Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 56, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.56 EXISTENCE OF GLOBAL WEAK SOLUTION TO TUMOR CHEMOTAXIS COMPETITION SYSTEMS WITH LOOP AND SIGNAL DEPENDENT SENSITIVITY SHANMUGASUNDARAM GNANASEKARAN, NAGARAJAN NITHYADEVI Abstract. This article examines the weak solution of a fully parabolic chemotaxis- competition system with loop and signal-dependent sensitivity. The system is subject to homogeneous Neumann boundary conditions within an open, bounded domain Ω ⊂ Rn, where n ≥ 1 and ∂Ω is smooth. We assume that the parameters in the system are positive constants. Additionally, the initial data (u10, u20, v10, v20) ∈ L2(Ω) × L2(Ω) × W 1,2(Ω) × W 1,2(Ω) are non-negative. The existence of a weak solution to the problem is established using energy inequality method. 1. Introduction This article shows the existence of weak solutions for a chemotaxis-competition system that features loop and signal dependent sensitivity. The system under con- sideration models the chemotactical communication within the tumor site, specifi- cally the EGF/CSF-1 paracrine invasion loop. u1t = d1∆u1 −∇ · (χ1(v1)u1∇v1)−∇ · (χ2(v2)u1∇v2) + δ1u1(1− u1 − a1u2), x ∈ Ω, t > 0, u2t = d2∆u2 −∇ · (ξ1(v1)u2∇v1)−∇ · (ξ2(v2)u2∇v2) + δ2u2(1− a2u1 − u2), x ∈ Ω, t > 0, v1t = d3∆v1 + α1u1 + β1u2 − γ1v1, x ∈ Ω, t > 0, v2t = d4∆v2 + α2u1 + β2u2 − γ2v2, x ∈ Ω, t > 0, ∂u1 ∂ν = ∂u2 ∂ν = ∂v1 ∂ν = ∂v2 ∂ν = 0, x ∈ ∂Ω, t > 0, u1(x, 0) = u10, u2(x, 0) = u20, v1(x, 0) = v10, v2(x, 0) = v20, x ∈ Ω, (1.1) where Ω ⊂ Rn, n ≥ 1 is an open bounded domain with smooth boundary ∂Ω and ∂ ∂ν indicate differentiation with respect to the outward normal on ∂Ω. The quantities u1(x, t) and u2(x, t) represent the densities of macrophages and tumor cells, respectively, while v1(x, t) and v2(x, t) represent the concentration of chemical signals secreted by u1 and u2, respectively. We assume the all parameters in the 2020 Mathematics Subject Classification. 35A01, 35D30, 92C17, 35Q92. Key words and phrases. Chemotaxis system; two species and two stimuli; weak solution; Lotka-Volterra competition. ©2024. This work is licensed under a CC BY 4.0 license. Submitted April 29, 2024. Published September 27, 2024. 1 2 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 equation are positive. Here, di (for i = 1, 2, 3, 4) denotes the diffusion coefficients. Meanwhile, δ1 and δ2 represent the growth rates of macrophages and tumor cells, respectively. The coefficient a1 describes the interaction among macrophages, while a2 describes the interaction among tumor cells. The parameters αi, βi, and γi (for i = 1, 2) represent the production rate of macrophages and tumor cells, respectively, while γi represents the decay of the chemical attractants. The initial conditions u10, u20, v10, and v20 satisfy u10, u20 ∈ L2(Ω), with u10, u20 ≥ 0 in Ω, v10, v20 ∈ W 1,2(Ω), with v10, v20 ≥ 0 in Ω. (1.2) The chemotactic sensitivity functions χi(vi) and ξi(vi), i = 1, 2 satisfy χi(vi), ξi(vi) ∈ L∞(Ω), with χi(vi), ξi(vi) > 0 in Ω. (1.3) The system (1.1) under consideration is a generalized version of the classic Keller- Segel chemotaxis system. Chemotaxis refers to the directional movement of micro- organisms in response to a chemical stimulus, and is involved in various biological processes such as disease progression, wound healing, neuron migration, and tumor invasion. Keller and Segel first introduced the original Keller-Segel system in 1970 [13], and since then, the theoretical analysis of Keller-Segel and its variants has been intensively studied due to its numerous applications in biology, medicine, and other sciences. For further insight into the applications of chemotaxis, [22] provides a comprehensive review. Many researchers have been attracted to the study of Keller-Segel chemotaxis systems, as evidenced by the reviews by Bellomo et al. [2], Horstmann [11], Lankeit and Winkler [16], and the references therein. For more information, see [1, 19, 20, 21, 25, 26, 34]. Recently, Wikler [33] discussed the following keller-segel system using some es- timates on the Neumann problem ut = ∇ · (D(v)∇u)− χ∇ · (uS(v)∇v) + ru− µu2, vt = ∆v − v + u. (1.4) When r ∈ R, D ∈ C2([0,∞)) and S ∈ C2([0,∞)) ∩W 1,∞((0,∞)), for any µ > 0, the authors provided a result on the global existence of classical solutions