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This paper considers the global dynamics of classical solutions to an initial-boundary value problem of the system of viscous balance laws arising from chemotaxis in one space dimension: u t − (u v) x = u x x + u (1 − u) , x ∈ (a , b) , t > 0 , v t − (u + v 2) x = v x x , x ∈ (a , b) , t > 0. The system of equations is supplemented with time-dependent influx boundary condition for u and homogeneous Dirichlet boundary condition for v. Under suitable assumptions on the dynamic boundary data, it is shown that classical solutions with generic initial data exist globally in time. Moreover, the solutions are shown to converge to the constant equilibrium (1 , 0) , as t → ∞. There is no smallness assumption on the initial data. This is the first rigorous mathematical study of the model subject to dynamic Neumann boundary condition, and generalizes previous works in content and technicality. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the full compressible Navier-Stokes-Maxwell system with a nonconstant background density in R 3. By means of suitable choosing of symmetrizers and weighted energy estimates with some new developments, we establish the global existence and uniqueness of the classical solution provided that the initial data are near this equilibrium. Furthermore, by using the spectrum analysis on the linearized homogeneous system of the full compressible Navier-Stokes-Maxwell equations and refining the convergence property, we obtain the time-algebraic convergence rates of the perturbed solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in H k (k ≥ 2) in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established. [ABSTRACT FROM AUTHOR]

While much literature on chemotaxis systems focuses on bounded domains, this paper emphasizes the global existence of classical solutions for three primary chemotaxis systems with a logistic source on R n. We present a unified proof demonstrating global existence of solutions can be deduced from their locally uniform boundedness in L p (R n) for some p > max ⁡ { 1 , n 2 }. We then provide sufficient conditions for the global existence and boundedness of classical solutions. Notably, our findings even improve several existing results for bounded domains. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem (IBVP) for the isentropic compressible Navier-Stokes equations (CNS) in the domain exterior to a ball in R d (d = 2 or 3) . When viscosity coefficients are given as a constant multiple of the mass density ρ , based on some analysis of the nonlinear structure of this system, we prove the global existence of the unique spherically symmetric classical solution for (large) initial data with spherical symmetry and far field vacuum in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained have the conserved total mass and finite total energy. ρ keeps positive in the domain considered but decays to zero in the far field, which is consistent with the facts that the total mass is conserved, and CNS is a model of non-dilute fluids where ρ is bounded away from the vacuum. To prove the existence, on the one hand, we consider a well-designed reformulated structure by introducing some new variables, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. On the other hand, it is observed that, for the spherically symmetric flow, the radial projection of the so-called effective velocity v = U + ∇ φ (ρ) (U is the velocity of the fluid, and φ (ρ) is a function of ρ defined via μ (ρ) : φ ′ (ρ) = 2 μ (ρ) / ρ 2 ), verifies a damped transport equation which provides the possibility to obtain its upper bound. Then combined with the BD entropy estimates, one can obtain the required uniform a priori estimates of the solution. It is worth pointing out that the framework on the well-posedness theory established here can be applied to the shallow water equations. [ABSTRACT FROM AUTHOR]

We consider the no-flux initial-boundary value problem for the growth-expansion model with chemotaxis in nutrient-replete environments in smoothly bounded domains Ω ⊂ ℝ n . It is shown that if n = 1 or if n = 2 under some structural assumptions on parameter functions therein, for any suitably regular initial data the problem admits a unique global bounded classical solution. For any dimensions n ≥ 3 , we also prove that the problem has a unique global classical solution which is bounded under a small assumption on the initial data. Moreover, we obtain that these solutions stabilize to a uniquely determined spatially uniform equilibrium. We also provide exponential rates of convergence of solutions in a special case. [ABSTRACT FROM AUTHOR]

This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from the Poiseuille flow of full Ericksen–Leslie model in the theory of nematic liquid crystals, which incorporates the crystal and liquid properties of the materials. The main difficulty comes from the degeneracy of the hyperbolic equation, which makes that the system is not continuously differentiable and then the classical methods for the strictly hyperbolic–parabolic coupled systems are invalid. With a choice of a suitable space for the unknown variable of the parabolic equation, we first solve the degenerate hyperbolic problem in a partial hodograph plane and express the smooth solution in terms of the original variables. Based on the smooth solution of the hyperbolic equation, we then construct an iterative sequence for the unknown variable of the parabolic equation by the fundamental solution of the heat equation. Finally, we verify the uniform convergence of the iterative sequence in the selected function space and establish the local existence and uniqueness of classical solutions to the degenerate coupled problem. [ABSTRACT FROM AUTHOR]

