Keller Segel System Research Articles

Maximal regularity and a singular limit problem for the Patlak\u2013Keller\u2013Segel system in the scaling critical space involving BMO

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter tau rightarrow infty . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces dot{B}^s_{q,sigma }({mathbb {R}}^n) and dot{F}^s_{q,sigma }({mathbb {R}}^n) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

Global weak solution in a p-Laplacian Keller–Segel system with nonlinear sensitivity and saturation effect

This paper deals with a p-Laplacian Keller–Segel chemotaxis system with nonlinear sensitivity and saturation effect under homogeneous Neumann boundary conditions in a smooth bounded domain. Under some suitable assumptions on these parameters and the initial data, we derive the uniform-in-time boundedness of global weak solutions for the system.

Unlimited growth in logarithmic Keller-Segel systems

The fully parabolic cross-diffusion system{εut=Δu−χ∇⋅(uv∇v),vt=Δv−v+u, is considered under homogneous Neumann boundary conditions in a ball Ω⊂Rn, n≥1, for ε∈(0,1) and χ>0. Previous literature has asserted global existence of certain generalized solutions whenever χ<n(n−2)+, but no result on any unboundedness phenomenon seems available; only in a parabolic-elliptic simplification, some solutions are known to blow up in finite time when n≥3 and χ>2nn−2.The present study examines the solution behavior in the singular limit ε↘0, and based on an essentially well-known result on finite-time blow-up in an accordingly obtained nonlocal scalar parabolic problem, under the assumptions that n≥3 and χ>nn−2 a statement on spontaneous emergence of arbitrarily large values of u for appropriately small ε is derived.

The one-dimensional stochastic Keller–Segel model with time-homogeneous spatial Wiener processes

Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak–Keller–Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak–Keller–Segel system perturbed by random processes.We study a stochastic version of the classical Patlak–Keller–Segel system under homogeneous Neumann boundary conditions on an interval O=[0,1]. In particular, let W1, W2 be two time-homogeneous spatial Wiener processes over a filtered probability space A. Let u and v denote the cell density and concentration of the chemical signal. We investigate the coupled system{du−(ruΔu−χdiv(u∇v))dt=u∘dW1,dv−(rvΔv−αv)dt=βudt+v∘dW2,with initial conditions (u(0),v(0))=(u0,v0). The positive terms ru and rv are the diffusivity of the cells and chemoattractant, respectively, the positive value χ is the chemotactic sensitivity, α≥0 is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given T>0, we will prove the existence of a martingale solution on [0,T].

On the attraction–repulsion chemotaxis system with volume-filling effect

In this paper, we consider the attraction–repulsion Keller–Segel system with volume-filling effect under homogeneous Neumann boundary conditions in a smooth boundary bounded domain with n ≥ 2. We study the global existence and asymptotic behavior of the classical solution to the system in various ranges of parameter values.

$L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation

$L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation

Time periodic strong solutions to the Keller-Segel system coupled to Navier-Stokes equation

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N ( N ≥ 4 ) . Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C ( R , L s , ∞ ( R N ) ) when S ( n ) = n − 1 2 . Next for the case S ( n ) = 0 , we study the mild solution and point out it can become strong solution in B C ( R , L s , ∞ ( R N ) ) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations.

The Keller-Segel system of parabolic-parabolic type in homogeneous Besov spaces framework

We show the existence and uniqueness of local strong solutions of Keller-Segel system of parabolic-parabolic type for arbitrary initial data in the homogeneous Besov space which is scaling invariant. We also construct global strong solutions for small initial data, where the solutions belong to the Lorentz space in time direction. The proof is based on the maximal Lorentz regularity theorem of heat equations.

