Keller Segel System Research Articles

Corners and collapse: Some simple observations concerning critical masses and boundary blow-up in the fully parabolic Keller–Segel system

Our main result shows that the mass 2π is critical for the minimal Keller–Segel system (⋆)ut=Δu−∇⋅(u∇v),vt=Δv−v+u,considered in a quarter disc Ω={(x1,x2)∈R2∣x1>0,x2>0,x12+x22<R2}, R>0, in the following sense: For all reasonably smooth nonnegative initial data u0,v0 with ∫Ωu0<2π, there exists a global classical solution to the Neumann initial boundary value problem associated to (⋆), while for all m>2π there exist nonnegative initial data u0,v0 with ∫Ωu0=m so that the corresponding classical solution of this problem blows up in finite time.At the same time, this gives an example of boundary blow-up in (⋆).Up to now, precise values of critical masses had been observed in spaces of radially symmetric functions or for parabolic–elliptic simplifications of (⋆) only.

Critical mass for Keller–Segel systems with supercritical nonlinear sensitivity

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity [Formula: see text] and [Formula: see text], posed on [Formula: see text] [Formula: see text] and subjected to homogeneous Neumann boundary conditions. It is well-known that [Formula: see text] is the critical exponent of the systems in the sense that all solutions exist globally if [Formula: see text] and there exist finite-time blowup solutions if [Formula: see text]. Here we consider the supercritical case [Formula: see text] and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass [Formula: see text] such that (1) for arbitrary nonincreasing nonnegative initial data [Formula: see text] with [Formula: see text] and [Formula: see text], the corresponding solution blows up in finite time if [Formula: see text], and if [Formula: see text] we can only prove that the solution blows up in finite time or infinite time; (2) for some nonincreasing nonnegative initial data with [Formula: see text], the corresponding solutions are globally bounded. Our results extend that of Winkler’s paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with [Formula: see text].

Solvability of the fractional hyperbolic Keller–Segel system

We study a new nonlocal approach to the mathematical modelling of the chemotaxis problem, which describes the random motion of a certain population due to a substance concentration. Considering the initial–boundary value problem for the fractional hyperbolic Keller–Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.

Global boundedness in a Keller-Segel system with flux limitation and logistic source

We consider the parabolic chemotaxis system{ut=Δu−∇⋅(uf(|∇v|2)∇v)+ru−μuγ,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, in a smooth bounded domain Ω⊂Rn(n≥1) with the homogeneous Neumann boundary conditions, where r,μ>0, γ>1 and the function f satisfiesf(ξ)=(1+ξ)−α2,for all ξ≥0, with α>0. In the case n≤2, it is shown that the corresponding initial value problem possesses a global bounded classical solution for any α,μ>0. In the case n≥3, if γ=2 and α=n−22n, there exists μ0>0 such that for any μ≥μ0, a global bounded classical solution exists.

Global Boundedness in a Logarithmic Keller–Segel System

In this paper, we propose a user-friendly integral inequality to study the coupled parabolic chemotaxis system with singular sensitivity under the Neumann boundary condition. Under a low diffusion rate, the classical solution of this system is uniformly bounded. Our proof replies on the construction of the energy functional containing ∫Ω|v|4v2 with v&gt;0. It is noteworthy that the inequality used in the paper may be applied to study other chemotaxis systems.

Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two

This paper deals with the parabolic-elliptic Keller–Segel system with weakly singular sensitivity and logistic source under the homogeneous Neumann boundary: , in a smooth bounded domain , with , . It is proved that the system possesses a globally bounded classical solution for with µ > 0 suitably large, without establishing the uniformly positive bound for v from below. In addition, we give the explicit expression of the upper bound for solution u with respect to the parameters via a recursive argument on α. This concludes that weakly singular sensitivity benefits to obtain the global boundedness of classical solution in dimension two.

Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a sphere that shrinks to a point. The singularity is in particular of type II, non self-similar and resembles a traveling wave imploding at the origin in renormalized variables. We show the stability of this dynamics among spherically symmetric solutions, and to our knowledge, this is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.

Particle approximation of the doubly parabolic Keller-Segel equation in the plane

In this work, we study a stochastic system of N particles associated with the parabolic-parabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any N≥2, we show tightness in N of its empirical measure, and that any weak limit point of this empirical measure, as N→∞, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.

Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity

The Cauchy problem in mathbb {R}^n is considered for the Keller–Segel system ut=Δu-∇·(u∇v),0=Δv+u,(⋆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \\Delta u - \ abla \\cdot (u\ abla v), \\\\ 0 = \\Delta v + u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data u_0 with non-integrable behavior at spatial infinity. It is shown that if u_0 is continuous and bounded, then (star ) admits a local-in-time classical solution, whereas if u_0(x)rightarrow +infty as |x|rightarrow infty , then no such solution can be found. Furthermore, a collection of three sufficient criteria for either global existence or global nonexistence indicates that with respect to the occurrence of finite-time blow-up, spatial decay properties of an explicit singular steady state plays a critical role. In particular, this underlines that explosions in (star ) need not be enforced by initially high concentrations near finite points, but can be exclusively due to large tails.

