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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]

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]

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]

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]

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]

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 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]

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]

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]

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]

In this paper, we shall study the following model: { N t = d N Δ N + λ N − N 2 − N P , x ∈ Ω , t > 0 , P t = Δ (d (S) P) + μ P − P 2 + γ N P , x ∈ Ω , t > 0 , S t = d S Δ S + τ N − η S , x ∈ Ω , t > 0 , ∂ N ∂ ν = ∂ P ∂ ν = ∂ S ∂ ν = 0 , x ∈ ∂ Ω , (N , P , S) (x , 0) = (N 0 , P 0 , S 0) (x) , x ∈ Ω , where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary. Based on semigroup estimates and Moser iteration, we first establish the existence of the global classical solution with a uniform-in-time bound. Moreover, we build the global stabilization of constant steady states by constructing suitable Lyapunov functionals and give the decay rates of solutions. Furthermore, we find that the anti-predation mechanism will generate steady state bifurcation. We further prove the global existence of non-constant positive steady state solutions under certain conditions. Finally, we numerically demonstrate spatially inhomogeneous coexistence patterns (i.e., non-constant positive steady state solutions) for the predator, the prey and the signal. [ABSTRACT FROM AUTHOR]

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm–Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods. [ABSTRACT FROM AUTHOR]

In this paper, we establish the uniqueness and existence of the classical solution to higher-order boundary value problems. Then, we give a new Lyapunov-type inequalities for higher order boundary value problems. Our study is based on Green’s functions corresponding to the five different types of two-point boundary value problems. In addition, some applications of the obtained inequalities are given. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for global-in-time existence of classical solutions for the fully parabolic chemorepulsion system posed on a convex, bounded three-dimensional domain. Our main result establishes global-in-time existence of regular nonnegative solutions provided tha... Our method is related to the Bakry-Émery calculation and appears to be new in this context. [ABSTRACT FROM AUTHOR]

We consider classical solutions to the chemotaxis system with logistic source, a u − μ u 2 , under nonlinear Neumann boundary conditions ∂ u ∂ ν = | u | p with p > 1 in a smooth convex bounded domain Ω ⊂ R n , where n ≥ 2. This paper aims to show that if p < 3 2 , and μ > 0 , n = 2 , or μ is sufficiently large when n ≥ 3 , then the parabolic-elliptic chemotaxis system admits a unique nonnegative global-in-time classical solution that is bounded in Ω × (0 , ∞). The similar result is also true if p < 3 2 , n = 2 , and μ > 0 or p < 7 5 , n = 3 , and μ is sufficiently large for the parabolic-parabolic chemotaxis system. [ABSTRACT FROM AUTHOR]

This paper deals with the singular chemotaxis-fluid system with indirect signal production or consumption n t + u ⋅ ∇ n = Δ n − χ ∇ ⋅ (n v ∇ v) ; v t + u ⋅ ∇ v = Δ v + g (v , w) ; w t + u ⋅ ∇ w = Δ w − w + n ; u t + (u ⋅ ∇) u = Δ u − ∇ P + n ∇ Φ ; ∇ ⋅ u = 0 , x ∈ Ω , t > 0 in a bounded and smooth domain Ω ⊂ R 2 with no-flux/no-flux/no-flux/Dirichlet boundary conditions, where Φ ∈ W 2 , ∞ (Ω). In the cases signal production (i.e., g (v , w) = − v + w) and consumption (i.e., g (v , w) = − v w), it is proved that for any appropriately regular initial data, the associated initial-boundary value problem possesses a global classical solution for any χ > 0. It is worth mentioning that the global existence of classical solution for χ ∈ (0 , 1) or generalized solutions for arbitrary χ > 0 was obtained in the previously known results on the corresponding system with direct signal production, and that only the global generalized solutions for arbitrary χ > 0 were established in a precedent concerning the corresponding system with direct signal consumption. In comparison to these results for the case of direct signal production or consumption, our results rigorously confirm that both the indirect signal production and consumption mechanisms authentically contribute to the global solvability of the two-dimensional singular chemotaxis-fluid system. [ABSTRACT FROM AUTHOR]