This paper concerns the homogeneous Neumann initial-boundary value problem for a two-species attraction-repulsion system with nonlinear signal production and logistic-type source. Under appropriate regularity assumptions on the initial data, using different methods we present the global existence and uniform boundedness of positive classical solutions in any dimensions for the parabolic-elliptic-parabolic-elliptic system and the fully parabolic system. Moreover, by constructing the appropriate energy functional, we further establish the global stabilities of nonnegative spatially homogeneous equilibria for a special case of the fully parabolic attraction-repulsion system. These results partially generalize and improve the previously known ones. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the long time existence of the classical solution to 2D one-fluid Euler-Poisson system of electrons with non-zero vorticity in the standard Sobolev spaces so that both the local solution and long time solution lie in the same Sobolev space framework without any extra restrictions of space-decay rates at infinity. It is shown that the lifespan of the classical solution is at least T ε , δ = min ⁡ { e ε − 2 | l n ε | − α , κ δ } , where ε > 0 is the size of the initial perturbed density and velocity, δ > 0 is the size of the initial vorticity, α > 0 is any fixed number, κ > 0 is a suitable constant. We decompose the Euler-Poisson system into three bilinear interaction parts: dispersion–dispersion, vorticity–dispersion, and vorticity–vorticity. Based on the Strichartz inequality, the related quartic energy estimates, the conservation of specific vorticity along the streamline and some delicate analysis, the precise bound of lifespan T ε , δ is obtained. By the way, for the 2D irrotational one-fluid Euler-Poisson system with small amplitude ε of the perturbed initial data in the Sobolev spaces, we obtain an interesting long time existence result with lifespan T ε ≥ e ε − 2 | l n ε | − α (α > 0 is any fixed constant). [ABSTRACT FROM AUTHOR]

We mainly study the negligibility of haptotaxis effect on global existence, boundedness and convergence of classical solutions to a four-component chemotaxis-haptotaxis system with indirect signal production: { u t = Δ u − χ ∇ ⋅ (u ∇ v) − ξ ∇ ⋅ (u ∇ w) in Ω × (0 , ∞) , v t = Δ v − v + m in Ω × (0 , ∞) , m t = Δ m − m + u in Ω × (0 , ∞) , w t = − v w + η w (1 − w) in Ω × (0 , ∞) , in a bounded and smooth domain Ω ⊂ R n (1 ≤ n ≤ 5) with nonnegative parameters χ , ξ , η , nonnegative regular initial data (u 0 , v 0 , m 0 , w 0) and homogeneous Neumann boundary conditions. When n ≤ 3 , it is proven that solutions to the corresponding Neumann initial-boundary value problem exist globally in time for any η ≥ 0 and remain bounded for small η ≥ 0 ; when n = 4 , η = 0 and Ω is a ball centered at the origin, global boundedness of radially symmetric solutions is shown under ‖ u 0 ‖ L 1 (Ω) < (8 π) 2 χ ; and when n ∈ { 4 , 5 } , η = 0 and χ = 0 , global boundedness of solutions is shown. In either case, the global bounded w -solution component is shown to decay exponentially in W 1 , ∞ (Ω) and the global bounded (u , v , m) -solution component converges exponentially in L ∞ (Ω) to that of the corresponding indirect chemotaxis-only model for suitably small χ ≥ 0. This project incorporates a biological mechanism modeling indirect signal production in chemotaxis-haptotaxis models and our results extend the previous work of Fujie & Senba (2017) [11] on indirect chemotaxis-only system (w ≡ 0) to indirect chemotaxis-haptotaxis one, revealing that the effect of haptotaxis can be neglected even in a chemotaxis system with indirect signal production. Furthermore, our findings demonstrate that indirect chemotaxis mechanism has positive effect on boundedness and enlarges critical blow-up mass comparing to the existing result of Jin & Xiang (2021) [19] on the corresponding direct chemotaxis-haptotaxis model, providing a further understanding about this indirect mechanism in chemotaxis-haptotaxis settings. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier-Stokes equations with Coulomb force [S. Liu, X. Xu and J. Zhang, Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier-Stokes-Poisson equations subject to large oscillations and non-flat doping profile, J. Differ. Equ. 269 (2020) 8468-8508] and compressible MHD equations [H. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal. 45 (2013) 1356-1387] to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity. [ABSTRACT FROM AUTHOR]

This paper deals with an initial-boundary value problem in two-dimensional smoothly bounded domains for the system n t + u · ∇ n = Δ n - ∇ · (n S (n) ∇ v) , x ∈ Ω , t > 0 , v t + u · ∇ v = Δ v - v + w , x ∈ Ω , t > 0 , w t + u · ∇ w = Δ w - w + n , x ∈ Ω , t > 0 , u t + κ (u · ∇) u + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ · u = 0 , x ∈ Ω , t > 0 , (∗) which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid, where κ ∈ R , the gravitational potential ϕ ∈ W 2 , ∞ (Ω) , and S (n) satisfies | S (n) | ≤ C S (1 + n) - α for all n ≥ 0 , C S > 0 and α > - 1. Under the boundary conditions (∇ n - n S (n) ∇ v) · ν = ∂ ν v = ∂ ν w = 0 , u = 0 , x ∈ ∂ Ω , t > 0 , it is shown in this paper that suitable regularity assumptions on the initial data entail the following: (i) If α > - 1 and κ = 0 , then the simplified chemotaxis-Stokes system possesses a unique global classical solution which is bounded. (ii) If α ≥ 0 and κ ∈ R , then the full chemotaxis-Navier–Stokes system admits a unique global classical solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the following two-species chemotaxis system with generalized volume-filling effect and general kinetic functions u t = ∇ · (D 1 (u) ∇ u) - ∇ · (χ 1 (u) ∇ w) + f 1 (u) - μ 1 a 1 u v , (x , t) ∈ Ω × (0 , ∞) , v t = ∇ · (D 2 (v) ∇ v) - ∇ · (χ 2 (v) ∇ w) + f 2 (v) - μ 2 a 2 u v , (x , t) ∈ Ω × (0 , ∞) , τ w t = Δ w - w + g 1 (u) + g 2 (v) , (x , t) ∈ Ω × (0 , ∞) , under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω ⊂ R n (n ≥ 1) , where a 1 , a 2 , μ 1 , μ 2 are positive constants. When the functions D i , S i , f i , g i (i = 1 , 2) belong to C 2 fulfilling some suitable hypotheses, we study the global existence and boundedness of classical solutions for the above system and find that under the case of τ = 1 or τ = 0 , either the higher-order nonlinear diffusion or strong logistic damping can prevent blow-up of classical solutions for the problem. In addition, when the functions are replaced to Lotka–Volterra competitive kinetic functional response term and linear signal generations, by constructing some appropriate Lyapunov functionals, we show that the solution convergences to the constant steady state in L ∞ (Ω) in the case of a 1 , a 2 ∈ (0 , 1) or a 1 ≥ 1 > a 2 > 0 under some more concise conditions than [2], which improved the existing conditions to some extent. [ABSTRACT FROM AUTHOR]

