On the existence of solutions for a parabolic-ellipticchemotaxis model with flux limitation and logistic source

Received: 15 August 2022 Revised: 26 December 2022 Accepted: 28 December 2022 DOI: 10.1002/mma.9050 RESEARCH ARTICLE On the existence of solutions for a parabolic-elliptic chemotaxis model with flux limitation and logistic source Silvia Sastre-Gomez1Jose Ignacio Tello2 1Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Seville, Spain 2Departamento de Matemáticas Fundamentales, Facultad de Ciencias, Universidad Nacional de Educación a Distancia, Madrid, Spain Correspondence Silvia Sastre-Gomez, Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Seville, Spain. Email: [email protected] Communicated by: M. A. Lachowicz Funding information Dirección General de Investigación Científica y Técnica, Grant/Award Number: Project MTM2017-83391-P DGICT Spain; Junta de Andalucía FQM-131; UCM, SPAIN, Grant/Award Number: GR58/08 Grupo 920894 In this article, we study the existence of solutions of a parabolic-elliptic system of partial differential equations describing the behaviour of a biological species “u” and a chemical stimulus “v” in a bounded and regular domain Ωof RN.The equation for uis a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as −𝜒div(u|∇v|p−2∇v), for p>1. The chemical substance distribution vsatisfies the elliptic equation −Δv+v=u. The evolution of uis also determined by a logistic type growth term 𝜇u(1−u). The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for p<3∕2 and any N≥2. KEYWORDS bounded solutions, chemotaxis, global existence of solutions MSC CLASSIFICATION 35A01, 35B09 1INTRODUCTION In the decade of 70s of the last century, Keller and Segel1,2 described chemotaxis phenomena for a biological species density “u” and a chemical substance concentration “v” using systems of partial differential equations. The Keller–Segel model presents a second order chemotaxis term with a linear dependence of the chemical flux “∇v”intheway −div(𝜒u∇v) where 𝜒is a given constant. After the model was proposed, many authors have studied such systems from a mathematical point of view, see for instance Horstmann3,4 and Hillen and Painter5for details concerning mathematical results of chemotaxis models with linear flux. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. © 2023 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd. Math Meth Appl Sci. 2023;46:9252–9267.wileyonlinelibrary.com/journal/mma 9252 Rivero et al6proposed a chemotaxis term with nonlinear dependence of the chemical flux, given in the form −𝜕 𝜕x(F(u,𝜕v 𝜕x)),F(u,𝜕v 𝜕x)=arctan [u 1+u 𝜕v 𝜕x] in a one-dimensional spatial domain. Saragosti et al7considered a flux limitation model of integral type in the form −div(u[v]),for u[v]=− 1 |V|∫V 𝑓(𝜈∇v)d𝜈, where Vdenotes the set of possible velocities 𝜈; see also Perthame et al.8 Chertock et al9consider the parabolic-parabolic system with a flux limitation defined by F(∇v)= ∇v √1+|∇v|2. Bianchi et al10,11 use such a nonlinear dependence to obtain a biomedical model. Bellomo and Winkler12,13 study the parabolic-elliptic