Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 09, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.09 GLOBAL BOUNDEDNESS IN AN INDIRECT CHEMOTAXIS-CONSUMPTION MODEL WITH SIGNAL-DEPENDENT DEGENERATE DIFFUSION CHUN WU Abstract. In this article, we consider the consumption chemotaxis system ut = ∆(uvα) + au− buγ , (x, t) ∈ Ω× (0,∞), vt = ∆v − uvw, (x, t) ∈ Ω× (0,∞), wt = −δw + u, (x, t) ∈ Ω× (0,∞), on a smooth bounded domain Ω ⊂ Rn, n ≥ 2 with homogeneous Neumann boundary conditions, where a > 0, b > 0, γ ≥ 2, and δ > 0. We shown that for sufficiently regular initial data, the associated initial-boundary value problem possesses global bounded classical solutions. 1. Introduction In 1971, Keller and Segel [16] proposed the model ut = ∆u− χ∇ · (u∇v) + f(u), x ∈ Ω, t > 0, vt = ∆v − uv, x ∈ Ω, t > 0, (1.1) to describe the movement of bacteria toward oxygen, and at the same time, oxygen is consumed by the bacteria, where u = u(x, t) denotes the density of the bacteria, where v = v(x, t) represents the oxygen concentration, and χ ∈ R represents the chemotactic sensitivity coefficient. In the above system and its various variants, we know that chemotaxis is the directed movement of cells or organisms in response to a concentration gradient of a chemical stimulus, and it plays a crucial role in a wide variety of biological processes. For f(u) = 0, Zhang and Li [54] proved that (1.1) admits a global classical solution (u, v) and that this solution converges exponentially to (ū, 0) as t → ∞ if either N ≤ 2 or χ ≤ 1 6(N + 1)∥v0∥L∞(Ω) , N ≥ 3, where ū0 := 1 |Ω| ∫ Ω u0. The global weak solution in a three-dimensional domain has been studied by Tao and Winkler [31]. In [31, 54], and Baghaei and Khelghati in [3] improved the result and showed that system (1.1) has a globally bounded classical 2020 Mathematics Subject Classification. 35K35, 35B40, 35A01, 92C17. Key words and phrases. Global boundedness; indirect chemotaxis-consumption; signal-dependent motility. ©2025. This work is licensed under a CC BY 4.0 license. Submitted November 6, 2024. Published January 22, 2025. 1 2 C. WU EJDE-2025/09 solution under the condition ∥v0∥L∞(Ω) ≤ π χ √ 2(N+1) . When we replace uv by uf(v), where f ∈ C1([0,∞)) is nonnegative with f(0) = 0, the global generalized solution was proved by Winkler [43] in any dimension. In the case of N ≥ 1, b > C1(N)∥χv0∥ 1 n L∞(Ω) + C2(N)∥χv0∥2NL∞(Ω), a global bounded classical solution of the system (1.1) was obtained by Lankeit and Wang [23] in 2017. In addition, the variant of (1.1) has been studied in [44, 45, 46, 47]. In regard to (1.1), the utilization of chemotaxis signal by cells may be more complex in realistic situations. The signal may originate from external substances produced indirectly, or consist of several signals produced by different mechanisms, as noted in [33]. In particular, a chemotaxis system with an indirect consumption of signals has been taken into consideration in [11]: ut = ∆u−∇ · (u∇v) + f(u), x ∈ Ω, t > 0, vt = ∆v − vw, x ∈ Ω, t > 0, wt = −δw + u, x ∈ Ω, t > 0. (1.2) When f(u) ≡ 0, assuming that n ≤ 2 or n ≥ 3 with ∥v0∥L∞(Ω) ≤ 1 3n , Fuest [11] proved that (1.2) admits a globally bounded classical solution, which converges to the constant steady state (ū0, 0, ū0/δ) as time goes to infinity. When f(u) = µu(1−u), µ > 0, if µ is suitably large, Li et al. [21] proved that (1.2) has a globally bounded classical solution. Many of the results related to the qualitative analysis of indirect signal mechanisms can be found in [5, 12, 19]. Recently, the research interest of scholars has gradually shifted to chemotaxis sys- tems with signal-dependent movements [4, 25]. For example, the following Keller- Segel production models with signal-dependent motility ut = ∆(γ(v)u) + f(u), x ∈ Ω, t > 0, vt = ∆v − v + u, x ∈ Ω, t > 0. (1.3) If f(u) = 0 and kγ ≤ γ(s) ≤ Kγ for all s ≥ 0, where kγ ,Kγ > 0, in the case of two dimensions, Tao and Winkler [34] proved that (1.3) has global bounded classical solutions; However, (1.3) has global weak solutions in the high-dimensional case. In particular, under the conditions γ(s) = c0/s k(c0, k > 0), if c0 is small enough, the existence of global classical solutions has been investigated in [53]. If γ(s) = s−α with α > 0, scholars have also obtained some results on the global existence of classical solutions [1, 8, 9, 15, 40]. If γ(s) = e−s for all s ≥ 0, [7, 14] show a phenomenon