In this article, a reaction–diffusion model of HIV immunity with chemotaxis and absorption effect is constructed. The paper proves the existence and boundedness of the global classical solution of this mode when the chemotactic coefficient is kept in a suitable range. Five equilibrium points are established based on the different ranges of the basic regeneration number, two immune reproduction numbers and the immune competitive reproduction number. The global asymptotic stability of each equilibrium point is established by constructing an appropriate Lyapunov function when the chemotactic coefficient is kept in a small interval. [ABSTRACT FROM AUTHOR]

This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source u t = Δ u v α + a u - b u l , x ∈ Ω , t > 0 , v t = Δ v - v w , x ∈ Ω , t > 0 , w t = - δ w + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ≥ 1 ). Here, the parameters a > 0 , b > 0 , α ≥ 1 , δ > 0 and l ≥ 2 . For all suitably regular initial data, if one of the following cases holds: l > 2 ; l = 2 , n ≤ 3 ; l = 2 , n ≥ 4 , and b is sufficiently large, then the corresponding initial boundary value problem possesses a global classical solution. [ABSTRACT FROM AUTHOR]

We consider the stochastic Cahn-Hilliard-Navier-Stokes with surfactant on a bounded domain, driven by a multiplicative noise. The resulting system of partial differential equations consists of a Navier-Stokes systems for the (average) velocity, a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth–order Cahn-Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier-Stokes system with a multiplicative noise of Gaussian type for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients in both two and three dimensional bounded domains. In the two dimensional case and when the viscosity of the mixture is assumed to be constant, we prove the pathwise uniqueness of the weak solution, and using the Yamada-Watanabe classical result to derive the existence of a strong solution (in probabilistic sense). [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel system with a fractional dissipation and advection by a weakly mixing flow (see Definition 2.5). Here the attractive kernel has a strong singularity, namely, the derivative appearing in the nonlinear term by singular integral. Without advection, the solution of system blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case that the order of the derivative of dissipative term is higher than that of nonlinear term. Since the attractive kernel has a strong singularity, the weakly mixing has both destabilizing effects and enhanced dissipation effects, which makes the problem more complicated and difficult. In this paper, we establish the $ L^\infty $-criterion and obtain the global $ L^\infty $ estimate of the solution by introducing some new ideas and techniques. [ABSTRACT FROM AUTHOR]

We are interested in the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Recently, the first result with the same formulation as in the higher dimensional case has been obtained if and only if the total integral of the initial speed is zero, namely Huygens' principle holds. In this paper, we extend the nonlinear term to the general form including the product type. Such model equations are extremely meaningful only in one space dimension because the most cases in higher dimensions possess the global-in-time existence of a classical solution in the general theory for nonlinear wave equations. It is also remarkable that our results on the lifespan estimates are partially better than those of the general theory. This fact tells us that there is a possibility to improve the general theory which was expected complete more than 30 years ago. [ABSTRACT FROM AUTHOR]

As a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high‐dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density ρ$\rho$ and the lower bound of the temperature θ$\theta$. Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux q$q$ and the temperature θ$\theta$ by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: κ(ρ,θ)⩾C(1+θβ)$\kappa (\rho,\theta)\geqslant C(1+\theta ^\beta)$ if ρ⩽ρ+$\rho \leqslant \rho _+$; κ(ρ,θ)⩽C(1+θβ)$\kappa (\rho,\theta)\leqslant C(1+\theta ^\beta)$ if ρ⩾ρ−>0$\rho \geqslant \rho _->0$. This can be viewed as the first result about the global classical large solutions of the radiation hydrodynamics model with some symmetry in the high‐dimensional space. [ABSTRACT FROM AUTHOR]