system for a chemotaxis term of the form −div(u𝑓(|∇v|)∇v) for 𝑓(|∇v|)= 𝜒 √1+|∇v|2 and a diffusive term −div (u∇u √u2+|∇u|2) to describe the evolution of the biological species u. For more information about this model, see also Chiyoda et al14 and Mizukami et al.15 In Bellomo and Winkler,12 for 𝜒<1andN≥2, the solutions are global and bounded if the initial mass is smaller than (𝜒2−1)−1 2 +. Using an energy method, Winkler16 has obtained blow up of solutions for 𝑓=𝑓(|∇v|)= 𝜒 (1+|∇v|2)𝛼 2 and N>2when 𝛼< N−2 2(N−1). More blow up results can be found in Marras et al.17 In this article, we consider a bounded N-dimensional domain, Ω, with regular boundary 𝜕Ωand denote by  nthe outward pointing normal vector on the boundary 𝜕Ω. The equation for vis restricted to the elliptic case, for simplicity, we assume that vsatisfies the Poisson equation and the system studied reads as follows: ut−Δu=−div(𝜒u|∇v|p−2∇v)+𝜇u(1−u)x∈Ω,t>0,(1.1) −Δv+v=ux∈Ω,t>0,(1.2) 𝜕u 𝜕 n=𝜕v 𝜕 n=0x∈𝜕Ω,t>0,(1.3) u(0,x)=u0(x)x∈Ω.(1.4) SASTRE-GOMEZ AND TELLO 9253 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License In Negreanu and Tello,18 the system is considered for 𝜇=0, that is, the logistic term does not appear and psatisfies p∈(1,∞),if N=1and p∈(1,N N−1),if N≥2, to obtain uniform bounds in L∞(Ω). The complementary case, for p∈(N∕(N−1),2),forN>2andΩdefined as the unit ball, presents blow up for some initial data when 𝜇=0; see Tello.19 The stationary case with flux limitation has been studied in Boccardo and Tello20;see also Boccardo and Orsina21 for the stationary case for the case p=2. Parabolic-elliptic systems of chemotaxis with logistic terms have been already studied in the literature. In Tello and Winkler,22 a comparison method is applied to proof the converge of the solution to the homogeneous steady state (see also Galakhov et al23 and references therein). In this article, we study the global existence of solutions in (0,T)for any T<∞under the following assumptions: Ωis an open and bounded domain with regular boundary 𝜕Ω,(1.5) {p<2,N=2, p∈(1,3∕2),N≥3,(1.6) and the initial data u0satisfy u0∈C2+𝛼(Ω),𝜕u0 𝜕 n=0,x∈𝜕Ω,(1.7) for some 𝛼∈(0,1). In this article, we prove the existence of weak solutions to problems (1.1)–(1.4) in the sense of the following definition. Definition 1.1. We say that (u,v)∈[L2(0,T∶H1(Ω)) ∩ H1(0,T∶(H1(Ω))′)]2is a weak solution to (1.1)-(1.4) if u∇v∈ [L1(Ω)]Nand for any 𝜑, 𝜓 ∈L2(0,T∶H1(Ω)) ∩ H1(0,T∶L2(Ω)),(u,v)satisfies ∫T 0∫Ω 𝜑utdxdt +∫T 0∫Ω ∇u∇𝜑dxdt =∫T 0∫Ω 𝜒u|∇v|p−1∇v∇𝜑dxdt +∫T 0∫Ω 𝜇u(1−u)𝜑dxdt,(1.8) ∫T 0∫Ω ∇v∇𝜓dxdt =∫T 0∫Ω (u−v)𝜓dxdt.(1.9) The existence of weak solutions given in the previous definition is enclosed in the following theorem. Theorem 1.1. Under assumptions (1.5)–(1.7), there exists at least a weak solution to (1.1) in (0,T)for any T <∞,in the sense of Definition 1.1. Moreover, there exists a constant c independent of T such that the solution satisfies ||u(t)||L∞(Ω) ≤c,for any t ∈(0,T). The article is organized as follows. In Section 2, we obtain some a priori estimates that are used in Section 3 to proof Theorem 1.1. We use an iterative method based in the Moser–Alikakos iteration that allows us to obtain explicit Lqestimates of the solutions. Then, we pass to the limit to get the boundedness of the solutions in L∞norm. Finally, using an approximated problem, we pass to the limit in the weak formulation to obtain the existence of weak solutions. 