of critical mass for (1.3) in the two-dimensional case. Further results on (1.3) can be found in [2, 10, 48]. If f(u) = µu(1 − u), µ > 0 and γ(s) satisfies γ(s) > 0, γ′(s) < 0 and lims→+∞ γ′(s)/γ(s) exists, in the two-dimensional settings, Jin et al. [13] proved the existence of a global classical solution to (1.3). Similar results in higher dimensions were proved by [22, 41]. For some other results on (1.3), see [6, 26, 27, 28]. On the other hand, if the signal is degraded rather than produced by the cells, the consumption of chemotaxis with signal-dependent motility has also been taken into account. ut = ∆(γ(v)u) + f(u), x ∈ Ω, t > 0, vt = ∆v − uv, x ∈ Ω, t > 0. (1.4) If f(u) = 0, γ is strictly positive on [0,∞), the existence of global bounded classical solutions has been shown by Li and Zhao [18], provided that ∥v0∥L∞(Ω) is sufficiently EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 3 small. In the case of n = 2, the smallness assumption of ∥v0∥L∞(Ω) was removed by Li and Winkler [20], who showed that (1.4) has global classical bounded solutions. When n ≥ 1, (1.4) possesses global very weak solutions for some weaker regularity properties on γ. If γ(s) = s−α, α > 0, Tao and Winkler [35] proved that there are global weak solutions to (1.4) provided that 2 ≤ n ≤ 5 and α > n−2 6−n . If γ(s) = sα, α > 0 for all s ≥ 0, there are some other results in [50, 52, 51]. When f(u) = au−bul, a > 0, b > 0, Wang [37] proved that if one of the following 3 conditions is satisfied: (i) n ≤ 2 and l > 1, (ii) n ≥ 3 and l > 2, (iii) n ≥ 3, l = 2 and b is sufficiently large; then there exists a bounded classical solution in (1.4), while in the case of n ≥ 3 and l ∈ (1, 2], (1.4) admits at least one global weak solution which becomes smooth after some waiting time. If γ(s) = sα, α ≥ 1, Wang established a global classical solution in [38]. Conversely, (1.4) admits at least one global weak solution in the case α > 0 if γ has rather mild regularities. Moreover, if γ is suitably smooth with α > 1, then the above weak solutions eventually become smooth. Some scholars have also studied the system (1.4) in other situations, we refer the reader to [24, 29, 39]. Recently, Wang [36] studied the chemotaxis system ut = ∆u− χ∇ · (u∇v) + η(u− um), x ∈ Ω, t > 0, vt = ∆v − uθvw, x ∈ Ω, t > 0, 0 = ∆w − w + uα, x ∈ Ω, t > 0 (1.5) and obtained bounded classical solutions of the corresponding initial value problem. With the above work as a motivation, in this article we consider the φ(v) = vα for α > 0, and the system ut = ∆(uvα) + au− buγ , x ∈ Ω, t > 0, vt = ∆v − uvw, x ∈ Ω, t > 0, wt = −δw + u, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x), w(x, 0) = w0(x) x ∈ Ω, (1.6) where Ω ⊂ Rn is a smooth and bounded domain, ∂ ∂ν denotes the derivative with respect the outer normal of ∂Ω, and α > 0, a > 0, b > 0, γ ≥ 2. The initial data satisfy u0 ∈ C0(Ω) with u0 ≥ 0 in Ω, v0, w0 ∈ W 1,∞(Ω) with v0 > 0, w0 ≥ 0 in Ω. (1.7) Our main goal is to study the initial boundary value problem of (1.6) and consider its global bounded solutions in the classical sense. This is stated in the following theorem. Theorem 1.1. Let Ω ⊂ Rn, n ≥ 1, be a bounded domain with smooth boundary, the parameters a, b, δ > 0 and α ≥ 1. Suppose that the initial data satisfy (1.7), if on of the following 3 cases holds: (i) γ > 2, (ii) n ≤ 3 and γ = 2, (iii) n ≥ 4, γ = 2 and b > ( n− 2 2 ) n+2 n (n− 1)−2/n(n+ √ n) 4 nα 2(n+2) n ∥v0∥ α(n+2)−2 n L∞(Ω) 4 C. WU EJDE-2025/09 + 22n+5(n+ √ n+ 1)n(n− 2 + √ n) n+2 2 δ n+2 2 (n+ 2)(n− 1) n 2 ∥v0∥L∞(Ω); then (1.6) has a global bounded classical solution (u, v, w) with u ∈ C0(Ω̄× [0,∞)) ∩ C2,1(Ω̄× (0,∞)), v ∈ ∩θ>nC 0([0,∞);W 1,θ(Ω)) ∩ C2,1(Ω̄× (0,∞)), w ∈ C0(Ω̄× [0,∞)) ∩ C0,1(Ω̄× (0,∞)). Remark 1.2. When n ≥ 2, reference [36] shows the existence of classical solutions for system (1.5). For (1.5), if the first and third equations are replaced by ut = ∆(uvα) + au − buγ , wt = −δw + u and θ = 1 respectively, this paper considers the global boundedness for an indirect chemotaxis-consumption model with signal- dependent degenerate diffusion. 