We introduce a novel gradient-based damping term into a Keller–Segel type taxis model with motivation from ecology and consider the following system equipped with homogeneous Neumann-boundary conditions:$ \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v)+a u^\alpha-b u^\beta-c|\nabla u|^\gamma & \text{ in } \Omega \times (0,T_{max}),\\ \tau v_t = \Delta v-v+u & \text{ in } \Omega \times (0,T_{max}). \end{cases} \quad\quad (\diamond) $The problem is formulated in a bounded and smooth domain $ \Omega $ of $ {\mathbb R}^N $, with $ N\geq 2 $, for some positive numbers $ a,b,c,\chi>0 $, $ \tau \in \{0,1\} $, $ \gamma\geq 1 $, $ \beta>\alpha\geq 1 $, and with $ {T_{max}}\in (0,\infty] $. As far as we know, Keller–Segel models with gradient-dependent sources are new in the literature and, accordingly, beyond giving a reasonable ecological interpretation, the objective of the paper is twofold:ⅰ) to provide a rigorous analysis concerning the local existence and extensibility criterion for a class of models generalizing problem ($\Diamond$), obtained by replacing $ a u^\alpha-b u^\beta-c|\nabla u|^\gamma $ with $ f(u)-g(\nabla u) $;ⅱ) to establish sufficient conditions on the data of problem ($\Diamond$) itself, such that it admits a unique classical solution $ (u,v) $ for $ {T_{max}} = \infty $ and with both $ u $ and $ v $ bounded.We handle ⅰ) whenever appropriately regular initial distributions $ u(x,0) = u_0(x)\geq 0 $, $ \tau v(x,0) = \tau v_0(x)\geq 0 $ are considered, $ f $ and $ g $ obey some regularity properties, and, moreover, some growth restrictions. Further, as to ⅱ), for the same initial data considered in the previous case, global boundedness of solutions is proven for any $ \tau\in \{0,1\} $, provided that $ \frac{2N}{N+1}<\gamma\leq 2 $. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global spherically symmetric solutions for the initial boundary value problem of a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of two-phase viscous compressible fluids. We prove the existence and uniqueness of global classical solution, weak solution and strong solution under the assumption of spherically symmetry condition for initial data ρ0 without vacuum state. [ABSTRACT FROM AUTHOR]

The article discusses the existence of weak solutions of the Kelvin–Voigt model in the context of the motion of water with added polymers. The model considers the delay in establishing equilibrium states due to relaxation time and internal reorganization processes. The study aims to prove the weak solvability of the initial–boundary value problem by introducing regular Lagrangian flows and defining weak solutions based on a priori estimates and the theory of the topological degree of contracting vector fields. The research was funded by the Russian Science Foundation and the author declares no conflicts of interest. [Extracted from the article]

In this work, we explore the Boltzmann equation with Fermi-Dirac statistics (briefly, BFD equation), which models the dilute gases when quantum effects are taking into consideration. Specifically, we firstly establish the uniform energy estimate and construct the global-in-time classical solutions of the scaled BFD equations under small assumption on the initial data. Then we use this uniform estimate to derive the acoustic limit from the scaled BFD equations within the context of classical solutions. Compared with our companion article Jiang and Zhou (2024) [23] , our uniform estimate in Knudsen number in this paper is crucial and hard to be obtained for the compressible Euler scaling and the acoustic limit is global-in-time, rather than almost global-in-time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial-boundary value problems of the compressible isentropic Navier-Stokes equations with density-dependent viscosity on two dimensional solid balls which was first introduced by Kazhikhov where shear viscosity μ is assumed to be constant and the bulk viscosity λ is a polynomial of density up to power β. Under the condition of β > 1 , we prove the global existence of the radially symmetric strong solutions to the Kazhikhov models under Dirichlet boundary conditions for arbitrary large initial smooth data. Moreover, the density is shown to be uniformly bounded with respect to time when β ∈ (max ⁡ { 1 , γ + 2 4 } , γ ]. This improves the previous result of [14,15] for general 2D domains where they require β > 4 / 3 to ensure global existence and is the first result concerning the global existence of classical solutions to the radially symmetric compressible Navier-Stokes equations in 2D solid balls under Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

We study global existence, boundedness, and convergence of nonnegative classical solutions to a Neumann initial‐boundary value problem for the following possibly cross‐diffusive SIS (susceptible–infected–susceptible) epidemic model with power‐like infection mechanism generalizing the standard mass action incidence: St=dSΔS+χ∇·(S∇I)−βSqIp+γI,x∈Ω,t>0,It=dIΔI+βSqIp−(γ+μ)I,x∈Ω,t>0$$\begin{equation*} {\left\lbrace\! \def\eqcellsep{&}\begin{array}{ll} S_t\,{=}\,d_S \Delta S+\chi \nabla \cdot (S \nabla I)-\beta S^{q} I^{p} +\gamma I,& x\in \Omega,\, t>0, \\[3pt] I_t\,{=}\,d_I\Delta I+\beta S^{q} I^{p} -(\gamma +\mu) I,& x\in \Omega,\, t>0 \end{array} \right.} \end{equation*}$$in a bounded smooth domain Ω⊂Rn(n≥1)$\Omega \subset \mathbb {R}^n \ (n\ge 1)$. The infection force of the form βSqIp$\beta S^q I^p$ with p,q>0$p, \, q>0$ is a natural extension of the classical mass action type βSI$\beta SI$, and the cross‐diffusive term χ∇·(S∇I)$\chi \nabla \cdot (S \nabla I)$ with χ≥0$\chi \ge 0$ describes the effect that susceptible individuals tend to move away from higher density of infected populations. Global existence and boundedness of classical solutions are established in certain parameter ranges, and threshold/nonthreshold long‐time behaviors of global bounded solutions are also detected. Our findings significantly improve and extend previous related studies. [ABSTRACT FROM AUTHOR]