2A PRIORI ESTIMATES FOR THE AUXILIARY PROBLEM We introduce the following auxiliary problem: unt −Δun=−div (𝜒un|∇vn|p−2∇vn 1+1 n|∇vn|p−1)+𝜇un(1−un)x∈Ω,t>0,(2.1) −Δvn+vn=unx∈Ω,t>0,(2.2) SASTRE-GOMEZ AND TELLO 9254 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 𝜕un 𝜕 n=𝜕vn 𝜕 n=0x∈𝜕Ω,t>0,(2.3) un(0,x)=u0(x)x∈Ω.(2.4) In this section, we give some estimates that will be useful to prove the existence of solution of (2.1)–(2.4). In particular, we obtain uniform bounds in L∞(Ω) for unand vn. In the following result, we obtain an upper bound of the total mass of un. Lemma 2.1. The total mass of the component unof the solution to (2.1) is bounded, ∫Ωundx ≤c1,with c1=∶ max {∫Ω u0dx,|Ω|}. Since the proof of this lemma is the same as in the case p=2, we omit the details. In the following result, we prove that unis nonnegative if the initial data u0is nonnegative. Lemma 2.2. The solution unto (2.1)–(2.4) with u0≥0satisfies un≥0. Proof. We first consider the auxiliary equation unt −Δun=−div (𝜒un|∇vn|p−2∇vn 1+1 n|∇vn|p−1)+𝜇un(1−(un)+)(2.5) and define Thas follows Th(s)={−h,if s≤−h, s,if −h<s<0, 0,if s≥0. We denote the primitive of Th,by𝛷h,whichisgivenby 𝛷h(s)=⎧ ⎪ ⎨ ⎪ ⎩ −hs −h2 2,if s≤−h, s2 2,if −h<s<0, 0,if s≥0, and satisfies 𝛷′ h=Th. We multiply (2.5) by Th(un)and integrate by parts to obtain d dt ∫Ω𝛷h(un)dx +∫−h<un<0|∇un|2dx =𝜒∫−h<un<0un|∇vn|p−2∇vn∇un 1+1 n|∇vn|p−1dx +𝜇∫ΩTh(un)un(1−(un)+)dx ≤𝜒n∫−h<un<0un|∇un|dx +𝜇∫ΩTh(un)undx and apply Hölder inequality, and after some computations, we get d dt∫Ω 𝛷h(un)dx +1 2∫−h<un<0|∇un|2dx ≤𝜒2 2n∫−h<un<0 u2 n+2𝜇∫Ω 𝛷h(un)dx, which implies d dt∫Ω 𝛷h(un)dx +1 2∫−h<un<0|∇un|2dx ≤ch2+2𝜇∫Ω 𝛷h(un)dx. We divide by hand take limits when h→0toget d dt∫Ω (−un)+dx ≤2𝜇∫Ω (−un)+dx. SASTRE-GOMEZ AND TELLO 9255 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License We apply Gronwall's lemma, and it results ∫Ω (−un)+dx ≤e2𝜇t∫Ω (−un(0))+dx. Since (−u0)+=0, we get un≥0. Since un≥0, we have that (un)+=unand unsatisfies (2.1). □ The following estimate will be useful to obtain Lqestimates for un. Lemma 2.3. Let p <3 2,N≥3and q satisfying assumptions max {1 2,N−1 2−N 4(p−1)}<q<N−1 2, then ∫Ω u2q ndx <c1 and ∫t 0∫Ω u2q+1 ndxds ≤c2t+c3, where c1=c1(q),c2=c2(q)and c3=c3(q)are independent of t. Proof. We multiply Equation (2.1) by |un|2q−2un, and after integration over Ω,weget d dt 1 2q∫Ω u2q ndx +2q−1 q2∫Ω|∇uq n|2dx =(2q−1)𝜒 q∫Ω uq n|∇vn|p−2∇vn 1+1 n|∇vn|p−1∇uq ndx +𝜇∫Ω u2q n(1−un)dx.(2.6) Notice that (2q−1)𝜒 q∫Ω uq n|∇vn|p−2∇vn 1+1 n|∇vn|p−1∇uq ndx ≤(2q−1) 2q2∫Ω|∇uq n|2dx +(2q−1)𝜒2 2∫Ω|∇vn|2p−2u2q ndx and thanks to Hölder inequality ∫Ωu2q n|∇vn|2p−2dx ≤[∫Ω|∇vn|2(p−1)(2q+1)dx]1 2q+1[∫Ωu(2q+1) ndx]2q 2q+1 ≤||∇vn||2(p−1) L2(p−1)(2q+1)(Ω)||un||2q L2q+1(Ω). The elliptic regularity of the problem gives the inequality, ||∇vn||L2(p−1)(2q+1)(Ω) ≤c||un||L2q+1(Ω) provided N(2q+1) N−2q−1>2(p−1)(2q+1), which is equivalent to N N−2q−1>2(p−1).