2. Preliminaries We first present a criterion for existence and extensibility of local solutions. Lemma 2.1. Let Ω ⊂ Rn (n ≥ 1) be a bounded domain with smooth boundary, and let a, b, δ, α > 0. Assume that (1.7) holds. Then model (1.6) possesses a nonnegative local classical solution (u, v, w) with u ∈ C0(Ω̄ × [0, Tmax)) ∩ C2,1(Ω̄ × (0, Tmax)), v ∈ ∩θ>nC 0([0, Tmax);W 1,θ(Ω))∩C2,1(Ω̄× (0, Tmax)) and w ∈ C0(Ω̄× [0, Tmax))∩ C0,1(Ω̄× (0, Tmax)). Moreover, if Tmax < ∞, then lim sup t↗Tmax ∥u(·, t)∥L∞(Ω) = ∞. Based on the established parabolic theory, the proof of the above lemma is similar to [13]. We omit it here. Lemma 2.2. Let the hypotheses of (1.7) hold. Then there exists C > 0 such that ∥u(·, t)∥L1(Ω) ≤ C for all t ∈ (0, Tmax), (2.1) 0 ≤ v ≤ ∥v0∥L∞(Ω) in Ω× (0, Tmax), (2.2)∫ t+ϵ t ∫ Ω uγ ≤ C for all t ∈ (0, Tmax − ϵ), (2.3) where ϵ = min{1, 1 2Tmax}. Proof. Integrating the first equation in (1.6) yields d dt ∫ Ω u = a ∫ Ω u− b ∫ Ω uγ ≤ a ∫ Ω u− b|Ω|1−γ (∫ Ω u )l for all t ∈ (0, Tmax), (2.4) upon an ODE comparison argument implies (2.1). By the non-negativity of u, v and w, using the maximum principle for the second equation of system (1.6) leads to (2.2). By integrating (2.4), we can easily obtain (2.3). □ Lemma 2.3 ([49, Lemma 3.4]). Assume p ≥ 2 and ϕ ∈ C2(Ω̄) is positive with ∂ϕ ∂ν on ∂Ω. Then∫ Ω ϕ−p−1|∇ϕ|p+2 ≤ (p+ √ n)2 ∫ Ω ϕ−p+3|∇ϕ|p−2|D2 lnϕ|2,∫ Ω ϕ−p+1|∇ϕ|p−2|D2ϕ|2 ≤ (p+ √ n+ 1)2 ∫ Ω ϕ−p+3|∇ϕ|p−2|D2 lnϕ|2. EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 5 Lemma 2.4 ([32]). Let T ∈ (0,∞], t0 ∈ (0, T ), λ > 0, k > 0. Assume that y : [0, T ) → [0,∞) is absolutely continuous and that y′(t) + λy(t) ≤ h(t) for all t ∈ (0, T ) with the nonnegative function h ∈ Ll loc([0, T )) fulfilling∫ t+t0 t h(s) ds ≤ k for all t ∈ (0, T − t0). Then y(t) ≤ max { y(0) + k, 2k + k λt0 } for all t ∈ (0, T ). Lemma 2.5 ([49, Lemma 3.5]). Suppose p ≥ 2 and µ > 0. Then there exists C = C(p, µ) > 0 such that for every positive ϕ ∈ C2(Ω̄) with ∂ϕ ∂ν = 0 on ∂Ω we have∫ ∂Ω ϕ−p+1|∇ϕ|p−2 ∂|∇ϕ|2 ∂ν ≤ µ ∫ Ω ϕ−p−1|∇ϕ|p+2 + µ ∫ Ω ϕ−p+1|∇ϕ|p−2|D2ϕ|2 + C ∫ Ω ϕ. Lemma 2.6. Assume that the initial value satisfies (1.7) and q ∈ [1, γ]. Then there exists C > 0 such that∫ Ω wq(·, t) ≤ C for all t ∈ (0, Tmax). (2.5) Proof. Using wq−1 to test the third equation of (1.6) and integrating gives 1 q d dt ∫ Ω wq = −δ ∫ Ω wq + ∫ Ω uwq−1 ≤ −δ ∫ Ω wq + δ 2 ∫ Ω wq + C ∫ Ω uq (2.6) for all t ∈ (0, Tmax) which implies (2.5) with (2.3) and Lemma 2.4. □ Lemma 2.7. Let T ∈ (0, Tmax). For some p > n 2 , there exists C1 > 0 such that ∥w(·, t)∥Lp(Ω) ≤ C1 for all t ∈ (0, Tmax). (2.7) Then there exists C2(T ) > 0 such that v(x, t) ≥ C2(T ) for all t ∈ (0, T ). (2.8) Proof. Let z(x, t) := ln 1 v(x,t) . Then, using the second equation in (1.6), we see that zt = ∆z − |∇z|2 + uw, x ∈ Ω, t > 0, ∂z ∂ν = 0, x ∈ ∂Ω, t > 0, z(x, 0) = z0(x) = ln 1 v0(x) , x ∈ Ω. (2.9) We use the variation-of-constants formula for the first equation of (2.9) to obtain z(·, t) ≤ et∆z0 + ∫ t 0 e(t−s)∆uw ds for all t ∈ (0, T ). 6 C. WU EJDE-2025/09 Then, by (2.7), (2.18) and the smoothing properties of the Neumann heat semigroup (et∆)t≥0 defined on Ω [42, Lemma 1.3], we can find constants c1, c2 > 0 such that ∥z(·, t)∥L∞(Ω) ≤ ∥et∆z0∥L∞(Ω) + ∫ t 0 ∥e(t−s)∆u(·, s)w(·, s)∥L∞(Ω) ds ≤ ∥z0∥L∞(Ω) + c1 ∫ t 0 {1 + (t− s)− n 2p }∥u(·, s)w(·, s)∥Lp(Ω) ds ≤ ∥z0∥L∞(Ω) + c1 ∫ t 0 {1 + (t− s)− n 2p }∥w(·, s)∥Lp(Ω)∥u(·, s)∥L∞(Ω) ds ≤ ∥z0∥L∞(Ω) + c2 ∫ T 0 (1 + σ− n 2p ) dσ for all t ∈ (0, T ). (2.10) From the definition of z, in view of p > n 2 and using (2.10), we can easily derive (2.8). □ Lemma 2.8. Let the initial value satisfy (1.7). If there exists C > 0 and q ≥ 1 such that q > n 2 and ∥u(·, t)∥Lq(Ω) ≤ C for all t ∈ (0, T ). (2.11) Then for all T ∈ (0, Tmax), there exists C(T ) > 0 such that ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥L∞(Ω) < C(T ) for all t ∈ (0, Tmax). (2.12) Proof. Testing the third equation in (1.6) with qwq−1 and applying Young’s in- equality to find a positive constant c1 such that d dt ∫ Ω wq = −qδ ∫ Ω wq + q ∫ Ω uwq−1 ≤ −qδ 2 ∫ Ω wq + c1 ∫ Ω uq for all t ∈ (0, Tmax). (2.13) From (2.11) and (2.13), it follows that for some c2 > 0,∫ Ω wq ≤ c2 for all t ∈ (0, Tmax). (2.14) From q > n/2, we have nq (n−q)+ > n. So, we can pick Θ > max{1, n 2 } such that nq (n−q)+ > 2Θ > n. By applying (2.14) and [17, Lemma 1.2], we conclude that there exists a positive constant c3 such that ∥∇v(·, t)∥L2Θ(Ω) ≤ c3 for all t ∈ (0, Tmax). Moreover, for arbitrary T ∈ (0, Tmax), (2.14) and Lemma 2.7 imply v > c4(T ) for some c4(T ) > 0 in Ω× (0, T ). For all p > 1, multiplying the first equation of (1.6) by pup−1 and employing Young’s inequality, we can arrive that d dt ∫ Ω up + cα4 (T )p(p− 1) 2 ∫ Ω up−2|∇u|2 + ∫ Ω up ≤ α2c5(T )p(p− 1) 2 ∫ Ω up|∇v|2 + (ap+ 1) ∫ Ω up (2.15) EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 7 for all t ∈ (0, Tmax), where c5(T ) = max{∥v0∥α−2 L∞(Ω), c α−2 4 (T )}. It is clear from the previously mentioned range of Θ that 2Θ Θ−1 < 2n (n−2)+ , using an Ehrling type inequality and Lemma 2.2, one can find c7 = c7(p) > 0 such that c5 ∫ Ω up|∇v|2 ≤ c5∥u p 2 ∥2 L 2Θ Θ−1 (Ω) ∥∇v∥2L2Θ(Ω) ≤ c5c 2 3∥u p 2 ∥2 L 2Θ Θ−1 (Ω) ≤ c4 ∫ Ω |∇u p 2 |2 + c7 (2.16) for all t ∈ (0, Tmax). By combining (2.15) and (2.16) and applying an ODE ar- gument, we conclude that there exists a constant c8 = c8(p, T ) > 0 such that ∥u(·, t)∥Lp(Ω) ≤ c8 for all t ∈ (0, Tmax). Using [17, Lemma 1.2] again, we can find that c9 > 0 such that ∥∇v∥L∞(Ω) ≤ c9 for all t ∈ (0, Tmax). (2.17) In accordance with [30, Lemma A.1], we can find a positive constant c10(T ) such that ∥u(·, t)∥L∞(Ω) ≤ c10(T ) for all t ∈ (0, Tmax). (2.18) Applying the variation-of-constants formula for w, we have w(·, t) = e−δtw0 + ∫ t 0 e−δ(t−s)u(·, s) ds, which implies from (2.18) that there exists c11(T ) > 0 such that ∥w(·, t)∥L∞(Ω) ≤ c11(T ) for all t ∈ (0, Tmax). (2.19) Thus, by (2.17)–(2.19) and the extensibility criterion from Lemma 2.1, the proof is complete. □ 3. Existence of global solutions In this section, we will provide a proof of the main theorem. Lemma 3.1. Assume that the initial value satisfies (1.7). If γ > 1, then for all p > 1, there exist C1, C2 > 0 such that d dt ∫ Ω up + p(p− 1) 2 ∫ Ω up−2vα|∇u|2 ≤ α2p(p− 1) 2 ∫ Ω upvα−2|∇v|2 + ap ∫ Ω up − bp ∫ Ω up+γ−1 for all t ∈ (0, Tmax) (3.1) and d dt ∫ Ω wp+1 + (p+ 1)δ 2 ∫ Ω wp+1 ≤ ( 2 δ )p ∫ Ω up+1 for all t ∈ (0, Tmax), (3.2) where K = max0≤s≤∥v0∥L∞(Ω) |ϕ′(s)|√ min0≤s≤∥v0∥L∞(Ω) ϕ(s) . 8 C. WU EJDE-2025/09 Proof. Multiplying the first equation of (1.6) by pup−1 and applying Young’s in- equality, we obtain d dt ∫ Ω up = −p(p− 1) ∫ Ω up−2vα|∇v|2 − αp(p− 1) ∫ Ω up−1vα−1∇u · ∇v + ap ∫ Ω up − bp ∫ Ω up+γ−1 ≤ −p(p− 1) 2 ∫ Ω up−2vα|∇v|2 + α2p(p− 1) 2 ∫ Ω upvα−2|∇v|2 + ap ∫ Ω up − bp ∫ Ω up+γ−1 for all t ∈ (0, Tmax), (3.3) which results in (3.1). The third equation in the system (1.6) is tested using (p+ 1)wp, then applying Young’s inequality to this resultant ensures that (3.2) is true. □ Lemma 3.2. Under the assumptions of (1.7), for all p ≥ 2, we have d dt ∫ Ω v−p+1|∇v|p + p(p− 1) ∫ Ω v−p+3|∇v|p−2|D2 ln v|2 ≤ p 2 ∫ ∂Ω v−p+1|∇v|p−2 · ∂|∇v|2 ∂v + p(p− 2 + √ n) ∫ Ω wv−p+2|∇v|p−2|D2v| (3.4) for all t ∈ (0, Tmax). Proof. Applying integration by parts to the second equation in (1.6) and using the well-known equation 2∇v · ∇∆v = ∆|∇v|2 − 2|D2v|2, we find that d dt ∫ Ω v1−p|∇v|p = p ∫ Ω v1−p|∇v|p−2∇v · ∇(∆v − uvw)− (p− 1) ∫ Ω v−p|∇v|p(∆v − uvw) = p 2 ∫ Ω v1−p|∇v|p−2(∆|∇v|2 − 2|D2v|2)− p ∫ Ω v1−p|∇v|p−2∇v · ∇(uvw) − (p− 1) ∫ Ω v−p|∇v|p∆v + (p− 1) ∫ Ω wv1−p|∇v|p = p(p− 1) ∫ Ω v−p|∇v|p−2∇v · ∇|∇v|2 − p ∫ Ω v1−p|∇v|p−2|D2v|2 − p(p− 2) 4 ∫ Ω v1−p|∇v|p−4|∇|∇v|2|2 − p(p− 1) ∫ Ω v−p−1|∇v|p+2 + p 2 ∫ ∂Ω v1−p|∇v|p−2 · ∂|∇v|2 ∂ν + p(p− 2) 2 ∫ Ω wv−p+2|∇v|p−4∇v · ∇|∇v|2 + p ∫ Ω wv−p+2|∇v|p−2∆v − (p− 1)2 ∫ Ω wv1−p|∇v|p. (3.5) EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 9 The pointwise identity [49, Lemma 3.2] and ∇|∇v|2 = 2D2v · ∇v imply that p(p− 1) ∫ Ω v−p|∇v|p−2∇v · ∇|∇v|2 − p ∫ Ω v1−p|∇v|p−2|D2v|2 − p(p− 2) 4 ∫ Ω v1−p|∇v|p−4|∇|∇v|2|2 − p(p− 1) ∫ Ω v−p−1|∇v|p+2 = −p(p− 1) ∫ Ω v−p+3|∇v|p−2( 1 v4 |∇v|4 − 1 v3 ∇v · ∇|∇v|2 + 1 v2 |D2v|2) = −p(p− 1) ∫ Ω v−p+3|∇v|p−2|D2 ln v|2. (3.6) Using the well-known inequality |∆v| ≤ √ n|D2v|, the sixth and seventh summands on the right hand side of (3.5) can be estimated as p(p− 2) 2 ∫ Ω wv−p+2|∇v|p−4∇v · ∇|∇v|2 + p ∫ Ω wv−p+2|∇v|p−2∆v ≤ p(p− 2 + √ n) ∫ Ω wv−p+2|∇v|p−2|D2v|. (3.7) Substituting (3.6) and (3.7) into (3.5) yields the result of this lemma. □ Lemma 3.3. Let γ > 1 and p > 1. If q ≥ 2 satisfies q < 2(p+ γ − 2). Then there exists a constant C > 0 such that∫ Ω u q+2 2 ≤ (q + 2)δ 8k1 · (δ 2 )q/2 · bp 4 ∫ Ω up+γ−1 + C for all t ∈ (0, Tmax), (3.8) where k1 is a positive constant. Proof. From q < 2(p + γ − 2), we have q+2 2 < p + γ − 1. Thus, we derive from q+2 2 + (p+ γ − 2− q 2 ) = p+ γ − 1 that p+ γ − 1− q + 2 2 = p+ γ − 2− q 2 > 0. Applying Young’s inequality to 8k1 (q+2)δ ( 2 δ ) q 2 ∫ Ω u q+2 2 , it is easy to see that for some c > 0, 8k1 (q + 2)δ ( 2 δ )q/2 ∫ Ω u q+2 2 ≤ bp 4 ∫ Ω up+γ−1 + c for all t ∈ (0, Tmax). (3.9) The proof is complete. □ Lemma 3.4. Let α ≥ 1, γ > 1 and p > 2 satisfying q > 2p γ−1 . Then one can find a positive constant C such that p(p− 1) 2 ( q(q − 1) 2p(p− 1)(q + √ n)2 )−2/q α 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q ≤ bp 4 ∫ Ω up+γ−1 + C (3.10) for all t ∈ (0, Tmax). Proof. From q > 2p γ−1 , we have q(γ − 1)− 2p > 0, then p+ γ − 1− p(q + 2) q = (γ − 1)q − 2p q > 0. 