We here investigate a modification of the compressible barotropic Euler system with friction, involving a fuzzy nonlocal pressure term in place of the conventional one. This nonlocal term is parameterized by ε > 0 and formally tends to the classical pressure when ε approaches zero. The central challenge is to establish that this system is a reliable approximation of the classical compressible Euler system. We establish the global existence and uniqueness of regular solutions in the neighborhood of the static state with density 1 and null velocity. Our results are demonstrated independently of the parameter ε , which enable us to prove the convergence of solutions to those of the classical Euler system. Another consequence is the rigorous justification of the convergence of the mass equation to various versions of the porous media equation in the asymptotic limit where the friction tends to infinity. Note that our results are demonstrated in the whole space, which necessitates to use the L 1 (R + ; B ˙ 2 , 1 σ (R d)) spaces framework. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and boundedness of a Neumann initial-boundary value problem for the following three-species predator-prey model with power-like inter-specific interaction mechanism and prey-taxis:$ \begin{align*} \begin{cases} u_t = d_1\Delta u+u(1-u)-b_1u^{1+\alpha}v^{1-\alpha}, \; &x\in{\Omega}, t>0, \\ v_t = d_2\Delta v-\nabla\cdot(\xi v\nabla u)+u^{1+\alpha}v^{1-\alpha}-b_2v^{1+\beta}w^{1-\beta}-\theta_1v, \; &x\in{\Omega}, t>0, \\ w_t = \Delta w-\nabla\cdot(\chi w\nabla v)+v^{1+\beta}w^{1-\beta}-\theta_2w, \; &x\in{\Omega}, t>0, \ \end{cases} \end{align*} $in a bounded smooth domain $ \Omega\subset \mathbb{R}^2 $. The inter-specific interaction of the forms $ b_1u^{1+\alpha}v^{1-\alpha} $ and $ b_2v^{1+\beta}w^{1-\beta} $ with $ \alpha\in (0, 1) $, $ \beta\in(0, 1) $ is a natural extension of the classical predator-prey types $ b_1uv $ and $ b_2vw $. By delicate coupling energy estimates, we first establish the global existence of classical solutions in two dimensional spaces for appropriate initial data. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional Nernst-Planck-Poisson-Navier-Stokes system in ℝ³. First, we obtain a priori estimates by using energy estimates. Then, we construct an iterative solution sequence by solving the approximate problem and obtaining the local existence and uniqueness of the classical solution. Finally, combining the local existence with a priori estimates, the global existence and uniqueness of the classical solution with small initial data are obtained. [ABSTRACT FROM AUTHOR]

This paper focuses on an attraction-repulsion chemotaxis system coupled with a logistic source,$ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+f(u),&\quad&x\in\Omega, t>0,\\ &0 = \Delta v-\beta v+\alpha u,&\quad&x\in\Omega, t>0,\\ &0 = \Delta w-\delta w+\gamma u,&\quad&x\in\Omega, t>0,\\ &\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0,&\quad&x\in\partial\Omega, t>0,\\ &u(x,0) = u_0(x),&\quad&x\in\Omega, \end{split} \right. \end{equation*} $in a smoothly bounded domain $ \Omega \subset \mathbb{R}^N $ ($ N\geq1 $), where $ f(u) \leq a - bu^r $ with $ a > 0 $, $ b > 0 $ and $ r \geq 1 $. The constants $ \chi $, $ \xi $, $ \alpha $, $ \beta $, $ \delta $ and $ \gamma $ are all positive. We establish the existence of a unique, globally bounded classical solution for this system under specific conditions: when $ N \leq 3 $ and $ r \geq 1 $, or when $ N \geq 4 $ and $ r = \frac{2N-2}{N} $, assuming that $ \xi\gamma = \chi\alpha $. By imposing additional assumptions on the coefficients, we also derive the asymptotic behavior of the solution. Notably, our proof technique relies heavily on the use of $ u \ln u $, a term typically employed in low-dimensional settings. We extend its applicability to arbitrary dimensions, thereby improving upon previous results reported in [17,33] which require $ r>\frac{2N-2}{N} $ for $ N\geq 1 $ and $ r = \frac{3}{2} $ specifically for $ N = 4 $. [ABSTRACT FROM AUTHOR]

We consider the quasilinear diffusion problem of u=(u1,...,um)$\bm {u}=(u^1,\ldots ,u^m)$u′+Π(t,x,u,Σu)Au=f(t,x,u,Σu)in]0,T[×Ω,u=0in]0,T[×Ωc,u(0,·)=u0(·)inΩ$$\begin{equation*} \hspace*{52pt}\begin{cases} \bm{u}'+\Pi(t,x,\bm{u},\Sigma \bm{u})\mathbb{A}\bm{u}=\bm{f}(t,x,\bm{u},\Sigma \bm{u})&\text{ in }]0,T[\times\Omega, \\ \bm{u}=\bm{0}&\text{ in }]0,T[\times\Omega^c, \\ \bm{u}(0,\cdot)=\bm{u}_0(\cdot)&\text{ in }\Omega \end{cases} \end{equation*}$$for an open set Ω⊂Rn$\Omega \subset \mathbb {R}^n$, u0∈H0s(Ω):=[H0s(Ω)]m$\bm {u}_0\in \mathbf {H}^s_0(\Omega):=[H^s_0(\Omega)]^m$, for 0

