Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 77, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.77 EXISTENCE OF GLOBAL SOLUTIONS FOR CROSS-DIFFUSION MODELS IN A FRACTIONAL SETTING JHEAN E. PÉREZ-LÓPEZ, DIEGO A. RUEDA-GÓMEZ, ÉLDER J. VILLAMIZAR-ROA Abstract. This article is devoted to the analysis of a fractional chemotaxis model in RN with a time fractional variation in the Caputo sense and a frac- tional spatial diffusion. This model encompasses the fractional Keller-Segel system [9] which describes the movement of living organisms towards higher concentration regions of chemical attractants, and a fractional Lotka-Volterra competition model [16] describing the competition interspecies in which one of the competing species avoids encounters with rivals by means of chemorepul- sion. We prove product estimates in Besov-Morrey spaces and derive global estimates for mild solutions of the fractional heat equation. We use these re- sults to prove the existence and uniqueness of global mild solutions for the differential system in a framework of Besov-Morrey spaces. 1. Introduction We consider a generic fractional chemotaxis model describing the evolution of two species subject to attraction and repulsion phenomena. This model includes the fractional Keller-Segel system which describes the movement of living organisms towards higher concentration regions of chemical attractants [9], and a fractional Lotka-Volterra competition model describing the competition interspecies to avoid encounters with rivals by means of chemorepulsion mechanism [16]. This model is composed of three coupled parabolic equations describing the interaction between the densities of competing species and the concentration of a chemical substance, which reads as follows cDα t n+ dn(−∆)θ/2n = χ∇ · (nG(v)) + a1n 2 − b1nm, cDα t m+ dm(−∆)θ/2m = a2m 2 − b2nm, cDα t v + dv(−∆)θ/2v = a3m+ b3n− γv, n(x, 0) = n0(x), m(x, 0) = m0(x), v(x, 0) = v0(x), (1.1) where the unknowns are n = n(x, t) and m = m(x, t), for x ∈ RN and t ∈ (0,∞), which denote the densities of competing species, while v = v(x, t) denotes the chemical signal. In (1.1), dn, dm, and dv are positive parameters representing the 2020 Mathematics Subject Classification. 35K55, 35Q35, 35Q92, 92C17. Key words and phrases. Keller-Segel system; Lotka-Volterra; cross-diffusion; Besov-Morrey spaces. ©2023. This work is licensed under a CC BY 4.0 license. Submitted August 6, 2022. Published November 10, 2023. 1 2 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 diffusion coefficients, a1, a2 ∈ R denote growth coefficients (a1, a2 > 0) or self- competition (a1, a2 < 0), a3, b1, b2, b3 and γ are positive parameters related to the population dynamics and the strength of competition, and χ denotes the chemotaxis coefficient. In (1.1) cDα t denotes the time fractional derivative operator of order α ∈ (0, 1) in the Caputo sense. We recall that for f ∈ C([0, T ];X), 0 < T ≤ ∞, such that I1−α t f ∈ W 1,1(0, T ;X), the Caputo fractional derivative of order α of f is defined by cDα t f(t) := d dt { I1−α t [f(t)− f(0)] } = d dt {∫ t 0 (t− τ)−α[f(τ)− f(0)] dτ } , where Iαt f denotes the Riemann-Liouville fractional integral of order α of f , defined by Iαt f(t) = 1 Γ(α) ∫ t 0 (t− τ)α−1f(τ) dτ, t ∈ [0, T ]. In addition, in (1.1), (−∆)θ/2, θ ∈ (0, 2], denotes the fractional Laplacian operator of order θ/2 defined by (−∆)θ/2f(x) = F−1(|ξ|θf̂(ξ))(x), where f̂(ξ) = F(f)(ξ) and F−1(f)(ξ) denote the Fourier transform and the inverse Fourier transform of f , respectively. Finally, G(v) is also a nonlocal term defined by G(v)(x) = ∇ ( (−∆)−θ1/2v ) (x), x ∈ RN , for θ1 ∈ [0, N), which can be alternatively represented by G(v) = K(x)∗ v, K(x) ∼ x |x|N−θ1 . If we take χ < 0, a1 = 0 and m = 0 in (1.1), we obtain the fractional version of the classical Keller-Segel system describing the movement of living organisms n towards higher concentration regions of chemical attractants v (cf. [1]). On the other hand, if we consider χ > 0 and b3 = 0 in (1.1), we obtain a fractional Lotka-Volterra competition model describing the competition interspecies n and m in which one of the competing species avoids encounters with rivals by means of chemorepulsion mechanism caused by the chemical signal