Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 81, pp. 1–10. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.81 FINITE-TIME BLOW-UP IN A QUASILINEAR FULLY PARABOLIC ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH DENSITY-DEPENDENT SENSITIVITY YUTARO CHIYO, TAKESHI UEMURA, TOMOMI YOKOTA Abstract. This article concerns the quasilinear fully parabolic attraction-repulsion chemotaxis system ut = ∇ · ((u+ 1)m−1∇u− χu(u+ 1)p−2∇v + ξu(u+ 1)p−2∇w), x ∈ Ω, t > 0, vt = ∆v + αu− βv, x ∈ Ω, t > 0, wt = ∆w + γu− δw, x ∈ Ω, t > 0 with homogeneous Neumann boundary conditions, where Ω ⊂ Rn (n ∈ {2, 3}) is an open ball, m, p ∈ R, χ, ξ, α, β, γ, δ > 0 are constants. The main result asserts finite-time blow-up of solutions to this system with some positive initial data when χα−ξγ > 0, p ≥ 2 and p−m > 2/n. 1. Introduction In this article we study the blow-up of solutions to the quasilinear fully parabolic attraction- repulsion chemotaxis system ut = ∇ · ((u+ 1)m−1∇u− χu(u+ 1)p−2∇v + ξu(u+ 1)p−2∇w), x ∈ Ω, t > 0, vt = ∆v + αu− βv, x ∈ Ω, t > 0, wt = ∆w + γu− δw, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = 0, x ∈ ∂Ω, t > 0, u(·, 0) = u0, v(·, 0) = v0, w(·, 0) = w0, x ∈ Ω, (1.1) where Ω = BR ⊂ Rn (n ∈ {2, 3}) is the open ball centered at the origin with radius R > 0; ν is the outward normal vector to ∂Ω; m, p ∈ R, χ, ξ, α, β, γ, δ > 0 are constants; u, v, w are unknown functions; u0, v0, w0 represent the initial data satisfying u0 ∈ C0(Ω), v0 ∈ W 1,∞(Ω), w0 ∈ W 1,∞(Ω), u0, v0, w0 > 0 in Ω. Both boundedness and blow-up of solutions are known as major topics in mathematical studies on chemotaxis systems. This article is devoted to the analysis of blow-up. Known results and purpose. We first recall studies on the quasilinear fully parabolic Keller- Segel system ut = ∇ · ((u+ 1)m−1∇u− χu(u+ 1)p−2∇v), vt = ∆v + αu− βv. Boundedness of solutions to this system was established. Indeed, Tao and Winkler [9] proved boundedness on convex domains under the assumption p −m < 2 n . Subsequently, the convexity assumption was removed in [6]. Regarding blow-up of solutions, there are several relevant studies. Cieślak and Stinner [3, 4] showed finite-time blow-up under the conditions n ≥ 2, p −m > 2/n, 2020 Mathematics Subject Classification. 35B44, 35K59, 35Q92, 92C17. Key words and phrases. Finite-time blow-up; quasilinear; attraction-repulsion; chemotaxis. ©2025. This work is licensed under a CC BY 4.0 license. Submitted May 1, 2025. Published August 6, 2025. 