in a two dimensional domain. The system consisting of two species chemotaxis with respect to two chemicals is expressed as follows ut = ∆u− χ∇ · (u∇v) , τvt = ∆v − v + w, wt = ∆w − ξ∇ · (w∇z) , τzt = ∆z − z + u. (1.5) Tao and Winkler [24] investigated system (1.5) under the conditions of τ = 0 and χ, ξ ∈ ±1. They established the existence of globally bounded classical solutions to (1.5) for both the attraction-repulsion case (χ = 1, ξ = −1) and double repulsion case (χ = ξ = −1). Furthermore, for the attraction-attraction case (χ = ξ = 1), they proved the global existence and boundedness of solutions to (1.5) if either m =∫ Ω u0+ ∫ Ω w0 is less than a certain threshold value in the two-dimensional space (n = 2), or if n ≥ 3 and the values of |u0|L∞(Ω) and |w0|L∞(Ω) are sufficiently small. Additionally, they showed that the system (1.5) exhibits a blow-up of solutions in finite time if either n = 2 and m is sufficiently large, or if n ≥ 3 and m > 0. Li EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 3 and Wang [18] extended the results of [24] to the fully parabolic case by presenting the unique global classical solution for the system in two dimensions under the condition that both m1 and m2 are small. Consider the chemotaxis system with two species and two chemicals, which in- cludes a logistic source term and is described by the following equations ut = d1∆u− χ1∇ · (u∇v) + µ1u(1− u− a1w), τvt = d2∆v − λ1v + α1w, wt = d3∆w − χ2∇ · (w∇z) + µ2w(1− a2u− w), τzt = d4∆z − λ2z + α2u. (1.6) Zhang et al. [35] studied the global existence and boundedness of solutions to (1.6) with τ = 0 and di = 1 (i = 1, 2, 3, 4) under smallness assumptions on the initial conditions and appropriate conditions on the strength of the damping death effects. They also established asymptotic stability when a1 ≥ 0 and a2 < 1. Tu et al. [30] studied the global bounded classical solution of the same system under the assumption that χi µi (i = 1, 2) are sufficiently small. They also showed that this solution converges exponentially to the steady state when a1, a2 ∈ (0, 1) and µ1, µ2 are sufficiently large. In the case where a1 ≥ 1 > a2 > 0 and µ2 is sufficiently large, the classical solution of the system converges to (0, 1, 1, 0) as t → ∞. Chunlai et al. [5] established the global-in-time solution of the system (1.6) using the eventual comparison approach and investigated the stability analysis of the system under suitable conditions. Wang and Mu [32] partially improved the results of [35] and [30] under suitable conditions on the parameters χi, µi, and ai (i = 1, 2). Zheng and Mu [36] investigated the global bounded classical solution of (1.6) for n = 2 and τ = 0 using a priori estimates and the Moser-Alikakos iteration tech- nique. Meanwhile, when τ = 1 and n ≥ 1, a globally bounded solution to system (1.6) is shown to exist by the authors, utilizing the maximal Sobolev regularity and semigroup technique. For the three-dimensional case, Li et al. [17] established the global boundedness of the classical solution if µi ≥ max{7χ2 i + 1, 51/2}, i = 1, 2, and proved that the solution exponentially converges to (1, 1, 1, 1) for large time, subject to the conditions that µ1 > chi22/8 and µ2 > chi21/8. Additionally, Black [3] examined the global existence of a bounded solu- tion for the two-dimensional Lotka-Volterra competitive system with an additional chemotactic influence, and established the asymptotic behavior of the solution for n ≥ 2. If µi/χ 2 i , i = 1, 2, are sufficiently large and a1, a2 < 1, any global solutions u ̸= 0 ̸= w of the system converge to the unique positive equilibrium point. Fur- thermore, the author also demonstrated that the solution of the system converges to (0, 1, 1, 0) as t → ∞ provided a1 ≥ 1, a2 < 1, and µ2 χ2 is sufficiently large. Finally, Pan et al. [23] studied the unique global bounded classical solution of (1.6) for n = 3 when µi, i = 1, 2, are sufficiently large. The system under consideration is a chemotaxis competition model featuring a loop structure, described by the set of equations u1t = d1∆u1 − χ11∇ · (u1∇v1)− χ12∇ · (u1∇v2) + µ1u1(1− u1 − a1u2), u2t = d2∆u2 − χ21∇ · (u2∇v1)− χ22∇ · (u2∇v2) + µ2u2(1− a2u1 − u2), τ1v1t = d3∆v1 + α11u1 + α12u2 − λ1v1, τ2v2t = d4∆v2 + α21u1 + α22u2 − λ2v2. (1.7) 4 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 