Keller Segel System Research Articles

Global existence, uniqueness and $$L^{\infty }$$-bound of weak solutions of fractional time-space Keller-Segel system

Global existence, uniqueness and $$L^{\infty }$$-bound of weak solutions of fractional time-space Keller-Segel system

Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture

Abstract The Neumann problem for the Keller-Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = − ∫ Ω u d x , is considered in n-dimensional balls Ω with n ⩾ 2 , with suitably regular and radially symmetric, radially nonincreasing initial data u 0. The functions D and S are only assumed to belong to C 2 ( [ 0 , ∞ ) ) and to satisfy D > 0 and S ⩾ 0 on [ 0 , ∞ ) as well as S ( 0 ) = 0 ; in particular, diffusivities with arbitrarily fast decay are included. In this general context, it is shown that it is merely the asymptotic behavior as ξ → ∞ of the expression I ( ξ ) := S ( ξ ) ξ 2 n D ( ξ ) , ξ > 0 , which decides about the occurrence of blow-up: Namely, it is seen that • if lim ξ → ∞ I ( ξ ) = 0 , then any such solution is global and bounded, that • if lim sup ξ → ∞ I ( ξ ) < ∞ and ∫ Ω u 0 is suitably small, then the corresponding solution is global and bounded, and that • if lim inf ξ → ∞ I ( ξ ) > 0 , then at each appropriately large mass level m, there exist radial initial data u 0 such that ∫ Ω u 0 = m , and that the associated solution blows up either in finite or in infinite time. This especially reveals the presence of critical mass phenomena whenever lim ξ → ∞ I ( ξ ) ∈ ( 0 , ∞ ) exists.

A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller–Segel model

Abstract We study a finite volume scheme approximating a parabolic-elliptic Keller–Segel system with power-law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^{\infty }(0,T;H^{1}(\varOmega ))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behavior of the error estimator under changes of $\gamma $.

Hele-Shaw flow as a singular limit of a Keller-Segel system with nonlinear diffusion

Hele-Shaw flow as a singular limit of a Keller-Segel system with nonlinear diffusion

Global solutions for time-space fractional fully parabolic Keller–Segel system

We show the existence of a global solution to time-space fractional fully parabolic Keller–Segel system: \left\{ \begin{align*}{}_{0}^{c}D_{t}^{\beta} u+(-\Delta)^{\alpha/2}u+\nabla\cdot(u \nabla v)&=0,&&x\in\mathbb{R}^{n},\ t>0,\\{}_{0}^{c}D^{\beta}_{t} v+(-\Delta)^{\alpha/2}v-u&=0,&&x\in\mathbb{R}^{n},\ t>0,\\ u(x,0)=u_{0}(x),\quad v(x,0)&=v_{0}(x),&&x\in\mathbb{R}^{n},\end{align*}\right. under the smallness condition on the initial data, where 0<\beta<1 , 1<\alpha\leq 2 and n\geq 2 , u and v denote the cell density and the concentration of the chemoattractant, respectively, and {}_{0}^{c}D^{\beta}_{t} denotes the Caputo fractional derivative of order \beta with respect to time t . The nonlocal operator (-\Delta)^{\alpha/2} , defined with respect to the space variable x , is known as the Laplacian of order \frac{\alpha}{2} . We establish the existence of weak solution to the above system by fixed-point arguments under suitable conditions on u_{0} and v_{0} .

Singular limit of a chemotaxis model with indirect signal production and phenotype switching

Abstract Convergence of solutions to a partially diffusive chemotaxis system with indirect signal production and phenotype switching is shown in a two-dimensional setting when the switching rate increases to infinity, thereby providing a rigorous justification of formal computations performed in the literature. The expected limit system being the classical parabolic–parabolic Keller–Segel system, the obtained convergence is restricted to a finite time interval for general initial conditions but valid for arbitrary bounded time intervals when the mass of the initial condition is appropriately small. Furthermore, if the solution to the limit system blows up in finite time, then neither of the two phenotypes in the partially diffusive system can be uniformly bounded with respect to the L 2-norm beyond that time.

On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller–Segel system

This paper is concerned with the fractional Keller–Segel system in temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag–Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller–Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition on the initial data.

Local and global solutions for a subdiffusive parabolic–parabolic Keller–Segel system

Local and global solutions for a subdiffusive parabolic–parabolic Keller–Segel system

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.