1 2 Y. CHIYO, T. UEMURA, T. YOKOTA EJDE-2025/81 and either m ≥ 1 or p ≥ 2. In the case that m, p ∈ R, blow-up was also established by Cieślak and Stinner [5] and Winkler [12] when p −m > 2 n . These results suggest that whether solutions remain bounded or blow-up is governed by the size relation between p−m and 2/n. Next, we refer to studies on the parabolic-elliptic-elliptic system related to (1.1), ut = ∇ · ((u+ 1)m−1∇u− χu(u+ 1)p−2∇v + ξu(u+ 1)q−2∇w), 0 = ∆v + αu− βv, 0 = ∆w + γu− δw, where q ∈ R. Concerning blow-up and boundedness of solutions to this system, the following four results were established in [2]. (I) If p < q, then for all nonnegative initial data u0 ∈ C0(Ω) the system possesses a unique global bounded classical solution. (II) If p = q and χα − ξγ < 0, then for all nonnegative initial data u0 ∈ C0(Ω) the system admits a unique global bounded classical solution. (III) If p > q, then there exist initial data such that the corresponding solutions blow up in finite time when n ≥ 3. (IV) If p = q and χα−ξγ > 0, then there exist initial data such that the corresponding solutions which blow up in finite time when n ≥ 3. These results mean that boundedness and blow-up are classified by the sizes of p, q, where the condition p = q is critical. However, the two-dimensional case is excluded in (III) and (IV), and the fully parabolic case is not covered. As to the fully parabolic case, there are several studies on finite-time blow-up when m = 1 and p = q = 2. Among others, Lankeit [7] investigated the most general case that δ ̸= β when n = 3, m = 1 and p = q = 2. Accordingly, the purpose of this paper is to establish finite-time blow-up of solutions in the fully parabolic case when n ≤ 3, m ≥ 1 and p = q ≥ 2. To this end, we formulate the problem by referring to the methods developed in [3] and [7]. Formulation and notation. We denote by (u, v, w) a local solution of (1.1), which will be given in Lemma 2.1, and by Tmax ∈ (0,∞] its maximal existence time. Putting z := χv − ξw and z0 := χv0 − ξw0, we rewrite (1.1) as ut = ∇ · ((u+ 1)m−1∇u− u(u+ 1)p−2∇z), x ∈ Ω, t ∈ (0, Tmax), zt = ∆z − δz + θu+ χ(δ − β)v, x ∈ Ω, t ∈ (0, Tmax), vt = ∆v + αu− βv, x ∈ Ω, t ∈ (0, Tmax), ∂u ∂ν = ∂z ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t ∈ (0, Tmax), u(·, 0) = u0, z(·, 0) = z0, v(·, 0) = v0, x ∈ Ω, (1.2) where θ := χα− ξγ. Also, with any fixed s0 > 1 we define several functions as follows: F̃(u, v, w) := 1 2 ∫ Ω |∇(χv − ξw)|2 + 1 2 ∫ Ω (χv − ξw)2 − θ ∫ Ω u(χv − ξw) + θ ∫ Ω G(u), (1.3) F(u, z) := 1 2 ∫ Ω |∇z|2 + 1 2 ∫ Ω z2 − θ ∫ Ω uz + θ ∫ Ω G(u), (1.4) G(s) := ∫ s s0 ∫ σ s0 (τ + 1)m−1 τ(τ + 1)p−2 dτdσ (s > 0), (1.5) D(u, z) := ∫ Ω (∆z − δz + θu)2 + θ ∫ Ω u(u+ 1)p−2 ∣∣∣ (u+ 1)m−1 u(u+ 1)p−2 ∇u−∇z ∣∣∣2, (1.6) H(s) := ∫ s 0 (σ + 1)m−p+1 dσ (s > 0). (1.7) EJDE-2025/81 ATTRACTION-REPULSION CHEMOTAXIS SYSTEMS 3 Main results. The first result asserts finite-time blow-up of classical solutions to (1.1). Theorem 1.1. Let Ω = BR ⊂ Rn (n ∈ {2, 3}, R > 0). Assume that χα − ξγ > 0, p ≥ 2 and p − m > 2 n . Let M > 0 and A > 0. Then there exists a constant K(M,A) > 0 such that if (u0, v0, w0) belongs to the