Does strong repulsion lead to smooth solutions in a repulsion-attraction chemotaxis system even when starting with highly irregular initial data

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arXiv:2210.12208v1 [math.AP] 21 Oct 2022

Does strong repulsion lead to smooth solutions in a

repulsion-attraction chemotaxis system even when starting with

highly irregular initial data?

Frederic Heihoﬀ∗

Institut für Mathematik, Universität Paderborn,

33098 Paderborn, Germany

Abstract

It has been well established that, in attraction-repulsion Keller–Segel systems of the form







ut= ∆u−χ∇ · (u∇v) + ξ∇ · (u∇w),

τ vt= ∆v+αu −βv,

τ wt= ∆w+γu −δw

in a smooth bounded domain Ω ⊆Rn,n∈N, with Neumann boundary conditions and parameters

χ, ξ ≥0, α, β, γ, δ > 0 and τ∈ {0,1}, ﬁnite-time blow-up can be ruled out in many scenarios given

suﬃciently smooth initial data if the repulsive chemotaxis is suﬃciently stronger than its attractive

counterpart. In this paper, we will go - in a sense - a step further than this by studying the same system

with initial data that could already be understood as being in a blown-up state (e.g. a positive Radon

measure for the ﬁrst solution component) and then ask the question whether suﬃciently strong repulsion

has enough of a regularizing eﬀect to lead to the existence of a smooth solution, which is still connected

to said initial data in a sensible fashion. Regarding this, we in fact establish that the construction of

such a solution is possible in the two-dimensional parabolic-parabolic system and the two- and three-

dimensional parabolic-elliptic system under appropriate assumptions on the interaction of repulsion and

attraction as well as the initial data.

Keywords: attraction-repulsion; Keller–Segel; measure-valued initial data; smooth solution

MSC 2020: 35Q92 (primary); 35K10; 35J15; 35K55; 35A09; 35B65; 92C17

1 Introduction

One of the fundamental questions in the mathematical analysis of partial diﬀerential equation models of

chemotaxis (cf. [2]) used in the analysis of various biological organisms is whether the struggle between the

forces of diﬀusion and chemotaxis ultimately leads to ﬁnite-time blow-up, regularization or even stabilization

of solutions. Importantly, the interest in the detailed analysis of solution behavior of such models does not

purely stem from mathematical curiosity about these often challenging systems, but also chieﬂy from a

desire to see whether the models appropriately represent the observed behavior of the biological organisms

that inspired them and thus could prove to be a useful tool in practical applications.

Most of the early analytical endeavors in this area, growing from the seminal paper by Keller and Segel

in 1970 (cf. [18]) and emboldened by the subsequent successful analysis of their proposed model (cf. [2]),

have to our knowledge focused on the interplay of the regularizing eﬀects of diﬀusion and the potentially

∗fheihoﬀ@math.uni-paderborn.de

destabilizing eﬀects of attractive chemotaxis in various scenarios. Though more recently in an eﬀort to

e.g. understand Alzheimer’s disease (cf. [29]) or describe quorum-sensing eﬀects observed in certain kinds

of bacteria (cf. [34]), variations of the original Keller–Segel model have been introduced, which not only

feature an attractive but also a repulsive chemical. A model of exactly this type will be the central object

of interest in this paper. More precisely, we will be analyzing the system of partial diﬀerential equations











ut= ∆u−χ∇ · (u∇v) + ξ∇ · (u∇w) on Ω ×(0,∞),

τvt= ∆v+αu −βv on Ω ×(0,∞),

τwt= ∆w+γu −δw on Ω ×(0,∞),

0 = ∇u·ν=∇v·ν=∇w·νon ∂Ω×(0,∞)

(1.1)

in a smooth bounded domain Ω ⊆Rn,n∈N, with parameters χ, ξ ≥0, α, β, γ, δ > 0 and τ∈ {0,1}.