v. In addition, taking α = 1, θ = 2 and θ1 = 0 in (1.1) we formally obtain, as particular cases, the classical Keller-Segel system (cf. [9]) and the non-fractional Lotka-Volterra model (cf. [16]). The fractional population model (1.1) is justified by the nonlocal behavior of the dynamics of the organisms. In fact, in several situations found in nature, organisms develop alternative search strategies, particularly when chemoattractants, food, or other targets are sparse or rare. Then, as pointed out in [5, 10], a good description of the trajectories of the population of organisms can be performed by using the so called Lévy flights in place of Brownian motion. We recall that Lévy flights have been considered in numerous biological contexts, including immune cells, ecol- ogy and human populations (see [6] and references for a deeper discussion). This consideration motivates the substitution of the classical diffusion in system (1.1) by a fractional diffusion. On the other hand, regarding the flux by chemotaxis (both attractive and repulsive), it is also relevant to consider the case where the attraction-repulsion source is replaced by a less singular interaction kernel. This last consideration has been pointed out in the analysis of the propagation of chaos for some aggregation-diffusion models [15]. Finally, taking into account that the behavior of most biological systems has memory properties, which are neglected when an integer-order time derivative is assumed, it is justified to consider a time EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 3 variation in a fractional framework, which introduces a nonlocal delay in time for the moving population [7]. The aim of this article is to analyze the existence and uniqueness of global solutions for the space-time fractional system (1.1) in the framework of critical Besov-Morrey spaces. To show the existence of global solutions of system (1.1) we first prove some product estimates in Besov-Morrey spaces and derive global estimates for the solutions of the linear fractional heat equation, which are key to apply the iterative contraction method. To the best of our knowledge, the complete fractional model (1.1) has not been analyzed in the literature. Some global existence results and long time behavior of solutions for the particular case of (1.1) assuming θ = 2, θ1 = 0, χ < 0, a1 = 0, m = 0, and α ∈ (0, 1), with small initial data in a different class of Besov-Morrey initial data, were obtained in [1]. Recently, some global existence results of the fractional chemotaxis-Navier-Stokes system with consumption, in the framework of Morrey spaces were obtained in [13]. The plan of this paper is as follows. In Section 2, we give preliminaries and state our main results. In Section 3, we prove some necessary lemmas and estimates in order to handle the system in our setting, and finally, in Section 4, we prove our existence and uniqueness result. 2. Functional setting and main results In this section we recall some preliminary results related to Morrey and homo- geneous Besov-Morrey spaces. Furthermore, we establish some essential linear and nonlinear estimates in our framework, including the continuity of the paraproduct and the Bony decomposition. For more details of Morrey spaces and homogeneous Besov-Morrey spaces the reader can see [8, 11, 14, 17]. Throughout this article, we denote by S and S ′ the Schwartz class and the set of tempered distributions over RN , respectively. As usual, F and F−1 denote the Fourier transform and its in- verse, respectively. We also denote by D(Ω) the set of C∞-functions with compact support defined in Ω ⊆ RN . Definition 2.1. Let 1 ≤ p ≤ ∞ and 0 ≤ λ < N . The Morrey space Mp,λ(RN ) is defined by the set Mp,λ(RN ) = {f ∈ Lploc(R N ) : ‖f‖Mp,λ := sup x0∈RN sup R>0 R−λ/p‖f‖Lp(B(x0,R)) <∞}, where B(x0, R) denotes the open ball in RN with center x0 and radius R. Note that Mp,0 = Lp and M∞,λ = L∞. Also, Lp ⊂ L(p,∞) ⊂ Ml,λ for all 0 < λ < N and 1 ≤ l ≤ p ≤ ∞ with N p = N−λ l , where L(p,∞) denotes the weak-Lp space. In addition, Hölder and Young inequalities hold true in Morrey spaces. That is, if 1 p3 = 1 p1 + 1 p2 , λ3 p3 = λ1 p1 + λ2 p2 , and 0 ≤ λi < N , i = 1, 2, 3, then ‖fg‖Mp3,λ3 ≤ ‖f‖Mp1,λ1 ‖g‖Mp2,λ2 , and ‖f ∗ g‖Mp,λ ≤ ‖f‖L1‖g‖Mp,λ , for 0 ≤ λ < N and 1 ≤ p ≤ ∞. (2.1) Next, we recall the Bernstein inequality inMp,λ-spaces. Let C = C(0, R1, R2) = {x ∈ RN : R1 ≤ |x| ≤ R2} and B = B(0;R) = {x ∈ RN : |x| ≤ R}, for 0 < R1 < R2 and R > 0. From now on, we denote supp f the support of f . 