set B(M,A) := { (u0, v0, w0) ∈ C0(Ω)×W 1,∞(Ω)×W 1,∞(Ω) : u0, v0 and w0 are radially symmetric and positive in Ω,∫ Ω u0 = M, ∥χv0 − ξw0∥W 1,2(Ω) ≤ A, F̃(u0, v0, w0) < −K(M,A) } , (1.8) then the solution (u, v, w) of (1.1) blows up in a finite time T in the sense that lim sup t↗T ∥u(·, t)∥L∞(Ω) = ∞. The second result guarantees that if the set B(M,A) defined in (1.8) is equipped with a suitable topology, then it is dense in the space of radially symmetric positive functions. Theorem 1.2. Let Ω = BR ⊂ Rn (n = 3, R > 0). Assume that χα − ξγ > 0, p ≥ 2 and p −m > 2 n . Let σ ∈ (1, 6 5 ). Then for all M > 0 and A > 0, the set B(M,A) defined in (1.8) is dense in the space Y := { (u, v, w) ∈ C0(Ω)×W 1,∞(Ω)×W 1,∞(Ω) : u, v and w are radially symmetric and positive in Ω } with respect to the topology in Lσ(Ω) × W 1,2(Ω) × W 1,2(Ω), that is, for all (u0, v0, w0) ∈ Y and all ε > 0 there exists (u0ε, v0ε, w0ε) ∈ B(M,A) such that ∥u0ε − u0∥Lσ(Ω) + ∥v0ε − v0∥W 1,2(Ω) + ∥w0ε − w0∥W 1,2(Ω) < ε, where the corresponding solution (uε, vε, wε) of (1.1) with initial data (uε, vε, wε)|t=0 = (u0ε, v0ε, w0ε) blows up in a finite time T in the sense that lim sup t↗T ∥uε(·, t)∥L∞(Ω) = ∞. Key for the proof. The proof relies on two key ingredients. The first one is to establish the following inequality given in Section 3, F(u, z) ≥ −c1 · (Dθ(u, z) + 1) ( ∃θ ∈ (1 2 , 1 )) , where c1 > 0. The second one is to establish the following inequality given in Section 4, d dt F(u, z) + 1 2 D(u, z) ≤ c2, where c2 > 0. By combining the above two inequalities, we obtain d dt ( − 1 c1 F(u, z)− 1 ) ≥ −c2 c1 + 1 2c1 ( − 1 c1 F(u, z)− 1 )1/θ + , which leads to the conclusion. 4 Y. CHIYO, T. UEMURA, T. YOKOTA EJDE-2025/81 2. Preliminaries We first state a result on local existence of classical solutions to (1.1). Lemma 2.1. Let Ω = BR ⊂ Rn (n ∈ {2, 3}, R > 0). Let q ∈ (n,∞). Then for all initial data (u0, v0, w0) ∈ C0(Ω)×W 1,∞(Ω)×W 1,∞(Ω), which is radially symmetric and positive in Ω, there exist Tmax = Tmax(u0, v0, w0) ∈ (0,∞] and a uniquely determined triplet (u, v, w) of radially symmetric and positive functions u ∈ C0([0, Tmax);C 0(Ω)) ∩ C2,1(Ω× (0, Tmax)), v, w ∈ C0([0, Tmax);W 1,q(Ω)) ∩ C2,1(Ω× (0, Tmax)) such that (u, v, w) is a classical solution of the system (1.1) in Ω×(0, Tmax), and that if Tmax < ∞, then lim sup t↗Tmax ∥u(·, t)∥L∞(Ω) = ∞. Moreover, ∫ Ω u(·, t) = ∫ Ω u0, (2.1)∫ Ω v(·, t) ≤ max {α β ∫ Ω u0, ∫ Ω v0 } , (2.2)∫ Ω w(·, t) ≤ max {γ δ ∫ Ω u0, ∫ Ω w0 } (2.3) for all t ∈ (0, Tmax). Proof. The existence of classical solutions is shown by fixed point arguments and parabolic reg- ularity theory (see e.g., [8, Lemma 2.1]. [1, Theorems 14.4, 14.6 and 15.6]), and the uniqueness is proved by the Gronwall-type argument. Also the positivity of solutions is obtained by the positivity of initial data and the