Unbounded mass radial solutions for the Keller–Segel equation in the disk

We consider the boundary value problem $$\begin{aligned}\left\{ \begin{array}{rcll} -\Delta u+ u -\lambda e^u&{}=&{}0,\ u>0 &{} \mathrm {in}\ B_1(0)\\ \partial _\nu u&{}=&{}0&{}\mathrm {on}\ \partial B_1(0), \end{array}\right. \end{aligned}$$whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here \(B_1(0)\) is the unit disk, \(\nu \) the outer normal to \(\partial B_1(0)\), and \(\lambda >0\) is a parameter. We show that, provided \(\lambda \) is sufficiently small, there exists a family of radial solutions \(u_\lambda \) to this system which blow up at the origin and concentrate on \(\partial B_1(0)\), as \(\lambda \rightarrow 0\). These solutions satisfy $$\begin{aligned}\lim _{\lambda \rightarrow 0} \frac{u_\lambda (0)}{|\ln \lambda |}=0\quad \text{ and }\quad 0<\lim _{\lambda \rightarrow 0} \frac{1}{|\ln \lambda |}\int _{B_1(0)}\lambda e^{u_\lambda (x)}dx<\infty , \end{aligned}$$having in particular unbounded mass, as \(\lambda \rightarrow 0\).

Blow-up in a quasilinear parabolic–elliptic Keller–Segel system with logistic source

This paper deals with the quasilinear parabolic–elliptic Keller–Segel system with logistic source, ut=Δ(u+1)m−χ∇⋅(u(u+1)α−1∇v)+λ(|x|)u−μ(|x|)uκ,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,where Ω≔BR(0)⊂Rn(n≥3) is a ball with some R>0; m>0, χ>0, α>0 and κ≥1; λ and μ are continuous nonnegative functions. About this problem, Winkler (2018) found the condition for κ such that solutions blow up in finite time when m=α=1. In the case that m=1 and α∈(0,1) as well as λ and μ are constants, some conditions for α and κ such that blow-up occurs were obtained in a previous paper (Tanaka and Yokota, 2020). Moreover, in the case that m≥1 and α=1 Black et al. (2021) showed that there exist initial data such that the corresponding solution blows up in finite time under some conditions for m and κ. The purpose of the present paper is to give conditions for m≥1, α>0 and κ≥1 such that solutions blow up in finite time.

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 \(L^\infty (\Omega )\).

Stability of constant steady states of a chemotaxis model

The Cauchy problem for the parabolic–elliptic Keller–Segel system in the whole n-dimensional space is studied. For this model, every constant A in {mathbb {R}} is a stationary solution. The main goal of this work is to show that A < 1 is a stable steady state while A > 1 is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.

Small-density solutions in Keller–Segel systems involving rapidly decaying diffusivities

Small-density solutions in Keller–Segel systems involving rapidly decaying diffusivities

Well-posedness and unconditional uniqueness of mild solutions to the Keller–Segel system in uniformly local spaces

Well-posedness and unconditional uniqueness of mild solutions to the Keller–Segel system in uniformly local spaces

Boundedness in a nonlinear attraction-repulsion Keller–Segel system with production and consumption

This paper is focused on the zero-flux attraction-repulsion chemotaxis model(◇){ut=∇⋅((u+1)m1−1∇u−χu(u+1)m2−1∇v in Ω×(0,Tmax),+ξu(u+1)m3−1∇w)vt=Δv−f(u)v in Ω×(0,Tmax),0=Δw−δw+g(u) in Ω×(0,Tmax), defined in Ω, which is a bounded and smooth domain of Rn, for n≥2, with χ,ξ,δ>0, m1,m2,m3∈R, and f(u) and g(u) reasonably regular functions generalizing the prototypes f(u)=Kuα and g(u)=γul, with K,γ>0 and appropriate α,l>0. Moreover Tmax is finite or infinite and (0,Tmax) stands for the maximal temporal interval where solutions to the related initial problem exist. Our main interest is to identify constellations of the impacts m1,m2 and m3 of the diffusion and drift terms, as well as of the growth l of the production g for the chemorepellent (i.e., w) and the rate α of the consumption f for the chemoattractant (i.e., v), which ensure boundedness of cell densities (i.e., u). Precisely, for any fixed α∈(0,12+1n) and l≥1, we prove that wheneverm1>min⁡{2m2+1−(m3+l),max⁡{2m2,n−2n}}, any sufficiently smooth initial data u(x,0)=u0(x)≥0 and v(x,0)=v0(x)≥0 produce a unique classical solution (u,v,w) to problem (◇) such that its life span Tmax=∞ and, moreover, u,v and w are uniformly bounded in Ω×(0,∞).