Espejo et al. [8] investigated the chemotaxis competition system with loop (1.7) in the case where µi = τi = λi = 0, i = 1, 2. They established the necessary and sufficient conditions for global existence and blow-up of solutions by adapting the second moment technique from [6] and [9]. This blow-up behavior of solutions mod- els the aggregation of tumor cells and macrophages. They also demonstrated the system has an energy structure and proved global existence using the logarithmic HLS-inequality. Tu et al. [27] studied the chemotaxis competition system with loop (1.7) with τ = 0 and established sufficient conditions for the existence of global solu- tion, as well as exponential convergence to the unique positive equilibrium point for sufficiently large µ1 and µ2. They also showed that if χ11/µ1, χ12/µ1, χ21/µ2, and χ22/µ2 are sufficiently small for n ≥ 2, then the system admits a globally bounded classical solution. When a1 > 1 and µ2 is sufficiently large, the solution converges to a semi-trivial equilibrium point, and this convergence is algebraic when a1 = 1. Tu et al. [28] further examined the global boundedness of the classical solution of the system in two dimensions for the case where τ = 1 and di = 1, i = 1, 2, 3, 4. They proved that the solution converges exponentially to the same point as in [27]. When µ1 and µ2 are sufficiently large, Tu et al. [29] obtained the global bounded classical solution of (1.7) in three dimensions. For more information, refer to [31]. The global existence of classical solutions to 1.7 with chemotaxis sensitivity function was studied by Gurusamy et al. [10]. Inspired by the aforementioned studies and their relevance in biological contexts, we investigate the system (1.1). By utilizing the energy estimates, we establish the global existence of weak solutions subject to suitable conditions on the parameters and non-negative initial data (u10, u20, v10, v20) ∈ L2(Ω) × W 1,2(Ω) × W 1,2(Ω), where Ω ⊂ Rn and n ≥ 1. Additionally, we present numerical simulations of the system (1.1) in two-dimensional domain. The structure of our article is as follows: In Section 2, we introduce some fun- damental inequalities and a key lemma, and we demonstrate the local existence of classical solutions. In Section 3, we focus on the global existence of solutions to the approximate system. In Section 4, some energy estimates are derived to support the weak solutions analysis. Section 5 discusses the weak solutions to the system (1.1). First we have the existence of global weak solutions. Theorem 1.1. Assume that Ω ⊂ Rn, n ≥ 1 is an open, bounded domain with smooth boundary. Let the initial data (u10, u20, v10, v20) ∈ L2(Ω)×L2(Ω)×W 1,2(Ω)× W 1,2(Ω) and assume that the functions χi(vi) and ξi(vi), for i = 1, 2 satisfy (1.3). Then, for any positive parameters, the system (1.1) admits at least one global weak solution. Definition 1.2. Let (u10, u20, v10, v20) ∈ L2(Ω)× L2(Ω)×W 1,2(Ω)×W 1,2(Ω) be nonnegative and the functions χi(vi) and ξi(vi), i=1,2 satisfy (1.3). We say that (u1, u2, v1, v2) of functions is a global weak solution of (1.1), if u1 ∈ L2 loc ( (0,∞);L2(Ω) ) , u2 ∈ L2 loc ( (0,∞);L2(Ω) ) , v1 ∈ L2 loc ( (0,∞);W 1,2(Ω) ) , v1 ∈ L2 loc ( (0,∞);W 1,2(Ω) ) EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 5 and satisfies − ∫ ∞ 0 ∫ Ω u1ϕt = d1 ∫ Ω u10ϕ0 − ∫ ∞ 0 ∫ Ω ∇u1 · ∇ϕ+ ∫ ∞ 0 ∫ Ω χ1(v1)u1∇v1∇ϕ + ∫ ∞ 0 ∫ Ω χ2(v2)u1∇v2∇ϕ+ δ1 ∫ ∞ 0 ∫ Ω u1(1− u1 − a1u2)ϕ, − ∫ ∞ 0 ∫ Ω u2ϕt = d2 ∫ Ω u20ϕ0 − ∫ ∞ 0 ∫ Ω ∇u2 · ∇ϕ+ ∫ ∞ 0 ∫ Ω ξ1(v1)u2∇v1∇ϕ + ∫ ∞ 0 ∫ Ω ξ2(v2)u2∇v2∇ϕ+ δ2 ∫ ∞ 0 ∫ Ω u2(1− u2 − a2u1)ϕ, − ∫ ∞ 0 ∫ Ω v1ϕt = d3 ∫ Ω v10ϕ0 − ∫ ∞ 0 ∫ Ω ∇v1 · ∇ϕ+ α1 ∫ ∞ 0 ∫ Ω u1ϕ+ β1 ∫ ∞ 0 ∫ Ω u2ϕ − γ1 ∫ ∞ 0 ∫ Ω v1ϕ, − ∫ ∞ 0 ∫ Ω v2ϕt = d4 ∫ Ω v20ϕ0 − ∫ ∞ 0 ∫ Ω ∇v2 · ∇ϕ+ α2 ∫ ∞ 0 ∫ Ω u1ϕ+ β2 ∫ ∞ 0 ∫ Ω u2ϕ − γ2 ∫ ∞ 0 ∫ Ω v2ϕ, for all ϕ ∈ C∞ 0 (Ω× [0,∞)). 