Global regularity and stability analysis of the Patlak–Keller–Segel system with flow advection in a bounded domain: A semigroup approach

We discuss the problem of suppression of singularity via flow advection in chemotaxis modeled by the Patlak–Keller–Segel (PKS) equations. It is well-known that for the system without advection, singularity of the solution may develop at finite time. Specifically, if the initial condition is above certain critical threshold, the solution may blow up in finite time by concentrating positive mass at a single point. In this work, we mainly focus on the global regularity and stability analysis of the PKS system in the presence of flow advection in a bounded domain Ω⊂Rd,d=2,3, by using a semigroup approach. We will show that the global well-posedness can be obtained as long as the semigroup generated by the associated advection–diffusion operator has a rapid decay property. We will also show that for cellular flows in rectangle-like domains, such property can be achieved by rescaling both the cell size and the flow amplitude. This is analogous to the result established by Iyer, Xu and Zlatoš (2021) on the torus Td,d=2,3.

Convergence analysis from the indirect signal production to the direct one

This paper aims to provide the rigorous convergence analysis of the chemotaxis system{∂nϵ∂t=Δnϵ−∇⋅(nϵ∇cϵ),τ1∂cϵ∂t=Δcϵ−cϵ+wϵ,τ2∂wϵ∂t=−wϵ+nϵ in a smoothly bounded planar domain Ω as τ1→0 and (or) τ2→0. Precisely, we reveal that actually the indirect signal properties are strong enough to allow for a uniformly global-in-time convergence in terms of τ1=τ2=ϵ↘0 and τ1=1,τ2=ϵ↘0 for ∫Ωnϵ(⋅,0)<4π, which correspond to the classical Keller-Segel systems with direct signal production, and of τ1=ϵ↘0,τ2=1 for suitable small initial data. As a byproduct, we show that for any p≥2, the Lp-norm of nϵ blows up if ∫Ωnϵ(⋅,0)∈(4π,∞)∖4πN, which improves the existing results that the solution is unbounded in the sense of L∞-norm of nϵ.

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.

Generalized solution and eventual smoothness in a logarithmic Keller–Segel system for criminal activities

This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well.

Existence of weak solutions for porous medium equation with a divergence type of drift term

We consider degenerate porous medium equations with a divergence type of drift terms. We establish existence of nonnegative $$L^{q}$$ -weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on q and m caused by nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider, if the drift term is divergence-free. Similar conditions of gradients of drift terms are also provided to ensure the existence of such weak solutions. Uniqueness results follow under an additional condition on the gradients of the drift terms with the aid of methods developed in Wasserstein spaces. One of our main tools is so called the splitting method to construct a sequence of approximated solutions, which implies, by passing to the limit, the existence of weak solutions satisfying not only an energy inequality but also moment and speed estimates. One of crucial points in the construction is uniform Hölder continuity up to initial time for homogeneous porous medium equations, which seems to be of independent interest. As an application, we improve a regularity result for solutions of a repulsive Keller-Segel system of porous medium type.

Convergence rate of solutions towards spiky steady state for the Keller–Segel system with logarithmic sensitivity

We study the large time behaviors of solutions to the Keller–Segel system with logarithmic singular sensitivity in the half space, where biological mixed boundary conditions are prescribed. The existence and asymptotic stability of spiky steady states of this system were proved by Carrillo et al. (2021). In this paper we obtain convergence rate of solutions towards the steady state under appropriate initial perturbations. The proofs are based on a Cole–Hopf type transformation and a weighted energy method, where the weights are artfully constructed.

Critical mass in a quasilinear parabolic-elliptic Keller-Segel model

The parabolic-elliptic Keller-Segel system{ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇v),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,(⋆) is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥3) with smooth boundary. Here ϕ and ψ are positive functions which satisfy ψ(s)/ϕ(s) grows like s2n as s→∞. We show that there exist m⁎,m⁎>0 such that if the total mass ∫Ωu0<m⁎, (⋆) admits a global bounded solution; if Ω is radially symmetric and ∫Ωu0>m⁎, we can find initial data u0(x)=u0(|x|) such that the corresponding solution must be unbounded.

Singular limit problem for the Keller–Segel system and drift-diffusion system in scaling critical Besov–Morrey spaces

A singular limit problem for the Cauchy problem of the Keller–Segel equation is considered in a critical function space based on Morrey spaces. A solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift-diffusion system of parabolic-elliptic type in the critical space strongly as the relaxation time tends to infinity. The key ingredient is a Banach space-valued Lorentz space in time. By the careful use of the real interpolation, a sharp estimate for the heat semigroup is obtained.

Global existence and uniform boundedness in a fully parabolic Keller–Segel system with non-monotonic signal-dependent motility

This paper is concerned with global solvability of a fully parabolic system of Keller–Segel-type involving non-monotonic signal-dependent motility. First, we prove global existence of classical solutions to our problem with generic positive motility function under a certain smallness assumption at infinity, which however permits the motility function to be arbitrarily large within a finite region. Then uniform-in-time boundedness of classical solutions is established whenever the motility function has strictly positive lower and upper bounds in any dimension N≥1, or decays at a certain slow rate at infinity for N≥2. Our results remove the crucial non-increasing requirement on the motility function in some recent work [10,13,16] and hence allow for both chemo-attractive and chemo-repulsive effect, or their co-existence in applications. The key ingredient of our proof lies in an important improvement of the comparison method developed in [10,16,18].

On the stability to Keller–Segel system coupled with Navier–Stokes equations in Besov–Morrey spaces

In this paper, we consider the Cauchy problem of Keller–Segel–Navier–Stokes equations in RN for N≥2. Under the small initial data assumption, we prove the asymptotic stability for this system in Besov–Morrey spaces by the structure of the model and semigroup estimates.