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Space-Time Analyticity of Weak Solutions to

Semilinear Parabolic Systems with Variable Coeﬃcients∗

Falko Baustian

and

Peter Tak´

aˇ

c

Institut f¨ur Mathematik, Universit¨at Rostock,

Ulmenstraße 69, Haus 3, D-18051 Rostock, Germany,

e-mail: peter.takac@uni-rostock.de

Web: https://www.mathematik.uni-rostock.de/struktur/lehrstuehle/angewandte-analysis/

January 5, 2021

Abstract

Analytic smooth solutions of a general, strongly parabolic 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) are investigated. 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 0< T ≤ ∞, it is proved that any Bs;p,p -type solution u:RN×(0, T )→

CMwith analytic initial data possesses a bounded holomorphic continuation u(x+ iy, σ + iτ) into a

complex domain in CN×Cdeﬁned by (x, σ)∈RN×(T0, T ), |y|< A0and |τ|< B0, where A0, B0>0 are

constants depending upon T0. 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.

Keywords: Space-time analyticity, parabolic PDE;

holomorphic semigroup, Besov space;

maximal regularity, Hardy space;

holomorphic continuation to a complex strip;

European option, bilateral counterparty risk.

2020 Mathematics Subject Classiﬁcation: Primary 35B65, 35K10;

Secondary 32D05, 91G40;

∗Research supported in part by Deutsche Forschungsgemeinschaft (D.F.G., Germany)

under Grant # TA 213/16–1.

1

arXiv:2101.00112v1 [math.AP] 31 Dec 2020

Contents

1 Introduction 3

2 Notation 9

3 Statement of the main result 11

4 Abstract Cauchy problem in an interpolation space 18

5 Analyticity in time for the abstract Cauchy problem 27

5.1 Auxiliary linear perturbation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Proof of Analyticity in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Analyticity in space for the Cauchy problem in RN×(0, T )40

7 Space-time analyticity for the Cauchy problem in RN×(0, T )53

8 Proofs of the main results, Theorem 3.4 and Proposition 3.5 55

9 An application to a Risk Model in Mathematical Finance 61

10 Historical remarks and comments 69

11 Discussion and possible generalizations 70

2

1 Introduction

In this article we investigate 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 u0belonging 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)]Mwhere Bs;p,p (RN) denotes the Besov space

Bs;p,p(RN)def

=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= 2m1−1

p∈(0,2m). It is deﬁned by real interpolation, e.g.,

in R. A. Adams and J. J. F. Fournier [1, Chapt. 7], §7.6–§7.23, pp. 208–221, A. Lunardi [65,

Chapt. 1], §1.2.2, pp. 20–25, or in H. Triebel [82, 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 )→CMhaving the maximal regularity

property (cf. A. Ashyralyev and P. E. Sobolevskii [9, Chapt. 3, pp. 21–36] and J. Pr¨

uss [74])

rather than weak L2-type solutions treated in P. Tak´

aˇ

c[80] 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 H. Amann [6, Chapt. III, §4, pp. 128–191], Ph. Cl´

ement

and S. Li [20], A. Lunardi [65, Chapt. 7, pp. 257–289], M. K¨

ohne,J. Pr¨

uss, and M. Wilke

[55], and H. Tanabe [79, Chapt. 5–6, pp. 117–229]. Our Cauchy problem has the following general

form for a semilinear 2mth-order parabolic problem,

(1.1)

∂u

∂t +Px, 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.

Here, ∂/∂x = (∂/∂x1, . . . , ∂/∂ xN) stands for the spatial gradient and ξ7→ P(x, t, ξ ) is a poly-

nomial of order 2min 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 RMor

CM) is assumed to be analytic in all variables x∈RN,t∈(0, T ), and X= (Xβ)|β|≤m∈RM˜

N

(or CM˜

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

∂xβ∈RM(or CM) for the (mixed) partial derivative

of uwith a multi-index β= (β1, . . . , βN)∈(Z+)Nof order |β|=β1+··· +βN,|β| ≤ m. Here,

Z+={0,1,2, . . . }and the Euclidean dimension of the m-jet Xequals to M˜

Nwith

(1.2) ˜

N=

m

X

k=0 X

|β|=kk

β=

m

X

k=0

Nkwhere k

βdef

=k!

β1!β2!. . . βN!.

As usual, RNand CN, respectively, denote the N-dimensional real and complex Euclidean spaces,

i = √−1, and M, N ∈Nwhere N={1,2,3, . . . }. We have identiﬁed Xβ=∂|β|u

∂xβ≡u(x, t) for

β= (0,0,...,0) of order |β|= 0.