4 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 Lemma 2.2 ([18, Lemma 2.1]). Let k be a nonnegative integer, 1 ≤ p ≤ ∞ and 0 ≤ λ < N . (i) If f ∈Mp,λ(RN ) and suppFf ⊂ τB(0;R) for some τ > 0 and R > 0, then sup |α|=k ‖∂αf‖Mp,λ ≤ Ck+1τk‖f‖Mp,λ , (2.2) for some positive constant C depending only on p, λ,R and N . (ii) If f ∈Mp,λ(RN ) and suppFf ⊂ τC for some τ > 0, then C−k−1τk‖f‖Mp,λ ≤ sup |α|=k ‖∂αf‖Mp,λ ≤ Ck+1τk‖f‖Mp,λ , (2.3) for some positive constant C depending only on p, λ,R1, R2 and N . Definition 2.3. Let C be the annulus {ξ ∈ RN : 3/4 ≤ |ξ| ≤ 8/3}. Consider the radial functions φ1 ∈ D(B(0, 4/3)) and φ2 ∈ D(C) valued in the interval [0, 1] and such that φ1(ξ) + ∑ j≥0 φ2(2−jξ) = 1,∀ξ ∈ RN , (2.4) ∑ j∈Z φ2(2−jξ) = 1,∀ξ ∈ RN \ {0} . (2.5) For each j ∈ Z, the homogeneous dyadic block ∆̇j and the homogeneous low- frequency cut-off operators Ṡj are defined as ∆̇jf(x) = F−1(φ2(2−jξ)F(f)(ξ)), Ṡjf(x) = F−1(φ1(2−jξ)F(f)(ξ)). We also denote ϕ̃j(ξ) = φ2(2−(j−1)ξ) + φ2(2−jξ) + φ2(2−(j+1)ξ) and C̃j = Cj−1 ∪ Cj ∪ Cj+1, where j ∈ Z and Cj = 2jC. Note that ϕ̃j = 1 in Cj . Definition 2.4. Let S ′h = {f ∈ S ′ : limj→−∞ Ṡjf = 0}. For s a real number, 1 ≤ p, q ≤ ∞ and 0 ≤ λ < N , the homogeneous Besov-Morrey space N s p,λ,q is the Banach space of all distributions f in S ′h such that ‖f‖N sp,λ,q := ‖2js‖∆̇jf‖Mp,λ ‖`q(Z) <∞. Definition 2.5. For s ∈ R, 1 ≤ p, q, r ≤ ∞ and 0 ≤ λ < N , the Banach space L̃r([0,∞);N s p,λ,q) is defined as the set of all tempered distributions f over RN × (0,∞) with limj→−∞ Ṡjf = 0 in Lr([0,∞);L∞(RN )) and such that ‖f‖ L̃r([0,∞);N sp,λ,q) := ‖2js‖∆̇jf‖Lr([0,∞);Mp,λ)‖`q <∞. (2.6) The following lemma corresponds to a Bernstein-type inequality in Morrey spaces. We omit its proof here because it is similar to the proof of Lemma 2.6 in [12]. In fact, it is sufficient to note that for f ∈ S ′ with supp f̂ ⊂ 2jC we have F(f) = ϕ̃j(ξ)F(f), this is, f = 2jn(ϕ̃0)̌(2j ·)∗f , and the function 2jn(ϕ̃0)̌(2j ·) and heat kernel g(x, 2−j) have a similar behavior. Lemma 2.6 (Bernstein-type inequality). If 1 ≤ p ≤ r ≤ ∞ and 0 ≤ λ < N , then ‖f‖Mr,λ ≤ C2j( N−λ p − N−λ r )‖f‖Mp,λ , for all f ∈Mp,λ such that supp f̂ ⊂ 2jC. EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 5 To establish the main existence result, we start by recalling the mild formulation of (1.1) in the fractional setting. System (1.1) is formally equivalent to the following integral formulation (from now on, without loss of generality we assume dn, dm, dv = 1): n(t) = Eα(−tα(−∆)θ/2)n0 + χ ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)∇ · (nG(v))(τ) dτ + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)(a1n 2 − b1nm)(τ) dτ, m(t) = Eα(−tα(−∆)θ/2)m0 + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)(a2m 2 − b2mn)(τ) dτ, v(t) =Eα(−tα((−∆)θ/2 − γ))v0 + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α((−∆)θ/2 − γ))(a3m+ b3n)(τ) dτ. (2.7) Here {Eα(·)}t≥0 and {Eα,α(·)}t≥0 denote the Mittag-Leffler families defined by Eα(−tα(−∆)θ/2) = ∫ ∞ 0 Mα(τ)Uθ(τt α) dτ, Eα(−tα((−∆)θ/2 + a)) = ∫ ∞ 0 Mα(τ)Uθ,a(τtα) dτ, Eα,α(−tα(−∆)θ/2) = ∫ ∞ 0 ατMα(τ)Uθ(τt α) dτ, Eα,α(−tα((−∆)θ/2 + a)) = ∫ ∞ 0 ατMα(τ)Uθ,a(τtα) dτ, where Uθ(t) and Uθ,a(t) are the fractional heat semigroup and the fractional damped heat semigroup, defined as Ûθ(t)f = e−t|ξ| θ f̂ and Uθ,a(t)f = eatUθ(t)f , respectively. The function Mα : C→ C is the Mainardi function defined by Mα(z) = ∞∑ n=0 zn n!