For the Cauchy problem of nonlinear elastic wave equations of three-dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in Agemi (Invent Math 142:225–250, 2000) and Sideris (Ann Math 151:849–874, 2000), independently. In this paper, we will consider the asymptotic behavior of global solutions. We first show that the global solution will scatter, i.e., it will converge to some solution of linear elastic wave equations as time tends to infinity, in the energy sense. We also prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically. The variational structure of the system will play a key role in our argument. [ABSTRACT FROM AUTHOR]

This paper deals with the following chemotaxis system: u t = ∇ · (γ (v) ∇ u - u ξ (v) ∇ v) + μ u (1 - u) , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary. Here, the positive function γ ∈ C 2 ([ 0 , + ∞)) satisfies γ ′ (s) < 0 and γ ′ ′ (s) ≥ 0 for all s ≥ 0 , also ξ (s) = - (1 - α) γ ′ (s) with α ∈ (0 , 1) . For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly in time bounded. This result is obtained for small initial data without any restriction on μ. The obtained result improves a recent result by Li and Lu (J Math Anal Appl 521:126902, 2023), which asserts the global existence of bounded classical solutions, provided that (γ ′ (s)) 2 γ ′ ′ (s) ≤ n 2 (n + 1) 3 and some conditions on initial data and μ. We should mention that in the special cases γ (s) = (1 + s) - k (k > 0) and γ (s) = e - χ s (χ > 0) , the result in Li and Lu (2023) is obtained under conditions on k and χ. But, our result is without any restriction on k and χ. [ABSTRACT FROM AUTHOR]

In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds, in a certain sense, to the Nash–Moser iterative scheme in combination with a new concept of "higher order averaging operators along highly fractal stochastic curves". This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations. [ABSTRACT FROM AUTHOR]

The attraction-repulsion chemotaxis system with logistic source $ \begin{equation*} \left\{ \begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot (u\nabla w)+u(\lambda_1-\mu_1 u^{r_1-1}+av),\quad x\in \Omega, t>0,\\ v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda_2-\mu_2 v^{r_2-1}-bu),\quad x\in \Omega, t>0,\\ w_t = \Delta w-w+v,\quad x\in \Omega, t>0,\\ 0 = \Delta z-z+u,\quad x\in \Omega, t>0\\ \end{array}\right. \;\;\;\;\;\;(*) \end{equation*} $ is considered in a bounded domain $ \Omega\subset\mathbb{R}^N (N\geq1) $ with smooth boundary, where $ \chi>0 $, $ \xi>0 $, $ \mu_1>0 $, $ \mu_2>0 $, $ \lambda_1>0 $, $ \lambda_2>0 $, $ a>0 $, $ b>0 $, $ r_1>1 $ and $ r_2>1 $ are given parameters. It is proved that under the assumption that$ \begin{align*} \min\bigl\{(r_1-1)(r_2-1), (r_1-1)\frac{2}{N}, (r_2-1)\frac{2}{N}, \frac{4}{N^2}\bigr\}>\frac{(N-2)_+}{N}, \end{align*} $for any sufficiently regular nonnegative initial data $ (u_0, v_0, w_0) $, the initial-value problem for $ (*) $ under homogeneous Neumann boundary conditions possesses one globally classical solution, which is uniformly bounded. This result significantly improves or extends previous results of [26], in which, the assumption$ \begin{align*} (r_1-1)(r_2-1)>\frac{(N-2)_+}{N} \end{align*} $is intrinsically required. [ABSTRACT FROM AUTHOR]

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics (0.1) { u t = Δ u − χ 1 ∇ ⋅ (u w ∇ w) + μ 1 u (1 − u − a 1 v) , t > 0 , x ∈ Ω , v t = Δ v − χ 2 ∇ ⋅ (v w ∇ w) + μ 2 v (1 − v − a 2 u) , t > 0 , x ∈ Ω , w t = Δ w − w + u + v , t > 0 , x ∈ Ω , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , t > 0 , x ∈ ∂ Ω , u (0 , x) = u 0 (x) , v (0 , x) = v 0 (x) , w (0 , x) = w 0 (x) , x ∈ Ω , where Ω ⊂ R N (N ≥ 2) is a bounded smooth convex domain, and the parameters χ 1 , χ 2 , μ 1 , μ 2 , a 1 and a 2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ 1 , χ 2 ∈ (0 , 1 2) for N = 2 , 3 or χ 1 , χ 2 ∈ (0 , N − 2 N − 1) for N ≥ 4. Moreover, if min ⁡ { μ 1 , μ 2 } > max ⁡ { χ 1 , χ 2 } 2 4 , we obtain the uniformly lower bound for w. Finally, when χ 1 , χ 2 are suitably small, it is proved that if 0 < a 1 , a 2 < 1 , then the solution (u , v , w) converges to (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , 2 − a 1 − a 2 1 − a 1 a 2 ) in L ∞ norm as t → ∞ ; if 0 < a 2 < 1 ≤ a 1 , then the solution (u , v , w) converges to (0 , 1 , 1) in L ∞ norm as t → ∞. [ABSTRACT FROM AUTHOR]

