This paper studies the following attraction-repulsion chemotaxis system involving nonlinear indirect repulsion-signal mechanism{ut=Δu−ξ∇·(u∇v)+χ∇·(u∇z)+u(a−buk),x∈Ω,t>0,0=Δv−v+uγ1,x∈Ω,t>0,zt=Δz−z+wγ2,x∈Ω,t>0,0=Δw−w+uγ3,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=∂z∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),z(x,0)=z0(x),x∈Ω,where Ω⊂Rn(n≥1) is a smoothly bounded domain and ξ, χ, a, b, k, γ1, γ2, γ3 > 0. If ξ and χ are small enough, it is shown that the global classical solution (u, v, z, w) exponentially converges to ((ab)1k,(ab)γ1k,(ab)γ2γ3k,(ab)γ3k) in L∞(Ω) as t → ∞. Finally, we present numerical simulations that not only support our theoretical results, but also involve new and interesting phenomena.

We study the repulsive chemotaxis-consumption system: u(t) = del (D(u)del u +uS(u)/v del v), 0 = Delta v-uv in bounded and smooth domains ohm subset of R-n (n >= 2), with no-flux and constant positive Dirichlet boundary conditions prescribed for u and v, respectively. Here D and S are suitably smooth and generalize the prototypes D(u) = (u + 1)(-alpha) and S(u) = (u + 1)(beta-1) with alpha, beta is an element of R. When ohm is aball, Wang and Winkler (2023) [20] established the finite-time blow-up of solutions for alpha > 0 and beta = 1. However, their proof cannot cover the seemingly inevitable blow-up for alpha > 0 and beta > 1, nor can it handle the possible finite-time blow-up in the more challenging case that the self-diffusion is relatively strong with alpha <= 0. Essentially relying on the analysis of a novel moment-like functional tailored to superlinear sensitivity, we prove in this paper that if beta is an element of [1, infinity) boolean AND (1-alpha, infinity) with alpha is an element of R, then for all initial data with sufficiently large mass, the corresponding initial-boundary value problem admits a finite-time blow-up solution. As opposed to the consideration for singularity formation, the global boundedness of solutions is also ascertained for beta < 1/2 + 1/n-alpha.

This paper studies the following attraction-repulsion chemotaxis system involving nonlinear indirect signal mechanism {u(t) = Delta u - xi del . (u del v) + chi del . (u del z) + f(u), x is an element of Omega,t> 0, 0=Delta v - v + u(gamma 1) , x is an element of Omega,t> 0, z(t) = Delta z - z + w(gamma 2) , x is an element of Omega,t> 0, 0=Delta w - w + u(gamma 3) , x is an element of Omega,t> 0, under homogeneous Neumann boundary conditions, where Omega subset of R-n(n >= 1) is a smoothly bounded domain and xi, chi, gamma(1), gamma(2), gamma(3) > 0. center dot When f equivalent to 0, it is shown that the solution of the above system is global and uniformly bounded if max{gamma(1), gamma(2)gamma(3)} < 2/n . Moreover, if max{gamma(1), gamma(2)gamma(3)} = 2/n , the boundedness of solution can be derived provided that the initial mass integral(Omega) u(0)(x)dx is small. center dot When f(u) <= u(a - bu(k)) with a, b, k > 0, it is proved that if one of the following conditions holds: (i) k > max{gamma(1), gamma(2)gamma(3)}, (ii) k = max{gamma(1), gamma(2)gamma(3)}, (iii) max{gamma(1), gamma(2)gamma(3)} < 2/n , then the solution is globally bounded in time provided that b is large enough. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.