maximum principle. Moreover, integrating the first, second and third equations in (1.1) over Ω, using the Neumann boundary conditions and applying an ODE comparison principle imply (2.1), (2.2) and (2.3). Finally, by uniqueness of solutions, we see that u, v, w are radially symmetric. □ We next give the property of the Neumann heat semigroup which will be used later. For the proof, see [10, Lemma 1.3]. Lemma 2.2. Let (et∆)t≥0 be the Neumann heat semigroup in Ω, and let λ1 > 0 denote the first nonzero eigenvalue of −∆ in Ω under Neumann boundary conditions. Then there exists C > 0 depending only on Ω such that if 1 ≤ q ≤ p ≤ ∞, then ∥et∆φ∥Lp(Ω) ≤ C(1 + t− n 2 ( 1 q − 1 p ))e−λ1t∥φ∥Lq(Ω) for all φ ∈ Lq(Ω) and t ∈ (0, Tmax). 3. Estimates for the Liapunov functional In this section we estimate the Liapunov functional F defined in (1.4) by the dissipation rate D defined in (1.6). To see this we define the set S(M,M̃,B, κ) := { (u, z) ∈ C1(Ω)× C2(Ω) : u, z are radially symmetric, u is positive,∫ Ω u = M, ∫ Ω |z| ≤ M̃, ∂z ∂ν = 0 on ∂Ω, |z(x)| ≤ B|x|−κ∀x ∈ Ω } for constants M,M̃,B > 0 and κ satisfying κ = 2 when n = 2, and κ > 1 when n = 3. The aim of this section is to establish the following proposition. EJDE-2025/81 ATTRACTION-REPULSION CHEMOTAXIS SYSTEMS 5 Proposition 3.1. Let n ∈ {2, 3}. Let κ = 2 when n = 2, and κ > 1 when n = 3. Assume that p ≥ 2 and p−m > 2 n . Let θ := 1 1 + n (2n+4)κ ∈ ( 1 2 , 1). Then there exists C(M,κ, s0) > 0 such that F(u, z) ≥ −C(M,κ, s0)(1 + M̃2 +B 2n+4 n+4 )(Dθ(u, z) + 1) holds for all (u, z) ∈ S(M, M̃,B, κ), where F and D are given in (1.4), (1.5) and (1.6) with any fixed s0 > 1. Proof. We first note that [3, Theorem 3.6] for n = 3 and [4, Lemma 2.6] for n = 2 are valid even if we do not assume z ≥ 0 (see [7, Appendix A]). Hence, to derive the desired inequality, we confirm the assumptions in these statements. Case n = 3. In this case the proof can be completed by verifying the existence of γ ∈ (0, n−2 n ) = (0, 1 3 ) and b, c > 0 such that H(s) ≤ γG(s) + b(s+ 1) for all s > 0, (3.1) s(s+ 1)p−2 ≥ cs for all s ≥ 0. (3.2) We first consider (3.1). By the definition of the function G (see (1.5)), we have G(s) = ∫ s s0 ∫ σ s0 (τ + 1)m−1 τ(τ + 1)p−2 dτdσ ≥ ∫ s s0 ∫ σ s0 (τ + 1)m−1 (τ + 1)(τ + 1)p−2 dτdσ = ∫ s s0 ∫ σ s0 (τ + 1)m−p dτdσ for all s > 0. In the case m− p+1 > 0, it follows from the definition of the function H (see (1.7)) that there exist c1, c2 > 0 such that G(s) ≥ 1 m− p+ 1 ∫ s s0 ( (σ + 1)m−p+1 − (s0 + 1)m−p+1 ) dσ = 1 m− p+ 1 {∫ s 0 (σ + 1)m−p+1 dσ − ∫ s0 0 (σ + 1)m−p+1 dσ − ∫ s s0 (s0 + 1)m−p+1 dσ } = 1 m− p+ 1 (H(s)− c1 − c2s), which means H(s) ≤ (m− p+ 1)G(s) + c1 + c2s for all s > 0. We now set γ := m− p+ 1 and b := max{c1, c2}. Then, since m− p+ 1 < n−2 n by the condition p−m > 2 n , we see that H(s) ≤ γG(s) + b(s+ 1) (3.3) for