We provide an approach of elastic flow with dynamic conditions at the knot points to resolve the path planning problem in the plane. The elastic flow is set up by a second-order semilinear parabolic system with nonlocal lower-order terms. We prove the existence of classical global solutions to the elastic flow, whose asymptotic limits provide classical solutions to the path planning problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial value problem for the 2D anisotropic micropolar Rayleigh–Bénard problem with horizontal dissipation. Based on the energy method and the bootstrapping argument, global classical solutions are proved under the assumptions of small initial data. Moreover, time-decay rates of the oscillation part of global classical solutions are obtained by combining the energy method and Poincaré type inequality. [ABSTRACT FROM AUTHOR]

We consider a -dimensional thermal–electrical model of semiconductor heating in an electric field. For the corresponding initial-boundary value problem, we prove the existence of a classical solution that cannot be continued in time and obtain sufficient conditions for the blow-up of the solution in a finite time. [ABSTRACT FROM AUTHOR]

We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such discrete-size fragmentation models, we allow the fragmentation coefficients to vary with time. By formulating the initial-value problem for the system as a non-autonomous abstract Cauchy problem, posed in an appropriately weighted $ \ell^1 $ space, and then applying results from the theory of evolution families, we prove the existence and uniqueness of physically relevant, classical solutions for suitably constrained coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we consider the predator-prey system with density-dependent motilities and pursuit-evasion interaction under homogeneous Neumann boundary conditions. The main obstacle of analysis comes from the term produced by pursuit-evasion interaction. With the $ L^p $-estimate techniques and Moser iteration, we show that the system possesses a global bounded classical solution. Furthermore, with the aid of Lyapunov functional, we establish the asymptotic behavior of solutions to this system under appropriate parameter conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction, where the heat flux is assumed to satisfy Cattaneo's law. When the initial perturbation is small and regular, we establish the global existence of the classical solution without any additional assumption on the relaxation time τ. Furthermore, assuming the initial perturbation belongs to a negative Sobolev space, we obtain the optimal time-decay rates of the high-order spatial derivatives of solutions by using a pure energy method instead of complicated spectral analysis. It is also observed that the heat flux, due to its damping structure, decays to the motionless state at a faster time-decay rate compared to density, velocity, and temperature. [ABSTRACT FROM AUTHOR]

We deal with the following predator-prey model involving nonlinear indirect chemotaxis mechanism { u t = Δ u + ξ ∇ ⋅ (u ∇ w) + a 1 u (1 − u r 1 − 1 − b 1 v) , x ∈ Ω , t > 0 , v t = Δ v − χ ∇ ⋅ (v ∇ w) + a 2 v (1 − v r 2 − 1 + b 2 u) , x ∈ Ω , t > 0 , w t = Δ w − w + z γ , x ∈ Ω , t > 0 , 0 = Δ z − z + u α + v β , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n (n ≥ 1 ), where the parameters ξ , χ , a 1 , a 2 , b 1 , b 2 , α , β , γ > 0 . It has been shown that if r 1 > 1 , r 2 > 2 and γ (α + β) < 2 n , then there exist some suitable initial data such that the system has a global classical solution (u , v , w , z) , which is bounded in Ω × (0 , ∞) . Compared to the previous contributions, in this work, the boundedness criteria are only determined by the power exponents r 1 , r 2 , α , β , γ and spatial dimension n instead of the coefficients of the system and the sizes of initial data. [ABSTRACT FROM AUTHOR]

This research paper introduces novel findings regarding a specific class of Kirchhoff problems. Building upon the theoretical groundwork established in a previous work by Ricceri [15], the paper centers its investigation on the Sierpiński gasket (V, |•|) situated in (Rn-1,- |•|), where n≥ 2. The intrinsic boundary of the Sierpiński gasket, denoted as Vo, comprises n corners. Under certain conditions, we prove that a Kirchhoff problem has two classical solutions in the space H0¹(V). [ABSTRACT FROM AUTHOR]

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta &#x0002B;{\left(-\Delta \right)}&#x0005E;s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}&#x0005E;s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N&#x0002B;4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]

This paper considers a density-suppressed motility model with a strong Allee effect under the homogeneous Neumman boundary condition. We first establish the global existence of bounded classical solutions to a parabolic–parabolic system over an |$N $| -dimensional |$\mathbf{(N\le 3)}$| bounded domain |$\varOmega $|⁠ , as well as the global existence of bounded classical solutions to a parabolic–elliptic system over the multidimensional bounded domain |$\varOmega $| with smooth boundary. We then investigate the linear stability at the positive equilibria for the full parabolic case and parabolic–elliptic case, respectively, and find the influence of Allee effect on the local stability of the equilibria. By treating the Allee effect as a bifurcation parameter, we focus on the one-dimensional stationary problem and obtain the existence of non-constant positive steady states, which corresponds to small perturbations from the constant equilibrium |$(1,1)$|⁠. Furthermore, we present some properties through theoretical analysis on pitchfork type and turning direction of the local bifurcations. The stability results provide a stable wave mode selection mechanism for the model considered in this paper. Finally, numerical simulations are performed to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