10 C. WU EJDE-2025/09 Thus, we apply Young’s inequality to p(p− 1) 2 ( q(q − 1) 2p(p− 1)(q + √ n)2 )−2/q α 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q we show that there exists c > 0 such that p(p− 1) 2 ( q(q − 1) 2p(p− 1)(q + √ n)2 )−2/q α 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q ≤ bp 4 ∫ Ω up+γ−1 + c for all t ∈ (0, Tmax). Therefore, we obtain the desired result. □ Lemma 3.5. Assume (1.7) holds, α ≥ 1, γ > 2. Then, for all p > 1 and q ≥ 2, one can find a positive constant C such that ∥u(·, t)∥Lp(Ω) ≤ C for all t ∈ (0, Tmax). (3.11) Proof. It follows from Lemms 3.1 and 3.2 that d dt (∫ Ω up + ∫ Ω v−q+1|∇v|q ) + ∫ Ω up + ∫ Ω v−q+1|∇v|q + q(q − 1) ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 ≤ α2p(p− 1) 2 ∫ Ω upvα−2|∇v|2 + (ap+ 1) ∫ Ω up − bp ∫ Ω up+l−1 + q(q − 2 + √ n) ∫ Ω wv−q+2|∇v|q−2|D2v| + q 2 ∫ ∂Ω v−q+1|∇v|q−2 · ∂|∇v|2 ∂ν + ∫ Ω v−q+1|∇v|q. (3.12) Using Young’s inequality and Lemma 2.3, we see that for any ζ > 0, α2p(p− 1) 2 ∫ Ω upvα−2|∇v|2 ≤ p(p− 1) 2 ζ q+2 2 ∫ Ω v−q−1|∇v|q+2 + p(p− 1) 2 ζ− q+2 q α 2(q+2) q ∫ Ω u p(q+2) q v α(q+2)−2 q ≤ p(p− 1)(q + √ n)2 2 ζ q+2 2 ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 + p(p− 1) 2 ζ− q+2 q α 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q . (3.13) By selecting ζ = ( q(q − 1) 2p(p− 1)(q + √ n)2 ) 2 q+2 EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 11 and applying (3.13), we estimate that α2p(p− 1) 2 ∫ Ω upvα−2|∇v|2 ≤ q(q − 1) 4 ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 + p(p− 1) 2 ( q(q − 1) 2p(p− 1)(q + √ n)2 )−2/qα 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q . (3.14) For any η1, η2 > 0, we can see from Young’s inequality that q(q − 2 + √ n) ∫ Ω wv−q+2|∇v|q−2|D2v| ≤ η1 ∫ Ω v−q+1|∇v|q−2|D2v|2 + η−1 1 q2(q − 2 + √ n)2 ∫ Ω w2v−q+3|∇v|q−2 ≤ η1(q + √ n+ 1)2 ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 + η−1 1 q2(q − 2 + √ n)2η q+2 q−2 2 ∫ Ω v−q−1|∇v|q+2 + η−1 1 q2(q − 2 + √ n)2η − q+2 4 2 ∫ Ω w q+2 2 v ≤ (q + √ n+ 1)2 { η1 + η−1 1 q2(q − 2 + √ n)2η q+2 q−2 2 }∫ Ω v−q+3|∇v|q−2|D2 ln v|2 + η−1 1 q2(q − 2 + √ n)2η − q+2 4 2 ∥v0∥L∞(Ω) ∫ Ω w q+2 2 . (3.15) If we let η1 = q(q − 1) 8(q + √ n+ 1)2 , η2 = ( (q − 1)2 64(q + √ n+ 1)4(q − 2 + √ n)2 ) q−2 q+2 , then (3.15) can be simplified as q(q − 2 + √ n) ∫ Ω wv−q+2|∇v|q−2|D2v| ≤ q(q − 1) 4 ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 + 8q/2q(q + √ n+ 1)q(q − 2 + √ n) q+2 2 ∥v0∥L∞(Ω) (q − 1)q/2 ∫ Ω w q+2 2 . (3.16) 12 C. WU EJDE-2025/09 Substituting (3.14) and (3.16) into (3.12) yields d dt (∫ Ω up + ∫ Ω v−q+1|∇v|q ) + ∫ Ω up + ∫ Ω v−q+1|∇v|q + q(q − 1) 2 ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 ≤ p(p− 1) 2 ( q(q − 1) 2p(p− 1)(q + √ n)2 )−2/qα 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q + k1 ∫ Ω w q+2 2 + q 2 ∫ ∂Ω v−q+1|∇v|q−2 · ∂|∇v|2 ∂ν + ∫ Ω v−q+1|∇v|q + (ap+ 1) ∫ Ω up − bp ∫ Ω up+γ−1, (3.17) where k1 = 8q/2q(q + √ n+ 1)q(q − 2 + √ n) q+2 2 ∥v0∥L∞(Ω) (q − 1)q/2 . Then, we estimate the second and third terms on the right-hand side of (3.17). By (2.2), Lemmas 2.3, 2.5 and Young’s inequality, we find that for any ϱ > 0, there exists a positive constant c1 such that q 2 ∫ ∂Ω v−q+1|∇v|q−2 · ∂|∇v|2 ∂ν ≤ ϱ ∫ Ω v−q−1|∇v|q+2 + ϱ ∫ Ω v−q+1|∇v|q−2|D2v|2 + c1 ∫ Ω v ≤ 2(q + √ n+ 1)2ϱ ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 + c1|Ω|∥v0∥L∞(Ω) (3.18) and ∫ Ω v−q+1|∇v|q ≤ ϱ q+2 q ∫ Ω v−q−1|∇v|q+2 + ϱ− q+2 2 ∫ Ω v ≤ (q + √ n)2ϱ q+2 q ∫ Ω v−q+3|∇v|q−2|D2 ln v|2 + ϱ− q+2 2 |Ω|∥v0∥L∞(Ω). (3.19) Because (ap+ 1) ∫ Ω up − bp ∫ Ω up+γ−1 ≤ −bp 2 ∫ Ω up+γ−1 + c2 (3.20) for some c2 > 0, by selecting a suitably small ϱ and substituting (3.18)–(3.20) into (3.17), we obtain d dt (∫ Ω up + ∫ Ω v−q+1|∇v|q ) + ∫ Ω up + ∫ Ω v−q+1|∇v|q ≤ p(p− 1) 2 ( q(q − 1) 2p(p− 1)(q + √ n)2 )−2/qα 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q + k1 ∫ Ω w q+2 2 − bp 2 ∫ Ω up+γ−1 + C. (3.21) EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 13 Multiplying the third equation of the system (1.6) by ( q2 + 1)wq/2 gives d dt ∫ Ω w q+2 2 + ( q 2 + 1)δ ∫ Ω w q+2 2 ≤ ( q 2 + 1) ∫ Ω wq/2u. (3.22) With the help of Young’s inequality, we have ( q 2 + 1) ∫ Ω wq/2u ≤ (q + 2)δ 4 ∫ Ω w q 2+1 + ( 2q (q + 2)δ )q/2 ∫ Ω u q 2+1 ≤ (q + 2)δ 4 ∫ Ω w q 2+1 + ( 2 δ )q/2 ∫ Ω u q 2+1. (3.23) Collecting (3.22) and (3.23), we obtain d dt ∫ Ω w q+2 2 + (q + 2)δ 4 ∫ Ω w q+2 2 ≤ ( 2 δ )q/2 ∫ Ω u q+2 2 . (3.24) Multiplying 8k1 (q+2)δ in the both sides of (3.24), we have 8k1 (q + 2)δ · d dt ∫ Ω w q+2 2 + 2k1 ∫ Ω w q+2 2 ≤ 8k1 (q + 2)δ ( 2 δ )q/2 ∫ Ω u q+2 2 . (3.25) Inserting (3.25) into (3.21), we see that d dt (∫ Ω up + ∫ Ω v−q+1|∇v|q + 8k1 (q + 2)δ ∫ Ω w q+2 2 ) + ∫ Ω up + ∫ Ω v−q+1|∇v|q + k1 ∫ Ω w q+2 2 ≤ p(p− 1) 2 ( q(q − 1) 2p(p− 1)(q + √ n)2 )−2/qα 2(q+2) q ∥v0∥ α(q+2)−2 q L∞(Ω) ∫ Ω u p(q+2) q + 8k1 (q + 2)δ ( 2 δ )q/2 ∫ Ω u q+2 2 − bp 2 ∫ Ω up+γ−1 + C, (3.26) where k1 is defined as in (3.17). Because 2(p+ γ − 2)− 2p γ − 1 = 2[(p+ γ − 2)(γ − 1)− p] γ − 1 = 2(γ − 2)(p+ γ − 1) γ − 1 > 0. Then, by selecting q ≥ 2, we have 2p γ − 1 < q < 2(p+ γ − 2). (3.27) Moreover, in Lemma 3.3, taking k = k1 and combining Lemma 3.4 with (3.26), we obtain d dt (∫ Ω up + ∫ Ω v−q+1|∇v|q + 8κ1 (q + 2)δ ∫ Ω w q+2 2 ) +min { 1, (q + 2)δ 8 }(∫ Ω up + ∫ Ω v−q+1|∇v|q + 8κ1 (q + 2)δ ∫ Ω w q+2 2 ) ≤ c0 for some c0 > 0. Thus, using a standard ODE comparison parameter gives (3.11). □ Lemma 3.6. Let γ = 2, n ≥ 4, p > 1 and assume that (1.7) holds, and b satisfies b > µ1(p, n)α 2(p+1) p ∥v0∥ α(p+1)−1 p L∞(Ω) + µ2(p, n, δ)∥v0∥L∞(Ω), (3.28) 14 C. WU EJDE-2025/09 where µ1(p, n) = (p− 1) p+1 p (2p+ √ n) 2 p (2p− 1)− 1 p , (3.29) µ2(p, n, δ) = 24p+4(2p+ √ n+ 1)2p(2p− 2 + √ n)p+1 (p+ 1)(2p− 1)pδp+1 . (3.30) Then one can find C > 0 such that ∥u(·, t)∥Lp(Ω) ≤ C for all t ∈ (0, Tmax). (3.31) Proof. Since γ = 2, from (3.29), (3.30) and setting q = 2p in (3.26), we deduce that d dt (∫ Ω up + ∫ Ω v1−2p|∇v|2p + 4k1 (p+ 1)δ ∫ Ω wp+1 ) + ∫ Ω up + ∫ Ω v1−2p|∇v|2p + k1 ∫ Ω wp+1 ≤ −p 2 { b− µ1(p, n)α 2(p+1) p ∥v0∥ α(p+1)−1 p L∞(Ω) − µ2(p, n)∥v0∥L∞(Ω) }∫ Ω up+1 + C (3.32) for all t ∈ (0, Tmax). Using (3.28), (3.32) can be reduced to d dt (∫ Ω up + ∫ Ω v1−2p|∇v|2p + 4k1 (p+ 1)δ ∫ Ω wp+1 ) +min { 1, (p+ 1)δ 4 }(∫ Ω up + ∫ Ω v1−2p|∇v|2p+ 4k1 (p+ 1)δ ∫ Ω wp+1 ) ≤ C. (3.33) Thus, a standard ODE comparison argument leads to (3.31). □ Lemma 3.7. Let γ = 2 and n = 2, 3, Then, for all T > 0, there exists C(T ) > 0 such that ∥u(·, t)∥L2(Ω) ≤ C for all t ∈ (0, Tmax). (3.34) Proof. Letting p = 2 in (3.1), we have d dt ∫ Ω u2 + ∫ Ω vα|∇u|2 + ∫ Ω u2 ≤ α2 ∫ Ω vα−2u2|∇v|2 + (2a+ 1) ∫ Ω u2. (3.35) For γ = 2, (2.5) and Lemma 2.7 show that for any T > 0, there exists a constant c1(T ) > 0 such that v ≥ c1(T ) in Ω× (0, T ). Thus (3.35) can be rewritten as d dt ∫ Ω u2 + cα1 (T ) ∫ Ω |∇u|2 + ∫ Ω u2 ≤ α2c2(T ) ∫ Ω u2|∇v|2 + (2a+ 1) ∫ Ω u2 (3.36) with c2(T ) := max { cα−2 1 (T ), ∥v0∥α−2 L∞(Ω) } . Using γ = 2, n = 2, 3, in conjunction with (2.5) and [17, Lemma 1.2], one can find c3 > 0 such that ∥v(·, t)∥W 1,4(Ω) ≤ c3 for all t ∈ (0, Tmax). (3.37) EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 15 Using n = 2, 3, (2.1), (3.37) and applying Hölder’s inequality and Ehrling type inequality, we estimate taht for some c4(T ) > 0, α2c2(T ) ∫ Ω u2|∇v|2 + (2a+ 1) ∫ Ω u2 ≤ α2c2(T )∥u∥2L4(Ω)∥∇v∥2L4(Ω) + (2a+ 1)∥u∥2L2(Ω) ≤ α2c2(T )c 2 3∥u∥2L4(Ω) + (2a+ 1)∥u∥2L2(Ω) ≤ cα1 (T ) 2 ∫ Ω |∇u|2 + c4(T ) (3.38) for all t ∈ (0, Tmax). Collecting (3.38) and (3.36), we estimate d dt ∫ Ω u2 + cα1 (T ) 2 ∫ Ω |∇u|2 + ∫ Ω u2 ≤ c4(T ) for all t ∈ (0, Tmax). (3.39) We complete the proof of (3.34) by using an ODE argument. □ Lemma 3.8. Let γ ≥ 1. Then, for all n ≥ 1, if one of the following 3 con- ditions is true: (i) l > 2; (ii) l = 2 and n ≤ 3; (iii) l = 2, n ≥ 4 and b > µ1( n 2 , n)α 2(n+2) n ∥v0∥ α(n+2)−2 n L∞(Ω) + µ2( n 2 , n, δ)∥v0∥L∞(Ω), where µ1 µ2 are defined as in (3.29) and (3.30). Then for all T ∈ (0, Tmax), one can find C(T ) > 0 such that ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥L∞(Ω) ≤ C(T ) for all t ∈ (0, T ). Proof. Let p ≥ 1 such that p > n 2 . For the parameter γ, the following three cases are discussed. If γ > 2, we know that ∥u(·, t)∥Lp(Ω) is bounded from Lemma 3.5. If γ = 2, from (2.1) and Lemma 3.7, we obtain that ∥u(·, t)∥L1(Ω) is bounded under the condition n = 1 and ∥u(·, t)∥L2(Ω) is bounded under the condition n = 2, 3. If γ = 2 and n ≥ 4, since p > n 2 , using Lemma 3.6 with respect to b, we also get that ∥u(·, t)∥Lp(Ω) is bounded for all t ∈ (0, Tmax). Finally, we use Lemma 2.8 to complete the proof. □ Now Theorem 1.1 follows directly from Lemmas 2.1 and 3.8. Acknowledgment. This work was supported by