all s > 0 with γ ∈ (0, n−2 n ). On the other hand, in the case m− p+ 1 ≤ 0 i.e. p−m ≥ 1, the definition of H implies H(s) = ∫ s 0 (σ + 1)m−p+1 dσ ≤ ∫ s 0 1 dσ = s for all s > 0. Thus, for all γ ∈ (0, n−2 n ), the inequality (3.3) with b = 1 holds. Hence we see that (3.1) holds in the case n = 3. We can also check (3.2) as s(s+ 1)p−2 ≥ s for all s ≥ 0 since p ≥ 2. Case n = 2. To complete the proof we verify the existence of b, c > 0 such that H(s) ≤ b s log s for all s ≥ s0, (3.4) s(s+ 1)p−2 ≥ cs for all s ≥ 0, (3.5) 6 Y. CHIYO, T. UEMURA, T. YOKOTA EJDE-2025/81 where s0 > 1 is a constant appearing in (1.5). To prove (3.4) we consider the two cases: m−p+2 ̸= 0 and m− p+ 2 = 0. From the definition of H, we infer that H(s) = ∫ s 0 (σ + 1)m−p+1 dσ = { (s+1)m−p+2−1 m−p+2 when m− p+ 2 ̸= 0, log(s+ 1) when m− p+ 2 = 0 = { (s+1)m−p+2−1 m−p+2 · log s s · s log s when m− p+ 2 ̸= 0, log(s+ 1) · log s s · s log s when m− p+ 2 = 0 (3.6) for all s ≥ s0. We first focus on the case m− p+ 2 ̸= 0. Dropping a negative term yields (s+ 1)m−p+2 − 1 m− p+ 2 · log s s ≤ { (s+1)m−p+2 m−p+2 · log s s when m− p+ 2 > 0, −1 m−p+2 · log s s when m− p+ 2 < 0 for all s ≥ s0. In the case m− p+ 2 > 0, since s ≥ s0 > 1, noting that (s+ 1)m−p+2 s ≤ (s+ s)m−p+2 s = 2m−p+2 sp−m−1 and p −m > 2 n = 1, we deduce from boundedness of the function log s s (s ≥ 1) that there exists b > 0 such that (s+ 1)m−p+2 − 1 m− p+ 2 · log s s ≤ b (3.7) for all s ≥ s0. Also, in the case m− p+2 < 0, since −1 m−p+2 > 0, we can derive (3.7) for all s ≥ s0 with some b > 0. On the other hand, in the case m− p+ 2 = 0 i.e. p−m = 2, we can prove that there exists b > 0 such that log(s+ 1) · log s s ≤ b (3.8) for all s ≥ s0. Combining (3.7) and (3.8) with (3.6) leads to (3.4). Finally, since (3.5) and (3.2) are identical, the proof is complete. □ 4. Proofs of Theorems 1.1 and 1.2 In this section we denote by (u, v, w) the local classical solution of (1.1) given in Lemma 2.1 and by Tmax ∈ (0,∞] its maximal existence time. To prove Theorems 1.1 and 1.2 we will establish two lemmas. The first one provides an L2-estimate for v. Lemma 4.1. Let n ∈ {2, 3}. Let M > 0 and A > 0. Then there is C = C(M,A) > 0 such that whenever 0 < u0 ∈ C0(Ω), ∫ Ω u0 ≤ M, ∥v0∥L2(Ω) ≤ A, all solutions of (1.1) satisfy ∥v(·, t)∥L2(Ω) ≤ C for all t ∈ (0, Tmax). (4.1) Proof. From the representation formula v(·, t) = et(∆−β)v0 + α ∫ t 0 e(t−s)(∆−β)u(·, s) ds, we employ Lemma 2.2 to find c1, c2 > 0 such that ∥v(·, t)∥L2(Ω) ≤ ∥et(∆−β)v0∥L2(Ω) + ∥∥∥α ∫ t 0 e(t−s)(∆−β)u(·, s) ds ∥∥∥ L2(Ω) ≤ c1e −βt∥v0∥L2(Ω) + c2 ∫ t 0 e−β(t−s)(1 + (t− s)− n 2 ( 1 1− 1 2 ))∥u(·, s)∥L1(Ω) ds for all t ∈ (0, Tmax). It follows from (2.1) and the assumption ∫ Ω u0 ≤ M that ∥u(·, s)∥L1(Ω) ≤ M EJDE-2025/81 ATTRACTION-REPULSION CHEMOTAXIS SYSTEMS 7 for all s ∈ (0, Tmax). Moreover, since −n 2 ( 1 