Herein, the ﬁrst equation models the movement of the organisms in question toward the attractive chemical

represented by vat rate χand away from the repulsive chemical represented by wat rate ξ. As seen in

the second and third equation, both chemicals are produced by the organisms at rates depending on the

choice of αand γ, respectively, and decay over time at rates βand δ, respectively. Notably, the second

and third equation can either be of parabolic or elliptic type depending on the choice of parameter τ. This

choice is generally interpreted as either the chemicals conforming to a time evolution at similar time scales

as the organisms in the parabolic case or the chemicals reacting almost immediately to changes in organism

concentration in the elliptic case.

Prior work. From an intuitive standpoint and in many cases very much by design, one would expect a

suﬃciently strong repulsive inﬂuence to counteract the aggregation behavior often underlying ﬁnite-time

blow-up and thus leading more readily to the global existence of classical solutions. In fact given regular

initial data, this intuition seems to be supported by prior mathematical analysis as it has been shown that,

if the repulsive taxis in the system (1.1) is strictly stronger than its attractive counterpart in the sense that

ξγ −χα > 0, then global classical solutions exist in two dimensions if τ= 1 and in arbitrary dimension if

τ= 0 (cf. [27], [38]). It has further been shown that under potentially additional parameter restrictions said

solutions are even globally bounded or exhibit certain large-time stabilization properties (cf. [15], [17], [25],

[28]). Conversely if attraction dominates over repulsion, the ﬁnite-time blow-up results already established

for attraction-only systems (cf. e.g. [31], [32], [41], [42]) seem to largely translate to the competition case

(cf. [16], [21], [38]). Naturally apart from system (1.1), which stays fairly close to the original Keller–Segel

system, many of its canonical variations have also been explored as well. To mention a few, there has been

some consideration of models, in which the attractant is consumed instead of produced (cf. [8], [9]), in which

the taxis mechanisms further interact with a logistic source term (cf. [23], [46]), or in which the movement

mechanisms feature some form of nonlinearity (cf. [7], [8], [26]). Apart from this, there has also been some

analysis of the interaction between attraction and consumption in the whole space case (cf. [33], [45]).

Let us further brieﬂy mention that there is another prominent setting which at times deals with interaction

of attraction and repulsion, namely predator-prey models. Here, the predators, which are generally modeled

by the ﬁrst of two diﬀusive equations, are attracted by the prey. The prey in turn, which is modeled by the

second of the aforementioned equations, is repelled by the predators. If both taxis mechanisms are present in

such a setting however, the situation seems to be much less clear cut than in the attraction-repulsion model

(1.1) as even the construction of (potentially only generalized) solutions seems to be rather challenging given

that cross-diﬀusion is present in both equations (cf. [11], [12], [39]). As such, most eﬀorts in this area focus

only on one of the two mechanisms and remove the other.

Main result. As the boundary between ﬁnite-time blow-up and global existence for the system (1.1) has

at this point been fairly thoroughly explored in the literature discussed above, we will in this paper go a

step further in a sense by addressing the following question: If the system (1.1) already starts in a state

resembling blow-up (e.g. a Dirac measure for the ﬁrst solution component), can suﬃciently strong repulsion

be enough of a regularizing inﬂuence counteracting attraction to still yield classical solutions, which remain

connected to the initial state in a reasonable way? Notably, not only is it well-documented that such an

immediate smoothing property is exhibited by the pure Neumann heat equation (cf. [14], [30]), but very

similar questions have been posed and positively answered for other cross-diﬀusion systems, providing us

with a degree of optimism regarding this type of inquiry. To be more explicit, construction of such solutions

has been accomplished in the parabolic-elliptic repulsive Keller–Segel system on two- and three-dimensional

domains (cf. [13]) as well as in the standard parabolic-parabolic attractive Keller–Segel system on one-

dimensional domains (cf. [44]) and on two-dimensional domains under the regularizing inﬂuence of a logistic

source term (cf. [22]). Without the regularizing aid of such a logistic source, mild solutions, which become

immediately bounded in L3

2(Ω), have also been constructed for the attractive system on two-dimensional

domains for measure-valued initial data with small mass or any L1(Ω) initial data (cf. [4]). There have

also been some similar discussions in the two-dimensional whole space (cf. [1], [3], [5], [36]), the radially

symmetric (cf. [40], [43]) as well as the toroidal case (cf. [37]).