2. Preliminaries and local solution In this section, we establish the existence of local solutions to the approximate system, a standard process that follows the principles outlined in [12]. System (1.1) is approximated for each ϵ ∈ (0, 1) by the system u1ϵt = d1∆u1ϵ −∇ · (χ1(v1ϵ)u1ϵ∇v1ϵ)−∇ · (χ2(v2ϵ)u1ϵ∇v2ϵ) + f1(u1ϵ, u2ϵ), u2ϵt = d2∆u2ϵ −∇ · (ξ1(v1ϵ)u2ϵ∇v1ϵ)−∇ · (ξ2(v2ϵ)u2ϵ∇v2ϵ) + f2(u1ϵ, u2ϵ), v1ϵt = d3∆v1ϵ + α1u1ϵ + β1u2ϵ − γ1v1ϵ, v2ϵt = d4∆v2ϵ + α2u1ϵ + β2u2ϵ − γ2v2ϵ, (2.1) with the source f1(u1ϵ, u2ϵ) = δ1u1ϵ(1− u1ϵ − a1u2ϵ)− ϵuq 1ϵ, f2(u1ϵ, u2ϵ) = δ2u2ϵ(1− u2ϵ − a2u1ϵ)− ϵuq 2ϵ. We introduce the non-negative approximate initial data u10ϵ, u20ϵ ∈ C0(Ω), with u10ϵ, u20ϵ ≥ 0 in Ω, v10ϵ, v20ϵ ∈ W 1,q(Ω), for some q > {2, n} with v10ϵ, v20ϵ ≥ 0 in Ω (2.2) that satisfy the following conditions u10ϵ → u10 in L2(Ω), u20ϵ → u20 in L2(Ω), v10ϵ → v10 in W 1,2(Ω), v20ϵ → v20 in W 1,2(Ω), 6 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 as ϵ → 0. Moreover, the chemotactic sensitivity functions χi(viϵ) and ξi(viϵ), i = 1, 2 are positive, non-decreasing and satisfy ∥χi(viϵ)∥L∞(Ω) ≤ C and ∥ξi(viϵ)∥L∞(Ω) ≤ C, χi(viϵ) → χi(vi) and ξi(viϵ) → ξi(vi) inL∞(Ω) a sϵ → 0, (2.3) where C > 0. Lemma 2.1 (Local solution). Suppose that Ω ⊂ Rn, n ≥ 1 is an open, bounded do- main with smooth boundary and q > max{2, n}. Assume that the functions χi(viϵ) and ξi(viϵ), i = 1, 2 satisfy (2.3) and the initial conditions (u10ϵ, u20ϵ, v10ϵ, v20ϵ) sat- isfy (2.2). Then there exists Tmax < ∞ such that (2.1) admits a unique solution (u1ϵ, u2ϵ, v1ϵ, v2ϵ) satisfies u1ϵ, u2ϵ ∈ C0 ( Ω× [0, Tmax ) ) ∩ C2,1 ( Ω× (0, Tmax) ) , v1ϵ, v2ϵ ∈ C0 ( Ω× [0, Tmax ) ) ∩ C2,1 ( Ω× (0, Tmax) ) ∩ L∞ loc( [0, Tmax ) ;W 1,q(Ω)). Proof. Standard techniques involving the Banach fixed point theorem and parabolic regularity theories can be utilized to derive the proof. For a detailed demonstration, please refer to [12]. Moreover, the non-negativity of the solution in Ω× (0, Tmax) is guaranteed by the maximum principle along with (2.2) . □ 3. Global solution To establish the weak solution of our system, we first demonstrate the existence of global solutions to the approximate system (2.1). Lemma 3.1. The solution (u1ϵ, u2ϵ, v1ϵ, v2ϵ) of (2.1) for every ϵ ∈ (0, 1) satisifes the following conditions ∫ Ω u1ϵ ≤ C, ∀t ∈ (0, Tmax,ϵ), (3.1)∫ Ω u2ϵ ≤ C, ∀t ∈ (0, Tmax,ϵ), (3.2)∫ Ω v1ϵ ≤ C, ∀t ∈ (0, Tmax,ϵ), (3.3)∫ Ω v2ϵ ≤ C, ∀t ∈ (0, Tmax,ϵ), (3.4)∫ Ω |∇v1ϵ|2 ≤ C, ∀t ∈ (0, Tmax,ϵ), (3.5)∫ Ω |∇v2ϵ|2 ≤ C, ∀t ∈ (0, Tmax,ϵ). (3.6) Moreover, we have∫ T 0 ∫ Ω u2 1ϵ + ϵ δ1 ∫ T 0 ∫ Ω uq 1ϵ ≤ C(T + 1), ∀t ∈ (0, Tmax,ϵ), (3.7)∫ T 0 ∫ Ω u2 2ϵ + ϵ δ2 ∫ T 0 ∫ Ω uq 2ϵ ≤ C(T + 1), ∀t ∈ (0, Tmax,ϵ), (3.8) where the constant C > 0. EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 7 Proof. Upon integrating the first equation in (2.1), we obtain d dt ∫ Ω u1ϵ = δ1 ∫ Ω u1ϵ(1− u1ϵ − a1u2ϵ)− ϵ ∫ Ω uq 1ϵ, (3.9) ≤ δ1 ∫ Ω u1ϵ − δ1 ∫ Ω u2 1ϵ, (3.10) and d dt ∫ Ω u2ϵ ≤ δ2 ∫ Ω u2ϵ − δ2 ∫ Ω u2 2ϵ. (3.11) The proof follows a similar approach as outlined in [10, lemma 1]. Next, we integrate the third equation in (2.1) over Ω, yielding d dt ∫ Ω v1ϵ = α1 ∫ Ω u1ϵ + β1 ∫ Ω u2ϵ − γ1 ∫ Ω v1ϵ. By utilizing (3.1) and (3.2), we can derive d dt ∫ Ω v1ϵ = −γ1 ∫ Ω v1ϵ + C, Now, applying ODE arguments, we deduce that∫ Ω v1ϵ ≤ max {∫ Ω v10ϵ, C } . Employing a similar procedure, we can derive equation (3.4). We then proceed by multiplying the third equation in (2.1) with −∆v1ϵ and integrating it over Ω, gives 1 2 d dt ∫ Ω |∇v1ϵ|2 + d3 ∫ Ω |∆v1ϵ|2 + γ1 ∫ Ω |∇v1ϵ|2 = − ∫ Ω (α1u1ϵ + β1u2ϵ)∆v1ϵ, (3.12) for all t ∈ (0, Tmax). Using Young’s inequality, we obtain 1 2 d dt ∫ Ω |∇v1ϵ|2 + d3 ∫ Ω |∆v1ϵ|2 + γ1 ∫ Ω |∇v1ϵ|2 ≤ d3 ∫ Ω |∆v1ϵ|2 + α2 1 2d3 ∫ Ω u2 1ϵ + β2 1 2d3 ∫ Ω u2 2ϵ, thus 1 2 d dt ∫ Ω |∇v1ϵ|2 + γ1 ∫ Ω |∇v1ϵ|2 ≤ α2 1 2d3 ∫ Ω u2 1ϵ + β2 1 2d3 ∫ Ω u2 2ϵ. For each ϵ ∈ (0, 1), we set yϵ(t) = α2 1 2d3δ1 ∫ Ω u1ϵ + β2 1 2d3δ2 ∫ Ω u2ϵ + 1 2 ∫ Ω |∇v1ϵ|2, ∀t ∈ (0, Tmax,ϵ). Therefore, yϵ(t) ′ + 2γ1yϵ(t) ≤ α2 1 2d3δ1 d dt ∫ Ω u1ϵ + β2 1 2d3δ2 d dt ∫ Ω u2ϵ + 1 2 d dt ∫ Ω |∇v1ϵ|2 + γ1α 2 1 d3δ1 ∫ Ω u1ϵ + γ1β 2 1 d3δ2 ∫ Ω u2ϵ + γ1 ∫ Ω |∇v1ϵ|2. 8 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 Using (3.10) and (3.11), we obtain y′ϵ(t) ′ + 2γ1yϵ(t) ≤ α2 1 d3 (1 2 + γ1 δ1 )∫ Ω u1ϵ + β2 1 d3 (1 2 + γ1 δ2 )∫ Ω u2ϵ. Now, using (3.1) and (3.2), one has yϵ(t) ′ + 2γ1yϵ(t) ≤ C, ∀t ∈ (0, Tmax). applying the ODE argument, yields yϵ(t) ≤ max { sup ϵ∈(0,1) yϵ(0), C 2γ1 } , ∀t ∈ (0, Tmax,ϵ), this proves (3.5). The proof for (3.6) follows a similar approach. We then proceed by integrating (3.9) with respect to time over (0, T ) and utilizing (3.1), which leads to∫ Ω u1ϵ(·, T ) + δ1 ∫ T 0 ∫ Ω u2 1ϵ + ϵ ∫ T 0 ∫ Ω uq 1ϵ ≤ ∫ Ω u1ϵ(·, 0) + δ1CT, ∀T ∈ (0, Tmax,ϵ). Hence, we obtain (3.7) from the above equation. Following the same argument, we can also obtain (3.8). This completes the proof. □ Lemma 3.2 (Global solution). Assume that the functions χi(viϵ) and ξi(viϵ), i = 1, 2 satisfy (2.3) and the initial conditions (u10ϵ, u20ϵ, v10ϵ, v20ϵ) satisfy (2.2). Then the solution of (2.1) remains global in time for any ϵ ∈ (0, 1). The proof of the above lemma is similar to [10, Theorem 1] and it is omited here. 4. Energy estimates We present a priori estimates that are necessary to establish the main results. Lemma 4.1. For each ϵ ∈ (0, 1), there exists a constant C > 0 such that for all T > 0, ∫ Ω v21ϵ(·, T ) + γ1 ∫ T 0 ∫ Ω v21ϵ ≤ C(T + 1), (4.1)∫ Ω v22ϵ(·, T ) + γ2 ∫ T 0 ∫ Ω v22ϵ ≤ C(T + 1). (4.2) Proof. We multiply the third equation in (2.1) by v1ϵ and integrate over Ω to obtain 1 2 d dt ∫ Ω v21ϵ + d3 ∫ Ω |∇v1ϵ|2 + γ1 ∫ Ω v21ϵ ≤ α1 ∫ Ω u1ϵv1ϵ + β1 ∫ Ω u2ϵv1ϵ, ≤ γ1 4 ∫ Ω v21ϵ + α2 1 γ1 ∫ Ω u2 1ϵ + γ1 4 ∫ Ω v21ϵ + β2 1 γ1 ∫ Ω u2 2ϵ, this gives d dt ∫ Ω v21ϵ + 2d3 ∫ Ω |∇v1ϵ|2 + γ1 ∫ Ω v21ϵ ≤ 2α2 1 γ1 ∫ Ω u2 1ϵ + 2β2 1 γ1 ∫ Ω u2 2ϵ, for all t > 0. Integrating with respect to time, yields∫ Ω v21ϵ(·, T ) + 2d3 ∫ T 0 ∫ Ω |∇v1ϵ|2 + γ1 ∫ T 0 ∫ Ω v21ϵ EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 9 ≤ ∫ Ω v21ϵ(·, 0) + 2α2 1 γ1 ∫ T 0 ∫ Ω u2 1ϵ + 2β2 1 γ1 ∫ T 0 ∫ Ω u2 2ϵ. Using (3.7) and (3.8), we obtain∫ Ω v21ϵ(·, T ) + 2d3 ∫ T 0 ∫ Ω |∇v1ϵ|2 + γ1 ∫ T 0 ∫ Ω v21ϵ ≤ C(T + 1), ∀T > 0. This completes the proof. □ Lemma 4.2. For each ϵ ∈ (0, 1), there exists a constant C > 0 such that for all T > 0, ∫ T 0 ∫ Ω |∆v1ϵ|2 ≤ C(T + 1), (4.3)∫ T 0 ∫ Ω |∆v2ϵ|2 ≤ C(T + 1), (4.4) Proof. From (3.12), we have 1 2 d dt ∫ Ω |∇v1ϵ|2 + d3 ∫ Ω |∆v1ϵ|2 + γ1 ∫ Ω |∇v1ϵ|2 = − ∫ Ω (α1u1ϵ + β1u2ϵ)∆v1ϵ, ≤ d3 2 ∫ Ω |∆v1ϵ|2 + α2 1 d3 ∫ Ω u2 1ϵ + β2 1 d3 ∫ Ω u2 2ϵ, for all t > 0. Hence d dt ∫ Ω |∇v1ϵ|2 + d3 ∫ Ω |∆v1ϵ|2 + 2γ1 ∫ Ω |∇v1ϵ|2 ≤ 2α2 1 d3 ∫ Ω u2 1ϵ + 2β2 1 d3 ∫ Ω u2 2ϵ. Integrating with respect to time, we infer that∫ Ω |∇v1ϵ(·, T )|2 + d3 ∫ T 0 ∫ Ω |∆v1ϵ|2 + 2γ1 ∫ T 0 ∫ Ω |∇v1ϵ|2 ≤ ∫ Ω |∇v1ϵ(·, 0)|2 + 2α2 1 d3 ∫ T 0 ∫ Ω u2 1ϵ + 2β2 1 d3 ∫ T 0 ∫ Ω u2 2ϵ. Using (3.7) and (3.8), we attain∫ Ω |∇v1ϵ(·, T )|2 + d3 ∫ T 0 ∫ Ω |∆v1ϵ|2 + γ1 ∫ T 0 ∫ Ω |∇v1ϵ|2 ≤ C(T + 1), for all T > 0. We can apply the same procedure as above to prove (4.4). This completes the proof. □ Lemma 4.3. For each ϵ ∈ (0, 1), there exists a constant C > 0 such that∫ T 0 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + δ1 ∫ T 0 ∫ Ω u2 1ϵ ln(1 + u1ϵ) + ϵ ∫ T 0 ∫ Ω uq 1ϵ ln(1 + u1ϵ) ≤ C(T + 1) (4.5) and ∫ T 0 ∫ Ω |∇u2ϵ|2 1 + u2ϵ + δ2 ∫ T 0 ∫ Ω u2 2ϵ ln(1 + u2ϵ) + ϵ ∫ T 0 ∫ Ω uq 2ϵ ln(1 + u2ϵ) ≤ C(T + 1), (4.6) 10 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 for all T > 0. Proof. By using ln(1 + u1ϵ) as a test function and applying the first equation in (2.1), we obtain d dt ∫ Ω ( (1 + u1ϵ) ln(1 + u1ϵ)− u1ϵ ) ≤ −d1 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + C ∫ Ω u1ϵ 1 + u1ϵ ∇u1ϵ · ∇v1ϵ + C ∫ Ω u1ϵ 1 + u1ϵ ∇u1ϵ · ∇v2ϵ + δ1 ∫ Ω u1ϵ(1− u1ϵ − a1u2ϵ) ln(1 + u1ϵ)− ϵ ∫ Ω uq 1ϵ ln(1 + u1ϵ), for all t > 0. It known that for all values of u1ϵ greater than 0, the inequality 0 ≤ ln(1 + u1ϵ) ≤ u1ϵ holds and d dt ∫ Ω ( (1 + u1ϵ) ln(1 + u1ϵ)− u1ϵ ) ≤ −d1 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + C ∫ Ω ( ln(1 + u1ϵ)− u1ϵ ) ∆v1ϵ + C ∫ Ω ( ln(1 + u1ϵ)− u1ϵ ) ∆v2ϵ + δ1 ∫ Ω u2 1ϵ − δ1 ∫ Ω u2 1ϵ ln(1 + u1ϵ)− ϵ ∫ Ω uq 1ϵ ln(1 + u1ϵ). Using Young’s inequality we obtain d dt ∫ Ω ( (1 + u1ϵ) ln(1 + u1ϵ)− u1ϵ ) ≤ −d1 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + C 2 ∫ Ω |∆v1ϵ|2 + C 2 ∫ Ω ( ln(1 + u1ϵ)− u1ϵ )2 + C 2 ∫ Ω |∆v2ϵ|2 + C 2 ∫ Ω ( ln(1 + u1ϵ)− u1ϵ )2 + δ1 ∫ Ω u2 1ϵ − δ1 ∫ Ω u2 1ϵ ln(1 + u1ϵ) − ϵ ∫ Ω uk 1ϵ ln(1 + u1ϵ). From the inequality (a− b)2 ≤ a2 + b2, we conclude that d dt ∫ Ω ( (1 + u1ϵ) ln(1 + u1ϵ)− u1ϵ ) ≤ −d1 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + C 2 ∫ Ω |∆v1ϵ|2 + C 2 ∫ Ω |∆v2ϵ|2 + C 2 ∫ Ω ln(1 + u1ϵ) 2 + C 2 ∫ Ω u2 1ϵ + C 2 ∫ Ω ln(1 + u1ϵ) 2 + C 2 ∫ Ω u2 1ϵ + δ1 ∫ Ω u2 1ϵ − δ1 ∫ Ω u2 1ϵ ln(1 + u1ϵ) − ϵ ∫ Ω uq 1ϵ ln(1 + u1ϵ), d dt ∫ Ω ( (1 + u1ϵ) ln(1 + u1ϵ)− u1ϵ ) ≤ −d1 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + C 2 ∫ Ω |∆v1ϵ|2 + C 2 ∫ Ω |∆v2ϵ|2 + (C + C + δ1) ∫ Ω u2 1ϵ − δ1 ∫ Ω u2 1ϵ ln(1 + u1ϵ)− ϵ ∫ Ω uq 1ϵ ln(1 + u1ϵ). EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 11 By integrating with respect to time and utilizing the fact that (1+u1ϵ) ln(1+u1ϵ)− u1ϵ > 0, it is possible to derive d1 ∫ T 0 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + δ1 ∫ T 0 ∫ Ω u2 1ϵ ln(1 + u1ϵ) + ϵ ∫ T 0 ∫ Ω uq 1ϵ ln(1 + u1ϵ) ≤ ∫ Ω ( (1 + u1ϵ0) ln(1 + u1ϵ0)− u1ϵ0 ) + C 2 ∫ T 0 ∫ Ω |∆v1ϵ|2 + C 2 ∫ T 0 ∫ Ω |∆v2ϵ|2 + (C + C + δ1) ∫ T 0 ∫ Ω u2 1ϵ. Using the previous Lemmas, we obtain d1 ∫ T 0 ∫ Ω |∇u1ϵ|2 1 + u1ϵ + δ1 ∫ T 0 ∫ Ω u2 1ϵ ln(1 + u1ϵ) + ϵ ∫ T 0 ∫ Ω uq 1ϵ ln(1 + u1ϵ) ≤ C(T + 1), for all T > 0. The same argument gives us that d2 ∫ T 0 ∫ Ω |∇u2ϵ|2 1 + u2ϵ + δ2 ∫ T 0 ∫ Ω u2 2ϵ ln(1 + u2ϵ) + ϵ ∫ T 0 ∫ Ω uq 2ϵ ln(1 + u2ϵ) ≤ C(T + 1), for all T > 0. This completes the proof. □ Lemma 4.4. For all values of ϵ ∈ (0, 1), there exists a positive constant C such that∥∥u1ϵ ∥∥ L4/3((0,T );W 1, 4 3 (Ω)) ≤ C(T + 1), ∥∥u2ϵ ∥∥ L4/3((0,T );W 1, 4 3 (Ω)) ≤ C(T + 1), (4.7) for all T > 0. Proof. Let ∫ T 0 ∫ Ω |∇u1ϵ|4/3 = ∫ T 0 ∫ Ω |∇u1ϵ|4/3 (1 + u1ϵ)2/3 (1 + u1ϵ) 2/3. Using the Young’s inequality, then (3.7) and (4.5) the above estimate yields∫ T 0 ∫ Ω |∇u1ϵ|4/3 ≤ ∫ T 0 ∫ Ω ( |∇u1ϵ|4/3 (1 + u1ϵ)2/3 )3/2 + 1 4 ∫ T 0 ∫ Ω (1 + u1ϵ) 2, ≤ ∫ T 0 ∫ Ω |∇u1ϵ|2 (1 + u1ϵ) + 1 4 ∫ T 0 ∫ Ω (1 + u1ϵ) 2 ≤ C(T + 1) . (4.8) Again, using the Young’s inequality, one obtains∫ T 0 ∫ Ω u 4/3 1ϵ ≤ ∫ T 0 ∫ Ω u2 1ϵ + 1 4 |Ω|T ≤ C(T + 1). Combining the preceding two estimates, we have established the proof for all T > 0. The same reasoning can be applied for u2ϵ. □ The following lemma is used for showing the strong compactness properties of the solution (u1ϵ, u2ϵ, v1ϵ, v2ϵ). Lemma 4.5. There exists C > 0 and let ϵ ∈ (0, 1) and p > 1 + n 2 , such that ∥∂u1ϵ ∂t ∥ L1 ( (0,T );(Wp,2 0 (Ω)) ′ ) ≤ C(T + 1), (4.9) ∥∂u2ϵ ∂t ∥ L1 ( (0,T );(Wp,2 0 (Ω)) ′ ) ≤ C(T + 1), (4.10) 12 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 ∥∂v1ϵ ∂t ∥ L2 ( (0,T );(W 1,2(Ω))′ ) ≤ C(T + 1), (4.11) ∥∂v2ϵ ∂t ∥ L2 ( (0,T );(W 1,2(Ω))′ ) ≤ C(T + 1), (4.12) for all T > 0. Proof. Multiply the first equation in (2.1) by ϕ ∈ C∞ 0 (Ω), and then integrate by parts to obtain∣∣ ∫ Ω u1ϵtϕ ∣∣ ≤ (∥∥∇ϕ ∥∥ L∞(Ω) + ∥∥ϕ∥∥ L∞(Ω) )( d1 ∫ Ω ∇u1ϵ +M1 ∫ Ω u1ϵ∇v1ϵ +M2 ∫ Ω u1ϵ∇v2ϵ + δ1 ∫ Ω u1ϵ + δ1 ∫ Ω u2 1ϵ + δ1a1 ∫ Ω u1ϵu2ϵ + ϵ ∫ Ω uq 1ϵ ) . Basic inequalities imply∣∣ ∫ Ω u1ϵtϕ ∣∣ ≤ ∥∥ϕ∥∥ W 1,∞(Ω) (3 4 ∫ Ω |∇u1ϵ|4/3 + d41|Ω| 4 + M1 2 ∫ Ω u2 1ϵ + M1 2 ∫ Ω |∇v1ϵ|2 + M2 2 ∫ Ω u2 1ϵ + M2 2 ∫ Ω |∇v2ϵ|2 + δ1 ∫ Ω u1ϵ + δ1 ∫ Ω u2 1ϵ + δ1a1 2 ∫ Ω u2 1ϵ + δ1a1 2 ∫ Ω u2 2ϵ + ϵ ∫ Ω uq 1ϵ ) . As a consequence of the embedding W p,2 0 (Ω) ↪→ W 1,∞(Ω) for p > 1 + n 2 , there exists a positive constant C such that ∥∥ϕ∥∥ W 1,∞(Ω) = ∥∥∇ϕ ∥∥ L∞(Ω) + ∥∥ϕ∥∥ L∞(Ω) ≤ C ∥∥ϕ∥∥ Wp,2 0 (Ω) . By utilizing the previous lemmas, we can apply the aforementioned inequality to obtain∫ T 0 ∥∥u1ϵt(·, t) ∥∥ (Wp,2 0 (Ω)) ′ ≤ 3 4 ∫ T 0 ∫ Ω ∣∣∇u1ϵ ∣∣4/3 + d41|Ω|T 4 + M1 2 ∫ T 0 ∫ Ω u2 1ϵ + M1 2 ∫ T 0 ∫ Ω |∇v1ϵ|2 + M2 2 ∫ T 0 ∫ Ω u2 1ϵ + M2 2 ∫ T 0 ∫ Ω |∇v2ϵ|2 + δ1 ∫ T 0 ∫ Ω u1ϵ + δ1 ∫ T 0 ∫ Ω u2 1ϵ + δ1a1 2 ∫ T 0 ∫ Ω u2 1ϵ + δ1a1 2 ∫ T 0 ∫ Ω u2 2ϵ + ϵ ∫ T 0 ∫ Ω uq 1ϵ ≤ C(T + 1). Similarly, we obtain ∫ T 0 ∥∥u2ϵt(·, t) ∥∥ (Wp,2 0 (Ω)) ′ ≤ C(T + 1). Choose ϕ ∈ W 1,2(Ω), test the third equation in (2.1) and applying the Hölder’s inequality infer that∫ Ω v1ϵtϕ ≤ d3 ∫ Ω ∇v1ϵ · ∇ϕ− γ1 ∫ Ω v1ϵϕ+ α1 ∫ Ω u1ϵϕ+ β1 ∫ Ω u2ϵϕ ≤ d3 (∫ Ω |∇v1ϵ|2 )1/2(∫ Ω |∇ϕ|2 )1/2 + γ1 (∫ Ω v21ϵ )1/2(∫ Ω ϕ2 )1/2 EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 13 + α1 (∫ Ω u2 1ϵ )1/2(∫ Ω ϕ2 )1/2 + β1 (∫ Ω u2 2ϵ )1/2(∫ Ω ϕ2 )1/2 ≤ ( d3 (∫ Ω |∇v1ϵ|2 )1/2 + γ1 (∫ Ω v21ϵ )1/2 + α1 (∫ Ω u2 1ϵ )1/2 + β1 (∫ Ω u2 2ϵ )1/2) ∥ϕ∥W 1,2(Ω). This implies ∥v1ϵt(·, t)∥2( W 1,2(Ω) )′ ≤ C ∫ Ω |∇v1ϵ|2 + C ∫ Ω v21ϵ + C ∫ Ω u2 1ϵ + C ∫ Ω u2 2ϵ. Integrating with respect to time, one obtains∫ T 0 ∥v1ϵt(·, t)∥2( W 1,2(Ω) )′ ≤ C ∫ T 0 ∫ Ω |∇v1ϵ|2 + C ∫ T 0 ∫ Ω v21ϵ + C ∫ T 0 ∫ Ω u2 1ϵ + C ∫ T 0 ∫ Ω u2 2ϵ, ≤ C(T + 1), for all T > 0. Similarly, we can show that∫ T 0 ∥v2ϵt(·, t)∥2( W 1,2(Ω) )′ ≤ C(T + 1), for all T > 0. This completes the proof. □ 5. Existence of weak solutions Next, as ϵ → 0, we proceed to passing the limits in order to construct a weak solution of (1.1). Lemma 5.1. There exist u1, u2, v1, v2 on Ω × (0,∞) and a sequence {ϵj}j∈N ⊂ (0, 1), with ϵj → 0 as j → ∞, such that u1ϵ → u1 in L2 loc(Ω× [0,∞)) and a.e in Ω× (0,∞), (5.1) ∇u1ϵ ⇀ ∇u1 in L 4/3 loc (Ω× [0,∞)), (5.2) ϵuq 1ϵ ⇀ 0 in L1 loc(Ω× [0,∞)), (5.3) u2 1ϵ ⇀ u2 1 in L1 loc(Ω× [0,∞)), (5.4) v1ϵ → v1 in L2 loc(Ω× [0,∞)) and a.e in Ω× (0,∞), (5.5) ∇v1ϵ ⇀ ∇v1 in L2 loc(Ω× [0,∞)), (5.6) χ1(v1ϵ)u1ϵ∇v1ϵ ⇀ χ1(v1)u1∇v1 in L1 