(2.7) SASTRE-GOMEZ AND TELLO 9256 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License We take qsuch that N[2p−3] 2(p−1)<2q+1 to get N−1 2−N 4(p−1)<q.(2.8) Thanks to (2.7) and (2.8), we have that q∈(max {1 2,N−1 2−N 4(p−1)},N−1 2). Then, after some computations, we get (2q−1)𝜒 q∫Ω uq n|∇vn|p−2∇vn∇uq ndx ≤(2q−1) 2q2∫Ω|∇uq n|2dx +c[∫Ω u2q+1 ndx]1+2(p−3 2) 2q+1 we replace the previous inequality in Equation (2.6) to get d dt 1 2q∫Ω u2q ndx +2q−1 2q2∫Ω|∇uq n|2dx ≤c[∫Ω u2q+1 ndx]1+2(p−3 2) 2q+1 −𝜇 2∫Ω u2q+1 ndx +c.(2.9) We apply Poincaré–Wirtinger inequality to the term ∫Ω|u2q n|dx in the following way ∫Ω|u2q n|dx ≤c∫Ω|∇uq n|2dx +c[∫Ω|un|qdx]2 then, 2q−1 2q2∫Ω|∇uq n|2dx ≥c∫Ω|u2q n|dx −c[∫Ω|un|qdx]2 . Since [∫Ω|un|qdx]2≤𝜖∫Ω|un|2q+1dx +c(𝜖) and [∫Ω|un|2q+1dx]1+2(p−3 2) 2q+1≤𝜖∫Ω|un|2q+1dx +c(𝜖). Thanks to (2.9), we have d dt 1 2q∫Ωu2q ndx +c∫Ωu2q ndx ≤−(𝜇 2−2𝜖)∫Ω|un|2q+1dx +c(𝜖).(2.10) We take 𝜖<𝜇∕4 and apply maximum principle for ODE to obtain ∫Ω u2q ndx ≤c1,∫t 0∫Ω u2q+1 ndxds ≤c2t+c3, where c1=c1(q),c2=c2(q)and c3=c3(q)are positive constants and independent of t.□ SASTRE-GOMEZ AND TELLO 9257 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Remark 2.1. In the previous lemma, the assumption p<3 2is used to obtain the estimate ||∇vn||L2(p−1)(2q+1)(Ω) ≤c||un||L2q+1(Ω) for any Nand any q<1 by using the elliptic regularity of the problem. In the result below, we give an L2q-estimate of unfor N=2. Lemma 2.4. Let N =2,p<2and q ∈(1,∞) then, ∫Ω u2q ndx <c4(q). and vn∈W1,∞(Ω), where c4(q)is independent of t. Proof. We proceed as in Lemma 2.3 to obtain d dt 1 2q∫Ω u2q ndx +2q−1 2q2∫Ω|∇uq n|2dx ≤c∫Ω u2q n|∇v|2(p−1)dx −𝜇 2∫Ω u2q+1 ndx +c.(2.11) Since ∫Ω u2q n|∇vn|2(p−1)dx ≤c||∇vn||2(p−1) L∞(Ω) ∫Ω u2q ndx, and thanks to the elliptic regularity, we know that ||∇vn||L∞(Ω) ≤c||un||L2+𝜖 q(Ω), and for q≥1, ||un||L2+𝜖 q(Ω) ≤c||un||L2q+𝜖(Ω) then d dt 1 2q∫Ω u2q ndx +2q−1 2q2∫Ω|∇uq n|2dx ≤c[∫Ω u2q+𝜖 ndx]2p−1+q 2q+𝜖 −𝜇 2∫Ω u2q+1 ndx +c.(2.12) Thanks to Gagliardo–Nirenberg inequality, we know that ||un||L2q+𝜖(Ω) =||uq n|| 1 q L2+𝜖 q(Ω) ≤c[||∇uq n|| a q L2(Ω)||uq n|| (1−a) q L1(Ω) +c||uq n||L1(Ω)]1 q(2.13) for asatisfying 1 2+𝜖 q =(1 2−1 N)a+(1−a), which is equivalent to a=2N(q+𝜖) (2q+𝜖)(N+2). For N=2andq=1, we have that a=1+𝜖 (2+𝜖). We replace q=1 into (2.13), and thanks to Lemma 2.1, we get ||un||2+𝜖 L2+𝜖(Ω) ≤c||∇un||1+𝜖 L2(Ω) +c. SASTRE-GOMEZ AND TELLO 9258 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Then, [∫Ω u2+𝜖 ndx]2p 2+𝜖≤c||∇un|| (1+𝜖)2p 2+𝜖 L2(Ω) +c, for 𝜖small enough, we get ||∇un|| (1+𝜖)2p 2+𝜖 L2(Ω) ≤𝛿||∇un||2 L2+c(𝛿) for any 𝛿>0. We replace into (2.12) to get d dt 1 2∫Ωu2 ndx +(1 2−𝛿)∫Ω|∇un|2dx ≤c−𝜇 4∫Ω|un|3dx. Standard computations show the uniform boundedness in time of ∫Ω u2 ndx. To end the proof, we take q∈(1,2)to obtain; thanks to Gagliardo–Nirenberg inequality that ||un||L2q+𝜖(Ω) =||uq n|| 1 q L2+𝜖 q(Ω) ≤c[||∇uq n||a L2(Ω)||uq n||1−a L2 q(Ω) +c||uq n||L1(Ω)]1 q(2.14) for asatisfying 1 2+𝜖 q =(1 2−1 N)a+q(1−a) 2 that is, q 2q+𝜖−q 2=(1 2−1 