Chemotaxis refers to the influence of chemical substances in the environment on the movement of motile species and serves as an important mechanism of cellular communication. Cells communicate with each other by secreting chemical substances, which in turn determine their movement and differentiation. In order to better understand the growth patterns and aggregation distribution of biological populations, an increasing number of scientists have begun to study chemotaxis and describe the observed phenomena by establishing mathematical models. Among these models, the most classical one is the Keller–Segel chemotaxis model. The first part of this paper mainly introduces the development process and current research status of chemotaxis models related to the content of this study, and provides a general overview of the main topics investigated in this paper. The second part of this paper, namely Chapter 2, studies the following chemotaxis mode (which is a parabolic–parabolic–elliptic system with nonlinear production terms and a Logistic-type source). We analyze the global boundedness of its solutions, where is a bounded domain and the system is subject to Neumann boundary conditions. Here, the nonlinear production terms of the attractive and repulsive chemical substances are described by and respectively. Moreover, the Logistic source in this model satisfies Finally, it is concluded that if that is, when either the Logistic source or the repulsive term dominates the attractive term, the solution is globally bounded. Moreover, in the three balanced cases or the boundedness of the solution depends on the magnitude of the corresponding coefficients. The third part of this paper mainly summarizes the main results of the study and provides a preliminary outlook on future research on chemotaxis models with nonlinear production terms.

We study the following chemotaxis-Navier–Stokes system with nonlinear production: n_t+u·∇ n=Δ n-∇· (nχ (n)∇ c) , c_t+u·∇ c=Δ c-c+n^β , u_t+(u·∇ )u=Δ u+∇ P+n∇Φ , and ∇· u=0 in a bounded domain Ω⊂ℝ^2 , where χ∈ C^2([0,∞ )) , β >0 , and Φ∈ W^2,∞(Ω ) . In our previous work [Z. Angew. Math. Phys. 75 (2024) 74], it was shown that if there exists a certain k>0 such that { χ (s)(s+1)^-1/2=O((s+1)^-k) as s→∞ for 0<β <1/2, χ (s)(s+1)^β -1=O((s+1)^-k) as s→∞ for β≥1/2, . then for any suitably smooth initial datum, the corresponding initial-boundary value problem possesses a unique globally bounded classical solution. Especially, in the case that β∈ (0,1/2) , the restriction on the growth of the sensitivity χ stems from the effect of fluid motion governed by the Navier–Stokes equations. In the present paper, we further indicate that when β∈ (0,1/2) , the solutions still remain globally bounded even if the cross-diffusion is intensified to satisfy χ (s)(s+1)^-1/2=o(1) as s→∞ . This improvement is obtained by more exhaustively controlling the destabilizing action of fluid-driven transport and so further reflects the underlying impact of fluid flow on global solvability.

In this paper we study the global boundedness of solutions to the quasilinear chemotaxis‐Stokes system with nonlinear production subject to no‐flux/no‐flux/Dirichlet boundary conditions in a smoothly bounded domain with , where , , and generalize the prototypes and for all with and . It is shown that if for , or for , then for any reasonably regular initial datum, the corresponding initial‐boundary value problem of () possesses a unique globally bounded classical solution. Compared to the global boundedness condition for in the relevant fluid‐free system [J. Differ. Equ. 268 6729–6777 (2020)], the present result indicates that the slow fluid motion described by the Stokes equation might possibly be adverse to the global solvability in the case that with . What is more, this paper provides for the quasilinear problems with general signal production a universal approach capable of deriving the boundedness of solutions in the optimal range of parameters.

Accurate pressure prediction in industrial compressed air pipeline networks is essential for ensuring dynamic equilibrium and enabling real-time operational scheduling. This paper presents a novel method, termed Decomposition and Attention Spatio-Temporal Graph Neural Network (DASTGNN), to address the challenges of pressure fluctuations induced by complex topological structures and variable end-user demands. The proposed framework incorporates a dual-channel temporal convolution module to enhance the extraction of nonlinear temporal dependencies, and introduces a sparse attention-based graph convolution mechanism to model spatial correlations among pipeline nodes. A hybrid graph structure with physical interpretability is constructed to support multi-step forecasting of outlet pressures at air compressor stations. Experimental validation on real-world data collected from a large-scale steel enterprise demonstrates that DASTGNN significantly outperforms several state-of-the-art baseline models in both predictive accuracy and long-horizon generalization. These results underscore the method’s potential for practical deployment in industrial energy systems.