1 − 1 2 ) > −1 from condition n < 4, we obtain supt>0 ∫ t 0 e−β(t−s)(1 + (t − s)− n 2 ( 1 1− 1 2 )) ds < ∞. Consequently, there exists c3 = c3(M,A) > 0 such that ∥v(·, t)∥L2(Ω) ≤ c3 for all t ∈ (0, Tmax), which means (4.1) holds. □ We next show that F (defined in (1.4)) satisfies an inequality that serves as a substitute for the Liapunov functional. Lemma 4.2. Let 0 < u0 ∈ C0(Ω), 0 < v0 ∈ W 1,∞(Ω), 0 < w0 ∈ W 1,∞(Ω). Assume that θ = χα− ξγ > 0. Then every solution (u, v, w) of (1.1) satisfies that if ∥v(·, t)∥L2(Ω) ≤ K for all t ∈ (0, Tmax) (4.2) with some K > 0, then d dt F(u, z) + 1 2 D(u, z) ≤ χ2(δ − β)2 2 K2, (4.3) where z = χv − ξw. Proof. By the definition of F , we have d dt F(u, z) = d dt {1 2 ∫ Ω |∇z|2 + 1 2 ∫ Ω z2 − θ ∫ Ω uz + θ ∫ Ω G(u) } = − ∫ Ω ∆z · zt + δ ∫ Ω zzt − θ ∫ Ω uzt − θ ∫ Ω utz + θ ∫ Ω G′(u)ut. (4.4) Here, by the first equation in (1.2) and integration by parts, we can rewrite the term ∫ Ω utz as∫ Ω utz = ∫ Ω z∇ · ((u+ 1)m−1∇u)− ∫ Ω z∇ · (u(u+ 1)p−2∇z) = − ∫ Ω (u+ 1)m−1∇u · ∇z + ∫ Ω u(u+ 1)p−2|∇z|2. (4.5) Next we consider ∫ Ω G′(u)ut. From the first equation in (1.2), integration by parts and the definition of G (see (1.5)), we see that∫ Ω G′(u)ut = ∫ Ω G′(u)∇ · ( (u+ 1)m−1∇u− u(u+ 1)p−2∇z ) = − ∫ Ω G′′(u)∇u · ( (u+ 1)m−1∇u− u(u+ 1)p−2∇z ) = − ∫ Ω (u+ 1)m−1 u(u+ 1)p−2 ∇u · ( (u+ 1)m−1∇u− u(u+ 1)p−2∇z ) = − ∫ Ω (u+ 1)2(m−1) u(u+ 1)p−2 |∇u|2 + ∫ Ω (u+ 1)m−1∇u · ∇z. (4.6) Here we compute∫ Ω (u+ 1)2(m−1) u(u+ 1)p−2 |∇u|2 = ∫ Ω u(u+ 1)p−2 ∣∣∣ (u+ 1)m−1 u(u+ 1)p−2 ∇u−∇z ∣∣∣2 + 2 ∫ Ω (u+ 1)m−1∇u · ∇z − ∫ Ω u(u+ 1)p−2|∇z|2. Substituting this to (4.6) and using (4.5), we obtain∫ Ω G′(u)ut = − ∫ Ω u(u+ 1)p−2 ∣∣∣ (u+ 1)m−1 u(u+ 1)p−2 ∇u−∇z ∣∣∣2 − ∫ Ω (u+ 1)m−1∇u · ∇z + ∫ Ω u(u+ 1)p−2|∇z|2 = − ∫ Ω u(u+ 1)p−2 ∣∣∣ (u+ 1)m−1 u(u+ 1)p−2 ∇u−∇z ∣∣∣2 + ∫ Ω utz. 8 Y. CHIYO, T. UEMURA, T. YOKOTA EJDE-2025/81 Combining this with (4.4) and integrating by parts yield d dt F(u, z) = ∫ Ω (−∆z + δz − θu)zt − θ ∫ Ω u(u+ 1)p−2 ∣∣∣ (u+ 1)m−1 u(u+ 1)p−2 ∇u−∇z ∣∣∣2. (4.7) Moreover, we see from the second equation in (1.2) that∫ Ω (−∆z + δz − θu)zt = − ∫ Ω (∆z − δz + θu)(∆z − δz + θu+ χ(δ − β)v) = −χ(δ − β) ∫ Ω v(∆z − δz + θu)− ∫ Ω (∆z − δz + θu)2. Inserting this into (4.7) and using the definition of D entail d dt ∫ Ω F(u, z) = −χ(δ − β) ∫ Ω v(∆z − δz + θu) − ∫ Ω (∆z − δz + θu)2 − θ ∫ Ω u(u+ 1)p−2 ∣∣∣ (u+ 1)m−1 u(u+ 1)p−2 ∇u−∇z ∣∣∣2 = χ(δ − β) ∫ Ω v(−∆z + δz − θu)−D(u, z), which means d dt ∫ Ω F(u, z) +D(u, z) = χ(δ − β) ∫ Ω v(−∆z + δz − θu). (4.8) Finally, applying the Young inequality and (4.2) as well as noting from the assumption θ > 0 that∫ Ω (∆z − δz + θu)2 ≤ D(u, z), we observe that χ(δ − β) ∫ Ω v(−∆z + δz − θu) ≤ 1 2 ∫ Ω (∆z − δz + θu)2 + χ2(δ − β)2 2 ∫ Ω v2 ≤ 1 2 D(u, z) + χ2(δ − β)2 2 K2, which along with (4.8) leads to the conclusion (4.3). □ We are now in a position