Keller Segel System Research Articles

Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source

We consider the fully parabolic Keller–Segel system with singular sensitivity and logistic-type source: ut=Δu−χ∇⋅(uv∇v)+ru−μuk, vt=Δv−v+u under the non-flux boundary conditions in a smooth bounded convex domain Ω⊂Rn, χ,r,μ>0, k>1. A global very weak solution for the system with n≥2 is obtained under one of the following conditions: (i) r>χ24 for 0<χ≤2, or r>max⁡{χ24(1−p02),χ−1} for χ>2 with p0=4(k−1)4+(2−k)kχ2 if k∈(2−1n,2]; (ii) χ2<min⁡{2(r+r2)k,4k(k−1)(k−2)} if k>2. Furthermore, this global very weak solution should be globally bounded in fact provided rμ and the initial data ‖u0‖L2(Ω),‖∇v‖L4(Ω) suitably small for n=2,3. In addition, if k>3(n+2)n+4 replaces k>2 in the condition (ii), the system admits globally bounded classical solutions. All these describe the influence of the exponent k>1 in the logistic-type source ru−μuk to the behavior of solutions for the considered fully parabolic Keller–Segel system with singular sensitivity.

Convergence of boundary layers for the Keller–Segel system with singular sensitivity in the half-plane

Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63]), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by ε≥0. Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with ε=0) plus the inner layer as ε→0, where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63], and open a new window in the future theoretical study of chemotaxis models.

Solvability of a Keller–Segel system with signal-dependent sensitivity and essentially sublinear production

ABSTRACT In this paper, we consider the zero-flux chemotaxis system in a smooth and bounded domain Ω of . The chemotactic sensitivity χ is a general nonnegative function from while g, the production of the chemical signal v, belongs to and satisfies , for all , and It is established that no chemotactic collapse for the cell distribution u occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data emanates a unique pair of global and uniformly bounded functions which classically solve the corresponding initial boundary value problem. Finally, we illustrate the range of dynamics present within the chemotaxis system by means of numerical simulations.

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.

Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system

We consider a parabolic-elliptic chemotaxis system generalizing \begin{document}$ \begin{align} & {{u}_{t}}=\nabla \cdot ({{(u+1)}^{m-1}}\nabla u)-\nabla \cdot (u{{(u+1)}^{\sigma -1}}\nabla v) & 0=\Delta v-v+u \end{align} $\end{document} in bounded smooth domains \begin{document}$ \Omega \subset \mathbb{R}^N $\end{document} , \begin{document}$ N\ge 3 $\end{document} , and with homogeneous Neumann boundary conditions. We show that ● solutions are global and bounded if \begin{document}$ \sigma ● solutions are global if \begin{document}$ \sigma\le 0 $\end{document} ● close to given radially symmetric functions there are many initial data producing unbounded solutions if \begin{document}$ \sigma>m-\frac{N-2}{N} $\end{document} . In particular, if \begin{document}$ \sigma\le 0 $\end{document} and \begin{document}$ \sigma>m-\frac{N-2}{N} $\end{document} , there are many initial data evolving into solutions that blow up after infinite time.

Decay in chemotaxis systems with a logistic term

This paper is concerned with a general fully parabolic Keller-Segel system, defined in a convex bounded and smooth domain $Ω$ of $\mathbb{R}^N, $ for N∈{2, 3}, with coefficients depending on the chemical concentration, perturbed by a logistic source and endowed with homogeneous Neumann boundary conditions. For each space dimension, once a suitable energy function in terms of the solution is defined, we impose proper assumptions on the data and an exponential decay of such energies is established.

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.

Existence and instability of traveling pulses of Keller–Segel system with nonlinear chemical gradients and small diffusions

In this paper, we consider a generalized model of Keller–Segel system with a nonlinear chemical gradient and small cell diffusion. The existence of the traveling pulses for such equations is established by the methods of geometric singular perturbation (GSP) and trapping regions from dynamical systems theory. By the technique of GSP, we show that the necessary condition for the existence of traveling pulses is that their limiting profiles with vanishing diffusion can only consist of the slow flows on the critical manifold of the corresponding algebraic-differential system. We also consider the linear instability of these pulses by the spectral analysis of the linearized operators.

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 .

Global existence and time decay estimate of solutions to the Keller–Segel system

Global existence and time decay estimate of solutions to the Keller–Segel system

A general existence result for stationary solutions to the Keller-Segel system

We consider the following Liouville-type PDE, which is related to stationary solutions of the Keller-Segel's model for chemotaxis: \begin{document}$\left\{ \begin{gathered} - \Delta u + \beta u = \rho \left( {\frac{{{e^u}}}{{\int_\Omega {{e^u}} }} - \frac{1}{{\left| \Omega \right|}}} \right)\;\;\;\;\;\;{\text{in}}\;\Omega \hfill {\partial _\nu }u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{on}}\;\partial \Omega \hfill \end{gathered} \right.,$ \end{document} where $\Omega \subset {\mathbb{R}^2}$ is a smooth bounded domain and $\beta, ρ$ are real parameters. We prove existence of solutions under some algebraic conditions involving $\beta, ρ$. In particular, if $\Omega$ is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.

Determination of blowup type in the parabolic–parabolic Keller–Segel system

This paper is concerned with a parabolic–parabolic Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{ll} u_t = \nabla \cdot ( \nabla u - u \nabla v ) &{} \quad \text{ in } \, \Omega \times (0,T), \\ v_t = \Delta v - \alpha v + u &{} \quad \text{ in } \, \Omega \times (0,T) \end{array} \right. \end{aligned}$$with a constant $$ \alpha \ge 0 $$ and nonnegative initial data in a smoothly bounded domain $$ \Omega \subset \mathbb {R}^2 $$ under the Neumann boundary condition or in $$ \Omega = \mathbb {R}^2 $$. It was introduced as a model of aggregation of bacteria, which is mathematically translated as finite-time blowup. A solution (u, v) is said to blow up at $$ t = T 0 $$, and type II otherwise. It was shown in Mizoguchi (J Funct Anal 271:3323–3347, 2016) that each blowup is type II in radial case. In this paper, we obtain the conclusion in general case.

