Quasilinear Keller Segel System Research Articles

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

AbstractRadially symmetric global unbounded solutions of the chemotaxis system $$\left\{ {\matrix{{{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (uS(u)\nabla v),} \hfill & {} \hfill \cr {0 = \Delta v - \mu + u,} \hfill & {\mu = {1 \over {|\Omega |}}\int_\Omega {u,} } \hfill \cr } } \right.$$ { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , 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 $$m + \lambda < 1 - {2 \over n}$$ m + λ < 1 − 2 n , 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 $${1 \over C} \cdot {(t + 1)^{{1 \over \lambda }}} \le ||u( \cdot ,t)|{|_{{L^\infty }(\Omega )}} \le C \cdot {(t + 1)^{{1 \over \lambda }}}\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,t > 0$$ 1 C ⋅ ( t + 1 ) 1 λ ≤ | | u ( ⋅ , t ) | | L ∞ ( Ω ) ≤ C ⋅ ( t + 1 ) 1 λ f o r a l l t > 0 as well as $$||u( \cdot \,,t)|{|_{{L^1}(\Omega \backslash {B_{{r_0}}}(0))}} \to 0\,\,\,{\rm{as}}\,\,t \to \infty \,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{r_0} \in (0,R)$$ | | u ( ⋅ , t ) | | L 1 ( Ω \ B r 0 ( 0 ) ) → 0 as t → ∞ for all r 0 ∈ ( 0 , R ) with some C > 0.

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.

Critical mass phenomena in higher dimensional quasilinear Keller–Segel systems with indirect signal production

In this paper, we deal with quasilinear Keller–Segel systems with indirect signal production, complemented with homogeneous Neumann boundary conditions and suitable initial conditions, where is a bounded smooth domain, and We show that in the case , there exists such that if either or , then the solution exists globally and remains bounded, and that in the case , if either or , then there exist radially symmetric initial data such that and the solution blows up in finite or infinite time, where the blow‐up time is infinite if . In particular, if , there is a critical mass phenomenon in the sense that is a finite positive number.

Strong periodic solutions to quasilinear parabolic equations: An approach by the Da Prato–Grisvard theorem

AbstractThis article develops an approach to unique, strong periodic solutions to quasilinear evolution equations by means of the classical Da Prato–Grisvard theorem on maximal ‐regularity in real interpolation spaces. The method is used to show that quasilinear Keller–Segel systems admit a unique, strong ‐periodic solution in a neighborhood of 0 provided the external forces are ‐periodic and satisfy certain smallness conditions. A similar assertion applies to a Nernst–Planck–Poisson type system in electrochemistry. The proof for the quasilinear Keller–Segel systems relies also on a new mixed derivative theorem in real interpolation spaces, that is, Besov spaces, which is of independent interest.

Exponential grow-up rates in a quasilinear Keller–Segel system

The chemotaxis system ( ⋆ ) u t = ∇ · ( D ( u ) ∇ u ) − ∇ · ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , is considered in a ball Ω = B R ( 0 ) ⊂ R n . It is shown that if S ∈ C 2 ( [ 0 , ∞ ) ) suitably generalizes the prototype given by S ( ξ ) = χ ξ + 1 , ξ ⩾ 0 , with some χ &gt; 0, and if diffusion is suitably weak in the sense that 0 &lt; D ∈ C 2 ( ( 0 , ∞ ) ) is such that there exist K D &gt; 0 and m ∈ ( − ∞ , 1 − 2 n ) fulfilling D ( ξ ) ⩽ K D ξ m − 1 for all ξ &gt; 0 , then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution ( u , v ) which blows up in infinite time and satisfies 1 C e χ t ⩽ ‖ u ( · , t ) ‖ L ∞ ( Ω ) ⩽ C e χ t for all t &gt; 0 . A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of (⋆).

Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems

The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text], [Formula: see text], where the positive function [Formula: see text] reflects suitably weak diffusion by satisfying [Formula: see text] for some [Formula: see text]. It is shown that whenever [Formula: see text] is positive and satisfies [Formula: see text] as [Formula: see text], one can find a suitably regular nonlinearity [Formula: see text] with the property that at each sufficiently large mass level [Formula: see text] there exists a globally defined radially symmetric classical solution to a Neumann-type boundary value problem for (⋆) which satisfies [Formula: see text]