loc(Ω× [0,∞)) (5.7) u2ϵ → u2 in L2 loc(Ω× [0,∞)) and a.e in Ω× (0,∞), (5.8) ∇u2ϵ ⇀ ∇u2 in L 4/3 loc (Ω× [0,∞)), (5.9) ϵuq 2ϵ ⇀ 0 in L1 loc(Ω× [0,∞)), (5.10) u2 2ϵ ⇀ u2 2 in L1 loc(Ω× [0,∞)), (5.11) u1ϵu2ϵ → u1u2 in L1 loc(Ω× [0,∞)), (5.12) v2ϵ → v2 in L2 loc(Ω× [0,∞)) and a.e in Ω× (0,∞), (5.13) 14 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 ∇v2ϵ ⇀ ∇v2 in L2 loc(Ω× [0,∞)), (5.14) χ2(v2ϵ)u1ϵ∇v2ϵ ⇀ χ2(v2)u1∇v2 in L1 loc(Ω× [0,∞)), (5.15) ξ1(v1ϵ)u2ϵ∇v1ϵ ⇀ ξ1(v1)u2∇v1 in L1 loc(Ω× [0,∞)), (5.16) ξ2(v2ϵ)u2ϵ∇v2ϵ ⇀ ξ2(v2)u2∇v2 in L1 loc(Ω× [0,∞)) (5.17) Proof. To prove the results, we simply take the subsequence ϵ := ϵj . Lemma 4.4 and 4.5 show the boundedness of {u1ϵ} in L4/3 ( (0, T );W 1, 43 (Ω) ) and {u1ϵt} in L1 ( (0, T ); (W p,2 0 (Ω))′ ) . Because of the embedding W 1, 43 (Ω) ↪→↪→ L4/3(Ω) ↪→( W p,2 0 (Ω) )′ , the Aubin-Lion’s lemma [4] yields a subsequence such that u1ϵ → u1 in L4/3(Ω×[0,∞)) as ϵ → 0 and this convergence is almost everywhere in Ω×(0,∞) for some u1 ∈ L4/3(Ω× [0,∞)). Furthermore, (3.7) and Egorov’s theorem yield a subsequence along which u1ϵ → u1, ensuring the boundedness of {u1ϵ} in L2 loc(Ω× [0,∞)) and allowing us to con- clude (5.2) from (4.8). Additionally, the sequence { u2 1ϵ } being equi-bounded and equi-integrable follows from (3.7) and (4.5). Applying the Dunford-Pettis theorem [7], a subsequence of { u2 1ϵ } is weakly convergent in L1 loc(Ω× [0,∞)) and hence ∥u1ϵ∥L2 loc(Ω×[0,∞)) → ∥u1∥L2 loc(Ω×[0,∞)) as ϵ → 0 by taking constant as test function. This result, along with the convergence u1ϵ → u1 in L2 loc(Ω× [0,∞)), allow us to achieve (5.1). Since the sequence {ϵuq 1ϵ} is equi-integrable from (4.5), uq 1ϵ weakly converges to 0 by applying the Dunford-Pettis theorem, which yields (5.3). Repeating the same arguments, we obtain (5.8)-(5.11) and the result (5.12) follows from the combination of (5.1) and (5.8). Furthermore, combining (3.5) and (4.1), we obtain that ∥v1ϵ∥L2((0,T );W 1,2(Ω)) is bounded for all T > 0. Along with a subsequence, (5.5) follows from previous argument and (4.12) using Aubin-Lion’s Lemma. At the same time, we can conclude (5.6). The same arguments are used to prove (5.13) and (5.14). Finally, the combination of (5.1) and (5.6), gives (5.7). The same arguments are used to prove the remaining results (5.15) - (5.17). This completes the proof. □ Lemma 5.2. (u1, u2, v1, v2) is a global weak solution to (1.1) in the sense of Defi- nition 1.2. Proof. Let ϕ ∈ C∞ 0 (Ω × [0,∞)) and test it in the approximate problem (2.1). Applying the convergence properties from Lemma 5.1 and passing the limits, we obtain the proof. □ Proof of Theorem 1.1. The proof follows by the combination of Lemma 5.1 and Lemma 5.2. □ 6. Conclusion This study provides the global existence and boundedness of weak solutions to the chemotaxis competition system with loop and signal dependent sensitivity based on the energy inequality method. Acknowledgments. The authors wish to thank the anonymous referee for her/his careful reading of the original manuscript and their comments that eventually led to an improved presentation. EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 15 References [1] Bai, X; Winkler, M; Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583. [2] Bellomo, N.; Bellouquid, A; Tao, Y.; Winkler, M.; Toward a mathematical theory of Keller- Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763. [3] Blac,. T; Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Continuous Dyn Syst Ser B. 22 (2017), 1253-1272. [4] Chen, X.; Jungel, A.; Liu, J. G; A note on Aubin–Lions–Dubinskii lemmas, Acta Appl. Math. 133 (1) (2014), 33–43 [5] Chunlai, M,; Mi, Y.; Zheng, P.; Global stability in a two-competing-species chemotaxis system with two chemicals, Differ. Integral Equ., 31 (2018), 547-558. [6] Conca, C.; Espejio, E.; Vilches, K.; Remarks on the blowup and global existence for a two species chemotaxis Keller-Segel system in R2, Eur. J. Appl. Math., 22 (2011), 553-580. [7] Dunford, N.; Schwartz, J. T.; Linear Operators. I. General Theory, with the assistance of W. G. Bade and R.G. Bartle, Pure Appl. Math., vol. 7, Interscience Publishers Ltd., New York, London, 1958. [8] Espejo, E.; Vilches, K.; Conca, C,; A simultaneous blow-up problem arising in tumor model- ing, J. Math. Biol., 79 (2019), 1357-1399. [9] Espejo, E.; Vilches, K, Conca, C; Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in R2, Eur. J. Appl. Math. 24 (2013), 297-313. [10] Gurusamy, A.; Gnanasekaran, S.; Nithyadevi, N.; Fully parabolic chemotaxis-competition system with loop and signal dependent sensitivity, J. Elliptic Parabol. 