Γ(1− α(1 + n)) , which verifies that Mα(τ) ≥ 0 for all τ ≥ 0 and∫ ∞ 0 τ rMα(τ) dτ = Γ(r + 1) Γ(1 + αr) , for −1 < r <∞. To analyze the initial value problem (2.7), and motivated by the intrinsic scaling of (1.1), we impose the condition θ1 = 2 − θ and consider the following time- dependent functional spaces. For 1 ≤ p < N − λ, 1 α < κ and s∗ = N−λ p , we define the Banach spaces X1 and X2 by X1 = L̃∞([0,∞);N s∗−θ p,λ,∞) ∩ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ), X2 = L̃∞([0,∞);N s∗ p,λ,∞), (2.8) 6 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 endowed with the corresponding norms ‖x‖X1 = ‖x‖ L̃∞([0,∞);N s∗−θp,λ,∞) + ‖x‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) , ‖x‖X2 = ‖x‖ L̃∞([0,∞);N s∗p,λ,∞) . (2.9) With this notation, we establish the notion of solution that we use. Definition 2.7. Let 1 ≤ p < N − λ, s∗ = N−λ p , and [n0,m0, v0] ∈ N s∗−θ p,λ,∞ × N s∗−θ p,λ,∞ × N s∗ p,λ,∞. A mild solution for (1.1) is a triple [n,m, v] ∈ X1 × X1 × X2, satisfying the integral system (2.7). Next, we state our main existence result. Theorem 2.8. Let 0 ≤ λ < N , 1 α < κ ≤ ∞, 1 < θ, 1 ≤ p < N−λ θ(2− 1 ακ ) , s∗ = N−λ p , [n0,m0, v0] ∈ N s∗−θ p,λ,∞ ×N s∗−θ p,λ,∞ ×N s∗ p,λ,∞. Then, there exists δ > 0 such that if ‖n0‖N s∗−θp,λ,∞ + ‖m0‖N s∗−θp,λ,∞ + ‖v0‖N s∗p,λ,∞ < δ, then problem (1.1) has a unique mild solution [n,m, v] in the class( C(R+;N s∗−θ p,λ,∞) ∩ X1 ) × ( C(R+;N s∗−θ p,λ,∞) ∩ X1 ) × C(R+;N s∗ p,λ,∞). Remark 2.9. (1) If in (1.1) we consider χ < 0, a1 = 0 and m = 0, we obtain the fractional Keller-Segel system. Thus, Theorem 2.8 provides the existence of global solution for the fractional Keller-Segel system in the class ( C(R+;N s∗−θ p,λ,∞)∩X1 ) × C(R+;N s∗ p,λ,∞) for small initial data [n0, v0] in N s∗−θ p,λ,∞ ×N s∗ p,λ,∞. (2) If in (1.1) we consider χ > 0 and b3 = 0, we obtain a fractional Lotka-Volterra competition model describing the competition interspecies n and m under a regime of chemorepulsion mechanism caused by the chemical signal v. Thus, Theorem 2.8 provides the existence of global solution for the fractional Lotka-Volterra system (under small initial data) in the class ( C(R+;N s∗−θ p,λ,∞) ∩ X1 ) × ( C(R+;N s∗−θ p,λ,∞) ∩ X1 ) × C(R+;N s∗ p,λ,∞). 3. Linear and nonlinear estimates The aim of this section is to derive estimates to be used in proving the existence of mild solutions. First we establish estimates for fractional heat semigroups, in Morrey spaces, acting on distributions whose Fourier transforms have support in an annulus. Lemma 3.1. There exist positive constants C1 and C2 such that for any 1 ≤ p ≤ ∞, 0 ≤ λ < N , a ∈ R, 0 < θ, and any positive real numbers t, τ , if suppFf ⊂ τC, then ‖Eα(−tα((−∆)θ/2 + a))f‖Mp,λ ≤ C2 ∫ ∞ 0 Mα(s)east α e−C1st ατθds‖f‖Mp,λ , (3.1) and if suppFf ⊂ τC , then ‖Eα,α(−tα((−∆)θ/2 + a))f‖Mp,λ ≤ C2 ∫ ∞ 0 sMα(s)east α e−C1st ατθds‖f‖Mp,λ . (3.2) EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 7 Proof. The proof follows the ideas in [2, Lemma 2.4] but adjusted to the context of Morrey spaces. Let φ ∈ D(RN \ {0}) be a function such that φ = 1 near the annulus C. Then, Eα(−tα((−∆)θ/2 + a))f = ∫ ∞ 0 Mα(s)east α F−1 ( φ(ξ/τ)e−st α|ξ|θF(f)(ξ) ) ds = ∫ ∞ 0 Mα(s)east α ( g(·, s 1 α t) ∗ f ) ds, (3.3) where g(x, t) = F−1(φ(ξ/τ)e−t α|ξ|θ ). The hypotheses on φ imply that ‖g(·, t)‖L1 ≤ C2e −C1t ατθ , ∀t > 0. (3.4) Thus, by applying the Young inequality (2.1) in (3.3), and using (3.4), we obtain ‖Eα(−tα((−∆)θ/2 + a))f‖Mp,λ ≤ C2 ∫ ∞ 0 Mα(s)east α e−C1st ατθds‖f‖Mp,λ , which proves (3.1). A similar argument proves (3.2). � The next lemma provides some inclusions involving Besov-Morrey spaces (see [11, 14]). Lemma 3.2. Suppose that 1 ≤ r ≤ ∞, s, s1, s2 ∈ R, 1 ≤ p, p1, p2 ≤ ∞, and 0 ≤ λ < N . (i) If 1 ≤ q1 ≤ q2 ≤ ∞, then L̃r([0,∞);N s p,λ,q1) ⊂ L̃r([0,∞);N s p,λ,q2). (ii) If p1 ≤ p2 and s2 − N−λ p2 = s1 − N−λ p1 , then L̃r([0,∞);N s1 p1,λ,q ) ⊂ L̃r([0,∞);N s2 p2,λ,q ). Proof. Part (i) follows from the definition of the norm (2.6) and the fact that lq1 ⊂ lq2 if q1 ≤ q2. To prove part (ii), from Lemma 2.6 we have ‖∆̇jf‖Mp2,λ ≤ C2 j ( N−λ p1 −N−λp2 ) ‖∆̇jf‖Mp1,λ . Taking the Lr-norm, multiplying by 2js2 and taking the lq-norm we obtain the result. � Next we recall estimates for multiplier operators with polynomial growth in the context Besov-Morrey spaces (see [11, 14]). Lemma 3.3. Let m ∈ R, 1 ≤ p <∞, 0 ≤ λ < N , and P (ξ) ∈ C [N/2]+1(RN \ {0}). Suppose that there exists A > 0 such that |∂ kP ∂ξk (ξ)| ≤ A|ξ|m−|k|, for all k ∈ (N∪{0})N with |k| ≤ [N/2] + 1 and |ξ| 6= 0. Then, for every g such that suppF(g) ⊂ 2jC we have that ‖F−1(P (ξ)F(g))‖Mp,λ ≤ CA2mj‖g‖Mp,λ . The following lemma is a consequence of the one above. 