3

As already indicated, we impose certain standard strong ellipticity and analyticity hypotheses

on the coeﬃcients of the partial diﬀerential operator Px, 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+iRNand 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 )×RM˜

N,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 X7→ 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., Px, t, 1

i

∂

∂x =−∆,

f(x, t;X) = 0, and M= 1); see e.g. F. John [50], Chapt. 7, Sect. 1, eq. (1.11), p. 209. The

heat equation case has been signiﬁcantly generalized in P. Tak´

aˇ

cet al. [81, Theorem 2.1, p. 429],

where only the leading coeﬃcients of the operator Px, 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 u0resembles more to a nonlocal version of the classical Cauchy-Kowalewski

theorem (F. 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 )→CMexists, 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

mand

s= 2m1−1

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., R. A. Adams and J. J. F. 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∈CM˜

Nwhose all 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 [81, Theorem 2.1, p. 429]. For instance, in a typical second-order

parabolic problem (i.e., eq. (1.1) with m= 1) we can allow for a reaction function fx, t;u,∂u

∂x

depending on uand its gradient ∂u/∂x (≡iDxu), besides the independent variables x∈RNand

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 P. Tak´

aˇ

c[80, Theorem 3.3, p. 59] where

the corresponding linear system is treated. In contrast to [81, Proposition A.4, p. 446], this means

that we cannot calculate the Green function for the Cauchy problem with the leading coeﬃcients

only,

(1.3)

∂u

∂t + (−1)mX

|α|=2m

P(α)(x, t)∂|α|u

∂xα=0for (x, t)∈RN×(0, T ) ;

u(x, 0) = u0(x) for x∈RN,

4

and then simply take advantage of the variation-of-constants formula [81, eq. (3.22), p. 437] in

order to obtain the solution of the original problem (1.1). Fortunately, the methods from [80],

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 u0are analytic. Here,

each P(α)(x, t) is an M×Mmatrix and recall that ∂|α|u/∂xα=∂|α|u

∂xα1

1... ∂xαN

N

denotes the (mixed)

partial derivative of u:RN×(0, T )→CMwith a multi-index α= (α1, . . . , αN)∈(Z+)Nof 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 [80,81].

As in [80,81], 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 ˜u: Ω →Cto

some complex domain Ω such that RN×(0, T )⊂Ω⊂CN×C,i.e., u= ˜u|RN×(0,T ), the restriction of ˜u

to RN×(0, T ). If the domain Ω is ﬁxed then the holomorphic extension ˜uof uto Ω is always unique,

see e.g. F. 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 ˜

uof uto some complex domain Ω (RN×(0, T )⊂Ω⊂CN×C).Due to 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 Szeg˝o spaces of

holomorphic functions) were used earlier in N. Hayashi [35,36,37,38].

Instead of using the Green function method (cf. [81]), we establish the existence of solutions

to the Cauchy problem (1.1) in a complex parabolic domain X(r)×[0, T ) in CN×Cwith initial

data u0from a space of holomorphic functions whose domain X(r)=RN+ iQ(r)is a tube in

CNwith base Q(r)= (−r, r)N, for some 0 < r < ∞, see P. Tak´

aˇ

c[80, eq. (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 u0be (complex) analytic in X(r).

In order 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 in one space

dimension (M=N= 1),

(1.4)

∂u

∂t =a(x, t)∂2u

∂x2+b(x, t)∂u

∂x +c(x, t)u+fx, t;u, ∂u

∂x

for (x, t)∈R1×(0, T ) ;

u(x, 0) = u0(x) for x∈R1.

We begin with the complexiﬁcations of the spatial and temporal variables, x∈R1and t∈

(0, T ), respectively: Given any real numbers 0 < r < ∞and 0 < T 0≤T < ∞, we introduce the

5

complex domains

X(r)def

={z=x+ iy∈C:|y|< r}=R+ i(−r, r),

∆ϑdef

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

∆(T0)

ϑ

def

= ∆ϑ∩ {t∈C: 0 <<et<T0}(1.6)

={t=eiθ∈C:|θ|< ϑ and 0 < < T0/cos θ},

and

∆T0,T

ϑ

def

= ∆(T)

ϑ∩ {t∈C:|=mt|< T 0·tan ϑ}=[

0≤ξ≤T−T0ξ+ ∆(T0)

ϑ

(1.7)

=[

0≤ξ≤T−T0{ξ+t0∈C:t0∈∆(T0)

ϑ}= [0, T −T0]+∆(T0)

ϑ

with the angle ϑ∈(0, π/2) given by tan ϑ=r/T 0. Of course, if T=T0then ∆T0,T

ϑ= ∆(T0)

ϑis an

open triangle. Clearly, we have

∆T0,T

ϑ=[

0<r≤T0

T(r)

r·cot ϑ, T =[

0<r≤T0

[(r·cot ϑ, T ) + i(−r, r)]

where

T(r)

T0,T

def

={t=σ+ iτ∈C:T0< σ < T and |τ|< r}= (T0, T ) + i(−r, r).(1.8)

We set T(r)

0,T = (0, T ) + i(−r, r) if T0= 0. The closures in Cof X(r), ∆ϑ, ∆(T0)

ϑ, ∆T0,T

ϑ, and T(r)

T0,T

are denoted by X(r), ∆ϑ, ∆(T0)

ϑ, ∆T0