This paper aims to provide the rigorous convergence analysis of the chemotaxis system { ∂ n ϵ ∂ t = Δ n ϵ − ∇ ⋅ (n ϵ ∇ c ϵ) , τ 1 ∂ c ϵ ∂ t = Δ c ϵ − c ϵ + w ϵ , τ 2 ∂ w ϵ ∂ t = − w ϵ + n ϵ in a smoothly bounded planar domain Ω as τ 1 → 0 and (or) τ 2 → 0. Precisely, we reveal that actually the indirect signal properties are strong enough to allow for a uniformly global-in-time convergence in terms of τ 1 = τ 2 = ϵ ↘ 0 and τ 1 = 1 , τ 2 = ϵ ↘ 0 for ∫ Ω n ϵ (⋅ , 0) < 4 π , which correspond to the classical Keller-Segel systems with direct signal production, and of τ 1 = ϵ ↘ 0 , τ 2 = 1 for suitable small initial data. As a byproduct, we show that for any p ≥ 2 , the L p -norm of n ϵ blows up if ∫ Ω n ϵ (⋅ , 0) ∈ (4 π , ∞) ∖ 4 π N , which improves the existing results that the solution is unbounded in the sense of L ∞ -norm of n ϵ. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis-(Navier-)Stokes system { n t + u ⋅ ∇ n = ∇ ⋅ (φ (c) ∇ n) − ∇ ⋅ (n χ (c) ∇ c) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + κ (u ⋅ ∇) u = Δ u − ∇ P + n ∇ ψ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , subject to no-flux boundary conditions in a bounded domain Ω ⊂ R 2 with smooth boundary, with a given smooth potential ψ : Ω → R and κ ∈ R. The density-dependent motility φ (c) and the chemotactic function χ (v) are assumed to satisfy some further technical conditions. It is shown that the problem possesses a global classical solution (n κ , c κ , u κ). Furthermore, we prove that these solutions stabilize uniformly to ( n 0 ‾ , 0 , 0) exponentially with respect to t , where n 0 ‾ : = 1 | Ω | ∫ Ω n 0. Finally, it is also shown that there exists a constant μ > 0 such that for all κ ∈ (0 , 1) ‖ n (κ) (⋅ , t) − n (0) (⋅ , t) ‖ L ∞ (Ω) + ‖ c (κ) (⋅ , t) − c (0) (⋅ , t) ‖ L ∞ (Ω) + ‖ u (κ) (⋅ , t) − u (0) (⋅ , t) ‖ L ∞ (Ω) ≤ C κ 1 3 e − μ t for all t > 0 , where C > 0 is a constant independent of κ and t. [ABSTRACT FROM AUTHOR]

We consider the chemotaxis system: ... under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, n ≥ 2, with smooth boundary. Here, the functions γ(v) and ξ(v) are as: γ(v) = (1+v)-k and ξ(v) = -(1 -- α)γ'(v), where k > 0 and α ∈ (0, 1). We prove that the classical solutions to the above system are uniformly-in-time bounded provided that k (1-- α) < 4/n+5 and the initial value v0 and µ satisfy the following conditions: ... This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023). [ABSTRACT FROM AUTHOR]

This article concerns the structure of the nonconstant steady states for a predator-prey model of Leslie-Gower type with Sigmoid functional and prey-taxis subject to the homogeneous Neumann boundary condition. The existence of bounded classical global solutions is discussed in bounded domains with arbitrary spatial dimension and any prey-taxis sensitivity coefficient. The local stability of the homogeneous steady state is analyzed to show that the prey-taxis sensitivity coefficient destabilizes the stability of the homogeneous steady state when prey defends. Then we study the existence and stability of the nonconstant positive steady state of the system over 1D domain by applying the bifurcation theory and present properties of local branches such as pitchfork and turning direction. Moreover, we discuss global bifurcation, homogeneous steady state solutions, nonconstant steady states solutions, spatio-temporal periodic solutions and spatio-temporal irregular solutions which demonstrate the coexistence and spatial distribution of prey and predator species. Finally, we perform numerical simulations to illustrate and support our theoretical analysis. [ABSTRACT FROM AUTHOR]

We prove a global version of the classical result that p -harmonic functions belong to W loc 2 , 2 for 1 < p < 3 + 2 n − 2. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6]. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and large time behavior of the classical solution for the non-isentropic compressible Navier-Stokes-Allen-Cahn (NSAC) system describing the mixture motion of a two-phase immiscible viscous compressible heat-conducting flow in R 3. We first establish the global well-posedness of the Cauchy problem to the non-isentropic compressible NSAC system when the initial data is a small regular perturbation near the constant state. Then by a delicate spectral analysis and the damping structure for the perturbation equation of the phase field, we prove that the phase field decays to the phase separation state at an exponential rate while the density, velocity and the temperature converge to the constant state at the optimal time rate (1 + t) − 3 / 4 in L 2 norm. Furthermore, for well-chosen initial data, we also establish the lower bounds of time decay rate. [ABSTRACT FROM AUTHOR]

