dWS

t ,

dVt = κ(θ − Vt) dt+ σV
√
Vt dWV

t ,

ρ dt = dWS
t dWV

t ,

(9.7)

the coefficients of the generator depend on the variables and we can no longer
calculate the Green function. In the volatility process (9.7)2 above, the parameters
κ, θ, and the volatility of volatility σV are positive constants and (WS

t )t>0 and
(WV

t )t>0 are one-dimensional Brownian motions correlated by a correlation factor
ρ ∈ [−1, 1] through eq. (9.7)3. This model has been treated recently in Salvador
and Oosterlee [76, 77].

Remark 9.1. Hypotheses (H1) and (H2) on the partial differential operator are
consistent with the hypotheses (H1) and (H2) in Takáč [82, p. 56]. As mentioned
above, the operator connected to the stochastic volatility model of Fouque, Papan-
icolaou, and Sircar [29, p. 47], which is parallel to (but not more general than) the
Heston model (9.7), has been studied in detail in [82, Sect. 8, pp. 74–83]. Vari-
ous other stochastic volatility models have been discussed in [82, Remark 8.7, pp.
82–83], as well. Hence, (H1) and (H2) are satisfied for these models; we refer the
reader to [82] for further details.

According to this remark, hypotheses (H1) and (H2) are fulfilled for (9.5) even
if we consider a stochastic volatility model, e.g., like (9.7), instead of (9.1). We
would like to give an example for suitable nonlinearities f (+) and f (−) that satisfy
the remaining hypothesis (H3) and approximate the functions u 7→ u+ : R → R
and u 7→ −u−, respectively.

Our example is motivated by Takáč [82, Example 8.2, pp. 79–80]. We define the
complex planar domains

∇(r)
ϑ := {ζ = ξeiθ + iη ∈ C : ξ ∈ R, η ∈ (−r, r), and |θ| < ϑ} , (9.8)



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 81

∇(r)
0 :={ζ = ξ + iη ∈ C : ξ ∈ R, η ∈ (−r, r)}

= ∩0<ϑ<π/2 ∇
(r)
ϑ = R + i(−r, r)

(9.9)

for r ∈ (0,∞) and 0 < ϑ < π/2 with their respective closures in C denoted by ∇̄(r)
ϑ

and ∇̄(r)
0 ; both contain the origin 0 ∈ C. For any given numbers r ∈ (0,∞) and

0 < ϑ < π/2, the domain ∇(r)
ϑ is of the form

∇(r)
ϑ = ∇(0)

ϑ + i(−r, r) = ∪η∈(−r,r)
(
iη +∇(0)

ϑ

)
⊂ C ,

where

∇(0)
ϑ := {ζ = ξeiθ ∈ C : ξ ∈ R and |θ| < ϑ} = (∆ϑ) ∪ (−∆ϑ) ∪ {0}

is a symmetric sector in C and the open sector ∆ϑ is defined as in (1.5). We notice

that ∇(r)
0 is a strip in C and X(r) = (∇(r)

0 )N ⊂ CN for every r ∈ (0,∞). At last,
we define the domain

O1 := C \ {iy : y ∈ (−∞,−1] ∪ [1,∞)}

that contains the closure ∇̄(r)
ϑ whenever 0 < r < 1 and 0 < ϑ < π/2. The definitions

of these domains follow [82, pp. 78–79].
We now give an example for functions f (+) and f (−) (approximating v 7→ v+

and v 7→ −v−, respectively) that are analytic in O1 and whose first derivatives are

bounded in ∇(r0)
ϑ0

, whenever r0 ∈ (0,∞) and 0 < ϑ0 < π/2.