Hasteheat boiler is an important part of dry quenching power generation, and its steam drum level cascade three impulse system is characterized by strong perturbation, non-linearity, strong coupling and multiple working conditions. In this paper, a reinforcement learning based parameter optimization of cascade control loop in the coke dry quenching system is proposed for such series control problems. Based on the uncertainty of the state combination between the control parameters of the cascade loop and the liquid level of the steam, this paper uses a BP neural network to model the controller parameters and optimization indexes, uses the NSGA-II algorithm to select the design factors of the system, transforms the PID parameter optimization problem into a combination optimization problem, and finally using the reinforcement learning mechanism gives the optimal control strategy under multiple operating conditions. The simulation experiments are carried out through the data of a 180t/h dry quench coke waste heat boiler steam drum level system, and the results show that compared with similar controllers, the control strategy proposed in this paper can significantly reduce the maximum overshooting amount with a small increment in the rise lime.

We study the following chemotaxis-Navier–Stokes system with general sensitivity and nonlinear production ⋆ { n_t+u·∇ n=Δ n-∇· (nf(n)∇ c), c_t+u·∇ c=Δ c-c+g(n), u_t+(u·∇ )u+∇ P=Δ u+n ∇ϕ , ∇· u=0 . in a bounded domain Ω⊂ℝ^2 , where the chemotaxis sensitivity function f∈ C^2([0,∞ )) satisfies that |f(s)|≤ K_f(1+s)^-α for all s≥ 0 with K_f>0 and α∈ℝ , and the signal production function g∈ C^1([0,∞ )) is such that 0≤ g(s)≤ K_g s(1+s)^β -1 for all s≥ 0 with K_g,β >0 . It is shown in this paper that for all reasonably regular initial data, the corresponding initial-boundary value problem of ( ⋆ ) possesses a unique globally bounded classical solution if α >1/2[(2β -1)_+-1] for 0<β <1 , or if α >β -1 for β≥ 1 . Our work is one of the few explorations involving chemotaxis-fluid models with nonlinear production mechanisms and greatly extends the global solvability result obtained in Black (Nonlinear Anal Real World Appl 31:593–609, 2016) only for the chemotaxis-Stokes variant of ( ⋆ ) with α =0 and the sublinear signal production of β∈ (0,1) .

We consider the chemotaxis-Navier–Stokes system with gradient-dependent flux limitation and nonlinear production: [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in a bounded domain [Formula: see text], where the flux limitation function [Formula: see text] and the signal production function [Formula: see text] generalize the prototypes [Formula: see text] and [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. For the linear production case of [Formula: see text], the global boundedness of solutions has been verified in the related literature for [Formula: see text]. In this paper, we expand to prove that the corresponding initial-boundary value problem possesses a unique globally bounded solution if [Formula: see text] for [Formula: see text], or if [Formula: see text] for [Formula: see text], which shows that when [Formula: see text], that is, the self-enhancement ability of chemoattractant is weak, the solutions still remain globally bounded even though the flux limitation is relaxed to permit proper [Formula: see text]; however, if [Formula: see text], it is necessary to impose the stronger flux limitation than that in the case [Formula: see text] to inhibit the possible finite-time blow-up. This seems to be the first result on the global solvability in the chemotaxis-Navier–Stokes model with nonlinear production.

This paper deals with a two-species chemotaxis model with two chemicals in R-2. In our previous work (2019 Nonlinearity 32 4762-78), the critical mass was obtained that the solutions exist globally if m(1)m(2) - 4 pi(m(1) + m(2)) < 0, and the finite time blow-up of solutions may occur if m(1)m(2) - 4 pi(m(1) + m(2)) > 0, where m(1) and m(2) describe the initial mass of the two species, respectively. In the present paper we furthermore determine that the critical situation belongs to the global existence case, namely, the system admits global solutions if m(1)m(2) - 4 pi(m(1) + m(2)) = 0. To apply the key logarithmic Hardy-Littlewood-Sobolev inequality of vector form for the critical case, we should establish a positive lower bound for L-1-norms of the two species in exterior domains {x is an element of R-2: vertical bar X vertical bar > R} uniformly for t > 0.

