Nonlinear Analysis: Real World Applications 54 (2020) 103090

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Nonlinear Analysis: Real World Applications

www.elsevier.com/locate/nonrwa

Existence of weak solutions to the Keller–Segel chemotaxis systemwith additional cross-diffusionGurusamy Arumugam a, André H. Erhardt b,∗, Indurekha Eswaramoorthy c,Balachandran Krishnan c

a Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Gandhinagar -382355, Gujarat, Indiab Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norwayc Department of Mathematics, Bharathiar University, Coimbatore - 641046, India

a r t i c l e i n f o

Article history:Received 12 July 2019Received in revised form 17 December2019Accepted 31 December 2019Available online xxxx

Keywords:Keller–Segel systemWeak solutionsFixed point theorem

a b s t r a c t

In this paper, we consider the Keller–Segel chemotaxis system with additionalcross-diffusion term in the second equation. This system is consisting of a fullynonlinear reaction–diffusion equations with additional cross-diffusion. We establishthe existence of weak solutions to the considered system by using Schauder’s fixedpoint theorem, a priori energy estimates and the compactness results.© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under

the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The term chemotaxis is used to denote cell movement towards or away from a chemical source, classifiedas positive and negative chemotaxis, respectively. According to the common, broad definition of chemotaxis,it is the study of any cell motion affected by a chemical gradient in a way that results in net propagationup a chemoattractant gradient or down a chemorepellent gradient. Typical examples include aggregationprocesses such as slime mold formation in Dictyostelium Discoideum [1] or pattern formation like in coloniesof Salmonella typhimurium [2], and also medically relevant processes such as tumour invasion [3,4] andself-organisation during embryonic development [5]. For some more detailed (biological and mathematicalmodelling) background we refer also to [6,7]. In 1970s, Keller and Segel [8] proposed a classical andfundamental mathematical systems to model chemotaxis. The general form of a chemotaxis system is givenby

ut = ∇ · (ψ(u, v)∇u− ϕ(u, v)∇v) + k3(u, v),vt = dv∆v + k4(u, v)u− k5(u, v)v,

(1.1)

∗ Corresponding author.E-mail address: andreerh@math.uio.no (A.H. Erhardt).

https://doi.org/10.1016/j.nonrwa.2020.1030901468-1218/© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license(http://creativecommons.org/licenses/by/4.0/).

2 G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090

where u denotes the density of cell population and v is the chemical attractant concentration. Moreover,ψ(u, v) represents the diffusivity of the cells, ϕ(u, v) denotes the chemotactic sensitivity, while k3(u, v)describes the cell growth and death. dv represents the diffusion coefficient of chemical attractant, whilek4(u, v) and k5(u, v) describe the production and degradation of the chemical signal.

After their pioneering works for chemotaxis the literature is extensive. In the last decades, there has beenconsiderable amount of work done in the analysis of various particular cases of system (1.1). The main focuson this model is its solvability.

In some special cases even more subtle analytical results on global existence of solutions of system (1.1)are available, such as the existence of weak solutions and their regularity proved by Bendahmane et al. in [9]for the case

ψ(u, v) = a(u), ϕ(u, v) = χuf(u), k3(u, v) = 0, k4(u, v) = α and k5(u, v) = β,

where α, β ≥ 0 and dv is a constant. Furthermore, f(u) is a density dependent probability that a cell inposition x at time t finds space in its neighbouring location. The cells are attracted by the chemical and χ

denotes their chemotactic sensitivity. In addition, the functions a(u) and f(u) are assumed to be sufficientlysmooth. In [10] Nakaguchi and Osaki studied global existence of solutions to a parabolic–parabolic systemfor chemotaxis with a logistic source in a two-dimensional domain, where the degradation order of the logisticsource is weaker than quadratic.

Bendahmane et al. [11] proved the existence and regularity of weak solutions for a fully parabolic modelof chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extranonlinearity represented by a p-Laplacian diffusion term. Wang et al. [12] extended a previous resultsfrom [13] on global existence of solutions to quasilinear chemotaxis models under the condition that m > 2− 2

n

can be relaxed to m > 2− 6n+4 in the case ψ(u, v) = D(u) ≥ cDu

m−1 for all u > 0 with some cD > 0, m > 1,dv = 1, ϕ(u, v) = u, k3(u, v) = k4(u, v) = 0 and k5(u, v) = u in dimension n ≥ 3.