Example 9.2. We consider the functions

f (+)(v) = v
[1
2

+
1

π
arctan(v)

]
, f (−)(v) = v

[1
2
− 1

π
arctan(v)

]
(9.10)

defined for every v ∈ R. We have chosen f (−) such that f (−)(v) = −f (+)(−v) and
f (+)(v) + f (−)(v) = v hold for all v ∈ R since the nonlinearities v+ and −v− in the
original equation (9.3) satisfy the same relations, v− = (−v)+ and v+ − v− = v,
respectively. In addition, by (9.10), we have

f (+)(v) =
1

π
v

∫ v

−∞

dt

1 + t2
and f (−)(v) =

1

π
v

∫ +∞

v

dt

1 + t2
(9.11)

defined for every v ∈ R, which yields

f (+)(v) > − 1

π
and f (−)(v) <

1

π
with the limits lim

v→∓∞
f (±)(v) = ∓ 1

π
,

respectively. We could immediately extend these two (real analytic) functions to
holomorphic (i.e., complex analytic) functions f (+), f (−) : C \ {−i, i} by replacing

the Lebesgue integrals
∫ v
−∞ . . . dt and

∫ +∞
v

. . . dt in (9.11) over the real domains

(−∞, v] and [v,+∞) in (9.11), respectively, by the complex path integrals along
some suitable (continuously differentiable) paths

γ+ : (−∞, 0]→ C \ {−i, i} and γ− : [0,+∞)→ C \ {−i, i}

connecting the points −∞ with v and v with +∞, respectively, whenever v ∈
C \ {−i, i}, where the paths γ+ and γ− do neither pass through nor wind around
the points ±i ∈ C, i.e., they have the winding numbers Indγ+(±i) = Indγ−(±i) = 0.
As a consequence, this extension procedure could produce multi-valued analytic
functions which is not desirable. Therefore, we prefer to perform this holomorphic



82 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

extension of the functions f (+) and f (−) below, by formulas in (9.12), as they meet
our current goals better.

We calculate the derivatives

(f (+)(v))′ =
1

2
+

1

π
arctan(v) +

1

π

v

1 + v2
> 0 ,

(f (−)(v))′ =
1

2
− 1

π
arctan(v)− 1

π

v

1 + v2
< 0 ,

with

(f (+)(v))′ + (f (−)(v))′ = 1, lim
v→∞

(f (+)(v))′ = lim
v→−∞

(f (−)(v))′ = 1,

lim
v→−∞

(f (+)(v))′ = lim
v→∞

(f (−)(v))′ = 0 .

By

(f (+)(v))′′ =
1

π

2

(1 + v2)2
> 0 and (f (−)(v))′′ = − 1

π

2

(1 + v2)2
< 0 ,

respectively, f (+) is a strictly monotone increasing and strictly convex function,
whereas f (−) is strictly monotone decreasing and strictly concave. We can use
Takáč [82, Example 8.2, pp. 79–80] and extend f (+) and f (−) to homolomorphic

functions f̃ (+) and f̃ (−) on the domain O1 via the formulas

f̃ (+)(z) = z
[1

2
+

i

2π
log
(1− iz

1 + iz

)]
= z
[1

2
+

1

π
arctan(z)

]
,

f̃ (−)(z) = z
[1

2
− i

2π
log
(1− iz

1 + iz

)]
= z
[1

2
− 1

π
arctan(z)

] (9.12)

for every z ∈ O1 = C \ {±iy : y ∈ [1,∞)}, thanks to the argument and logarithm
formulas

arg(1 + iy) = arctan(y) for y ∈ R and log
(1− iz

1 + iz

)
= −2i · arctan(z)

for z ∈ O1. The extensions f̃ (+) and f̃ (−) have the restrictions f̃ (±)|R = f (±) to
the real axis R, respectively, and they are holomorphic on the domain O1 since
the argument restriction arg

(
1−iz
1+iz

)
∈ (−π, π) holds for z ∈ O1. We refer to [82,

Eqs. (76)–(78), p. 79] for further discussion of the behavior of log( 1−iz
1+iz ). As a

consequence of the arguments in [82, Example 8.2, pp. 79–80], we obtain (f̃ (+)(z))′+