As such in an eﬀort to expand on this exact area of inquiry and in many ways as a direct extension

of the results presented in [13], [27] as well as [38], the main result of this paper is the construction of

classical solutions to (1.1) with measure-valued initial data for the ﬁrst solution component and, in the

parabolic-parabolic case, appropriately regular initial data for the second and third solution component

under suﬃciently strong repulsion in the two-dimensional parabolic-parabolic and two- as well as three-

dimensional parabolic-elliptic case. More precisely, we will prove the following result:

Theorem 1.1. Let Ω⊆Rn,n∈N, be a bounded domain with a smooth boundary, χ, ξ ≥0and α, β, γ, δ > 0

as well as τ∈ {0,1}. Let further u0∈ M+(Ω) be an initial datum with m:=u0(Ω) >0, where M+(Ω)

is the set of positive Radon measures with the vague topology. If τ= 1, let further v0, w0∈W1,r(Ω) with

r∈(6

5,2) be some additional nonnegative initial data. If

τ= 1, n= 2 and ξγ −χα ≥0or (S1)

τ= 0, n= 2 and ξγ −χα ≥ −CS2

mor (S2)

τ= 0, n= 3 and ξγ −χα ≥0as well as u0∈Lκ(Ω) with some κ∈(1,2),(S3)

where CS2 >0is a constant only depending on the domain Ω, then there exist nonnegative functions

u∈C2,1(Ω×(0,∞)) and v, w ∈C2,τ (Ω ×(0,∞)) solving (1.1) classically and attaining their initial data in

the following fashion:

u(·, t)→u0in M+(Ω),(1.2)

v(·, t)→v0in W1,r(Ω) if τ= 1,(1.3)

w(·, t)→w0in W1,r(Ω) if τ= 1 (1.4)

as tց0, where we interpret the functions u(·, t),t > 0, as the positive Radon measures u(x, t)dxwith dx

being the standard Lebesgue measure on Ω.

Approach. As is not uncommon, the construction of our desired solution will be based on approximating

them by a family of solutions (uε, vε, wε)ε∈(0,1), for which global existence is much easier to establish. To

this end, we will spend the next section approximating our initial data by smooth functions in a fashion

convenient for later arguments and then prove that with such smooth initial data classical solutions to (1.1)

exist globally as an extension of the arguments presented in [27] and [38]. Additionally in this section, we

also introduce the functions zε:=ξwε−χvεfor each ε∈(0,1), which allow us to transform the system (1.1)

to the system (2.7), because the second system will prove more convenient for some of the later arguments.

The remainder of this paper will then be devoted to deriving uniform a priori bounds for exactly these

approximate solutions to facilitate a central compact embedding argument, which will serve as the source

of our actual solutions, as well as to ensure that the thus constructed solutions have our desired continuity

properties at t= 0.

Naturally any substantial a priori estimates have to necessarily decay toward t= 0 if they are to be uniform

in the approximation parameter εas our initial data are very irregular. As such, the a priori estimates

serving as the linchpins of all further considerations will roughly have the form

G(uε(·, t), zε(·, t)) ≤Ct−λ(1.5)

or Zt

sλG(uε(·, s), zε(·, s)) ds≤C(1.6)

for some λ > 0, where Gis some key norm or functional and the parameter λrepresents how severely the

estimate decays close to t= 0. The derivation of these estimates will take place in Section 3 and, while most

of the arguments afterward will handle both the parabolic-parabolic and parabolic-elliptic cases in a fairly

integrated fashion, said derivation will use very diﬀerent methods depending on the value of τ.