Moreover, Wang et al. [14] dealt with the global existence and boundedness of solutions for the quasilinearchemotaxis system in the case when ψ(u, v) ≥ D0(u+1)m−1 for all u ≥ 0, ϕ(u, v) = uχ(v), k4(u, v) = −f(v)and k3(u, v) = k5(u, v) = 0. In addition, the given functions D(s), χ(s) and f(s) are supposed to besufficiently smooth for all s ≥ 0 and that f(0) = 0.

Shangerganesh et al. [15] analysed the existence and uniqueness of a weak solution of the strongly coupledchemotaxis model with Dirichlet boundary conditions. In [16] Khelghati and Baghaei studied the classicalsolutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source that are uniformly in-time-bounded without any restrictions on m and b. Moreover, they assumed that ψ(u, v) = D(u) > 0 foru ≥ 0, D(u) ≥ CDu

m−1 for u > 0 and k3(u, v) = g(u), g(0) ≥ 0, g(u) ≤ a − buγ , u > 0, dv = 1,k4(u, v) = k5(u, v) = 1, vt = 0, with constants CD > 0, m ≥ 1, a ≥ 0, b > 0 and γ > 2. For the caseψ(u, v) = 1, ϕ(u, v) = χu, k3(u, v) = g(u) = Au− buα with α > 1, A ≥ 0 and b > 0, k4(u, v) = k5(u, v) = 1,dv = 1 and vt = 0.

Furthermore, in [17] the existence of weak solutions to the following parabolic–elliptic systemut = ∆u− ∇ · (u∇v) + κu− µu2,

0 = ∆v + u− v,

under homogeneous Neumann boundary conditions was established for arbitrary µ > 0, while classicalsmooth solutions exist only for µ > n−2

n . In addition to it Winkler introduced in [18] the concept of very weaksolutions and established (as a special case) the global existence of very weak solutions for any nonnegativeinitial data u0 ∈ L1(Ω). Notice that his result is more general since instead of the logistic source termκu − µu2 he considers a more general logistic source term g(u) ≈ κu − µuα, α > 2 − 1

n , κ ≥ 0 andµ > 0. Moreover, boundedness properties of the constructed solutions are studied. Then, in [19] Viglialoro

G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090 3

proved the existence of very weak solutions to the corresponding parabolic–parabolic logistic Keller–Segelsystem with smooth initial data and α ≥ 1. Furthermore, very recently Winkler used the concept of veryweak solutions to investigate a general class of systems involving superlinear degradation in [20], which findsapplication to logistic Keller–Segel systems.

In [21] Lankeit proved the existence of global weak solutions to the chemotaxis systemut = ∆u− ∇ · (u∇v) + κu− µu2,

vt = ∆v + u− v,(1.2)

under homogeneous Neumann boundary conditions in a smooth bounded convex domain Ω ⊂ Rn, n ≥ 3, forarbitrarily small values of µ > 0. He also showed that in the three-dimensional setting, after some time, thesesolutions become classical solutions, provided that κ is not too large. Moreover, in [22] the authors establishthe existence of a global bounded classical solution for suitably large µ and prove that for any µ > 0 thereexists a weak solution. These are showcases that the concept of (very) weak solutions can be used to provemore general problems in the sense that one does not have to impose stronger assumptions on the system.

Furthermore, in [23] Winkler proved that system (1.1) has unique globally bounded classical solutionunder suitable assumptions on the initial data, provided ψ(u, v) = 1, k3(u, v) = 0, k4(u, v) = k5(u, v) = 1,dv = 1, ϕ(u, v) = χ(v), where 0 < χ(v) ≤ χ0/(1 + αv)k with some α > 0 and k > 1 for any χ0 > 0. Fujieet al. [24] extended the results from [23] to the singular case 0 < χ(v) ≤ χ0/v

k for some χ0 > 0 and k ≥ 1.In [25] Fujie solved the open problem of uniform-in-time boundedness of solutions for χ <

√2/n, which was

conjectured by Winkler [26].Moreover, Zheng et al. [27] showed the unique global classical solution is uniformly bounded in Ω ×(0,∞)

for quasilinear parabolic–elliptic chemotaxis system with signal dependent sensitivity under the assumptionsψ(u, v) = φ(u) ≥ c0(u + 1)m−1 for all u ≥ 0 with 0 < c0 ≤ 1, m ∈ R, ψ(u, v) = uχ(v), 0 < χ(v) ≤ χ0/v

k

with k ≥ 1 and χ0 > 0, k3(u, v) = 0, vt = 0, k4(u, v) = k5(u, v) = 1, dv = 1. He also proved nonnegativeglobal-in-time bounded weak solution when ψ(u, v) = φ(u) is degenerate.