(f̃ (−)(z))′ = 1 for z ∈ O1 together with limits

(f̃ (+)(z))′ → 1 as |z| → ∞ with z ∈ C, <ez > 0,

(f̃ (−)(z))′ → 0 as |z| → ∞ with z ∈ C, <ez > 0,

(f̃ (+)(z))′ → 0 as |z| → ∞ with z ∈ C, <ez < 0,

(f̃ (−)(z))′ → 1 as |z| → ∞ with z ∈ C, <ez < 0.

(9.13)

The domain O1 = C\±i[1,∞) contains the strip R× i(−r0, r0) for every 0 < r0 < 1

and the imaginary parts of (f̃ (+)(z))′ and (f̃ (−)(z))′ are uniformly bounded for
|=mz| < r0. Consequently, it suffices to verify (9.13) only for z = <ez = x > 0 and
x < 0, respectively, by Cauchy’s integral theorem applied to the integral formula
for the function arctan(z) in O1.



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 83

Finally, in order to approximate the functions u 7→ u+ : R → R and u 7→ −u−,
we take ε ∈ (0, 1) small enough and use the functions

f̃ (+)
ε (z) := εf̃ (+)(

z

ε
) = z

[1

2
+

i

2π
log
(1− i(z/ε)

1 + i(z/ε)

)]
,

f̃ (−)
ε (z) := εf̃ (−)(

z

ε
) = z

[1

2
− i

2π
log
(1− i(z/ε)

1 + i(z/ε)

)]
,

(9.14)

for every z ∈ Oε := C \ {±iy : y ∈ [ε,∞)} = εO1, respectively. Notice that

f̃
(+)
ε (z) + f̃

(−)
ε (z) = z. In particular, for every u ∈ R we obtain the approximation

f̃ (+)
ε (u)→ u+ and f̃ (−)

ε (u)→ −u− as ε→ 0 + .

This convergence is uniform on any compact interval [−R,R] ⊂ R with 0 < R <
+∞.

Example 9.3. Another example for real analytic functions f (+) and f (−) could be
obtained by means of Takáč [82, Example 8.3, p. 80]. For this purpose, we could
consider the real analytic functions

f (±)(v) =
1

2
v ± 1

2
log(cosh(v)) for every v ∈ R ,

which have similar properties as the functions defined in (9.10).

We wish to apply our main result, Theorem 3.4, to the semilinear inital value
problem in (9.5) where we choose f (+) and f (−) as in (9.10), f̃ (+) and f̃ (−) as in

(9.12), and f̃
(+)
ε and f̃

(−)
ε as in (9.14), respectively. On the right-hand side of (9.5)

we replace the (nonlinear) functions v̂ 7→ v̂+ : R→ R and v̂ 7→ −v̂−, respectively, by

the pair of functions v̂ 7→ f̃
(+)
ε (v̂) : R→ R and v̂ 7→ f̃

(−)
ε (v̂) defined in (9.14). Here,

we take ε ∈ (0, 1) small enough, but fixed. The initial data in (9.5) are given by a

payoff function Ĥ ∈ Bs;p,p(RN ) with p > 2 + N
m = 3 and s = 2m

(
1− 1

p

)
= 2
(
1− 1

p

)
(M = N = m = 1). We assume that these initial data possess a holomorphic

extension H̃ : X(κ0) → C1 from R1 to the complex domain X(κ0) ⊂ C1, for some
κ0 ∈ (0, r0], such that the function

H̃(·+ iy) : x 7→ H̃(x+ iy) : R1 → C1

belongs to Bs;p,p(R1) for each y ∈ Q(κ0) and has finite norm N(κ0)(H̃) <∞ which
has been defined in (3.10). In the case of a simple European call or put option, i.e.,