Prevention of infinite-time blowup by slightly super-linear degradation in a Keller–Segel system with density-suppressed motility

An initial-Neumann boundary value problem for a Keller–Segel system with density-suppressed motility and source terms is considered. Infinite-time blowup of the classical solution was previously observed for its source-free version when dimension N⩾2 . In this work, we prove that with any source term involving a slightly super-linear degradation effect on the density, of a growth order of slog⁡s at most, the classical solution is uniformly-in-time bounded when N⩽3 , thus preventing the infinite-time explosion detected in the source-free counter-part. The cornerstone of our proof lies in an improved comparison argument and a construction of an entropy inequality.

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ℝ d for d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝ d . We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

Local existence and global boundedness for a chemotaxis system with gradient dependent flux limitation

In this paper, we investigate the following Keller–Segel system with flux limitation [Formula: see text] under no-flux boundary conditions in a ball [Formula: see text]), where [Formula: see text] is positive and [Formula: see text] with [Formula: see text]. It is proved that the problem (⋆) possesses a unique classical solution that can be extended in time up to a maximal [Formula: see text]. Moreover, it is shown that the above solution is global and bounded when either [Formula: see text] if [Formula: see text] and [Formula: see text], or [Formula: see text] if [Formula: see text] and [Formula: see text]. We point out that when [Formula: see text], our result is consistent with that of [N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436–473].

Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System

Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system We consider the critical mass case ∫R2u0(x)dx=8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{{\\mathbb {R}}^2} u_0(x)\\, \ extrm{d}x = 8\\pi $$\\end{document}, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function u0∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0^*$$\\end{document} with mass 8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi $$\\end{document} such that for any initial condition u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0$$\\end{document} sufficiently close to u0∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0^*$$\\end{document} and mass 8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi $$\\end{document}, the solution u(x, t) of (∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) is globally defined and blows-up in infinite time. As t→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow +\\infty $$\\end{document} it has the approximate profile u(x,t)≈1λ2(t)Ux-ξ(t)λ(t),U(y)=8(1+|y|2)2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u(x,t) \\approx \\frac{1}{\\lambda ^2(t)} U\\left( \\frac{x-\\xi (t)}{\\lambda (t)} \\right) , \\quad U(y)= \\frac{8}{(1+|y|^2)^2}, \\end{aligned}$$\\end{document}where λ(t)≈clogt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (t) \\approx \\frac{c}{\\sqrt{\\log t}}$$\\end{document}, ξ(t)→q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi (t)\\rightarrow q$$\\end{document} for some c>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c>0$$\\end{document} and q∈R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in {\\mathbb {R}}^2$$\\end{document}. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).

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.

On asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic manifolds

In this article, we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of the semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality.

Well‐posedness and time decay of fractional Keller–Segel–Navier‐Stokes equations in homogeneous Besov spaces

AbstractIn this paper, we consider the parabolic–elliptic Keller–Segel system, which is coupled to the incompressible Navier–Stokes equations through transportation and friction. It is shown that when the system is diffused by Lévy motion, the well‐posedness of the mild solution to the corresponding Cauchy problem in homogeneous Besov spaces is established by means of the Banach fixed point theorem. Furthermore, we prove the Lorentz regularity in time direction and the maximal regularity of solutions. In addition, we obtain the additional regularity and explore the time decay property of global mild solutions.