The authors of [28] studied the global existence of solution for the quasilinear chemotaxis system withDirichlet boundary condition and also established that the blow-up properties of the solution depend only onthe first eigenvalue. Moreover, Stinner and Winkler [29] showed the global existence of nonnegative radiallysymmetric solutions to

ut = ∆u− χ∇ ·(uv

∇v),

vt = ∆v + u− v,(1.3)

where χ > 0. Additionally, Lankeit and Winkler designed a novel concept of generalised solvability andproved global solvability for large nonradial data [30].

Finally, we want to highlight that very recently Lankeit and Lankeit extended the study of the twoprevious chemotaxis systems (1.2) and (1.3) and established the existence of classical solutions andgeneralised global solutions, respectively, to the following system

ut = ∆u− χ∇ ·(uv

∇v)

+ κu− µu2,

vt = ∆v − uv,

for suitable choices of χ, κ and µ, see [31,32].The literature related to Keller–Segel chemotaxis system is very rich and the given overview does not

cover everything. Nevertheless, we believe that this overview gives a brief insight in the development andimportance of this topic.

During the past decades, the main issue of the investigation of system (1.1) is whether the chemotaxismodel allows for a chemotactic collapse, that is, if it possesses solutions that blow up in infinite or infinite time. However, in this paper we will prove global existence of solutions to system (1.4) (from below)

4 G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090

under the assumption (1.8) which rules out the chemotactic collapse. This work adds the significance bythe introduction of cross-diffusion terms into classical diffusion systems as it allows the mathematical modelto capture much more important features of many phenomena in physics, biology, ecology and engineeringsciences.

In general, the chemotaxis model has become one of the best-studied models in mathematical biologyand the main issue of the investigation was whether the models allow for solutions that blow-up or existglobally. Moreover, other contributions in this direction is also tremendous and there is a fairly rich literatureaddressing boundedness and blow-up issues in such systems of both parabolic–parabolic and parabolic–elliptic type (see [11,33–39]) and the references therein. In addition to that for stability and pattern formationone can refer [40,41] and the references therein.

In this paper, we consider a special case of the classical Keller–Segel chemotaxis system for the cell densityu and the concentration of the chemical signal v with an additional cross-diffusion term δ∆u with δ > 0,which reads as follows:

∂tu = ∇ · (d1∇u− ϕ(u, v)∇v) + h(u, v)∂tv = d2∆v + δ∆u+ g(u, v)

on QT = Ω × (0, T ), (1.4)

where T > 0 is a fixed time and Ω is a bounded domain in Rn, n ≥ 2 with smooth boundary ∂Ω ,and ψ(u, v) = d1 and ϕ(u, v) = χ u

γ+v . The positive constant χ is called the chemotactic coefficient, d1,d2 > 0 are the diffusion coefficients and γ ≥ 1. Furthermore, we also assume the logistic growth termh(u, v) = u(1 − u) and g(u, v) = αu − βv. The nonnegative constant α is called the creation rate and βu

the chemical degradation with β > 0. Additionally, system (1.4) is supplemented by homogeneous Neumannboundary conditions

∂u

∂η= ∂v

∂η= 0 on ΓT := ∂Ω × (0, T ), (1.5)

where η is the exterior unit normal to ∂Ω ; a no-flux boundary condition is implied on ∂Ω such that theecosystem is closed to the exterior environment and with the initial conditions

u(x, 0) = u0(x)∈ L2(Ω), v(x, 0) = v0(x)∈ L2(Ω) for all x ∈ Ω . (1.6)

This model is a special case of the general chemotaxis system with a cross-diffusion term δ∆u, where δ > 0.This term describes the movement of chemical signal along cell concentration towards or away from the celldensity. Biologically, it models some diffusive effects in the equation of the chemical signal due to variationsof the cell density, a kind of feedback effects. Additionally, for the classical Keller–Segel system

ut = ∇ · (∇u− χu∇v)vt = ∆v − v + u

in Ω × (0, T )

under homogeneous Neumann conditions and χ > 0 one may expect a finite-time or infinite-time blow-up ofsolutions, cf. e.g [42,43]. For more details on blow-up we refer to the survey paper of Lankeit and Winkler [44].