Ĥ(x) = (ex − K)+ or Ĥ(x) = (K − ex)+, x ∈ R1, respectively, one may use the

functions f̃
(+)
ε and f̃

(−)
ε in order to find the desired holomorphic extension H̃ of Ĥ

that satisfies the hypotheses required in Theorem 3.4, part (iii).
According Example 9.2, the nonlinearity

f(v̂) = −(1−RB)λBf
(−)(v̂)− (1−RC)λCf

(+)(v̂)− sF f (+)(v̂)

on the right-hand side of (9.5) possesses a holomorphic extension

f̃ε(v̂) = −(1−RB)λB f̃
(−)
ε (v̂)− (1−RC)λC f̃

(+)
ε (v̂)− sF f̃ (+)

ε (v̂) (9.15)

for all v̂ ∈ Oε, where Oε = ε · O1 = C \ {iy : y ∈ (−∞,−ε] ∪ [ε,∞)}.
Now let us recall the definition of the complex planar domain ∇(r)

ϑ ⊂ C in (9.8)

for r ∈ (0,∞) and 0 < ϑ < π/2 with the closure ∇̄(r)
ϑ in C. We have ∇̄(r)

ϑ ⊂ Oε
whenever 0 < r < ε and 0 < ϑ < π/2. Moreover, both f̃ε and its complex



84 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

derivative f̃ ′ε are uniformly bounded in ∇̄(r)
ϑ . Thus, fixing any number ε ∈ (0, 1)

and taking r ∈ (0, ε), we observe that both f̃ε and f̃ ′ε are uniformly bounded

in ∇̄(r)
ϑ and, consequently, also in the complex strip X(r), X(r) ⊂ ∇̄(r)

ϑ ⊂ Oε.
We stress that the number ε ∈ (0, 1) may be chosen arbitrarily small in order to
achieve a sufficiently precise approximation of the reaction function f = f(v̂) by

the holomorphic function f̃ε = f̃ε(v̂) as desribed in Example 9.2 above. Naturally,
the choice of a smaller number ε ∈ (0, 1) diminishes the width of the strip X(r)

according to 0 < r < ε.

We have f̃
(±)
ε ∈ A(Oε) and f ′ is bounded in ∇(r0)

ϑ0
for every r0 ∈ (0, ε/2) and

0 < ϑ0 < π/2. The technical estimate (3.4) is trivially satisfied, owing to f̃
(±)
ε (0) =

0.
Following the discussion in Section 6, we recall that the (unique) strict solution is

restricted to the bounded open set U ⊂ Bs;p,p(RN ) defined in (6.3), which is, due to
the continuous Sobolev imbedding Bs;p,p(RN ) ↪→ L∞(RN ), bounded in L∞(RN ),
as well. Hence, it is convenient to loosen Hypothesis (H3) in the sense that we
replace the complex plane C in the assumptions by smaller domains. In particular,
the function f̃ε in (9.15) fulfills Hypothesis (H3) with such weakened assumptions.
Since all requirements are satisfied, we can apply Theorem 3.4 to the initial value
problem (9.5) and obtain the real analyticity of the solution.

Indeed, we apply our main result, Theorem 3.4, to the inital value problem (9.5),
where we choose f (+) and f (−) as follows: We replace the functions f (+) and f (−),

respectively, by their complexifications f̃
(+)
ε (z) and f̃

(−)
ε (z), respectively, defined in

formulas (9.14) for z ∈ Oε = ε · O1. Here, ε > 0 is as small as needed. The initial

data is given by the payoff function Ĥ ∈ Bs;p,p(RN ) for p > 3 and s = 2(1− 1/p).

The partial differential operator Ãt defined in (9.6) satisfies (H1) and (H2) (with

N = 1). If we replace Ãt by the generator of a stochastic volatility process like
(9.7), then (H1) and (H2) (with N = 2) are still fulfilled, according to Remark 9.1.