In this paper, we study the following initial-boundary value problem of a three-species spatial food chain model { u t = d 1 Δ u + u (1 − u) − b 1 u v , x ∈ Ω , t > 0 v t = d 2 Δ v − ∇ ⋅ (ξ v ∇ u) + u v − b 2 v w − θ 1 v , x ∈ Ω , t > 0 w t = Δ w − ∇ ⋅ (χ w ∇ v) + v w − θ 2 w , x ∈ Ω , t > 0 in a bounded domain Ω ⊂ R 2 with smooth boundary and homogeneous Neumann boundary conditions, where all parameters are positive constants. By the delicate coupling energy estimates, we first establish the global existence of classical solutions in two dimensional spaces for appropriate initial data. Moreover by constructing Lyapunov functionals and using LaSalle's invariance principle, we establish the global stability of the prey-only steady state, semi-coexistence and coexistence steady states. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator-prey model with density-dependent prey-taxis and stage structure for the predator. We establish the existence of classical solutions with uniform-in-time bound in a one-dimensional case. In addition, we prove that the solution stabilizes to the prey-only steady state under some conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to a chemotaxis system with signal-dependent motility under homogeneous Neumann boundary conditions in a bounded domain. We prove that this problem possesses a global classical solution which is uniformly bounded under weaker conditions than that obtained by Lv and Wang (2020). The findings of our study demonstrate the presence of a consistent decay rate, effectively ruling out the occurrence of blow-up phenomena in the system across all spatial dimensions. [ABSTRACT FROM AUTHOR]

The quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization (⁎) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n S (x , n , c) ∇ c) − n m , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + m , x ∈ Ω , t > 0 , m t + u ⋅ ∇ m = Δ m − n m , x ∈ Ω , t > 0 , u t + κ (u ⋅ ∇) u + ∇ P = Δ u + (n + m) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 is considered in a bounded domain Ω ⊂ R 3 with smooth boundary, where ϕ ∈ W 2 , ∞ (Ω) , S denotes a given tensor-valued function fulfilling | S (x , n , c) | ≤ S 0 (c) (1 + n) − α with α ≥ 0 , and S 0 is a nonnegative nondecreasing function. It is shown that if α > 0 , then for any κ ∈ R , the corresponding initial-boundary value problem (⁎) possesses at least one global weak solution, which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. Additionally, we also show that, after some time, these solutions become classical solutions, that is, there exists a positive constant T such that (n , c , m , u) is smooth in Ω × (T , ∞). Finally, we can also prove that when t → ∞ , any such solution (n , c , m , u) would stabilize to the spatially uniform equilibrium (n ˆ , m ˆ , m ˆ , 0) in L ∞ (Ω) , where n ˆ = 1 | Ω | { ∫ Ω n 0 − ∫ Ω m 0 } + and m ˆ = 1 | Ω | { ∫ Ω m 0 − ∫ Ω n 0 } +. The results can provide an important reference for solving the existence of classical solutions of three-dimensional Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

We investigate the formation of singularities in one-dimensional hyperbolic compressible Navier-Stokes equations, a model proposing a relaxation leading to a hyperbolization through a nonlinear Cattaneo law for heat conduction as well as through the constitutive Maxwell type relations for the stress tensor. By using the entropy dissipation inequality, which gives the lower energy estimates of the local solutions without any smallness condition on initial data, and by constructing some useful averaged quantities we show that there are in general no global C 1 solutions for the studied system with some large initial data. This appears as a remarkable contrast to the situation without relaxation, i.e. for the classical compressible Navier-Stokes equations, where global large solutions exist. It also contrasts the fact that for the linearized system associated to the classical resp. relaxed compressible Navier-Stokes equations, the qualitative behavior is exactly the same: exponential stability in bounded domains and polynomial decay without loss of regularity for the Cauchy problem. [ABSTRACT FROM AUTHOR]