However, from a biological point of view a blow-up of solutions is not realistic. Therefore, one has tointroduce ‘something’ realistic to prevent such blow-up. This might be achieved by introducing a certainlogistic growth or an additional cross-diffusion. In our study we consider both, indeed the main focus is onthe additional cross-diffusion term.

There are some known facts about system (1.4) in presence of the cross-diffusion term δ∆u. Let us mentionbriefly some interesting literatures. Hittmeir and Jungel [45] were the first who introduced a particular caseof system (1.4) in 2 −D with ϕ(u, v) = χu, d1 = χ = 1, g(u, v) = µu− v with µ > 0, and the logistic growthterm h(u, v) = 0 in order to avoid the blow-up phenomena.

Xiang [46] studied the existence of global weak solutions to system (1.4) when ϕ(u, v) = χu and his resultsprovided global dynamics and insights on how the biological parameters, especially, the cross-diffusion,

G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090 5

affects the pattern formations. Meyries [47] investigated the existence of travelling waves for a model similarto system (1.4).

The analysis of system (1.4) with diffusion matrix

A(u, v) =

⎛⎝d1 −χ u

γ + v

δ d2

⎞⎠ , (1.7)

faces a number of mathematical challenges. First of all, we can rewrite system (1.4) as

∂t

(uv

)= ∇ ·

(A(u, v)∇

(uv

))+ F(u, v),

whereF(u, v) :=

(F1(u, v)F2(u, v)

)=

(h(u, v)g(u, v)

)=

(u(1 − u)αu− βv

).

Note that system (1.4) is strongly coupled so that standard tools, like maximum principles and regularitytheory, generally do not apply, cf. [48]. Many existence results for similar parabolic rely on Amann’stheorem [49, Section 1] for general reaction–diffusion system. However, in our situation this theorem is notdirectly applicable due to (1) the choice of our initial values u0, v0 ∈ L2(Ω), (2) the additional cross-diffusionterm, and (3) since the diffusion matrix A(u, v) is not positive definite.

Hence, we apply fixed point method for system (1.4), cf. [50]. For using this concept, the diffusion matrixA(u, v) of system (1.4) should be symmetric, positive definite and uniformly bounded. However, in our case,the matrix is non-symmetric and in fact, it is uniformly bounded only when certain L∞ bounds are available.

Nevertheless, we are able to give an existence result for weak solutions of (1.4) under the conditions thatthere exists a positive constant K, such that

d := mind1, d2 > δ

2 + χ

2γK. (1.8)

Additionally, we supposed that γ ≥ 1 and χ, d1, d2 and δ are positive constants.Under this hypotheses, it turns out that the diffusion matrix is positive definite. The lack of uniform

upper bounds for the diffusion terms is compensated by a weaker definition of solution, cf. Definition 1.1.Before we state our main results, let us introduce the definition of weak solutions to system (1.4).

Definition 1.1. A pair of nonnegative functions (u, v) is said to be a weak solution of (1.4)–(1.6) if andonly if

u, v ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1(Ω)), ut, vt ∈ L2(0, T ; (W 1,∞(Ω))′),such that for all test functions φ, ζ ∈ L2(0, T ;W 1,∞(Ω)) the following holds true:∫ T

0⟨ut, φ⟩dt+

∫∫QT

(d1∇u− χ

u

γ + v∇v

)· ∇φdxdt =

∫∫QT

F1(u, v)φdxdt

and ∫ T

0⟨vt, ζ⟩dt+

∫∫QT

(d2∇v + δ∇u) · ∇ζdxdt =∫∫

QT

F2(u, v)ζdxdt,

where ⟨·, ·⟩ denotes the duality pairing between (W 1,∞(Ω))′ and W 1,∞(Ω), and T ∈ (0,∞].

Now, we state our main theorem as follows.

Theorem 1.1. Assume that (1.8) holds. If u0, v0 ∈ L2(Ω), then problem (1.4)–(1.6) possesses a weaksolution.

The outline of this article is as follows: In Section 2, we establish the existence of weak solutions to theregularised system (2.2) by using Schauder’s fixed point theorem. Then, in Section 3, we prove the existenceof weak solutions to (1.4)–(1.6) by letting regularisation parameter ε → 0.

6 G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090

2. Existence of weak solutions

In this section, we will prove the existence of weak solutions to (1.4)–(1.6) satisfying (1.8). In order toprove the main result, we will first prove the existence of weak solutions to the regularised system (2.2) of(1.4)–(1.6) which can be obtained by using Schauder’s fixed-point theorem, a priori estimates and using thecompactness results. We start by introducing for a small number ε > 0 the following:⎧⎪⎨⎪⎩

Fi,ε := Fi

1 + ε|Fi|for i = 1, 2,

fε(a) := a

1 + ε|a|and b+ := max(0, b) for any a, b ∈ R.