Existence and asymptotic behaviour for the time‐fractional Keller–Segel model for chemotaxis

AbstractOne of the most important systems for understanding chemotactic aggregation is the Keller–Segel system. We consider the time‐fractional Keller–Segel system of order . We prove an existence result with small initial data in a class of Besov–Morrey spaces. Self‐similar solutions are obtained and we also show an asymptotic behaviour result.

Well-Posedness of the Keller–Segel System in Fourier–Besov–Morrey Spaces

In this note, we investigate the Cauchy problem for Keller–Segel system with fractional diffusion for the initial data (u_0,v_0) in the critical Fourier–Bessov–Morrey spaces \mathcal{FN}_{q,\mu,r}^{2-2\alpha+d-\frac{d-\mu}{q}}(\mathbb R^d)\times \mathcal{FN}_{q,\mu,r}^{2-\alpha+d-\frac{d-\mu}{q}}(\mathbb R^d) with 1 &lt; \alpha\le 2 . The global well-posedness with a small initial data of the solution to Keller–Segel system of double-parabolic type is established.

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.

Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics with Mildly Saturated Chemotactic Sensitivity

This work is concerned with an extension of the May-Nowak ODE model for virus dynamics to cases in which diffusive motion of cells as well as cross-diffusive movement of healthy cells toward infected individuals becomes relevant. Specifically, the parabolic system ⋆ $$ \left \{ \textstyle\begin{array}{l} u_{t} = \Delta u - \nabla \cdot (uf(u) \nabla v ) - uw + \kappa - u,\\ v_{t} = \Delta v - v +uw, \\ w_{t} = \Delta w - w + v, \end{array}\displaystyle \right . $$ is considered for $\kappa \ge 0$ and $f\in C^{2}([0,\infty ))$ generalizing the prototypical chemotactic sensitivity function given by $f(s)=(1+s)^{-\alpha }$ , $s\ge 0$ , for suitable $\alpha \in \mathbb{R}$ . The main results assert global existence of bounded classical solutions to a corresponding no-flux initial-boundary value problem in smoothly bounded $n$ -dimensional domains whenever $n\le 3$ and $$\begin{aligned} \alpha > \left \{ \textstyle\begin{array}{l@{\quad }l} -1 & \mbox{if } n=1, \\ \frac{n-2}{n-1} & \mbox{if } n\in \{2,3\}. \end{array}\displaystyle \right . \end{aligned}$$ In particular, this shows that in the case $n\le 2$ , a respective condition on $\alpha $ , known to be essentially optimal with regard to global boundedness for classical Keller-Segel systems with corresponding taxis saturation, remains unchanged even in the context of the superlinear—and hence potentially more destabilizing—signal production mechanism in (⋆).

Blow-up Profiles for the Parabolic–Elliptic Keller–Segel System in Dimensions $${n\geq 3}$$ n ≥ 3

We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In view of the biological background of this system and of its mass conservation property, blowup is usually interpreted as a phenomenon of concentration or aggregation of the bacterial population. Understanding the asymptotic behavior of solutions at the blowup time is thus meaningful for the interpretation of the model. Under mild assumptions on the initial data, for $n\ge 3$, we show that the final profile satisfies $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$, with convergence in $L^1$ as $t\to T$. This is in sharp contrast with the two-dimensional case, where solutions are known to concentrate to a Dirac mass at the origin (plus an integrable part). We also obtain refined space-time estimates of the form $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type~I blowup solutions. Previous work had shown that radial, self-similar blowup solutions (which satisfy the above estimates) exist in dimensions $n\ge 3$ and do not exist in dimension $2$. Our results thus reveal that the final profile displayed by these special solutions actually corresponds to a much more general phenomenon.

Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?

A class of Keller-Segel-Stokes systems generalizing the prototype \[ \left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& \Delta n - \nabla \cdot \Big(n(n+1)^{-\alpha}\nabla c\Big), c_t + u\cdot\nabla c &=& \Delta c-c+n, u_t +\nabla P &=& \Delta u + n \nabla \phi + f(x,t), \qquad \nabla\cdot u =0, \end{array} \right. \qquad \qquad (\star) \] is considered in a bounded domain $\Omega\subset R^3$, where $\phi$ and $f$ are given sufficiently smooth functions such that $f$ is bounded in $\Omega\times (0,\infty)$. It is shown that under the condition that \[ \alpha>\frac{1}{3}, \] for all sufficiently regular initial data a corresponding Neumann-Neumann-Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption $\alpha>\frac{1}{2}$. In view of known results on the existence of exploding solutions when $\alpha<\frac{1}{3}$, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate $\frac{1}{3}$, as previously found for the fluid-free counterpart of ($\star$), remains essentially unaffected by fluid interaction of the type considered here.

Properties of blow-up solutions and their initial data for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper is concerned with blow-up solutions to the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type{ut=∇⋅(∇um−uq−1∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂RN (N≥3), m≥1, q≥2. As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q>m+2N ([5]). In the parabolic–elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic–parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic–parabolic case.

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system{ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω⊂R4 with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0).Global existence and boundedness of solutions were shown if ‖u0‖L1(Ω)<(8π)2/χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖u0‖L1(Ω)>(8π)2/χ. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8π/χ-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R4.