8 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 Lemma 3.4. Let m, s ∈ R, 1 ≤ p < ∞, 0 ≤ λ < N , 1 ≤ q, r ≤ ∞ and P (ξ) ∈ C [N/2]+1(RN \ {0}). Suppose that there exists A > 0 such that |∂ kP ∂ξk (ξ)| ≤ A|ξ|m−|k|, for all k ∈ (N ∪ {0})N with |k| ≤ [N/2] + 1 and |ξ| 6= 0. Then the multiplier operator P (D) defined by P (D)g = F−1(P (ξ)F(g)(ξ)) is bounded from N s p,λ,q to N s−m p,λ,q . Moreover, ‖P (D)g‖ L̃r([0,∞);N s−mp,λ,q) ≤ CA‖g‖ L̃r([0,∞);N sp,λ,q) . Lemma 3.5. If f be a distribution in S ′h, s < 0, 1 ≤ p, q ≤ ∞, 0 ≤ λ < N and 1 ≤ r ≤ ∞, then f belongs to the Besov-Morrey space L̃r([0,∞);N s p,λ,q) if and only if (2js‖Ṡjf‖Lr([0,∞);Mp,λ))j∈Z ∈ `q. Moreover, C1‖f‖L̃r([0,∞);N sp,λ,q) ≤ ‖(2js‖Ṡjf‖Lr([0,∞);Mp,λ))j‖`r ≤ C2‖f‖L̃r([0,∞);N sp,λ,q) , where the positive constants C1 and C2 depend only on N, s. Proof. This proof is inspired in the proof of [2, Proposition 2.33] adapted to the context of L̃r([0,∞);N s p,λ,q) spaces. After taking the norm Lr([0,∞);Mp,λ) in the localization operator, we obtain 2js‖∆̇jf‖Lr([0,∞);Mp,λ) ≤ 2js ( ‖Ṡjf‖Lr([0,∞);Mp,λ) + ‖Ṡj+1f‖Lr([0,∞);Mp,λ) ) = 2js‖Ṡjf‖Lr([0,∞);Mp,λ) + 2−s2(j+1)s‖Ṡj+1f‖Lr([0,∞);Mp,λ), which implies the left-hand side inequality. On the other hand, we have 2js‖Ṡjf‖Lr([0,∞);Mp,λ) ≤ 2js ∑ j′≤j−1 ‖∆̇j′f‖Lr([0,∞);Mp,λ) = ∑ j′≤j−1 2(j−j′)s2j ′s‖∆̇j′f‖Lr([0,∞);Mp,λ), and thus, using that s < 0 and applying the Young inequality in the framework of lr-spaces, we conclude the result. � The lemma below contains a criterion for the limit of a series to belong to a class of homogeneous Besov-Morrey space. Its proof follows the ideas in [2, Lemma 2.23]. Lemma 3.6. Let C′ be an annulus, s ∈ R, 0 ≤ λ < N , and 1 ≤ p, q, r ≤ ∞ such that s < N/p, or s = N/p with q = 1. Let (fj)j∈Z be a sequence of smooth functions such that suppFfj ⊂ 2jC′ and ‖(2js‖fj‖Lr([0,∞);Mp,λ))j∈Z‖`q <∞. Then ∑ j∈Z fj converges in S ′ to some f in L̃r([0,∞);N s p,λ,q); moreover, there exists a constant C = C(s) > 0 such that ‖f‖ L̃r([0,∞);N sp,λ,q) ≤ C‖(2js‖fj‖Lr([0,∞);Mp,λ))j∈Z‖`q . EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 9 Proof. Considering the annulus C in Definition 2.3, there exists k0 ∈ N such that 2j ′C ∩ 2jC′ = ∅ provided that |j′ − j| ≥ k0. Therefore, ∆̇j′fj = 0 for |j′ − j| ≥ k0, and ‖∆̇j′f‖Lr([0,∞);Mp,λ) = ∥∥ ∑ |j′−j| 0, 0 ≤ λ < N , and 1 ≤ p, q, r ≤ ∞ be such that s < N/p, or s = N/p with q = 1. If (fj)j∈Z be a sequence of smooth functions such that suppFfj ⊂ 2jB and ‖(2js‖fj‖Lr([0,∞);Mp,λ))j∈Z‖`q <∞, then ∑ j∈Z fj converges in S ′ to some f in L̃r([0,∞);N s p,λ,q). Moreover, there exists a constant C = C(s) > 0 such that ‖f‖ L̃r([0,∞);N sp,λ,q) ≤ C‖(2js‖fj‖Lr([0,∞);Mp,λ))j∈Z‖`q . The paraproduct of v by u denoted by Tuv is the bilinear operator Tuv := ∑ j∈Z Ṡj−1u∆̇jv. The remainder of u and v, denoted by R(u, v), is the bilinear operator R(u, v) := ∑ |i−j|≤1 ∆̇iu∆̇jv. Formally, using the operators Tuv and R(u, v), we can express the product uv by means of the Bony decomposition uv = Tuv + Tvu+R(u, v). In the next two lemmas we provide continuity properties for the paraproduct and remainder term on Besov-Morrey spaces, which can be seen as extensions from the corresponding ones in Besov spaces found in [2, Theorems 2.47 and 2.52]. Lemma 3.8. Let s, σ ∈ R with σ > 0, 0 ≤ λ < N , and let 1 ≤ p, q, q1, q2, r, r1, r2 ≤ ∞ with 1 q = 1 q1 + 1 q2 and 1 r = 1 r1 + 1 r2 . (i) If s < N p , or s = N p with q = 1, then, there exists a positive constant C > 0 such that ‖Tuv‖L̃r([0,∞);N sp,λ,q) ≤ C‖u‖ L̃r1 ([0,∞);L∞) ‖v‖ L̃r2 ([0,∞);N sp,λ,q) . 10 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 (ii) If s−σ < N p , or s−σ = N p with q = 1, then, there exists a positive constant C > 0 such that ‖Tuv‖L̃r([0,∞);N s−σp,λ,q) ≤ C‖u‖ L̃r1 ([0,∞);N−σ∞,λ,q1 ) ‖v‖ L̃r2 ([0,∞);N sp,λ,q2 ) . Proof. From Definition 2.3, the support of F(Ṡj−1u∆̇jv) is contained in 2j(C+B). In addition, there is an integer k0 such that ∆̇j′(Ṡj−1u∆̇jv) = 0 provided that j′ > j + k0. Thus, it is sufficient to estimate ‖Ṡj−1u∆̇jv‖Lr([0,∞);Mp,λ). From the definition of the operators Ṡj , it holds ‖Ṡj−1u‖L∞ ≤ C‖u‖L∞ . (3.5) Thus, using Lemma 3.6 and the Hölder inequality in Lr, we conclude that Tuv ∈ L̃r([0,∞);N s p,λ,q), and the inequality in the part (i) of the lemma. On the other hand, from Lemma 3.5 it follows that ‖Ṡj−1u‖Lr1 ([0,∞);L∞) ≤ C −z cjq12−jz‖u‖Lr1 ([0,∞);N z∞,λ,q1 ), (3.6) for all j ∈ Z and z < 0, where (cj,q1)j∈Z has norm 1 in `q1(Z). Again, Lemma 3.6 and the Hölder inequality