(2.1)

Then, let us define the following regularised system:ut − ∇ · (a11∇u+ a12∇v) = F1,ε(u+, v+)vt − ∇ · (a21∇v + a22∇u) = F2,ε(u+, v+)

on QT , (2.2)

where the diffusion matrix A is given by

A(u+, v+) =(a11 a12a21 a22

)=

⎛⎝d1 −χ f+ε (u)

γ + f+ε (v)

δ d2

⎞⎠ . (2.3)

We supplement system (2.2) with no-flux boundary condition (1.5) and the initial data (1.6). Moreover, wehave

0 ≤ f+ε (s) = s+

1 + εs+ ≤ mins+, ε−1

for any s ∈ R

and f+ε (s) → s+ pointwise in R as ε → 0. Hence, we can conclude that there exists a positive constant K,

such thatf+

ε (u)γ + f+

ε (v)≤ 1γf+

ε (u) = 1γ

u+

1 + εu+ ≤ 1γ

minu+, ε−1

≤ Kγ

=⇒ −χ f+ε (u)

γ + f+ε (v)

≥ −χ

γK

and thus, it follows that

ξTAξ =d1ξ21 + d2ξ

22 +

(δ − χ

f+ε (u)

γ + f+ε (v)

)ξ1ξ2 ≥

(d− δ

2 − χ

2γK) (

ξ21 + ξ2

2)

≥ 0 (2.4)

for any ξ = (ξ1, ξ2) ∈ R2, where we used (1.8) in combination with a·b = 12 (a+ b)2− 1

2 (a2+b2) ≥ − 12 (a2+b2)

or −a·b = 12 (a− b)2 − 1

2 (a2 +b2) ≥ − 12 (a2 +b2) for all a, b ∈ R. Therefore, the matrix A(u+, v+) is uniformly

positive definite, provided (1.8) is fulfilled. Note that we will frequently use (2.4) to prove the existence andnonnegativity of weak solutions. Additionally, we want to mention that similar holds true if we considerbounded nonnegative functions d1(u, v) and d2(u, v) which are sufficiently smooth.

Remark 2.1. If condition (1.8) fails, we may loose the ellipticity condition (2.4) which leads to unboundedsolutions in finite time that is called blow-up. Thus, the assumption (1.8) allows us to actually prove theexistence of weak solutions. The case ϕ(u, v) = δ would also imply that the diffusion matrix A(u, v) isuniformly positive definite.

Our next aim is to establish an existence result to the fixed problem. Here, we omit the dependence of thesolutions on the parameter ε. We prove, for each fixed ε > 0, the existence of solutions to the fixed problem(2.2) by applying the Schauder’s fixed-point theorem. Since we use Schauder’s fixed-point theorem, we needto introduce the following closed subset of the Banach space L2(QT ):

S =U = (u, v) ∈ L2(QT ) : ∥U∥L∞(0,T ;L2(Ω))∩L2(0,T ;H1(Ω)) ≤ δ

, (2.5)

G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090 7

where δ > 0 is a constant. With (u, v) ∈ S fixed, let (u, v) be the unique solution of the system⎧⎨⎩ut = ∇ ·(d1∇u− χ

f+ε (u)

γ + f+ε (v)

∇v)

+ F1,ε(u+, v+)

vt = ∇ · (d2∇v + δ∇u) + F2,ε(u+, v+)on QT . (2.6)

The existence of a unique solution is guaranteed by [51, Chapter VII] due to (2.4) and Aij ∈ L∞(QT ) andFi,ε(u+, v+) ∈ L2(QT ), 1 ≤ i, j ≤ 2, cf. also [52].

Next, we introduce a map T : S → S such that T (u, v) = (u, v), where (u, v) solves (2.6). By using theSchauder’s fixed-point theorem, we prove that the map T has a fixed point for (2.6). First, we show thatthe map T is continuous. For this, let (un, vn)n be a sequence in S. Furthermore, let (u, v) ∈ S be such that(un, vn)n → (u, v) in L2(QT ) as n → ∞. Moreover, we define (un, vn) = T (u, v). The goal is to show that(un, vn) converges to T (u, v) in L2(QT ). To this end, we need the following lemma:

Lemma 2.1. The solution (un, vn) to system (2.6) satisfies(i) the sequence (un, vn)n is bounded in L2(0, T ;H1(Ω)) ∩ L∞(0, T ;L2(Ω)),(ii) the sequence (un, vn)n is relatively compact in L2(QT ).

