Global Classical Solutions Research Articles

Global classical solutions of 2D Keller–Segel–Navier–Stokes equations with logistic source

Global classical solutions of 2D Keller–Segel–Navier–Stokes equations with logistic source

Global Classical Solution to the Strip Problem of 2D Compressible Navier–Stokes System with Vacuum and Large Initial Data

Global Classical Solution to the Strip Problem of 2D Compressible Navier–Stokes System with Vacuum and Large Initial Data

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

Dynamic behavior in a pursuit-evasion system with signaling mechanism

We mainly focus on the predator-prey problem associated with indirect predator taxis{ut=Δu+χ1∇⋅(u∇w)+u(1−μ1u−αv),vt=Δv−χ2∇⋅(ψ(v)∇u)+v(−1−μ2v+βu),0=Δw+v−w in a smooth bounded domain Ω⊂Rn(n≤3) with homogeneous Neumann boundary conditions, where the parameters χ1, χ2, μ1, μ2, α and β are positive. The chemotaxis sensitivity function with/without “volume-filling” effect ψ(v)=v1+σv, where σ≥0 is a constant. Compared with the well-known predator-prey system, the discussion of signaling mechanism w forms a novel feature of our system. For the case σ=0, we identify a positive constant χ0 such that the above system possesses global classical solutions with relative bound if χ2<χ0; For the case σ>0, we also show that the classical solutions of the system are globally bounded without any restrictions on system parameters. Furthermore, utilizing the Lyapunov functionals, we establish stability for the prey-only situation, the steady state of coexistence is also investigated.

Global Existence in a Predator-Prey Model with Nonlinear Indirect Chemotaxis Mechanism

One of the fundamental processes in ecology is the interaction between predator and prey. Predator-prey interactions refer to the relative changes in population density of two species as they share the same environment and one species preys on the other. There are many studies global existence or blow-up of solutions on the predator-prey model. Our this paper related to the predator-prey model with nonlinear indirect chemotaxis mechanism under homogeneous Neumann boundary conditions. We establish the global existence and boundedness of classical solutions of our problem by using parabolic regularity theory. Namely, firstly we show that u and υ boundedness in L^p for some p&amp;gt;1, then we obtain the L^∞-bound of u and υ by using Alikakos-Moser iteration. Thus, it is proved that the model has a unique global classical solution under suitable conditions on the parameters in a smooth bounded domain.

Study on a growth-expansion model with chemotaxis in nutrient-replete environments

We consider the no-flux initial-boundary value problem for the growth-expansion model with chemotaxis in nutrient-replete environments in smoothly bounded domains [Formula: see text]. It is shown that if [Formula: see text] or if [Formula: see text] 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 [Formula: see text], 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.

Global classical solutions to a multidimensional radiation hydrodynamics model with symmetry and large initial data

AbstractAs 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 and the lower bound of the temperature . 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 and the temperature by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: if ; if . 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.

Suppression of blowup by slightly superlinear degradation in a parabolic–elliptic Keller–Segel system with signal-dependent motility

In this paper, we consider an initial–Neumann boundary value problem for a parabolic–elliptic Keller–Segel system with signal-dependent motility and a source term. Previous research has rigorously shown that the source-free version of this system exhibits an infinite-time blowup phenomenon when dimension N≥2. In the current work, when N≤3, we establish uniform boundedness of global classical solutions with an additional source term that involves slightly super-linear degradation effect on the density, of a maximum growth order slogs, unveiling a sufficient blowup suppression mechanism. The motility function considered in our work takes a rather general form compared with recent works (Fujie and Jiang, 2020; Lyu and Wang, 2023) which were restricted to the monotone non-increasing case. The cornerstone of our proof lies in deriving an upper bound for the second component of the system and an entropy-like estimate, which are achieved through tricky comparison skills and energy methods, respectively.

Classical and generalized solutions of an alarm-taxis model

In bounded, spatially two-dimensional domains, the system complemented with initial and homogeneous Neumann boundary conditions, models the interaction between prey (with density u), predator (with density v) and superpredator (with density w), which preys on both other populations. Apart from random motion and prey-tactical behavior of the primary predator, the key aspect of this system is that the secondary predator reacts to alarm calls of the prey, issued by the latter whenever attacked by the primary predator. We first show in the pure alarm-taxis model, i.e. if ξ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi = 0$$\\end{document}, that global classical solutions exist. For the full model (with ξ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi > 0$$\\end{document}), the taxis terms and the presence of the term -a2uw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-a_2 uw$$\\end{document} in the first equation apparently hinder certain bootstrap procedures, meaning that the available regularity information is rather limited. Nonetheless, we are able to obtain global generalized solutions. An important technical challenge is to guarantee strong convergence of (weighted) gradients of the first two solution components in order to conclude that approximate solutions converge to a generalized solution of the limit problem.