in Lr-spaces imply that Tuv ∈ L̃r([0,∞);N s−σ p,λ,q); moreover, by using (3.6) we obtain the inequality in the part (ii) of the lemma. � Lemma 3.9 ([4]). Let 0 ≤ λ, λ1, λ2 < N , s1, s2 ∈ R and p, p1, p2, q, q1, q2, r, r1, r2 ∈ [1,∞] satisfying s1+s2 > 0, 1 p = 1 p1 + 1 p2 , λp = λ1 p1 + λ2 p2 , 1 q = 1 q1 + 1 q2 , and 1 r = 1 r1 + 1 r2 . If s1 + s2 < N p , or s1 + s2 = N p with q = 1, then there exists a constant C > 0 such that ‖R(u, v)‖ L̃r([0,∞);N s1+s2 p,λ,q ) ≤ C‖u‖ L̃r1 ([0,∞);N s1p1,λ1,q1 ) ‖v‖ L̃r2 ([0,∞);N s2p2,λ2,q2 ) . Proof. From Definition 2.3, we have the support of F (∑ |ν|≤1∆̇j−νu∆̇jv ) is con- tained in 2jB. Moreover, there is an integerN0 such that ∆̇j′ (∑ |ν|≤1∆̇j−νu∆̇jv ) = 0 for all j′ > j +N0. Applying Hölder’s inequality we have 2j ′(s1+s2)‖∆̇jR(u, v)‖Lr([0,∞);Mp,λ) ≤ C2j ′(s1+s2) ∑ |ν|≤1, j≥j′−N0 ‖∆̇j−νu∆̇jv‖Lr([0,∞);Mp,λ) ≤ C2j ′(s1+s2) ∑ |ν|≤1, j≥j′−N0 ( ‖∆̇j−νu‖Lr1 ([0,∞);Mp1,λ1 ) × ‖∆̇jv‖Lr2 ([0,∞);Mp2,λ2 ) ) ≤ C ∑ |ν|≤1, j≥j′−N0 ( 2−(j−j′)(s1+s2)2(j−ν)s1 × ‖∆̇j−νu‖Lr1 ([0,∞);Mp1,λ1 )2 js2‖∆̇jv‖Lr2 ([0,∞);Mp2,λ2 ) ) . Note that s1 + s2 > 0. From Lemma 3.7, R(u, v) ∈ L̃r([0,∞);N s1+s2 p,λ,q ). Applying the discrete Hölder and Young inequalities, we conclude the proof of the lemma. � EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 11 4. Existence results First we derive global estimates for mild solutions of the fractional heat equa- tion and then obtain a solution to (2.7) using an iterative scheme. Let us rewrite equations (2.7) in the form n(x, t) = Eα(−tα(−∆)θ/2)n0 + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)(f1 + f2 + f3)(τ) dτ, m(x, t) = Eα(−tα(−∆)θ/2)m0 + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)(f4 + f5)(τ) dτ, v(x, t) = Eα(−tα((−∆)θ/2 − γ))v0 + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α((−∆)θ/2 − γ))(f6 + f7)(τ) dτ, (4.1) where f1 = χ∇ · (nG(v)), f2 = a1n 2, f3 = −b1nm, f4 = a2m 2, f5 = −b2mn, f6 = a3m, f7 = b3n. (4.2) Proposition 4.1. Let s ∈ R, 0 < θ,1 ≤ p, q ≤ ∞, 1 ≤ κ ≤ r ≤ ∞, 1 α < r ≤ ∞, a ≥ 0, and 0 ≤ λ < N . If v0 ∈ N s p,λ,q and f(x, t) ∈ L̃κ([0,∞);N s−θ(1− 1 ακ ) p,λ,q ), then v(x, t) = Eα(−tα((−∆)θ/2 − a))v0 + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α((−∆)θ/2 − a))f(x, τ) dτ satisfies ‖v(x, t)‖ L̃r([0,∞);N s+ θ αr p,λ,q ) ≤ C ( ‖v0‖N sp,λ,q + ‖f‖ L̃κ([0,∞);N s−θ(1− 1 ακ ) p,λ,q ) ) , for some C > 0. Proof. Applying ∆̇j to the fractional heat equation cDα t v + (−∆)θ/2v = −av + f with initial data v(0) = v0, we obtain cDα t ∆̇jv + (−∆)θ/2∆̇jv = −a∆̇jv + ∆̇jf, ∆̇jv|t=0 = ∆̇jv0. (4.3) The solution to this equation is given by ∆̇jv = Eα(−tα((−∆)θ/2 − a))∆̇jv0 + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α((−∆)θ/2 − a))∆̇jf dτ, t > 0. (4.4) 12 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 Since the Fourier transform of ∆̇jv0 and ∆̇jf are supported in the annulus 2jC, using Lemma 3.1 and the Young inequality in Lr, we have ‖Eα(−tα(−∆)θ/2 − a)∆̇jv0‖Lr [0,∞) (Mp,λ) ≤ C2‖ ∫ ∞ 0 Mα(s)e−ast α e−C1st α2θj‖∆̇jv0‖Mp,λ ds‖Lr(0,∞) ≤ C2 ∫ ∞ 0 Mα(s)‖e−C1st α2θj‖∆̇jv0‖Mp,λ ‖Lr(0,∞)ds ≤ C2 ∫ ∞ 0 Mα(s) (∫ ∞ 0 e−C1st α2θj )1/r ‖∆̇jv0‖Mp,λ ds ≤ C22− θj αr ∫ ∞ 0 s− 1 αrMα(s)ds‖∆̇jv0‖Mp,λ ≤ C2− θj αr ‖∆̇jv0‖Mp,λ , (4.5) and∥∥∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α((−∆)θ/2 − a))∆̇jf dτ ∥∥ Lr [0,∞) (Mp,λ) ≤ C2 ∫ ∞ 0 sMα(s)‖ ∫ t 0 (t− τ)α−1e−as(t−τ)αe−C1s(t−τ)α2θj‖∆̇jf‖Mp,λ dτ‖Lr(0,∞)ds ≤ C2 ∫ ∞ 0 sMα(s) (∫ ∞ 0 e−C1st α2θjκ∗t(α−1)κ∗ )1/κ∗ ‖∆̇jf‖Lκ [0,∞) (Mp,λ)ds ≤ C ∫ ∞ 0 sMα(s) ( s2θj )− 1 ακ∗− (α−1) α ds‖∆̇jf‖Lκ [0,∞) (Mp,λ) ≤ C ∫ ∞ 0 sMα(s)s− 1 αr−1+ 1 ακ 2− θj αr−θj(1− 1 ακ )ds‖∆̇jf‖Lκ [0,∞) (Mp,λ) ≤ C2− θj αr−θj(1− 1 ακ ) ∫ ∞ 0 s− 1 αr+ 1 ακMα(s)ds‖∆̇jf‖Lκ [0,∞) (Mp,λ) ≤ C2− θj αr−θj(1− 1 ακ )‖∆̇jf‖Lκ [0,∞) (Mp,λ), (4.6) where κ∗ is such that 1 + 1/r = 1/κ∗ + 1/κ. Thus, from (4.5)-(4.6) we obtain ‖∆̇jv‖Lr [0,∞) (Mp,λ) ≤ C ( 2−j θ αr ‖∆̇jv0‖Mp,λ + 2−j θ αr 2−jθ(1− 1 ακ )‖∆̇jf‖Lκ [0,∞) (Mp,λ) ) . (4.7) Therefore, multiplying both sides of (4.7) by 2j θ αr 2js and taking the `q-summation, we obtain ‖v‖ L̃r([0,∞);N s+ θ αr p,λ,q ) ≤ C ( ‖v0‖N sp,λ,q + ‖f‖ L̃κ([0,∞);N s−θ(1− 1 ακ ) p,λ,q ) ) , where C > 0 is a constant. The proof is complete. � Lemma 4.2. If 0 ≤ λ < N , 1 ≤ p < ∞, 1 α < κ ≤ ∞, 1 < θ, p < N−λ θ(2− 1 ακ ) , and fj = 1, . . . , 7 are defined as in (4.2), then the