Slow Grow-up in a Quasilinear Keller–Segel System

In a ball Ω=BR(0)⊂Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega =B_R(0)\\subset \\mathbb {R}^n$$\\end{document}, n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 2$$\\end{document}, 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}$$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \ abla \\cdot \\big ( D(u) \ abla u \\big ) - \ abla \\cdot \\big ( uS(u)\ abla v\\big ), \\\\ 0 = \\Delta v - \\mu + u, \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}is considered under no-flux boundary conditions, with a focus on nonlinearities S∈C2([0,∞))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\in C^2([0,\\infty ))$$\\end{document} which exhibit super-algebraically fast decay in the sense that with some KS>0,β∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_S>0, \\beta \\in [0,1)$$\\end{document} and ξ0>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi _0>0$$\\end{document}, S(ξ)>0andS′(ξ)≤-KSξ-βS(ξ)forallξ≥ξ0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} S(\\xi )>0 \\quad \ ext{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi ) \\qquad \ ext{ for } \ ext{ all } \\xi \\ge \\xi _0. \\end{aligned}$$\\end{document}It is, inter alia, shown that if furthermore D∈C2((0,∞))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D\\in C^2((0,\\infty ))$$\\end{document} is positive and suitably small in relation to S by satisfying ξS(ξ)D(ξ)≥KSDξλforallξ≥ξ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \ ext{ for } \ ext{ all } \\xi \\ge \\xi _0 \\end{aligned}$$\\end{document}with some KSD>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{SD}>0$$\\end{document} and λ>2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda >\\frac{2}{n}$$\\end{document}, then throughout a considerably large set of initial data, (⋆\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\star $$\\end{document}) admits global classical solutions (u, v) fulfilling z(t)C≤‖u(·,t)‖L∞(Ω)≤Cz(t)forallt>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t) \\qquad \ ext{ for } \ ext{ all } t>0, \\end{aligned}$$\\end{document}with some C=C(u,v)≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=C^{(u,v)}\\ge 1$$\\end{document}, where z denotes the solution of z′(t)=z2(t)·S(z(t)),t>0,z(0)=ξ0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad t>0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$\\end{document}which is seen to exist globally, and to satisfy z(t)→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z(t)\\rightarrow +\\infty $$\\end{document} 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}. As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in (⋆\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\star $$\\end{document}) at logarithmic and doubly logarithmic rates, respectively.

Asymptotic behavior of a quasilinear Keller–Segel system with signal-suppressed motility

This paper is concerned with the density-suppressed motility model: \(u_{t}=\Delta \left( \displaystyle \frac{u^m}{v^\alpha }\right) +\beta uf(w), v_{t}=D\Delta v-v+u, w_{t}=\Delta w-uf(w)\) in a smoothly bounded convex domain \(\Omega \subset {{\mathbb {R}}}^2\), where \(m>1\), \(\alpha>0, \beta >0\) and \(D>0\) are parameters, the response function f satisfies \(f\in C^1([0,\infty )), f(0)=0, f(w)>0\) in \((0,\infty )\). This system describes the density-suppressed motility of Eeshcrichia coli cells in the process of spatio-temporal pattern formation via so-called self-trapping mechanisms. Based on the duality argument, it is shown that for suitable large D the problem admits at least one global weak solution (u, v, w) which will asymptotically converge to the spatially uniform equilibrium \((\overline{u_0}+\beta \overline{w_0},\overline{u_0}+\beta \overline{w_0},0)\) with \(\overline{u_0}=\frac{1}{|\Omega |}\int _{\Omega }u(x,0)dx \) and \(\overline{w_0}=\frac{1}{|\Omega |}\int _{\Omega }w(x,0)dx \) in \(L^\infty (\Omega )\).

On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems

ABSTRACT Solutions to the chemotaxis system in a ball , , wherein and are given parameters with m−q>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in for some . For radially symmetric solutions, we show that, if u is only bounded in and the technical condition is fulfilled, then, for any , there is C>0 with denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any , then u would already be bounded in for some .

Boundedness in a three-dimensional Keller–Segel–Stokes system with subcritical sensitivity

We consider a quasilinear Keller–Segel system with density-dependent migration rates, coupled to the incompressible Stokes equations through transport and buoyancy. By means of an apparently novel approach based on certain conditional estimates for the taxis gradient and the fluid field, for diffusion rates asymptotically controllable by power-tape majorants and minorants a result on global existence and boundedness is derived under an essentially optimal condition on the strength of cross-diffusion relative to diffusion.

Strong time periodic solutions to Keller-Segel systems: An approach by the quasilinear Arendt-Bu theorem

It is shown that the classical as well as quasilinear Keller-Segel systems with non-degenerate diffusion possess for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution. The proof given relies on the existence of strong T-periodic solutions for the linearized system, its characterization in terms of maximal Lp-regularity of the underlying operator and a quasilinear version of the Arendt-Bu Theorem. The latter is of independent interest and yields the existence of strong T-periodic solutions to general quasilinear evolution equations under suitable conditions on the operators and the forcing terms.

Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity

The paper should be viewed as complement of an earlier result in [ 10 ]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form \begin{document}$ 1/u $\end{document} all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [ 10 ] deals with the Jager-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [ 10 ], being a main step of the method, works only for the Jager-Luckhaus modification. It does not seem to be applicable in the usual version of the parabolic-elliptic Keller-Segel system. The present paper fulfils this gap and deals with the case of the usual parabolic-elliptic version. To handle it we establish a new Lyapunov-like functional (it is related to what was done in [ 10 ]), which leads to global existence of the initial-boundary value problem for any initial mass.

Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities

The chemotaxis system(⋆){ut=∇⋅(D(u,v)∇u)−∇⋅(S(u,v)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, along with initial conditions involving suitably regular and nonnegative data.It is firstly asserted that if the positive smooth function D decays at most algebraically with respect to u, then for any smooth nonnegative and bounded S fulfilling a further mild assumption especially satisfied when S≡S(u) with S(0)=0, (⋆) possesses a globally defined classical solution.If Ω is a ball, then under appropriate assumptions on D and S generalizing the prototypical choices in(⋆⋆)D(u,v)=(u+1)m−1andS(u,v)=u(u+1)σ−1,u≥0,v≥0, with m∈R and σ∈R such that(⋆⋆⋆)m−n−2n<σ≤0, the phenomenon of infinite-time blow-up is next shown to occur for all initial data within a set B of functions which inter alia is found to be dense in the set of all radially symmetric and suitably regular positive functions on Ω‾.Up to equality in (⋆⋆⋆) thereby covering the largest possible range of nonpositive σ for the appearance of unbounded solutions, this extends previous findings on blow-up in infinite time which in the context of (⋆⋆) were limited to a smaller parameter region, and which were restricted to mere existence results without information on the richness of B.

Boundedness of solutions to the critical fully parabolic quasilinear one‐dimensional Keller–Segel system

AbstractIn this paper we consider a one‐dimensional fully parabolic quasilinear Keller–Segel system with critical nonlinear diffusion. We show uniform‐in‐time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well‐known Lyapunov functional and a recently introduced new Lyapunov‐like functional in .

Convergence to equilibria of global solutions to a degenerate quasilinear Keller–Segel system

This paper is devoted to the longtime behavior of global weak solutions to initial-boundary value problems of the following degenerate quasilinear Keller–Segel system $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u^m-\nabla \cdot (u\nabla v), &{}\quad x\in \Omega ,\;t>0\\ v_t=\Delta v-v+\gamma u, &{}\quad x\in \Omega ,\;t>0. \end{array}\right. }\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (0.1) \end{aligned}$$ Here $$\Omega \subset \mathbb {R}^n$$ is a bounded domain, $$\gamma >0$$ and $$m>\max \{1,2-\frac{2}{n}\}$$ . First, we establish the existence of global weak solution which additionally satisfies an energy dissipation inequality. Thanks to the energy inequality, we prove that the global weak solution will converge to an equilibrium as time goes to infinity if $$m\ge 2$$ and $$\gamma <\frac{m}{m-1} M_0^{m-2}$$ , where $$M_0$$ is the average mass of cells. The proof is based on an application of a slightly modified Lojasiewicz–Simon inequality of non-smooth type, where the requirement on compactness of the trajectory is weakened compared with those in Feireisl et al. (J Differ Equ 236:551–569, 2007), Jiang and Zhang (Asymptot Anal 65:79–102, 2009). Moreover, in the special case $$m=2$$ and $$\gamma <2$$ , convergence toward trivial solution $$(M_0,\gamma M_0)$$ is verified by an alternative straightforward way.

Blow-up profiles and refined extensibility criteria in quasilinear Keller–Segel systems

In this work we consider the system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)inΩ×(0,∞)vt=Δv−v+uinΩ×(0,∞), for a bounded domain Ω⊂Rn, n≥2, where the functions D and S behave similarly to power functions. We prove the existence of classical solutions under Neumann boundary conditions and for smooth initial data. Moreover, we characterise the maximum existence time Tmax of such a solution depending chiefly on the relation between the functions D and S: We show that a finite maximum existence time also results in unboundedness in Lp-spaces for smaller p∈[1,∞).

Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity

The Neumann initial-boundary value problem for the chemotaxis system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u,(⋆)is considered in a bounded domain , , with smooth boundary. In compliance with refined modeling approaches, the diffusivity function D therein is allowed to decay considerably fast at large densities, where a particular focus will be on the mathematically delicate case when D(s) decays exponentially as . In such situations, namely, straightforward Moser-type recursive arguments for the derivation of estimates for u from corresponding Lp bounds seem to fail. Accordingly, results on global existence, and especially on quantitative upper bounds for solutions, so far mainly concentrate on cases when D decays at most algebraically, and hence are unavailable in the present context.This work develops an alternative approach, at its core based on a Moser-type iteration for the quantity , to establish global existence of classical solutions for all reasonably regular initial data, as well as a logarithmic upper estimate for the possible growth of as , under the assumptions that with some K1 > 0, K2 > 0, and we have for all , and that the size of S relative to D can be estimated according to for all with some K3 > 0 and .Making use of the fact that this allows for certain superalgebraic growth of , as a particular consequence of this and known results on nonexistence of global bounded solutions we shall see that in the prototypical case when and for all and some positive α and β, the assumptions that and that0\\quad \\quad \ ext{and}\\quad \\quad \\left\\{\\begin{array}{*{35}{l}} \\alpha \\in \\left(\\frac{\\beta}{2},\\beta \\right)\\quad \\quad & \ ext{if}~n=2, \\\\ \\alpha \\in \\left(\\frac{\\beta}{2},\\beta \\right]\\quad \\quad & \ ext{if}~n\\geqslant 3, \\end{array}\\right. \\end{eqnarray} ?>β>0and{α∈(β2,β)if n=2,α∈(β2,β]if n⩾3,warrant the existence of classical solutions which are global but unbounded, and for which this infinite-time blow-up is slow in the sense that the corresponding grow-up rate is at most logarithmic.To the best of our knowledge, this inter alia seems to constitute the first quantitative information on a blow-up rate in a parabolic Keller–Segel system of type () for widely arbitrary initial data, hence independent of a particular construction of possibly non-generic exploding solutions.

Global bounded solutions in a two-dimensional quasilinear Keller–Segel system with exponentially decaying diffusivity and subcritical sensitivity

The quasilinear chemotaxis system (∗){ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂R2 with smooth boundary.It is shown that if D and S are sufficiently smooth nonnegative functions on [0,∞) satisfying K1e−β−s≤D(s)≤K2e−β+sfor all s≥0 with some K1>0,K2>0, β+>0 and β−≥β+, then whenever S satisfies the condition of subcritical growth relative to D given by S(s)D(s)≤K3sαfor all s≥0 with some K3>0 and α∈(0,1), for all suitably regular nonnegative initial data the corresponding initial–boundary value problem for (⋆) possesses a global classical solution for which the component u is bounded in Ω×(0,∞).

Stabilization in a higher-dimensional quasilinear Keller–Segel system with exponentially decaying diffusivity and subcritical sensitivity

The quasilinear chemotaxis system (⋆){ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, with smooth boundary, where the focus is on cases when herein the diffusivity D(s) decays exponentially as s→∞.It is shown that under the subcriticality condition that (0.1)S(s)D(s)≤Csαfor all s≥0 with some C>0 and α<2n, for all suitably regular initial data satisfying an essentially explicit smallness assumption on the total mass ∫Ωu0, the corresponding Neumann initial–boundary value problem for (⋆) possesses a globally defined bounded classical solution which moreover approaches a spatially homogeneous steady state in the large time limit. Viewed as a complement of known results on the existence of small-mass blow-up solutions in cases when in (0.1) the reverse inequality holds with some α>2n, this confirms criticality of the exponent α=2n in (0.1) with regard to the singularity formation also for arbitrary n≥2, thereby generalizing a recent result on unconditional global boundedness in the two-dimensional situation.As a by-product of our analysis, without any restriction on the initial data, we obtain boundedness and stabilization of solutions to a so-called volume-filling chemotaxis system involving jump probability functions which decay at sufficiently large exponential rates.

New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models

We carry on our studies related to the fully parabolic quasilinear Keller–Segel system started in [6] and continued in [7]. In the above mentioned papers we proved finite-time blowup of some radially symmetric solutions to the quasilinear Keller–Segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds. On the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time. The present paper is devoted to looking for critical exponents distinguishing between those two behaviors. Moreover, we apply our results to the so-called volume filling models with a power-type jump probability function. The most important consequence of our investigations of the latter is a critical mass phenomenon found in dimension 2. Namely we find a value m⁎ such that when the solution to the two-dimensional volume filling Keller–Segel system starts with mass smaller than m⁎, then it is bounded, while for some initial data with mass exceeding m⁎ solutions are unbounded, though being defined for any time t>0.