to prove Theorems 1.1 and 1.2. Proof of Theorem 1.1. Let n ∈ {2, 3}. Assume that χα − ξγ > 0, p ≥ 2 and p − m > 2 n . Let M,A > 0 and κ = 2 when n = 2, and κ > 1 when n = 3. Then, using [7, Lemma 4.4], we see that there exist M̃,B > 0 such that if u0, v0, w0 are radially symmetric and satisfy 0 < u0 ∈ C0(Ω), 0 < v0 ∈ W 1,∞(Ω), 0 < w0 ∈ W 1,∞(Ω),∫ Ω u0 = M, ∥v0∥L1(Ω) ≤ A, ∥z0∥L1(Ω) ≤ A, ∥∇z0∥L2(Ω) ≤ A with z0 = χv0 − ξw0, then the corresponding solution (u, z, v) of (1.2) satisfies (u(·, t), z(·, t)) ∈ S(M,M̃,B, κ) for all t ∈ (0, Tmax) by Lemma 2.1 and [7, Lemma 4.2]. Let us show that there exists K(M,A) > 0 such that if F(u0, z0) < −K(M,A), then the solution (u, z, v) blows up in finite time. It follows from Proposition 3.1 that there exists c1 = c1(M,M̃,B, κ, s0) > 0 such that F(u, z) ≥ −c1 · (Dθ(u, z) + 1), where θ ∈ ( 12 , 1), which is rewritten as D(u, z) ≥ ( − 1 c1 F(u, z)− 1 )1/θ + . (4.9) Also, by Lemmas 4.1 and 4.2, we obtain d dt F(u, z) + 1 2 D(u, z) ≤ c2 := χ2(δ − β)2 2 K2, EJDE-2025/81 ATTRACTION-REPULSION CHEMOTAXIS SYSTEMS 9 which means d dt F(u, z) ≤ c2 − 1 2 D(u, z). Combining (4.9) with this inequality yields d dt ( − 1 c1 F(u, z)− 1 ) ≥ −c2 c1 + 1 2c1 ( − 1 c1 F(u, z)− 1 )1/θ + . We now set y(t) := − 1 c1 F(u, z)− 1, and then we have y′(t) ≥ 1 2c1 y(t) 1/θ + − c2 c1 . Also, letting K(M,A) = c1 ( ( 2c2c1 )θ + 1 ) , we see that if F(u0, z0) < −K(M,A), then y(0) = − 1 c1 F(u0, z0)− 1 > K(M,A) c1 − 1 = (2c2 c1 )θ , which leads to 1 2 y(0) 1 θ + − c2 c1 > 0. Consequently, y blows up in finite time. Therefore, F(u, z) and thus u must blow up in finite time, which completes the proof of Theorem 1.1. □ Proof of Theorem 1.2. This result is derived from an argument similar to that in the proofs of [11, Lemma 6.1] and [7, Theorem 1.1]. □ Acknowledgments. The authors would like to thank the anonymous referees for their comments on improving this paper. Y. Chiyo was partially supported by JSPS KAKENHI grant number JP24K22838. T. Yokota was partially supported by JSPS KAKENHI grant number JP21K03278. References [1] H. 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Pures Appl., 100 (2013), no. 5, 748–767. [12] M. Winkler; Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities. J. Differential Equations, 266 (2019), no. 12, 8034–8066. Yutaro Chiyo Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162- 8601, Japan Email address: ycnewssz@gmail.com 10 Y. CHIYO, T. UEMURA, T. YOKOTA EJDE-2025/81 Takeshi Uemura Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162- 8601, Japan Email address: uemura.tus@gmail.com Tomomi Yokota Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162- 8601, Japan Email address: yokota@rs.tus.ac.jp 1. Introduction 2. Preliminaries 3. Estimates for the Liapunov functional 4. Proofs of Theorems 1.1 and 1.2 Acknowledgments References