Proof. Multiplying the first equation in system (2.6) by un and integrate over Ω , we derive at∫Ω

untundx−∫Ω

∇ · (d1∇un)un − ∇ ·(χ

f+ε (u)

γ + f+ε (v)

∇vn

)undx =

∫Ω

F1,εundx.

Integration by parts yields

12

ddt

∫Ω

|un|2dx+∫Ω

d1|∇un|2 −(χ

f+ε (u)

γ + f+ε (v)

∇vn

)· ∇undx =

∫Ω

F1,εundx. (2.7)

Similarly, we derive from the second equation in (2.6) multiplying by vn the following:

12

ddt

∫Ω

|vn|2dx+∫Ω

d2|∇vn|2dx+∫Ω

δ∇un · ∇vndx =∫Ω

F2,εvndx. (2.8)

Adding (2.7) and (2.8), by the definition and boundedness of Fi,ε, cf. (2.1), and using (2.4), we get

ddt

∫Ω

(|un|2 + |vn|2

)dx+

(d− δ

2 − χ

2γK) ∫

Ω

(|∇un|2 + |∇vn|2

)dx ≤ C

∫Ω

(|un|2 + |vn|2

)dx (2.9)

for some constant C > 0. Using Gronwall’s inequality in (2.9) and taking sup over (0, T ), we obtain that

supt∈(0,T )

∫Ω

(|un|2 + |vn|2)dx ≤ exp(∫ t

0Cds

) [∫Ω

(|u0|2 + |v0|2)dx]

≤ exp(CT )(∥u0∥2L2(Ω) + ∥v0∥2

L2(Ω)),(2.10)

which proves the first part of (i). Furthermore, from (2.9)–(2.10), we can conclude that(d− δ

2 − χ

2γK) ∫

Ω

(|∇un|2 + |∇vn|2)dx ≤ exp(CT )(∥u0∥2L2(Ω) + ∥v0∥2

L2(Ω)).

Now, integrating over (0, T ), we have∫∫QT

(|∇un|2 + |∇vn|2

)dxdt ≤ exp(CT )

d− δ2 − χ

2γ K

∫ T

0(∥u0∥2

L2(Ω) + ∥v0∥2L2(Ω))dt

≤ T exp(CT )d− δ

2 − χ2γ K

(∥u0∥2L2(Ω) + ∥v0∥2

L2(Ω)),(2.11)

8 G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090

which proves (i). Next, we multiply the first equation of (2.6) by φ ∈ L2(0, T ;H1(Ω)) and integrate over QT

to get ∫ T

0⟨unt , φ⟩dt

≤∫∫

QT

d1∇un · ∇φdxdt +

∫∫QT

(χ

f+ε (u)

γ + f+ε (v)

∇v)

· ∇φdxdt

+∫∫

QT

F1,ε(u+, v+)φdxdt .

(2.12)

Then, using the boundedness of f+ε and Fi,ε for i = 1, 2, and (2.11), there exists a constant C =

C(ε, d1, χ, γ) > 0 such that we can conclude from (2.12) that∫ T

0⟨unt , φ⟩dt

≤ C∥φ∥L2(0,T ;H1(Ω)). (2.13)

Similarly, multiplying the second equation in (2.6) by ζ ∈ L2(0, T ;H1(Ω)) and integrating over QT , we get∫ T

0⟨vnt , ζ⟩dt

≤∫∫

QT

d2∇v · ∇ζdxdt +

∫∫QT

δ∇u · ∇ζdxdt

+∫∫

QT

F2,ε(u+, v+)ζdxdt .

(2.14)

Again, using the boundedness of f+ε and Fi,ε for i = 1, 2, and (2.11), there exists a constant C = C(ε, d2, δ) >

0 such that Eq. (2.14) becomes ∫ T

0⟨vnt , ζ⟩dt

≤ C∥ζ∥L2(0,T ;H1(Ω)). (2.15)

Finally, (ii) is a consequence of the boundedness of (un, vn)n, cf. (i), and the uniform bounds (2.13) and(2.15) of (un, vn)n in (L2(0, T ; (H1(Ω))′))2.