In a bounded smooth domain Ω ⊂ R 3 and with a positive parameter χ > 0 , this paper is devoted to investigating the global boundedness and asymptotic behavior of solutions to a immune chemotaxis system. By establishing the uniform boundedness of L ∞ -norm for immune cells and the uniform boundedness of L q -norm with q ∈ (3 , ∞) for the chemokines gradient, it is possible to identify a smallness condition on chemosensitivity χ that ensures global existence of classical solution to the model in a three-dimensional bounded domain. Moreover, to get exponential decay rate the smallness of (‖ M 0 ‖ L ∞ , ‖ A 0 ‖ L ∞) is pivotal, we shall see that the system still admits an energy type inequality, and an analysis thereof will show that any nontrivial solution approaches exponentially fast with respect to the topology in L ∞ (Ω) . [ABSTRACT FROM AUTHOR]

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter δ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as δ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model. [ABSTRACT FROM AUTHOR]

A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing "third-party" molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system (4) has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system (3) from L ρ to L 2 ρ ( ρ ≥ 1 ) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N ≤ 3 . Lastly, by using various analytical methods, a threshold value ξ 00 (ξ 00 < 0) is found, such that positive constant steady state (u ∗ , v ∗ , w ∗) becomes unstable when ξ < ξ 00 . Our results show that the haptotaxis plays a crucial role in determining the stability to the model (3), that is, it can have a destabilizing effect. [ABSTRACT FROM AUTHOR]

The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, $ \begin{equation} \begin{cases} u_t = \Delta u-\chi_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) , \quad &x\in \Omega\cr v_t = \Delta v-\chi_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u), \quad &x\in \Omega\cr 0 = \Delta w-\mu w +\nu u+ \lambda v, \quad &x\in \Omega \cr \frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = \frac{\partial w}{\partial n} = 0, \quad &x\in\partial\Omega, \end{cases} \end{equation}~~~~(1) $ where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, and $ \chi_i $, $ a_i $, $ b_i $, $ c_i $ ($ i = 1, 2 $) and $ \mu, \, \nu, \, \lambda $ are positive constants. In [25], among others, we proved that for any given nonnegative initial data $ u_0, v_0\in C^0(\bar\Omega) $ with $ u_0+v_0\not \equiv 0 $, (1) has a unique globally defined classical solution $ (u(t, x;u_0, v_0), v(t, x;u_0, v_0), w(t, x;u_0, v_0)) $ with $ u(0, x;u_0, v_0) = u_0(x) $ and $ v(0, x;u_0, v_0) = v_0(x) $ in any space dimensional setting with any positive constants $ \chi_i, a_i, b_i, c_i $ ($ i = 1, 2 $) and $ \mu, \nu, \lambda $. In this paper, we assume that the competition in (1) is weak in the sense that $ \frac{c_1}{b_2}<\frac{a_1}{a_2}, \quad \frac{c_2}{b_1}<\frac{a_2}{a_1}. $ Then (1) has a unique positive constant solution $ (u^*, v^*, w^*) $, where$ u^* = \frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2}, \quad v^* = \frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, \quad w^* = \frac{\nu}{\mu}u^*+\frac{\lambda}{\mu} v^*. $We obtain some explicit conditions on $ \chi_1, \chi_2 $ which ensure that the positive constant solution $ (u^*, v^*, w^*) $ is globally stable, that is, for any given nonnegative initial data $ u_0, v_0\in C^0(\bar\Omega) $ with $ u_0\not \equiv 0 $ and $ v_0\not \equiv 0 $, $ \lim\limits_{t\to\infty}\Big(\|u(t, \cdot;u_0, v_0)-u^*\|_\infty +\|v(t, \cdot;u_0, v_0)-v^*\|_\infty+\|w(t, \cdot;u_0, v_0)-w^*\|_\infty\Big) = 0. $ [ABSTRACT FROM AUTHOR]

In this paper, we consider a three component quasilinear chemotaxis system for alopecia areata$ \begin{equation*} \begin{cases} u_t = \nabla\cdot(D_1(u)\nabla u)-\chi_1\nabla\cdot(S_1(u)\nabla w)+w-\mu_1u^{\gamma_1}, &x\in\Omega,t>0,\\ v_t = \nabla\cdot(D_2(v)\nabla v)-\chi_2\nabla\cdot(S_2(v)\nabla w)+w+ruv-\mu_2v^{\gamma_2},&x\in\Omega,t>0,\\ w_t = \Delta w+u+v-w, &x\in\Omega,t>0, \end{cases} \end{equation*} $in a smoothly bounded domain $ \Omega\subset\mathbb{R}^n(n\geq1) $ with Neumman boundary conditions, where parameters $ \chi_i,\mu_i\; (i = 1,2) $ and $ r $ are positive. The functions $ D_i(\cdot) $ and $ S_i(\cdot) $ belong to $ C^2 $ satisfying $ D_i(s)\ge(s+1)^{\alpha_i} $ and $ S_i(s)\le s(s+1)^{\beta_i-1} $ with $ \alpha_i,\beta_i\in \mathbb{ R} $ for all $ s\ge0 $ and $ i = 1,2 $. We study the global boundedness of classical solutions existing without any further restrictions on the size of system parameters in two cases: (i) both the diffusion and the logistic damping balance the cross-diffusion; (ii) the logistic damping inhibits the cross-diffusion. Those results not only extend the existing results by Xu (JMAA, 2023), but also draw some new conclusions. [ABSTRACT FROM AUTHOR]