7 (2021) 727–746. [11] Horstmann, D.; From 1970 until present: the Keller-Segel model in chemotaxis and its con- sequences I, Jahresberichte DMV. 105 (2003), 103-165. [12] Horstmann, D.; Winkler, M.; Boundedness vs. blow–up in a chemotaxis system, J. Differ Equ., 215 (2005), 52–107. [13] Keller, E. F.; Segel, L. A.; Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [14] Knutsdottir, H.; Palsson, E.; Edelstein-Keshet, L.; Mathematical model of macrophage- facilitated breast cancer cells invasion, J. Theor. Biol., 357 (2014), 184-199. [15] Lankeit, J.; Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158–1191. [16] Lankeit, J.; Winkler, M.; Facing Low Regularity in Chemotaxis Systems, Jahresbericht der Deutschen Mathematiker Vereinigung, 122 (2020), 35-64. [17] Li, D., Mu, C.; Lin, K.; Wang, L.; Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions, Z. Angew. Math. Phys., 68 (2017), 56. [18] Li, X.; Wang, Y.; Boundedness in a two-species chemotaxis parabolic system with two chem- icals, Discrete Continuous Dyn. Syst. Ser B., 22 (2017), 2717-2729. [19] Lin, K.; Mu, C.; Zhong, H.; A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. with Appl., 75 (2018), 837-849. [20] Mizukami, M.; Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Continuous Dyn. Syst. Ser B, 22 (2017), 2301-2319. [21] Mizukami, M.; Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Continuous Dyn. Sys.t Ser S, 13 (2020), 269-278. [22] Painter, T. H. K.; Volume-filling and quorum-sensing in models for chemosensitive move- ment, Can. Appl. Math. Q., 10 (2002), 501-543. [23] Pan, X.; Wang, L.; Zhang, J.; Wang, J.; Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020), 26. [24] Tao, Y.; Winkler, M.; Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Continuous Dyn. Syst. Ser B, 20 (2015), 3165-3183. [25] Tello, J. I.; Winkler, M.; A Chemotaxis System with Logistic Source, Commun. Partial Differ. Equ., 32 (2007), 849-877. 16 S. GNANASEKARAN, N. NITHYADEVI EJDE-2024/56 [26] Tello, J. I.; Winkler, M.; Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. [27] Tu, X.; Mu, C.; Qiu, S.; Global asymptotic stability in a parabolic-elliptic chemotaxis system with competitive kinetics and loop, J. Appl. Anal., 101 (2022), 1532-1551. [28] Tu, X.; Mu, C.; Qiu, S.; Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Analysis, 198 (2020), 111923. [29] Tu, X.; Mu, C.; Qiu, S.; Yang, L.; Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020) 185. [30] Tu, X.; Mu, C.; Zheng, P.; Lin, K.; Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Continuous Dyn. Syst. Ser S, 38 (2018), 3617-3636. [31] Tu, X.; Tang, C. L.; Qiu, S.; The phenomenon of large population densities in a chemotaxis competition system with loop, J. Evol. Equ., 21 (2021), 1717–1754. [32] Wang, L.; Mu, C.; A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Continuous Dyn. Syst. Ser B, 25 (2020) 4585-4601. [33] Winkler, M; A result on parabolic gradient regularity in Orlicz spaces and application to absorption-induced blow-up prevention in a Keller–Segel-type cross-diffusion system, Inter- national Mathematics Research Notices. 19 (2023), 16336-16393. [34] Zhang, Q.; Li, Y.; Global solutions in a high-dimensional two-species chemotaxis model with Lotka-Volterra competitive kinetics, J. Math. Anal., 467 (2018), 751-767. [35] Zhang, Q.; Liu, X.; Yang, X.; Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J Math Phys, 58 (2017), 111504. [36] Zheng, P.; Mu, C.; Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177. Shanmugasundaram Gnanasekaran Department of Mathematics, Easwari Engineeering College, Chennai, TN 600089, India. Department of Mathematics, National Institute of Technology Tiruchirappalli, Tiruchi- rappalli - 620015, India Email address: dr.sakar.mat@gmail.com Nagarajan Nithyadevi Department of Applied Mathematics, Bharathiar University, Coimbatore, TN 641046, India Email address: nithyadevin@buc.edu.in 1. Introduction 2. Preliminaries and local solution 3. Global solution 4. Energy estimates 5. Existence of weak solutions 6. Conclusion Acknowledgments References