Thus, from Lemma 2.1, we can conclude that there exist functions (un, vn) ∈ (L2(0, T ;H1(Ω)))2 suchthat (up to a subsequence)

(un, vn) → (u, v) in(L2(QT )

)2 strongly,

which shows the continuity of T on S. Further, from Lemma 2.1, we can conclude that T (S) is bounded in

X = u ∈ L2(0, T ;H1(Ω)) : ut ∈ L2(0, T ; (H1(Ω))′).

Moreover, since T (S) is bounded and the embedding X → L2(QT ) is compact (by the results of [53]), weknow that, T is compact. Thus, T satisfies all assumptions of Schauder’s fixed-point theorem. Therefore,the operator T has a fixed point (uε, vε) such that T (uε, vε) = (uε, vε). Then, there exists a solution (uε, vε)of ∫ T

0⟨uεt , φ⟩dt+

∫∫QT

(d1∇uε − χ

f+ε (u)

γ + f+ε (v)

∇v)

· ∇φdxdt =∫∫

QT

F1,ε(u+ε , v

+ε )φdxdt (2.16)

and ∫ T

0⟨vεt , ζ⟩dt+

∫∫QT

d2∇vε · ∇ζdxdt+∫∫

QT

δ∇u · ∇ζdxdt =∫∫

QT

F2,ε(u+ε , v

+ε )ζdxdt (2.17)

for all φ, ζ ∈ L2(0, T ;H1(Ω)). Furthermore, the solution (uε, vε) is nonnegative and satisfies certainregularity estimate as stated in the following lemma.

G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090 9

Lemma 2.2. Assume that condition (1.8) holds. If the initial conditions u0 ∈ L2(Ω) and v0 ∈ L2(Ω) arepositive, then the solution (uε, vε) is nonnegative. Moreover, there exist constants C1, C2, C3, C4 > 0 notdepending on ε such that

∥uε∥L∞(0,T ;L2(Ω)) + ∥vε∥L∞(0,T ;L2(Ω)) ≤ C1, (2.18)∥F1,ε(uε, vε)∥L1(QT ) + ∥F2,ε(uε, vε)∥L1(QT ) ≤ C2, (2.19)∥∇uε∥L2(QT ) + ∥∇vε∥L2(QT ) ≤ C3, (2.20)∥uεt∥L2(0,T ;(W 1,∞(Ω))′) + ∥vεt∥L2(0,T ;(W 1,∞(Ω))′) ≤ C4. (2.21)

Proof. Choosing φ = −u−ε = uε−|uε|

2 in (2.16) and integrate over Ω instead QT . Using [54, Proposition1.2] yields

12

ddt

∫Ω

|u−ε |2dx+

∫Ω

(d1∇u−

ε − χf+

ε (uε)γ + f+

ε (vε)∇v−

ε

)· ∇u−

ε dx =∫Ω

F1,ε(u+ε , v

+ε )u−

ε dx. (2.22)

Similarly, in (2.17), we take ζ = −v−ε = vε−|vε|

2 and integrate over Ω instead of QT to obtain

12

ddt

∫Ω

|v−ε |2dx+

∫Ω

(d2∇v−ε + δ∇u−

ε ) · ∇v−ε dx =

∫Ω

F2,ε(u+ε , v

+ε )v−

ε dx. (2.23)

Adding (2.22) and (2.23), using (2.4) and the definition of Fi,ε, we conclude that

12

ddt

∫Ω

(|u−

ε |2 + |v−ε |2

)dx+

(d− δ

2 − χ

2γK) ∫

Ω

(|∇u−

ε |2 + |∇v−ε |2

)dx ≤ C

∫Ω

(|u−

ε |2 + |v−ε |2

)dx

for some constant C > 0. Utilising Gronwall’s lemma, we get∫Ω

(|u−

ε |2 + |v−ε |2

)dx ≤ 0 (2.24)

for a.e. t ∈ (0, T ). This yields the nonnegativity of uε and vε. Additionally, for i = 1, 2, we have

|Fi,ε(uε, vε)| ≤ C2

(|uε|2 + |vε|2

)(2.25)

for some constant C2 > 0. Now, using (2.18) in (2.25), we get (2.19). Because of the weak lowersemicontinuity properties of norms, estimates (2.9) and (2.10) hold for uε and vε, respectively. Thus, weobtain (2.18) and (2.20). Finally, using Holder’s inequality, (2.18) and (2.20), we can deduce from (2.16) forall φ ∈ L2(0, T ;W 1,∞(Ω)) that