following estimates hold (i) ‖f1‖ L̃κ([0,∞);N s∗−θ(2− 1 ακ ) p,λ,∞ ) ≤ C‖n‖X1 ‖v‖X2 , (ii) ‖f2‖ L̃κ([0,∞);N s∗−θ(2− 1 ακ ) p,λ,∞ ) ≤ C‖n‖2X1 , EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 13 (iii) ‖f3 + f5‖ L̃κ([0,∞);N s∗−θ(2− 1 ακ ) p,λ,∞ ) ≤ C‖n‖X1 ‖m‖X1 , (iv) ‖f4‖ L̃κ([0,∞);N s∗−θ(2− 1 ακ ) p,λ,∞ ) ≤ C‖m‖2X1 , (v) ‖f6 + f7‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ≤ C(‖m‖X1 + ‖n‖X1 ). Proof. We start by analyzing (i). From Lemma 3.4 we obtain ‖∂j([G(v)]jn)‖ L̃κ([0,∞);N s∗−θ(2− 1 ακ ) p,λ,∞ ) ≤ C‖n[G(v)]j‖ L̃κ([0,∞);N s∗+1−θ(2− 1 ακ ) p,λ,∞ ) , j = 1, 2, 3. Now, from the Bony decomposition we have [G(v)]jn = T[G(v)]jn+ Tn[G(v)]j +R([G(v)]j , n), j = 1, 2, 3. Taking into account that s∗ = N−λ p , using Lemmas 3.8, 3.2 and 3.4, we estimate ‖Tn[G(v)]j‖ L̃κ([0,∞);N s∗−θ+1−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖n‖ L̃κ([0,∞);N −θ(1− 1 ακ ) ∞,λ,∞ ) ‖[G(v)]j‖L̃∞([0,∞);N s∗−θ+1 p,λ,∞ ) ≤ C‖n‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ‖v‖ L̃∞([0,∞);N s∗p,λ,∞) ≤ C‖n‖X1‖v‖X2 , (by the condition κ > 1 α ) and ‖T[G(v)]jn‖ L̃κ([0,∞);N s∗−θ+1−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖[G(v)]j‖L̃∞([0,∞);N−(θ−1) ∞,λ,∞ ) ‖n‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖[G(v)]j‖L̃∞([0,∞);N s ∗−(θ−1) p,λ,∞ ) ‖n‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖v‖ L̃∞([0,∞);N s∗p,λ,∞) ‖n‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖n‖X1 ‖v‖X2 , (by the condition θ > 1). Moreover, from Lemmas 3.9 and 3.2 we obtain ‖R([G(v)]j , n)‖ L̃κ([0,∞);N s∗−θ+1−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖n‖ L̃κ([0,∞);N −θ(1− 1 ακ ) ∞,λ,∞ ) ‖[G(v)]j‖L̃∞([0,∞);N s ∗−θ+1) p,λ,∞ ) ≤ C‖n‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ‖v‖ L̃∞([0,∞);N s∗p,λ,∞) ≤ C‖n‖X1‖v‖X2 . Therefore, ‖f1‖ L̃κ([0,∞);N s∗−θ(2− 1 ακ ) p,λ,∞ ) ≤ C‖n‖X1‖v‖X2 . (4.8) Now we obtain the estimate for f2. From Lemmas 3.8 and 3.2 we have ‖Tnn‖ L̃κ([0,∞);N s∗−θ−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖n‖ L̃κ([0,∞);N −θ(1− 1 ακ ) ∞,λ,∞ ) ‖n‖ L̃∞([0,∞);N s∗−θp,λ,∞) ≤ C‖n‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ‖n‖ L̃∞([0,∞);N s∗−θp,λ,∞) ≤ C‖n‖X1 ‖n‖X1 . 14 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 On the other hand, from Lemmas 3.9 and 3.2, and using the condition s∗ − θ − θ(1− 1 ακ ) > 0, we obtain ‖R(n, n)‖ L̃κ([0,∞);N s∗−θ−θ(1− 1 ακ ) p,λ,∞ ) ≤ C‖n‖ L̃κ([0,∞);N −θ(1− 1 ακ ) ∞,λ,∞ ) ‖n‖ L̃∞([0,∞);N s∗−θp,λ,∞) ≤ C‖n‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ‖n‖ L̃∞([0,∞);N s∗−θp,λ,∞) ≤ C‖n‖X1 ‖n‖X1 . Here we have used that p < N−λ θ(2− 1 ακ ) to guarantee that s∗ − θ − θ(1 − 1 ακ ) > 0. Note that the conditions for κ and θ imply that 1 − θ(2 − 1 ακ ) < 0 and therefore s∗ − θ + 1− θ(1− 1 ακ ) < N p . In conclusion, there exists a positive constant C such that ‖f2‖ L̃κ([0,∞);N s∗−θ(2− 1 ακ ) p,λ,∞ ) ≤ C‖n‖X1‖n‖X1 . (4.9) The proof of (iii) and (iv) for f3, f4, and f5 follows similarly. Finally, for f6 and f7, we have ‖f6 + f7‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) ≤ C(‖n‖X1 + ‖m‖X1 ). (4.10) � 4.1. Proof of Theorem 2.8. Motivated by [3], we consider the following iterative scheme whose limit will give the global mild solution for the system (1.1) in the sense of Definition 2.7: n(1) = Eα(−tα(−∆)θ/2)n0, m(1) = Eα(−tα(−∆)θ/2)m0, v(1) = Eα(−tα((−∆)θ/2 − γ))v0, n(k+1) = n(1) + χ ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)∇ · (n(k)G(v(k)))(τ) dτ + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)(a1n (k)n(k) − b1n(k)m(k))(τ) dτ, m(k+1) = m(1) + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2) × (a2m (k)m(k) − b1m(k)n(k))(τ) dτ, v(k+1) = v(1) + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α((−∆)θ/2 − γ)) × (a3m (k+1) + b3n (k+1))(τ) dτ. Applying Proposition 4.1 and Lemma 4.2 to the above equality, we obtain the following estimates: ‖n(k+1)‖X1 ≤ C ( ‖n0‖N s∗−θp,λ,∞ + ‖n(k)‖X1 ‖v(k)‖X2 + ‖n(k)‖2X1 + ‖n(k)‖X1 ‖m(k)‖X1 ) , ‖m(k+1)‖X1 ≤ C ( ‖m0‖N s∗−θp,λ,∞ + ‖m(k)‖2X1 + ‖n(k)‖X1 ‖m(k)‖X1 ) , ‖v(k+1)‖X2 ≤ C ( ‖v0‖N s∗p,λ,∞ + ‖n(k+1)‖X1 + ‖m(k+1)‖X1 ) . (4.11) EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 15 For small initial data, the sequence [n(k),m(k), v(k)] is uniformly bounded in the space X := X1 ×X1 ×X2 with the norm ‖[n(k),m(k), v(k)]‖X = ‖n(k)‖X1 + ‖m(k)‖X1 + ‖v(k)‖X2 . In fact, from (4.11) we obtain ‖[n(k+1),m(k+1), v(k+1)]‖X = ‖n(k+1)‖X1 + ‖m(k+1)‖X1 + ‖v(k+1)‖X2 ≤ C ( A0 + 4‖[n(k),m(k), v(k)]‖2X ) , (4.12) where A0 = ‖n0‖N s∗−θp,λ,∞ + ‖m0‖N s∗−θp,λ,∞ + ‖v0‖N s∗p,λ,∞ and C is a positive constant. Let δ > 0 be small enough such that if ‖n0‖N s∗−θp,λ,∞ + ‖m0‖N s∗−θp,λ,∞ + ‖v0‖N s∗p,λ,∞ < δ, then 1 − 16A0C 2 > 0, and consider the smallest root R of 4CR2 − R + CA0 = 0; that is, R = 1− √ 1− 16A0C2 8C . (4.13) Thus, if ‖[n(k),m(k), v(k)]‖X ≤ R, (4.14) then ‖[n(k+1),m(k+1), v(k+1)]‖X ≤ R, (4.15) which implies that [n(k),m(k), v(k)], k ∈ N, is uniformly bounded in X . Next, we bound the difference vector[ n(k+1) − n(k),m(k+1) −m(k), v(k+1) − v(k) ] . Noting that n(k+1) − n(k) = χ ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)∇ · (n(k)G(v(k)))(τ) dτ − ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)∇ · (n(k−1)G(v(k−1)))(τ) dτ + ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2)(a1n (k)n(k) − b1n(k)m(k))(τ) dτ − ∫ t 0 (t− τ)α−1Eα,α(−(t− τ)α(−∆)θ/2) × ( a1n (k−1)n(k−1) − b1n(k−1)m(k−1) ) (τ) dτ, (4.16) by using Proposition 4.1 and Lemma 4.2, we obtain ‖n(k+1) − n(k)‖X1 ≤ C ( ‖n(k−1)‖X1‖v(k−1) − v(k)‖X2 + ‖n(k−1) − n(k)‖X1‖v(k)‖X2 + ‖n(k−1)‖X1‖n(k−1) − n(k)‖X1 + ‖n(k−1) − n(k)‖X1‖n(k)‖X1 + ‖m(k−1)‖X1 ‖n(k−1) − n(k)‖X1 + ‖m(k−1) −m(k)‖X1 ‖n(k)‖X1 ) ≤ CR‖[n(k−1) − n(k),m(k−1) −m(k), v(k−1) − v(k)]‖X . (4.17) Similarly we have ‖m(k+1) −m(k)‖X1 ≤ CR‖[n(k−1) − n(k),m(k−1) −m(k), v(k−1) − v(k)]‖X . (4.18) 16 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 On the other hand, ‖v(k+1) − v(k)‖X2 ≤ C ( ‖n(k+1) − n(k)‖X1 + ‖m(k+1) −m(k)‖X1 ) . (4.19) From estimates (4.17) and (4.19) we obtain ‖[n(k+1) − n(k),m(k+1) −m(k), v(k+1) − v(k)]‖X ≤ 4CR‖[n(k−1) − n(k),m(k−1) −m(k), v(k−1) − v(k)]‖X . (4.20) Thus, choosing R as in (4.13) such that R < 1 4C (reducing δ, if necessary), we see that [n(k),m(k), v(k)] is a Cauchy sequence in X . The limit [n,m, v] is a mild solution in X for system (4.1). The proof of uniqueness follows by using arguments similar to those in the proof of inequality (4.20), and therefore we omit the details. This proves Theorem 2.8. Acknowledgments. The authors were supported by Vicerrectoŕıa de Investigación y Extensión of Universidad Industrial de Santander, Colombia, project “Análisis teórico y numérico de sistemas diferenciales que modelan la dinámica de células tu- morales en el cerebro”, code 3704. The authors would like to thank the anonymous referees for their useful comments and suggestions. References [1] J. Azevedo, C. Cuevas, E. 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Ferreira; On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier– Stokes equations. 615 J. Math. Pures Appl. 105(2), (2016) 228–247. [13] M. A. Fontecha, E. J. Villamizar-Roa; Global existence and asymptotic behavior of solutions for a fractional chemotaxis-Navier-Stokes system. Dyn. Partial Differ. Equ. 19 (2022), no. 4, 285–309. [14] A. L. Mazzucato; Besov-Morrey spaces: function space theory and applications to non-linear pde. Trans. Amer. Math. Soc., 355 (2003), no. 4, 1297-1364. EJDE-2023/77 NONLINEAR POPULATION MODEL WITH CROSS-DIFFUSION 17 [15] S. Salem; Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases, J. Math. Pures Appl.132 (2019) 79–132. [16] J. I. Tello, D. Wrzosek; Inter-species competition and chemorepulsion, J. Math. Anal. Appl. 459 (2018), no. 2, 1233-1250. [17] M. E. Taylor; Analysis on Morrey spaces and applications to Navier-Stokes and other evolu- tion equations. Communications in Partial Differential Equations, 17 (1992), no. 9-10, 1407- 1456. [18] J. Xu, Y. Tan; The well-posedness of the surface quasi-geostrophic equations in the Besov- Morrey spaces. Nonlinear Analysis: Theory, Methods and Applications, 92 (2013), 60-71. Jhean E. Pérez-López Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia Email address: jelepere@uis.edu.co Diego A. Rueda-Gómez Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia Email address: diaruego@uis.edu.co Élder J. Villamizar-Roa Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia Email address: jvillami@uis.edu.co 1. Introduction 2. Functional setting and main results 3. Linear and nonlinear estimates 4. Existence results 4.1. Proof of Theorem ?? Acknowledgments References