This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation \begin{equation}\label{ellip}\tag{$\ast$} Δu-βu+λ\frac{e^u}{\int_Ωe^u}=0~\mathrm{in}~Ω, \end{equation} where $Ω$ is a bounded domain in $\mathbb{R}^2$ and $β, λ$ are positive parameters. The above equation arises as the stationary problem of the well-known classical Keller-Segel model describing chemotaxis. For equation \eqref{ellip} with Neumann boundary condition, we establish an integral inequality and prove that the solution of (\ref{ellip}) is unique if $0<λ\leq 8π$ and $u$ satisfies some symmetric properties. While for \eqref{ellip} with Dirichlet boundary condition, the same uniqueness result is obtained without symmetric condition by a different approach inspired by some recent works [19,21]. As an application of the uniqueness results, we prove that the radially symmetric solution of the classical Keller-Segel system with subcritical mass subject to Neumann boundary conditions will converge to the unique constant equilibrium as time tends to infinity if $Ω$ is a disc in two dimensions. As far as we know, this is the first result that asserts the exact asymptotic behavior of solutions to the classical Keller-Segel system with subcritical mass in two dimensions.
The focus of this paper is on solutions to a two-dimensional Keller-Segel system containing sub-logistic sources. We show that the presence of sub-logistic terms is adequate to prevent blow-up phenomena even in strongly degenerate Keller-Segel systems. Our proof relies on several techniques, including parabolic regularity theory in Orlicz spaces, variational arguments, interpolation inequalities, and the Moser iteration method.
We construct radial self-similar solutions of the, so called, minimal parabolic-elliptic Keller--Segel model in several space dimensions with radial, nonnegative initial conditions with are below the Chandrasekhar solution -- the singular stationary solution of this system.
This paper generalizes and extends to the case of nonlinear effects and logistic perturbations some results recently developed in the literature where, for the linear counterpart and in absence of logistics, criteria toward boundedness for an attraction-repulsion Keller-Segel system with double saturation are derived.
We study a new nonlocal approach to the mathematical modelling of the Chemotaxis problem, which describes the random motion of a certain population due 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.
In this paper we study the initial boundary value problem for the system $u_t-Δu^m=-\mbox{div}(u^{q}\nabla v),\ v_t-Δv+v=u$. This problem is the so-called Keller-Segel model with nonlinear diffusion. Our investigation reveals that nonlinear diffusion can prevent overcrowding. To be precise, we show that solutions are bounded as long as $m>q>0$, thereby substantially generalizing the known results in this area. Furthermore, our result seems to imply that the Keller-Segel model can have bounded solutions and blow-up ones simultaneously.
We introduce the planar helical flows on three dimensional torus and study the dissipation enhancement of such flows. We then use such flows as transport flows to solve the three dimensional advective Kuramoto-Sivashinsky and Keller-Segel equations. The global well-posedness of the Kuramoto-Sivashinsky equation is achieved when the linearized operator does not have growing mode in the direction orthogonal to the flow. The global classical solution of the three dimensional Keller-Segel is ensured for any size of the torus with arbitrarily large initial data.
We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the existence of radial global-in-time solutions in terms of suitable Morrey norms are derived.
In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with fractional dissipation and the additional mixing effect of advection by an incompressible flow. Under suitable mixing condition on the advection, we study well-posedness of solution with large initial data. We establish the global $L^\infty$ estimate of the solution through nonlinear maximum principle, and obtain the global classical solution.
We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the blowup of radially symmetric solutions in terms of suitable Morrey spaces norms are derived.
The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living t…
We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.
This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{align*} &u_t = Δu - χ\nabla \cdot (uS(v)\nabla v), &v_t = Δv - v + u, \end{align*} where $χ>0$ and $S$ is a given function generalizing the sensitivity $S(s)=\frac{1}{(a+s)^{k}}$, $k>1$, $a\ge 0$, and shows exponential convergence of global classical solutions under an additional smallness condition condition for $χ>0$.
We consider a Liouville-type PDE on a smooth bounded planar domain, which is related to stationary solutions of the Keller-Segel's model for chemotaxis. We prove existence of solutions under some algebraic conditions on the parameters. In particular, if the domain is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.
The existence of generalised global supersolutions with a control upon the total muss is established for the parabolic-parabolic Keller-Segel system with logarithmic sensitivity for any space dimension. It is verified that smooth supersolutions of this sort are actually classical solutions. Unlike the previously existing constructions, neither is the chemotactic sensitivity coefficient required to be small, nor is it necessary for the initial data to be radially symmetric. Keywords: chemotaxis; generalised supersolution; global existence; logarithmic sensitivity
We study stationary solutions to the Keller--Segel equation on curved planes. We prove the necessity of the mass being $8 π$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case. Furthermore, we provide a correspondence between stationary solutions to the static Keller--Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily c…
We show that the Keller-Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to find a myriad of symmetric and asymmetric stationary states whose stability properties are mostly studied via free energy decreasing numerical schemes. The metastability behavior and staircased free energy decay are also illustrated via these numerical simulations.
We prove the uniqueness of Hölder continuous weak solutions via duality argument and vanishing viscosity method for the Keller-Segel system of porous medium type equations coupled to the Stokes system in dimensions three. An important step is the estimate of the Green function of parabolic equations with lower order terms of variable coefficients, which seems to be of independent interest.
We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $\mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.
The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyper-bolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently. We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemot…
We give a simple proof, relying on a {\it two-particles} moment computation, that there exists a global weak solution to the $2$-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure $f_0$ such that $f_0(\mathbb{R}^2)< 8 π$.
In 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 [3].
In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $\rr^2$, if the total mass of the cells is strictly less than $8π$, classical solutions exist for any finite time, and their $H^s$-Sobolev norms are almost uniformly bounded in time. For the radially symmetric solutions, this $8π$-mass threshold is critical. On the torus $\mathbb{T}^2$, the solutions are uniformly bounded in time under the same mass constraint.
This paper is concerned with the global boundedness and blowup of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility function decays exponentially, then a critical mass phenomenon similar to the minimal Keller-Segel model will arise. That is there is a number $m_*>0$, such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e. $L^1$-norm of the initial value of cell density) is less than $m_*$, while the solution may blow up if the…
In this paper, we study the global stability of classical solutions to a Keller--Segel equations in scaling-invariant spaces. We prove that for any given $0<\mathcal{M}<1+λ_1$ with $λ_1$ being the first eigenvalue of Neumann Laplacian, the initial--boundary value problem of the Keller--Segel system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to $(\mathcal{M},\mathcal{M})$ in the norm of $L^{d/2}(Ω)\times \dot{W}^{1,d}(Ω)$ and satisfies a natral average mass condition. Our proof is based on the perturbation theory of semigroups a…
This note is devoted to several inequalities deduced from a special form of the logarithmic Hardy-Littlewood-Sobolev, which is well adapted to the characterization of stationary solutions of a Keller-Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass.