For the nonlinearity f(v̂) in (9.5), extended in (9.15) as f̃ε(v̂) for v̂ ∈ Oε, we have

f̃ε ∈ A(Oε) and f̃ ′ε is bounded in ∇(r0)
ϑ0

for every r0 ∈ (0, ε/2) and 0 < ϑ0 < π/2.

The technical estimate (3.4) is trivially satisfied, owing to f̃
(±)
ε (0) = 0.

10. Historical remarks and comments

The questions we studied in this paper are clearly related to the classical Cauchy-
-Kowalewski theorem (John [50], Chapt. 3, Sect. 3(d), pp. 73–77). It has been
known since the work by Holmgren [43] that even the heat equation (in one space
dimension(!)) has solutions that are not real analytic in the time variable (cf.
Bilodeau [12, pp. 124–125]). This phenomenon is due to a possibly very rapid
growth of the solutions as the spatial variable x ∈ R escapes to ±∞; to eliminate
it one needs to restrict the function space, where the solutions are considered at
each time moment t ∈ R+, in order to prevent a too rapid growth of the solutions
as x→ ±∞. This is precisely what has been done also in our present article.

Here, the emphasis is on the analytic dependence in time t and the Cauchy prob-
lem (1.1) is viewed as an evolutionary equation in some suitable function space, e.g.,
L2(R) or L2(RN ). Consequently, the solution is viewed as a vector-valued function
u : (0, T ) → L2(RN ) and, thus, regularity results (including analyticity results)
have been obtained in this setting. The interested reader is referred to Takáč [82,



EJDE-2021/SI/01 SPACE-TIME ANALYTICITY OF WEAK SOLUTIONS 85

Sect. 9, pp. 83–85] for a number of pertinent references and their description; for
example, Kato and Tanabe [53], Komatsu [56], Massey III [66], Masuda [68], and
in particular Tanabe [81] and the references therein.

Investigation of the smoothing (or regularizing) effect in evolutionary equations
of parabolic type has a long history; see e.g. Eidel′man [25], Friedman [30, 31, 32],
Pazy [72], and Tanabe [81] and numerous references therein. Analytic smooth-
ing (or regularizing) effects, similar to those treated in our present article, in the
space (x) and/or time (t) variable(s), have been obtained somewhat later, begin-
ning with the theory of analytic semigroups (in an abstract Banach space), see
e.g. the monographs by Kato [52], Lions [62], Pazy [72], and Tanabe [81], and ap-
plying (extending) it to nonautonomous analytic evolutionary equations, see e.g.
Kato and Tanabe[53], Komatsu [56], Masuda [68], and Tanabe [81]. Evolution-
ary equations exhibiting analytic smoothing effects may be split into the following
two classes: dissipative and dispersive. Again, we refer to [82, Sect. 9, pp. 83–85]
for greater details about these two classes. The results for dissipative evolutionary
equations establish only analyticity with respect to the time variable t ∈ (0, T ) ⊂ R.
Hayashi and Kato [39] establish an analogous time-analyticity result for the non-
linear Schrödinger equation (NLS). The early (general) treatments on the analytic
smoothing effect with respect to the space variable x ∈ RN are given in Kahane
[51] and Foias and Temam [27, 28].

Finally, we mention the analyticity results by Komatsu [54] obtained for solu-
tions to elliptic and parabolic problems in a bounded spatial domain Ω ⊂ RN (with
analytic boundary ∂Ω). Analyticity in the space variable x and 2-nd Gevrey class
regularity (weaker than analyticity) in the time variable t are established in Cav-
allucci [19, Teorema 6.1, p. 166] for linear parabolic equations. Some results about
the analyticity of solutions of nonlinear parabolic systems, which are related to
ours, are stated in Friedman [30, Theorems 3 and 4] without proofs, and for linear
elliptic systems in Morrey, Jr., and Nirenberg [71]. For the Navier-Stokes equations,
such analyticity results have been established in Masuda [69] and, with respect to
the space variable x ∈ RN only, earlier in Kahane [51] and Masuda [67]. These
results state local analyticity of infinitely differentiable solutions without any de-
scription of their domain of holomorphy (i.e., domain of complex analyticity). Our
present article provides such description in Theorem 3.4 and so do Refs. [14, 15].
More results of global nature on the space analyticity can be found in Bardos and
Benachour [11] and Grujić and Kukavica [34].