We study the chemotaxis-(Navier–)Stokes system modeling coral fertilization: $$n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c)-nm$$ , $$c_t+u\cdot \nabla c=\Delta c-c+m$$ , $$m_t+u\cdot \nabla m=\Delta m-nm$$ , $$u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \phi $$ and $$\nabla \cdot u=0$$ in a bounded and smooth domain $$\Omega \subset \mathbb {R}^2$$ , where $$\kappa \in \mathbb {R}$$ , $$\phi \in W^{2,\infty }(\Omega )$$ , and $$S\in C^2({\bar{\Omega }}\times [0,\infty )^2;\mathbb {R}^{2\times 2})$$ satisfies $$|S(x,n,c)|\le S_0(c)(1+n)^{-\alpha }$$ for all $$(x,n,c)\in {\bar{\Omega }}\times [0,\infty )^2$$ with $$\alpha \in \mathbb {R}$$ and the function $$S_0:[0,\infty )\rightarrow [0,\infty )$$ nondecreasing. Under the relatively weak destabilizing action of cross-diffusion for $$\alpha \ge 0$$ , the global boundedness of classical solutions was obtained in Espejo and Winkler (Nonlinearity 31:1227–1259, 2018) and Li (Differ Equ 267:6290–6315, 2019). In this paper, we show that even if n|S| with $$-\frac{1}{2}<\alpha <0$$ bears a superlinear growth of n, the corresponding initial-boundary value problem (with any $$\kappa \in \mathbb {R}$$ ) still possesses a global classical solution emanating from any suitably smooth initial data. Moreover, when $$\kappa =0$$ , this solution is globally bounded.

In this paper we study the fully parabolic chemotaxis system with p-Laplacian diffusion and logistic-type source: $$u_{t}=\nabla \cdot (|\nabla u|^{p-2}\nabla u)-\nabla \cdot (S(u)\nabla v)+f(u)$$ , $$v_{t}=\Delta v-v+u$$ , in a bounded domain $$\Omega \subset {\mathbb {R}}^n(n\ge 1)$$ with $$p>1$$ , subject to the non-flux boundary, where $$S\in C^2([0,\infty ))$$ satisfies $$0\le S(s)\le b_0(s+1)^\beta $$ with $$b_0>0, \beta \in \mathbb {R}$$ , and the logistic-type source $$f(s)\le b-\mu s^r$$ for $$b\ge 0$$ with $$\mu >0$$ and $$r\ge 1$$ . We obtain the global boundedness of weak solutions, where the aggregation is dominated by the logistic-type source, the p-Laplacian diffusion, and the cooperation of the logistic source and the p-Laplacian diffusion, respectively.

In this paper we study the chemotaxis–Stokes system with slow p-Laplacian diffusion and rotation: nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−∇⋅(nS(x,n,c)⋅∇c), ct+u⋅∇c=Δc−nc, ut+∇P=Δu+n∇ϕ+f(x,t) and ∇⋅u=0 in a bounded domain Ω⊂R3 with p>2, subject to the Neumann–Neumann–Dirichlet boundary conditions, where ϕ:Ω̄→R, f:Ω̄×[0,∞)→R3 and S:Ω̄×[0,∞)2→R3×3 are given sufficiently smooth functions with f bounded in Ω×(0,∞), |S(x,n,c)|≤S0(c)(1+n)−α for (x,n,c)∈Ω̄×[0,∞)2 with α≥0, and nondecreasing function S0:[0,∞)→[0,∞). It is proved that the problem possesses a globally bounded weak solution provided α+43p>259 and 11p+6α+2αp>23. This extends the current global boundedness result by Tao and Li (2020), where the case of α=0 was well solved. It is mentioned that, without constructing coupled energy functionals, the technique used in the present paper is somewhat different.

In this paper we study the asymptotic behaviors of global solutions to the fully parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇v)+ru−μu1+σ, vt=Δv−v+uγ, subject to the homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥2), where parameters μ,σ,γ>0, r∈R, and the nonlinearity D,S∈C2([0,∞)) are supposed to generalize the prototypesD(u)≥a0(u+1)−α,0≤S(u)≤b0u(u+1)β−1 with a0,b0>0 and α,β∈R. We first consider the case of r>0 and provide a boundedness result under α+β+γ<2n, or β+γ<1+σ, or β+γ=1+σ with large μ>0. The main result is concerned with the asymptotic stability when damping effects of logistic source are strong enough. Specifically, there is μ0>0 independent of initial data, such that the bounded classical solution (u,v) satisfies (u,v)→((rμ)1σ,(rμ)γσ) in L∞(Ω) exponentially under conditions of μ>μ0 and r>0. For the case of r<0, the trivial constant equilibria in the model is obtained in a priori way, that is, any bounded solution (u,v) satisfies (u,v)→(0,0) in L∞(Ω) exponentially, regardless of the size of μ>0.