N)a−qa 2 since N=2, we have q 2q+𝜖−q 2=− qa 2 and then (2(q−1)+𝜖 2q+𝜖)=a then, for 𝜖small enough and qclose to 1, and since p<2, we have that a(2p−1+q 2q+𝜖)<2. We replace into (2.12) to obtain d dt 1 2q∫Ω u2q ndx +2q−1 2q2∫Ω|∇uq n|2dx ≤c[∫Ω|∇uq n|2dx]𝛽 −𝜇 2∫Ω u2q+1 ndx +c for 𝛽<2. We proceed as before to get ∫Ω u2q ndx <c for some q>1. We apply elliptic regularity to second equation and get that vn∈W1,∞(Ω).Then,theterm ∫Ω u2q n|∇vn|2(p−1)≤c∫Ω u2q n≤c(𝜖)+𝜖∫Ω u2q+1 n SASTRE-GOMEZ AND TELLO 9259 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License which implies, for 𝜖≤𝜇 4 d dt 1 2q∫Ω u2q ndx +2q−1 2q2∫Ω|∇uq n|2dx ≤c−𝜇 4∫Ω u2q+1 ndx. Since ∫Ω u2qdx ≤∫Ω u2q+1dx +|Ω|, we get d dt 1 2q∫Ω u2q ndx +2q−1 2q2∫Ω|∇uq n|2dx +𝜇 4∫Ω u2q ndx ≤c+|Ω|. Maximum principle ends the proof. □ In the following lemma, we obtain an estimate of unand vn,forN≥3. Lemma 2.5. Let N ≥3and p <3 2; then, for any s <∞,wehavethat ∫Ω us n≤c5 and vn∈W2,N+1(Ω) ∩ W1,∞(Ω), where c5is independent of s and t. Proof. Thanks to Lemma 2.3, we have that unis uniformly bounded in Lq(Ω) for q<N−1; then we have that, thanks to the elliptic regularity, vn∈W2,q(Ω) for q<N−1, which implies vn∈W1,Nq N−q(Ω) for any q∈(1,N−1). In particular, we have that, for any p<3∕2,q<N−1 and close to N−1, the inequality 2(p−1)(2q+1)<Nq N−q is satisfied, and then, the term ||∇vn||L2(p−1)(2q+1)(Ω) is bounded. Then we proceed as in the case N=2 (see Lemma 2.4) to get that unis uniformly bounded in Ls(Ω) for any s<N(N−1) 2(p−1)−1. Since p<3 2,wegetthatunis uniformly bounded in Ls(Ω) for any s<N(N−1)−1 and we deduce, in view of N≥3thatun∈LN+1(Ω). Elliptic regularity implies that vn∈W2,N+1(Ω), which is included in W1,∞(Ω) and the proof ends. □ The following result gives a uniform bound of unwhen N≥3andp<3∕2. Lemma 2.6. Let N ≥2and p <3 2; then, we have that ||un||L∞(Ω) ≤c6 where c6is independent of t. Proof. Thanks to Lemmas 2.4 and 2.5, we have that vn∈W1,∞(Ω).Wetakeu2q−1 nas test function in the weak formulation of (2.1)–(2.4) to get, after some computations, that 1 2q d dt∫Ω u2q ndx +2q−1 2q2∫Ω|∇uq n|2dx ≤||∇vn||2(p−1) L∞(Ω) 𝜒2(2q−1) 2∫Ω u2q ndx +𝜇∫Ω u2q ndx −𝜇∫Ω u2q+1 ndx.(2.15) SASTRE-GOMEZ AND TELLO 9260 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 29. Negreanu M, Tello JI, Vargas AM. A note on a periodic Parabolic-ODE chemotaxis system. Appl Math Lett. 2020;106:106351. 30. Stampacchia G, Le problèmedeDirichletpourleséquationselliptiquesdusecondordreàcoefficientsdiscontinus. Ann Inst Fourier (Grenoble). 1965;15:189-258. doi:10.5802/aif.204 31. Tao Y, Winkler M. Persistence of mass in a chemotaxis system with logistic source. J Differ Equ. 2015;259:6142-6161. How to cite this article: Sastre-Gomez S, Tello JI. On the existence of solutions for a parabolic-elliptic chemotaxis model with flux limitation and logistic source. Math Meth Appl Sci. 2023;46(8):9252-9267. doi:10.1002/ mma.9050 SASTRE-GOMEZ AND TELLO 9267 10991476, 2023, 8, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.9050 by Universidad De Sevilla, Wiley Online Library on [11/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License