Maximal Lorentz regularity for the Keller–Segel system of parabolic–elliptic type

Maximal Lorentz regularity for the Keller–Segel system of parabolic–elliptic type

The Keller\u2013Segel system on bounded convex domains in critical spaces

Consider the classical Keller–Segel system on a bounded convex domain varOmega subset {mathbb {R}}^3. In contrast to previous works it is not assumed that the boundary of varOmega is smooth. It is shown that this system admits a local, strong solution for initial data in critical spaces which extends to a global one provided the data are small enough in this critical norm. Furthermore, it is shown that this system admits for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution.

On the parabolic‐elliptic Keller–Segel system with signal‐dependent motilities: A paradigm for global boundedness and steady states

This paper is concerned with a parabolic‐elliptic Keller–Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel to describe the aggregation phase of Dictyostelium discoideum cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of solutions in any dimensions with suitable general conditions on the signal‐dependent motility functions, which are applicable to a wide class of motility functions. The existence/nonexistence of non‐constant steady states is studied and abundant stationary profiles are found. Some open questions are outlined for further pursues. Our results demonstrate that the global boundedness and profile of stationary solutions to the Keller–Segel system with signal‐dependent motilities depend on the decay rates of motility functions, space dimensions and the relation between the diffusive and chemotactic motilities, which makes the dynamics immensely wealthy.

Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic\u2013elliptic Keller\u2013Segel systems

We study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider JLut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-1|Ω|∫Ωu+u\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\nabla \\cdot ((u+1)^{m-1}\\nabla u -u \\nabla v) + \\lambda u - \\mu u^{1+\\kappa }, \\\\ 0 = \\Delta v - \\frac{1}{|\\Omega |} \\int \\limits _\\Omega u + u \\end{array}\\right. } \\end{aligned}$$\\end{document}and PEut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-v+u,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\nabla \\cdot ((u+1)^{m-1}\\nabla u -u \\nabla v) + \\lambda u - \\mu u^{1+\\kappa }, \\\\ 0 = \\Delta v - v + u, \\end{array}\\right. } \\end{aligned}$$\\end{document}where lambda and mu are given spatially radial nonnegative functions and m, kappa > 0 are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for m,kappa leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant lambda , mu > 0, we find that there are initial data which lead to blow-up in (JL) if 0≤κ<min12,n-2n-(m-1)+ifm∈2n,2n-2nor0≤κ<min12,n-1n-m2ifm∈0,2n,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} 0 \\le \\kappa&< \\min \\left\\{ \\frac{1}{2}, \\frac{n - 2}{n} - (m-1)_+ \\right\\}&\\quad \\text {if } m\\in \\left[ \\frac{2}{n},\\frac{2n-2}{n}\\right) \\\\ \\text { or }\\quad 0 \\le \\kappa&<\\min \\left\\{ \\frac{1}{2},\\frac{n-1}{n}-\\frac{m}{2}\\right\\}&\\quad \\text {if } m\\in \\left( 0,\\frac{2}{n}\\right) , \\end{aligned}$$\\end{document}and in (PE) if m in [1, frac{2n-2}{n}) and 0≤κ<min(m-1)n+12(n-1),n-2-(m-1)nn(n-1).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} 0 \\le \\kappa < \\min \\left\\{ \\frac{(m-1) n + 1}{2(n-1)}, \\frac{n - 2 - (m-1) n}{n(n-1)} \\right\\} . \\end{aligned}$$\\end{document}

Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary

In this paper, we investigate a two-species competitive chemotaxis Keller-Segel system equipped with a free boundary. The local existence and global existence of classical solutions are established, and then the asymptotic behavior of the solutions is described. Some spreading-vanishing dichotomies have been obtained both for the strong competition case: 0<a1<1≤a2, and weak competition case: 0<a1, a2<1. Finally, an obstructed spreading case has been found, which could happen due to the effect of the chemotaxis.