the Natural Science Foundation Project of Chongqing, China (Grant No. CSTB2024NSCQ-MSX0220). The author is very grateful to the anonymous reviewers and editors for their valuable comments that led to a substantial improvement of this manuscript. References [1] J. Ahn, C. Yoon; Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing. Nonlinearity., 32 (2019), 1327-1351. [2] M. Burger, P. Laurençot, A. Trescases; Delayed blow-up for chemotaxis models with local sensing. J. Lond. Math. Soc., 103(2021), 1596-1617. [3] K. Baghaei, A. Khelghati; Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant. C. R. Acad. Sci. Paris, Ser. I., 355 (2017), 633-639. [4] X. Fu, L. Tang, C. Liu, J. Huang, T. Hwa, P. Lenz; Stripe formation in bacterial systems with density-suppressed motility. Phys. Rev. Lett., 108 (2012), 198102. [5] K. Fujie, T. Senba; Application of an Adams type inequality to a two-chemical substances chemotaxis system. J. Differ. Equ., 263 (2017), 88-148. [6] K. Fujie, J. Jiang; Global existence for a kinetic model of pattern formation with density- suppressed motilities. J. Differ. Equ., 269 (2020), 5338-5378. 16 C. WU EJDE-2025/09 [7] K. Fujie, J. Jiang; Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities. Calc. Var. Partial Differ. Equ., 60 (2021), 92. [8] K. Fujie, J. Jiang; Boundedness of classical solutions to a degenerate Keller-Segel type model with signal-dependent motilities. Acta Appl. Math.,176 (2021), 3. [9] K. Fujie, T. Senba; Global boundedness of solutions to a parabolic-parabolic chemotaxis sys- tem with local sensing in higher dimensions. Nonlinear Anal., 35 (2022), 3777-3811. [10] K. Fujie, T. Senba; Global existence and infinite time blow-up of classical solutions to chemo- taxis systems of local sensing in higher dimensions. Nonlinear Anal., 222 (2022), 112987. [11] M. Fuest; Analysis of a chemotaxis model with indirect signal absorption. J. Differ. Equ., 267 (2019), 4778-4806. [12] B. Hu, Y. Tao; To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production. Math. Models Methods Appl. Sci., 26 (2016), 2111-2128. [13] H. Y. Jin, Y. J. Kim, Z. A. Wang; Boundedness, stabilization, and pattern formation driven by density-suppressed motility. SIAM J. Appl. Math., 78 (2018), 1632-1657. [14] H. Y. Jin, Z. A. Wang; Critical mass on the Keller-Segel system with signal-dependent motil- ity. Proc. Am. Math. Soc., 148 (2020), 4855-4873. [15] J. Jiang, P. Laurençot; Global existence and uniform boundedness in a chemotaxis model with signal-dependent motility. J. Differ. Equ., 299 (2021), 513-541. [16] E. F. Keller, L. A. Segel; Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol.,30 (1971), 377-380. [17] R. Kowalczyk, Z. Szyma’nska; On the global existence of solutions to an aggregation model. J. Math. Anal. Appl.,343 (2008), 379-398. [18] D. Li, J. Zhao; Global boundedness and large time behavior of solutions to a chemotaxis consumption system with signal-dependent motility, Z. Angew. Math. Phys., 72 (2021), Paper No. 57, 20 pp. [19] D. Li, Z. Li, J. Zhao; Boundedness and large time behavior for a chemotaxis system with signal-dependent motility and indirect signal consumption. Nonlinear Anal. Real World Appl., 64 (2022), 103447. [20] G. Li, M. Winkler; Refined regularity analysis for a Keller-Segel-consumption system involv- ing signal-dependent motilities, Appl. Anal., 103 (2024), 45-64. [21] Y. Liu, Z. Li, J. Huang; Global boundedness and large time behavior of a chemotaxis system with indirect signal absorption. J. Differ. Equ., 269 (2020), 6365-6399. [22] Z. Liu, J. Xu; Large time behavior of solutions for density-suppressed motility system in higher dimensions. J. Math. Anal. Appl., 475 (2019), 1596-1613. [23] J. Lankeit, Y. L. Wang; Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discret. Contin. Dyn. Syst. Ser. A., 37 (2017), 6099- 6121. [24] J. Lee, C. Yoon; Existence and asymptotic properties of aerotaxis model with the Fokker- Planck type diffusion, Nonlinear Anal. Real World Appl., 71 (2022), 103758. [25] C. Liu, X. Fu, et al.; Sequential establishment of stripe patterns in an expanding cell popula- tion. Science, 334 (2011), 238-241. [26] W. Lv, Q. Wang; An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization. Proc. R. Soc. Edinb. A., 151 (2021), 821-841. [27] W. Lv, Q .Wang; Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evol. Equ. Control Theory., 10 (2021), 25-36. [28] W. Lyu, Z. Wang; Logistic