Boundedness in a Chemotaxis-May-Nowak model for virus dynamics with gradient-dependent flux limitation

This paper investigates an extension of the May-Nowak ODE model for virus dynamics with gradient-dependent flux limitation of cross diffusion. In particular, we consider the associated no-flux initial–boundary value problem (0.1)ut=Δu−∇⋅(uf(|∇v|2)∇v)+κ−u−uw,vt=Δv−v+uw,wt=Δw−w+vin a smoothly bounded domain Ω⊂Rn(n≤3), where the parameter κ≥0. The prototypical chemotactic sensitivity function f∈C2([0,∞)) is given by f(ξ)=(1+ξ)−α,ξ≥0 with some α∈R. It is proved that whenever α∈R,ifn=1,α>n−22(n−1),ifn={2,3},global classical solutions to (0.1) exist and are uniformly bounded. Such result consists with that in [Winkler (2022), Proposition 1.2] when n≤3, which shows that the effect of gradient-dependent flux limitation in weakening the cross-diffusion term remains unchanged even in the context of nonlinear signal production mechanism.

Global classical solutions to an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source

Global classical solutions to an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source

Global well-posedness and asymptotic behaviour for a reaction–diffusion system of competition type

We analyse a reaction–diffusion system describing the growth of microbial species in a model of flocculation type that arises in biology. A generalized model is formulated on a one dimensional bounded domain with feed terms at one end of the interval. Existence of global classical positive solutions is proved under general growth assumptions, with polynomial flocculation and deflocculation rates that guarantee uniform sup norm bounds for all time t obtained by an Lp−energy functional estimate. We also show finite time blow up can occur when the yield coefficients are large enough. Also, using arguments relying on the spectral and fixed theory, we show persistence and existence of nonhomogeneous population steady-states. Finally, we present some numerical simulations to show the combined effects of motility coefficients and the flocculation–deflocculation rates on the coexistence of species.

Global smooth solutions of multidimensional compressible Euler equations with critical time depending damping and vorticity

The authors (Hou and Yin, 2017 [9]; Hou et al., 2018 [10]) proved that the 2D and 3D irrotational compressible Euler equations with critical time depending damping admit global classical solutions. In this paper, we will study this problem with non-zero vorticity and show the global existence of smooth solutions to the 2D and 3D compressible Euler equations with critical time depending damping. They used the Lorentz vector field and their method fails in the presence of the vorticity. The main ingredients in this paper are the auxiliary energies which will make up for the loss of the Lorentz vector field and the improved decay estimates of the energy which can be achieved by a new multiplier.

Global Classical Large Solutions for the Radiation Hydrodynamics Model in Unbounded Domains

Global Classical Large Solutions for the Radiation Hydrodynamics Model in Unbounded Domains

Multidimensional Hyperbolic Equations with Nonlocal Potentials: Families of Global Solutions

In spaces of arbitrary dimensions, hyperbolic differential–difference equations with potentials undergoing translations along arbitrary spatial directions are investigated. The fundamental novelty of this study is that, unlike previous investigations, no restrictions are imposed on the symbol of the differential–difference operator contained in the equation. For the investigated equation, we succeed to explicitly construct a multiparametric family of classical global solutions.

On the global existence of solutions to a chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source

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.

Global Classical Solution and Asymptotic Behavior to a Kind of Linearly Degenerate Quasilinear Hyperbolic System

Global Classical Solution and Asymptotic Behavior to a Kind of Linearly Degenerate Quasilinear Hyperbolic System

Regularizing effect in a Keller–Segel system with density-dependent sensitivity for Lp-initial data of cell density

The fully parabolic Keller–Segel system{ut=Δu−χ∇⋅(u(u+1)α−1∇v)inΩ×(0,∞),vt=Δv−v+uinΩ×(0,∞) is considered in a bounded domain Ω⊂RN (N∈N) with smooth boundary, where χ,α∈R. It is shown that an associated Neumann initial-boundary value problem admits a global classical solution for initial data with low regularity, in particular, for Lp-initial data (p>max⁡{1,α,NN+1(α+1)}) of u when α<2N and when α=2N and the initial mass is small. This extends a well-known result on global solvability for reasonably regular initial data.

Global boundedness and asymptotic behavior in a double haptotaxis model for oncolytic virotherapy

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy{ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v−κv,wt=DwΔw−ξw∇⋅(w∇v)−w+ρuz,zt=DzΔz−z−ρuz+βw, in a smooth bounded domain Ω⊂R2, with parameters Dw>0, Dz>0, ξw≥0, ρ≥0, κ>0 and β≥0. This system describes the 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 bounded in (L∞(Ω))4. Moreover, it is shown that this bounded solution can approach a spatially constant equilibrium in the large time limit under the additional assumption that 0≤β<1.

Global Classical Solutions to a Predator-Prey Model with Nonlinear Indirect Chemotaxis Mechanism

Global Classical Solutions to a Predator-Prey Model with Nonlinear Indirect Chemotaxis Mechanism