In this paper we study the quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source: ut=∇⋅(D(u)∇u−S(u)∇v)+f(u), vt=Δv−a1v+b1w, wt=Δw−a2w+b2u, under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥1), where ai,bi>0 (i=1,2), D,S∈C2([0,∞)) and f:R→R is a smooth function generalizing the logistic source f(s)=b−μsr for all s≥0 with b≥0, μ>0 and r≥1. We obtain the global boundedness of solutions in four cases: (i) the self-diffusion dominates the cross-diffusion; (ii) the logistic source suppresses the cross-diffusion; (iii) the logistic dampening balances the cross-diffusion with μ>0 suitably large; (iv) the self-diffusion and the logistic source both balance the cross-diffusion to some extent with μ>0 arbitrary. As corollaries, we also consider the global boundedness of solutions for the quasilinear attraction-repulsion chemotaxis model with logistic source: u˜t=∇⋅(D(u˜)∇u˜)−χ∇⋅(u˜∇z)+ξ∇⋅(u˜∇w˜)+f(u˜), zt=Δz−ρz+ηu˜, w˜t=Δw˜−δw˜+γu˜, where χ,η,ξ,γ,ρ,δ>0.

In this paper we study the fully parabolic chemotaxis system with logistic-type source and nonlinear production: ut=Δu−χ∇⋅(u∇v)+f(u), vt=Δv−v+g(u), subject to the non-flux boundary conditions with a smooth and bounded domain Ω⊂Rn (n≥1), and the nonnegative initial data u0∈C0(Ω̄) and v0∈W1,∞(Ω), where the sensitivity χ>0, the logistic-type source f(s)≤s−μsα for s≥0 with α>1, μ>0, and the nonlinear production g(s)≤s(s+1)β−1 for s≥0 with β>0. It is obtained that the problem possesses a globally bounded classical solution if the self-restriction mechanism of the logistic-type source dominates the nonlinear production that β<α−1, or β=α−1 with μ>0 sufficiently large. This extends the current global existence result by Nakaguchi and Osaki (2018). The situation with the nonlinear production controlled by the linear diffusion that β∈(0,2n) is considered as well, which is parallel to the model with nonlinear diffusion and subcritical chemotactic sensitivity without logistic source studied by Tao and Winkler (2012).

In this paper we study the global existence of solutions to the fully parabolic chemotaxis system: ut=Δu−χ∇⋅(uv∇v)+f(u), vt=Δv−v+u in a smooth bounded domain Ω⊂Rn (n≥3) subject to the non-flux boundary conditions, where χ>0 and the logistic function f∈C1[0,∞) satisfies f(s)≤r−μsγ with r≥0 and γ,μ>0. It is shown that the problem possesses a global and classical solution as long as γ>2. Moreover, the global existence of the weak solution is also established provided that f(s)=rs−μs2 with any r,μ>0.

In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: u(t) = del . (D(u) del u - S(u) del v), tau(1)v(t) = Delta v - a(1)v + b(1)w, tau(2)w(t) = Delta w - a(2)w + b(2)u, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n (n >= 1), where tau(i), a(i), b(i) > 0 (i = 1, 2) are constants, and the diffusivity D and the density-dependent sensitivity S satisfy D (s) >= a(0)(s + 1)(-alpha) and 0 <= S(s) <= b(0)(s + 1)(beta) for all s >= 0 with a(0), b(0) > 0 and alpha, beta is an element of R. It is proved that if alpha + beta < 3 and n = 1, or alpha + beta < 4/n with n >= 2, for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: u(t) = del . (D(u) del u) - chi del . (u del z) + xi del . (u del w), z(t) = Delta z - rho z + mu u, w(t) = Delta w - delta w + gamma u, where chi, mu, xi, gamma, rho, delta > 0, and the diffusivity D fulfills D(s) >= c(0) (s + 1)(M-1) for any s >= 0 with c(0) > 0 and M is an element of R. As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. chi mu = xi gamma), the solution is globally bounded if M > -1 and n = 1, or M > 2 - 4/n with n >= 2. This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.