11. Discussion and possible generalizations

In contrast to the analytic smoothing results established in Takáč [82, Theorem
3.3, p. 59] (for a linear parabolic problem) and Takáč et al. [83, Theorem 2.1,
p. 429] (for a semilinear parabolic problem), in the present work we have focused
on preserving the spatial analyticity of the initial data, u0, for all times t ∈ [0, T ] as
long as a (global) weak solution u ∈ C ([0, T ]→ Bs;p,p(R)) to the Cauchy problem
(1.1) exists, that is, loosely written, u(·, 0) = u0 is spatially analytic (at t = 0)
implies u(·, t) is spatially analytic at all times t ∈ (0, T ], even for all t ∈ (0, T + T1]
with some T1 > 0 small enough.

However, also a spatial analytic smoothing result analogous to those in [82,
Theorem 3.3, p. 59] and [83, Theorem 2.1, p. 429] should hold in our present
setting in the Besov space Bs;p,p(R), by arguments similar to those used in [83,



86 F. BAUSTIAN, P. TAKÁČ EJDE/SI/01

pp. 434–435], proof of Lemma 3.4. The Banach contraction principle can then be
used in analogy with [83, pp. 437–438], Step 4 in the proof of Theorem 3.1. This
approach requires separation of the linear part of the Cauchy problem (1.1) (cf. [82])
followed by an application of the Banach contraction principle to the full semilinear
parabolic problem in (1.1) (cf. [83, Theorem 2.1, p. 429]).

Acknowledgments. This work was partially supported by a grant from Deutsche
Forschungsgemeinschaft (DFG, Germany), Grant no. TA 213/16-1, and by the
Federal Ministry of Education and Research of Germany under grant No. 57063847
(D.A.A.D. Program “PPP”), the latter within a common exchange program be-
tween the Czech Republic and Germany. The authors would like to express their
sincere thanks to Professors Herbert Amann (University of Zürich, Switzerland),
Philippe Clément (Delft University of Technology, The Netherlands), and Alessan-
dra Lunardi (University of Parma, Italy) for their kind help with the literature on
the maximal regularity and analyticity questions for linear parabolic problems.

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Falko Baustian

Institut für Mathematik, Universität Rostock, Ulmenstraße 69, Haus 3, D-18051 Ros-
tock, Germany

Email address: falko.baustian@uni-rostock.de

Peter Takáč
Institut für Mathematik, Universität Rostock, Ulmenstraße 69, Haus 3, D-18051 Ros-

tock, Germany
https://www.mathematik.uni-rostock.de/struktur/lehrstuehle/angewandte-analysis/

Email address: peter.takac@uni-rostock.de


	1. Introduction
	2. Notation
	3. Statement of main results
	3.1. Hypothesis

	4. Abstract Cauchy problem in an interpolation space
	4.1. Hypothesis

	5. Analyticity in time for the abstract Cauchy problem
	5.1. Auxiliary linear perturbation results
	5.2. Proof of analyticity in time
	5.3. Hypothesis

	6. Analyticity in space for the Cauchy problem in RN(0,T)
	6.1. Hypothesis
	6.2. Hypothesis
	6.3. Hypothesis

	7. Space-time analyticity for the Cauchy problem in RN(0,T)
	8. Proofs of main results
	9. An application to a risk model in mathematical finance
	10. Historical remarks and comments
	11. Discussion and possible generalizations
	Acknowledgments

	References