damping effect in chemotaxis models with density-suppressed motility. Adv. Nonlinear Anal., 12 (2023), 336-355. [29] W. Lv; Global existence for a class of chemotaxis-consumption systems with signal-dependent motility and generalized logistic source, Nonlinear Anal. Real World Appl., 56 (2020) 103160, 13 pp. [30] Y. Tao, M. Winkler; Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations., 252 (2012), 692-715. [31] Y. Tao, M. Winkler; Eventual smoothness and stabilization of large-data solutions in a three- dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ., 252 (2012), 2520-2543. [32] Y. Tao, M. Winkler; Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016) Art. 138, 23 pp. EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 17 [33] Y. Tao, M. Winkler; Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production. J. Eur. Math. Soc., 19 (2017), 3641-3678. [34] Y. Tao, M. Winkler; Effects of signal-dependent motilities in a Keller-Segel-type reaction- diffusion system. Math. Models Methods Appl. Sci., 27 (2017), 1645-1683. [35] Y. Tao, M. Winkler; Global solutions to a Keller-Segel-consumption system involving sin- gularly signal-dependent motilities in domains of arbitrary dimension, J. Differential Equa- tions., 343 (2023), 390-418. [36] C. J. Wang, Z. H. Zheng; Global boundedness for a chemotaxis system involving nonlinear indirect consumption mechanism, DCDS-B., 29(5) (2024), 2141-2157. [37] L. Wang; Global dynamics for a chemotaxis consumption system with signal-dependent motil- ity and logistic source, J. Differential Equations., 348 (2023), 191-222. [38] L. Wang; Global solutions to a chemotaxis consumption model involving signal-dependent degenerate diffusion and logistic-type dampening, arXiv:2304.02915. [39] L. Wang, R. Huang; Global classical solutions to a chemotaxis consumption model involving singularly signal-dependent motility and logistic source, Nonlinear Anal. Real World Appl., 80 (2024), 104174. [40] Z. Wang; On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness and steady states. Math. Methods Appl. Sci., 44 (2021), 10881-10898. [41] J. Wang, M. Wang; Boundedness in the higher-dimensional Keller-Segel model with signal- dependent motility and logistic growth. J. Math. Phys., 60 (2019), 011507. [42] M. Winkler; Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segal model, J. Differ. Equ., 248 (2010), 2889-2905. [43] M. Winkler; Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities. SIAM J. Math. Anal., 47 (2015), 3092-3115. [44] M. Winkler; Asymptotic homogenization in a three-dimensional nutrient taxis system involv- ing food-supported proliferation. J. Differ. Equ.,263 (2017), 4826-4869. [45] M. Winkler; Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption. J. Differ. Equ., 264 (2018), 2310-2350. [46] M.Winkler; Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement. J. Differ. Equ., 264 (2018), 6109-6151. [47] M. Winkler; A three-dimensional Keller-Sege-Navier-Stokes system with logistic source: global weak solutions and asymptotic stabilization. J. Differ. Equ., 276 (2019), 1339-1401. [48] M. Winkler; Can simultaneous density-determined enhancement of diffusion and cross- diffusion foster boundedness in Keller-Segel type systems involving signal-dependent motili- ties? Nonlinearity, 33 (2020), 6590-6623. [49] M. Winkler; Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities. Discret. Contin. Dyn. Syst. Ser. B., 27 (2022), 6565-6587. [50] M. Winkler; Application of the Moser-Trudinger inequality in the construction of global solutions to a strongly degenerate migration model, B. Math. Sci., 13 (2023), 2250012. [51] M. Winkler; Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration-consumption interaction, Z. Angew. Math. Phys., 74 (2023), Paper No. 32, 20 pp. [52] M. Winkler; A degenerate migration-consumption model in domains of arbitrary dimension, Adv. Nonlinear Stud., 24 (2024), 592-615. [53] C. Yoon, Y. J. Kim; Global existence and aggregation in a Keller?Segel model with Fokker- Planck diffusion. Acta App. Math., 149 (2017), 101-123. [54] Q. Zhang, Y. Li; Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant. J. Math. Phys., 56 (2015), 081506. Chun Wu School of Mathematics Science, Chongqing Normal University, Chongqing 401331, China Email address: wuchun@cqnu.edu.cn 1. Introduction 2. Preliminaries 3. Existence of global solutions Acknowledgment References