We will consider a type of elementary fuzzy partial differential equation that we wish to solve. The classical solution and the extension solution are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the small-convection limit of chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source { n t κ + u κ ⋅ ∇ n κ = Δ n κ − χ ∇ ⋅ (n κ ∇ log c κ) + f (n κ) , x ∈ Ω , t > 0 , c t κ + u κ ⋅ ∇ c κ = Δ c κ − c κ + n κ , x ∈ Ω , t > 0 , u t κ + κ (u κ ⋅ ∇) u κ = Δ u κ + ∇ P κ + n κ ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u κ = 0 , x ∈ Ω , t > 0 , in a bounded convex domain Ω ⊆ R 2 with smooth boundary, where κ ∈ R , f (s) = μ 1 s − μ 2 s λ , λ > 1 , and ϕ : Ω → R is a given smooth potential with second-order partial derivatives. When the chemotaxis sensitivity χ satisfies the appropriate conditions, it is proved that the unique global classical solutions (n κ , c κ , u κ) will stabilize to (n 0 , c 0 , u 0) as κ → 0 . [ABSTRACT FROM AUTHOR]

This paper deals with the Keller-Segel(-Navier)-Stokes system with indirect signal production (⋆) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n ∇ v) + r n − μ n 2 , v t + u ⋅ ∇ v = Δ v − v + w , w t + u ⋅ ∇ w = Δ w − w + n , u t + κ (u ⋅ ∇) u = Δ u − ∇ P + n ∇ Φ , ∇ ⋅ u = 0 in a bounded and smooth domain Ω ⊂ R N (N = 2 , 3) with no-flux boundary for n , v , w and no-slip boundary for u , where r ∈ R , μ ≥ 0 , κ ∈ { 0 , 1 } and Φ ∈ W 2 , ∞ (Ω). In the case without logistic source (r = μ = 0), it is proved that for all suitably regular initial data, the associated initial-boundary value problem for the spatially two-dimensional Navier-Stokes system (⋆) admits a globally bounded classical solution. This result improves and extends the previously known ones. We point out that the same result to the corresponding two-dimensional Navier-Stokes system with direct signal production holds necessarily imposing some saturated chemotactic sensitivity, logistic damping or small total initial population mass. In the case coupled with logistic source (r ∈ R , μ > 0), it is shown that for any reasonably regular initial data, the corresponding initial-boundary value problem for the spatially three-dimensional Stokes system (⋆) possesses a globally bounded classical solution, and that this solution stabilizes toward the corresponding spatially homogeneous equilibrium with the explicit convergence rates for the cases r < 0 , r = 0 and r > 0. We underline that the global boundedness of classical solution to the corresponding three-dimensional Stokes system with direct signal production was obtained only for μ ≥ 23 (or sublinear signal production), and that the convergence result to the corresponding system with direct signal production was established only for r = 0 and μ ≥ 23. Our results rigorously confirm that the indirect signal production mechanism genuinely contributes to the global boundedness of classical solution to the Keller-Segel(-Navier)-Stokes system. [ABSTRACT FROM AUTHOR]

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy in two-dimensional domains, accounting for interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic viruses. It is rigorously proved that an associated no-flux initial-boundary value problem has a unique global classical solution which is uniformly bounded under suitable assumptions. Moreover, it is shown that a bounded solution can approach spatially constant equilibrium in the large time limit. [ABSTRACT FROM AUTHOR]

We study Boltzmann-Fermi-Dirac equation when quantum effects are taken into account in dilute gas dynamics. By employing new estimates on trilinear terms in collision integral, we prove the global existence of the classical solution to Boltzmann-Fermi-Dirac equation near equilibrium. Furthermore, the limit from Boltzmann-Fermi-Dirac equation to incompressible Navier-Stokes-Fourier equations is justified rigorously, which was formally derived in the thesis of Zakrevskiy [48]. [ABSTRACT FROM AUTHOR]

In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on R 3 × [ 0 , T ] with any T ∈ (0 , ∞) , provided that the initial magnetic field in the L 3 -norm and the initial density are suitably small. Note that the first result is obtained under the condition of ρ 0 ∈ L γ ∩ W 2 , q with q ∈ (3 , 6) and γ ∈ (1 , 6) . It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., ρ 0 ∉ L 1 ). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial value problem for the Euler–Riesz system, where the interaction forcing is given by ∇ (− Δ) s ρ for some − 1 < s < 0 , with s = − 1 corresponding to the classical Euler–Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler–Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler–Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler–Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler–Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support. [ABSTRACT FROM AUTHOR]

In this paper, we consider planar magnetohydrodynamics (MHD) system when the viscous coefficients and heat conductivity depend on specific volume v and temperature 𝜃. For technical reasons, the viscous coefficients and heat conductivity are assumed to be proportional to h(v)𝜃α where h(v) is a non-degenerate smooth function satisfying some additional conditions. We prove the existence and uniqueness of the global-in-time classical solution to the Cauchy problem with general large initial data when |α| is sufficiently small and the coefficient of magnetic diffusion ν is suitably large. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity. [ABSTRACT FROM AUTHOR]