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

AbstractIn this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

Complete infinite-time mass aggregation in a quasilinear Keller–Segel system

Radially symmetric global unbounded solutions of the chemotaxis system {ut=∇⋅(D(u)∇u)−∇⋅(uS(u)∇v),0=Δv−μ+u,μ=1|Ω|∫Ωu,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ {\\matrix{{{u_t} = \ abla \\cdot (D(u)\ abla u) - \ abla \\cdot (uS(u)\ abla v),} \\hfill & {} \\hfill \\cr {0 = \\Delta v - \\mu + u,} \\hfill & {\\mu = {1 \\over {|\\Omega |}}\\int_\\Omega {u,} } \\hfill \\cr } } \\right.$$\\end{document} are considered in a ball Ω = BR(0) ⊂ ℝn, where n ≥ 3 and R > 0.Under the assumption that D and S suitably generalize the prototypes given by D(ξ) = (ξ + ι)m−1 and S(ξ) = (ξ + 1)−λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling m+λ<1−2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m + \\lambda < 1 - {2 \\over n}$$\\end{document}, a considerably large set of initial data u0 is found to enforce a complete mass aggregation in infinite time in the sense that for any such u0, an associated Neumann type initial-boundary value problem admits a global classical solution (u, v) satisfying 1C⋅(t+1)1λ≤||u(⋅,t)||L∞(Ω)≤C⋅(t+1)1λforallt>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${1 \\over C} \\cdot {(t + 1)^{{1 \\over \\lambda }}} \\le ||u( \\cdot ,t)|{|_{{L^\\infty }(\\Omega )}} \\le C \\cdot {(t + 1)^{{1 \\over \\lambda }}}\\,\\,\\,{\\rm{for}}\\,\\,{\\rm{all}}\\,\\,t > 0$$\\end{document} as well as ||u(⋅,t)||L1(Ω\\Br0(0))→0ast→∞for allr0∈(0,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$||u( \\cdot \\,,t)|{|_{{L^1}(\\Omega \\backslash {B_{{r_0}}}(0))}} \ o 0\\,\\,\\,{\\rm{as}}\\,\\,t \ o \\infty \\,\\,\\,{\\rm{for}}\\,\\,{\\rm{all}}\\,\\,{r_0} \\in (0,R)$$\\end{document} with some C > 0.

Classical and Quantum Mechanical Models of Many-Particle Systems

The workshop focused on the collective behavior of many-particle systems in various application fields: physics (gas dynamics, plasmas, quantum mechanics), mathematical biology (cell mobility, evolution of trait-structured species), and social sciences (wealth distribution). This includes famous models such as the Boltzmann equation of gas dynamics, Vlasov equation for plasmas, Fokker–Planck equations, Smoluchowski and related equations, Keller–Segel system of chemotaxis.

A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects

The quasilinear Keller–Segel system ut=∇·(D(u)∇u)-∇·(S(u)∇v),vt=Δv-v+u,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t=\ abla \\cdot (D(u)\ abla u) - \ abla \\cdot (S(u)\ abla v), \\\\ v_t=\\Delta v-v+u, \\end{array}\\right. \\end{aligned}$$\\end{document}endowed with homogeneous Neumann boundary conditions is considered in a bounded domain Ω⊂Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset {\\mathbb {R}}^n$$\\end{document}, n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 3$$\\end{document}, with smooth boundary for sufficiently regular functions D and S satisfying D>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D>0$$\\end{document} on [0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,\\infty )$$\\end{document}, S>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S>0$$\\end{document} on (0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0,\\infty )$$\\end{document} and S(0)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(0)=0$$\\end{document}. On the one hand, it is shown that if SD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{S}{D}$$\\end{document} satisfies the subcritical growth condition S(s)D(s)≤Csαforalls≥1withsomeα<2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{S(s)}{D(s)} \\le C s^\\alpha \\qquad \ ext{ for } \ ext{ all } s\\ge 1 \\qquad \ ext{ with } \ ext{ some } \\alpha < \\frac{2}{n} \\end{aligned}$$\\end{document}and C>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C>0$$\\end{document}, then for any sufficiently regular initial data there exists a global weak energy solution such that esssupt>0‖u(t)‖Lp(Ω)<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${ \\mathrm{{ess}}} \\sup _{t>0} \\Vert u(t) \\Vert _{L^p(\\Omega )}<\\infty $$\\end{document} for some p>2nn+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p > \\frac{2n}{n+2}$$\\end{document}. On the other hand, if SD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{S}{D}$$\\end{document} satisfies the supercritical growth condition S(s)D(s)≥csαforalls≥1withsomeα>2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{S(s)}{D(s)} \\ge c s^\\alpha \\qquad \ ext{ for } \ ext{ all } s\\ge 1 \\qquad \ ext{ with } \ ext{ some } \\alpha > \\frac{2}{n} \\end{aligned}$$\\end{document}and c>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c>0$$\\end{document}, then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value α=2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha = \\frac{2}{n}$$\\end{document} for n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 3$$\\end{document}, without any additional assumption on the behavior of D(s) as s→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\rightarrow \\infty $$\\end{document}, in particular without requiring any algebraic lower bound for D. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type Q(s)=exp(-sβ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q(s) = \\exp (-s^\\beta )$$\\end{document}, s≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\ge 0$$\\end{document}, for global solvability the exponent β=n-2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta = \\frac{n-2}{n}$$\\end{document} is seen to be critical.