For the parabolic-elliptic case, we essentially adapt methods found in [13] to the system (1.1). At its core, the

argument boils down to testing the ﬁrst equation in (1.1) with up−1

εand then after two partial integrations

directly replacing the resulting ∆vεand ∆wεterms by lower order expressions using the elliptic second and

third equations in (1.1). Applying carefully chosen interpolation inequalities as well as elliptic regularity

theory to this then allows us to derive a diﬀerential inequality with super-linear decay for RΩup

ε, which is

suﬃcient to yield an estimate of the type (1.5) for RΩup

εwith any p∈(1,∞).

In the parabolic-parabolic case, our approach hinges on the use of the well-known energy-type functional

F(t):=ζRΩuεln(uε)+ 1

2RΩ|∇zε|2with ζ:=ξγ −χα. But as this functional cannot necessarily be uniformly

bounded at t= 0 due to our initial data potentially not having ﬁnite energy, we further decouple it from the

initial data by multiplying it with tλ,λ > 0. Using testing based methods, analysis of this functional then

not only yields an estimate of type (1.5) for the functional itself but crucially also a bound of type (1.6)

for the dissipative terms RΩ|∆zε|2and ζRΩ|∇√uε|2. Notably while the ﬁrst set of bounds could also be

achieved by a similar super-linear decay approach as described in the previous paragraph, the latter bounds

seem to be much more conveniently accessible by analyzing the aforementioned time-dampened version of

the functional F. And importantly, it is in fact exactly said latter bounds that will allow us to prove our

desired continuity properties at t= 0 in this scenario, as well as allow us to derive a set of uniform bounds

for RΩu2

εaway from t= 0 to give us a similar starting point to the parabolic-elliptic case for the next

section.

In Section 4, we then use the uniform bounds for RΩun

εaway from t= 0, which we have at this point

established in all scenarios, as the basis for a bootstrap argument taking us all the way to uniform bounds

for the ﬁrst solution components in C2+θ,1+ θ

2(Ω ×[t0, t1]) and uniform bounds for the second and third

solution components in C2+θ,τ +θ

2(Ω ×[t0, t1]) with t1> t0>0. We do this mostly using the variation-of-

constants representation of the involved equations or corresponding elliptic regularity theory as well as fairly

standard Hölder regularity theory from [19], [24] as well as [35]. Due to the compact embedding properties

of Hölder spaces this immediately allows us to construct our desired solutions (u, v, w) as limits of the thus

far discussed approximate ones and argue that they classically solve (1.1) as the resulting strong convergence

properties safely transfer any solution properties from the approximate solutions to their limits.

It thus only remains to be shown that said solutions are connected to our initial data in the fashion outlined

in (1.2)–(1.4), which will be the main subject of Section 5. To do this for the ﬁrst solution component, we

essentially start by proving that the approximate solutions are uniformly continuous at t= 0 in the sense

of (1.2). We accomplish this by showing that the space-time integral Rt

0kuε(·, s)∇zε(·, s)kL1(Ω) dsrelated to

the chemotaxis mechanism becomes uniformly small as tgoes to zero in all scenarios. That it is possible to

prove such a property chieﬂy depends on the degradation toward zero of our estimates derived in Section

2 to be suﬃciently benign. Notably, the magnitude of the degradation parameters naturally depends on

the regularity of our initial data in the sense that better regularity leads to smaller degradation toward

zero and thus the derivation of the aforementioned bound is, however indirect, the source of our initial

data regularity needs in Theorem 1.1. We then use said property combined with the ﬁrst equation in (1.1)

and the fundamental theorem of calculus to gain our desired uniform continuity property, which by virtue

of the already established convergence properties translates immediately to our actual solutions. Using

a convenient property of our initial data approximation, a similar argument built on semigroup methods

grants us properties (1.3) and (1.4) in the parabolic-parabolic case.