This article concerns the chemotaxis-growth system with indirect signal production ... on a smooth bounded domain Ω ⊂ Rn (n ≥ 1) with homogeneous Neumann boundary condition, where the parameters μ; δ > 0. It is proved that if n ≤ 2 and μ > 0, for all suitably regular initial data, this model possesses a unique global classical solution which is uniformly-in-time bounded. While in the case n ≥ 3, we show that if μ is suffciently large, this system possesses a global bounded solution. Furthermore, the large time behavior and rates of convergence have also been considered under some explicit conditions. [ABSTRACT FROM AUTHOR]

The multicritical point principle is one of the interesting theoretical possibilities that can explain the fine-tuning problems of the universe. It simply claims that "the coupling constants of a theory are tuned to one of the multicritical points, where some of the extrema of the effective potential are degenerate." One of the simplest examples is the vanishing of the second derivative of the effective potential around a minimum. This corresponds to the so-called classical conformality, because it implies that the renormalized mass m 2 vanishes. More generally, the form of the effective potential of a model depends on several coupling constants, and we should sweep them to find all the multicritical points. We study the multicritical points of a general scalar field ϕ at one-loop level under the circumstance that the vacuum expectation values of the other fields are all zero. For simplicity, we also assume that the other fields are either massless or so heavy that they do not contribute to the low-energy effective potential of ϕ. This assumption makes our discussion very simple because the resultant one-loop effective potential is parametrized by only four effective couplings. Although our analysis is not completely general because of the assumption, it can still be widely applicable to many models of the Coleman–Weinberg mechanism and its generalizations. After classifying the multicritical points at low-energy scales, we will briefly mention the possibility of criticalities at high-energy scales and their implications for cosmology. [ABSTRACT FROM AUTHOR]

The extended Drinfel'd algebra (ExDA) is the underlying symmetry of non-Abelian duality in the low-energy effective theory of string theory. Non-Abelian U -dualities in maximal supergravities have been studied well, but there has been no study on non-Abelian dualities in half-maximal supergravities. We construct the ExDA for half-maximal supergravities in d ≥ 4. We also find an extension of the homogeneous classical Yang–Baxter equation in these theories. [ABSTRACT FROM AUTHOR]

We present a series of new gauge-invariant quantities in Witten's open string field theory. They are defined for a given set of open string states which satisfy the physical state condition around a classical solution. For known classical solutions, we demonstrate that these gauge-invariant quantities compute on-shell tree-level scattering amplitudes around the corresponding D-brane configuration. [ABSTRACT FROM AUTHOR]

We study a thermofield double type entangled state on two disjoint universes A and B , where one of the universes is asymptotically flat containing a black hole. As we increase the entanglement temperature, this black hole receives back-reaction from the stress–energy tensor of the state. This results in lengthening of the wormhole region in the black hole interior, and decreasing of its horizon area, both of which are key features of an evaporating black hole. We then compute the entanglement entropy in universe A through the island formula, and argue that it naturally follows the Page curve of an evaporating black hole in flat space. We also study the effects of local operations in the gravitating universe with the black hole. We find that they accelerate the evaporation of the black hole, thereby disrupting the entanglement between the two universes. Furthermore, we observe that, depending on whether the operation can be regarded as a local operation and classical communication or not, the behavior of the entanglement entropy changes. In particular, when the operation is made neither in the entanglement wedge of the radiation system nor that of the black hole, the transition between the island phase and the no-island phase can happen multiple times. [ABSTRACT FROM AUTHOR]

The article presents a study that explores about the global minimization via classical tunneling assisted by collective force field formation. It mentions the mechanism as appealing for its physical relevance in nanoscale physics and for its possible applications in optimization, Monte Carlo schemes, and machine learning.

In this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity. [ABSTRACT FROM AUTHOR]

In this article, we study further applications of the Schwarzschild–Finsler–Randers (SFR) model which was introduced in a previous work Triantafyllopoulos et al. (Eur Phys J C 80(12):1200, 2020). In this model, we investigate curvatures and the generalized Kretschmann invariant which plays a crucial role for singularities. In addition, the derived path equations are used for the gravitational redshift of the SFR-model and these are compared with the GR model. Finally, we get some results for different values of parameters of the generalized photonsphere of the SFR-model and we find small deviations from the classical results of general relativity (GR) which may be ought to the possible Lorentz violation effects. [ABSTRACT FROM AUTHOR]