∫ T

0⟨uεt , φ⟩dt

≤ C∥∇uε∥L2(QT )∥∇φ∥L2(QT )

+ C ′∥uε∥L∞(0,T ;L2(Ω))∥∇vε∥L2(QT )∥∇φ∥L2(0,T ;L∞(Ω))

+ C ′′∥uε∥L∞(0,T ;L2(Ω))∥uε∥L2(QT )∥φ∥L2(0,T ;L∞(Ω))

+ C ′′′∥vε∥L∞(0,T ;L2(Ω))∥vε∥L2(QT )∥φ∥L2(0,T ;L∞(Ω))

≤ C ′′′′∥φ∥L2(0,T ;W 1,∞(Ω))

(2.26)

for some constant C,C ′, C ′′, C ′′′, C ′′′′ > 0 independent of ε. From this, the following bound

∥uεt∥L2(0,T ;(W 1,∞(Ω))′) ≤ C ′′′ (2.27)

derives. Similarly, we can conclude from the weak formulation (2.17) that we have for all ζ ∈ L2(0, T ;W 1,∞(Ω))

∫ T

0⟨vεt , ζ⟩dt

≤ C ′′′′∥ζ∥L2(0,T ;W 1,∞(Ω)). (2.28)

10G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 103090

From this, we deduce the bound∥vεt∥L2(0,T ;(W 1,∞(Ω))′) ≤ C ′′′′. (2.29)

Adding (2.27) and (2.29), we derive (2.21). This completes the proof of the lemma.

3. Proof of Theorem 1.1

In this section, we prove the existence of weak solutions to (1.4)–(1.6), since until now we only showedthat approximate system (2.2) admits a solution (uε, vε), which is nonnegative. The next goal is to sendthe regularisation parameter ε to zero in sequences of such solution to obtain weak solutions of the originalsystem (1.4)–(1.6).

From Lemma 2.2 and Aubin’s lemma, we can conclude that there exist limit functions (u, v) such that asε → 0 the following convergences (up to a subsequence) hold:⎧⎪⎨⎪⎩

(uε, vε) → (u, v) a.e. in QT and strongly in L2(QT ), weakly in L2(0, T ;H1(Ω)),Fi,ε(uε, vε) → Fi(u, v) a.e. in QT and strongly in L1(QT ),(uεt , vεt) → (ut, vt) weakly in L2(0, T ; (W 1,∞(Ω))′).

(3.1)

Furthermore, we have

∥fε(uε) − u∥L2(QT ) = ∥fε(uε) − uε + uε − u∥L2(QT )

≤√

2∥uε − u∥L2(QT ) +√

2 ∥fε(uε) − uε∥L2(QT )

=√

2∥uε − u∥L2(QT ) +√

2ε ∥fε(uε)∥L2(QT ) −→ 0 as ε → 0,(3.2)

which impliesfε(uε) → u a.e. in QT and strongly in L2(QT ).

In addition, we can conclude using Holder’s inequality the following

∥fε(uε) − u∥L1(QT ) ≤√

|QT |∥fε(uε) − u∥L2(QT ) −→ 0 as ε → 0,

which implies fε(uε) → u a.e. in QT and strongly in Lp(QT ) for all 1 ≤ p ≤ 2. Finally, by passingto the limits as ε → 0 in the weak formulation (2.16)–(2.17) with φ ∈ L2(0, T ;W 1,∞(Ω)) and ζ ∈L2(0, T ;W 1,∞(Ω)), we could obtain in this way that the limit (u, v) is a solution of the system (1.4)–(1.6)satisfying (1.8) in the sense of Definition 1.1.

4. Conclusions

In this paper, we considered a Keller–Segel chemotaxis system with additional cross-diffusion and provedthe existence of weak solutions to the regularised system of (2.2) by using Schauder’s fixed point theorem.Then, the existence of weak solutions to system (1.4)–(1.6) satisfying (1.8) has been achieved by a prioriestimates and compactness arguments.

Acknowledgements

G.A. and B.K. are supported by Defence Research and Development Organisation, Government of India,Grant Number: DLS/86/50011/DRDO-BU Center Phase II. Furthermore, the authors wish to thank theanonymous referee for the careful reading of the original manuscript and the comments that eventually ledto an improved presentation. Open access funding was provided by the University of Oslo.

G. Arumugam, A.H. Erhardt, I. Eswaramoorthy et al. / Nonlinear Analysis: Real World Applications 54 (2020) 10309011

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