2 Regularized initial data and approximate solutions

From here on out, we ﬁx the system parameters χ, ξ ≥0, α, β, γ, δ > 0 and τ∈ {0,1}as well as a domain

Ω⊆Rn,n∈N, with a smooth boundary for the remainder of the paper. We further ﬁx some initial data

u0∈ M+(Ω) with m:=u0(Ω) >0, where M+(Ω) is the set of positive Radon measures with the vague

topology, as well as nonnegative v0, w0∈W1,r(Ω) with r∈(6

5,2) if τ= 1. Moreover if u0is additionally

assumed to be an element of Lκ(Ω) for some κ∈(1,2), we also ﬁx this parameter κ. Otherwise, we let κbe

equal to 2 so κis deﬁned in all scenarios for convenience of notation in some later arguments.

To construct the solutions laid out in Theorem 1.1, we will use a family of approximate solutions, which will

later be argued to converge to our desired solutions. This section will thus be devoted to the construction

of said approximate solutions and to facilitate this we will begin by approximating our potentially highly

irregular initial data by smooth functions, which will in fact be the only regularization necessary.

As such, we now ﬁx a family of approximate positive initial data (u0,ε)ε∈(0,1) ⊆C∞(Ω) such that

u0,ε →u0in M+(Ω) as εց0 as well as ZΩ

u0,ε =u0(Ω) = mfor all ε∈(0,1),(2.1)

where we interpret the functions u0,ε as the positive Radon measures u0,ε(x)dxwith dxbeing the standard

Lebesgue measure on Ω. For a discussion of how this can be achieved, we refer the reader to e.g. [13, Remark

2.2]. If u0is additionally an element of Lκ(Ω), we let u0,ε :=eε∆u0∈C∞(Ω) for all ε∈(0,1) instead,

where (et∆)t≥0is the Neumann heat semigroup on Ω. By the continuity, positivity and mass conservation

properties of said semigroup, this not only directly ensures the previously prescribed properties but also

that

u0,ε →u0in Lκ(Ω) as εց0 (2.2)

as well. Similarly if τ= 1, we let

v0,ε :=eε(∆−β)v0:=e−εβeε∆v0∈C∞(Ω) and w0,ε :=eε(∆−δ)w0:=e−εδ eε∆w0∈C∞(Ω) (2.3)

for all ε∈(0,1). These approximate functions are nonnegative due to the maximum principle and have the

following convergence property due to the continuity of the semigroup at t= 0:

v0,ε →v0and w0,ε →w0in W1,r(Ω) as εց0.(2.4)

As a convenient by-product of this construction, we also gain that

ZΩ

v0,ε ≤ZΩ

v0and ZΩ

w0,ε ≤ZΩ

w0(2.5)

again due to the mass conservation property of the Neumann heat semigroup.

As it will not only be a useful tool in arguing that our approximate solutions are in fact global, but will also

play a key part in many of our later derivations of a priori estimates, we will now introduce the following

transformation for classical solutions to (1.1): For any such solution (u, v, w), we let

z(x, t):=ξw(x, t)−χv(x, t) and ζ:=ξγ −χα ∈Ras well as σ:=χ(β−δ)∈R(2.6)

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arXiv:2210.12208v1[math.AP]21Oct2022Doesstrongrepulsionleadtosmoothsolutionsinarepulsion-attractionchemotaxissystemevenwhenstartingwithhighlyirregularinitialdata?FredericHeihoﬀ∗InstitutfürMathematik,UniversitätPaderborn,33098Paderborn,GermanyAbstractIthasbeenwellestablishedthat,inattraction-repulsio...

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Does strong repulsion lead to smooth solutions in a repulsion-attraction chemotaxis system even when starting with highly irregular initial data

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