Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.     

Time-decay Estimates for Keller–Segel System

For the Cauchy problem to Keller–Segel system, we show well-posedness and time-decay estimates in the critical scaling-invariant Besov spaces by using Littlewood–Paley analysis together with the decay estimates of heat kernels.     

Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

We investigate global strong solution to a 3-dimensional parabolic–hyperbolic system arising from the Keller–Segel model. We establish the global well-posedness and asymptotic behavior in the energy functional setting. Precisely speaking, if the initial difference between cell density and its mean is small in L²$L^2$, and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H¹$H^1$, then they remain to be small in L²×H¹$L^2\times H^1$ for all time. Moreover, if the mean value of the initial cell density is smaller than some constant, then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity. The proof relies on an application of Fourier analysis to a linearized parabolic–hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration.     

Blowup of solutions to generalized Keller–Segel model

The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.     

Blow-up for a three dimensional Keller–Segel model with consumption of chemoattractant

We investigate blow-up properties for the initial-boundary value problem of a Keller–Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller–Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4], [5] from the whole space ${\mathbb{R}}^3$ to the case of bounded smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Lower global blow-up estimate on ${6n6}_{L^{\infty }(\mathrm{\Omega })}$ is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.     

Propagation of chaos for the Keller–Segel equation over bounded domains

In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions.     

Boundedness of the attraction–repulsion Keller–Segel system

This paper considers the initial–boundary value problem of the attraction–repulsion Keller–Segel model describing aggregation of Microglia in the central nervous system in Alzheimer's disease due to the interaction of chemoattractant and chemorepellent. If repulsion dominates over attraction, we show the global existence of classical solution in two dimensions and weak solution in three dimensions with large initial data.     

Interfaces for 1-D degenerate Keller–Segel systems

Let us consider the Keller–Segel system of degenerate type (KS) _m with m >1 below. We prove the property of finite speed of propagation for weak solutions u with a certain regularity. Moreover, we investigate the interface curve which separates the regions and . Concretely, we characterize the interface curve as the solution of a certain ordinary differential equation associated with (KS) _m .     

Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

In this paper we study the long time asymptotic behavior for a class of diffusion–aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak–Keller–Segel problem with critical power m=2−2/d$m=2-2/d$, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial solutions with subcritical mass converge to a self-similar dissipating solution algebraically fast. For non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small. We also apply the mass comparison method to another aggregation model with repulsive–attractive interaction, and prove that radial solutions converge to the stationary solution exponentially fast.     

Global well-posedness and blow-up for the 2-D Patlak–Keller–Segel equation

In this paper, we prove that for every nonnegative initial data in $L^1({\mathbb{R}}^2)$, the Patlak–Keller–Segel equation is globally well-posed if and only if the total mass $M\le 8\pi$. Our proof is based on some monotonicity formulas of nonnegative mild solutions.     

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.     

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

We study the chemotaxis effect vs. logistic damping on boundedness for the well-known minimal Keller–Segel model with logistic source:

$\{\begin{array}{ccc}\hfill & u_t=\mathrm{?}?(\mathrm{?}u?\chi u\mathrm{?}v)+u?\mu u^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^2$ with $\chi ,\mu >0$, nonnegative initial data $u_0$, $v_0$, and homogeneous Neumann boundary data. It is well known that this model allows only for global and uniform-in-time bounded solutions for any $\chi ,\mu >0$. Here, we carefully employ a simple and new method to regain its boundedness, with particular attention to how upper bounds of solutions qualitatively depend on χ and μ. More, precisely, it is shown that there exists $C=C(u_0,v_0,\mathrm{\Omega })>0$ such that

${6u(?,t)6}_{L^{\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\chi K(\chi ,\mu )N(\chi ,\mu )]$

and

${6v(?,t)6}_{W^{1,\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\frac{{\chi }^{\frac{8}{3}}}{\mu }{K}^{\frac{8}{3}}(\chi ,\mu )]=:CN(\chi ,\mu )$

uniformly on $[0,\infty )$, where

$K(\chi ,\mu )=M(\chi ,\mu )E(\chi ,\mu ),M(\chi ,\mu )=1+\frac{1}{\mu }+\sqrt{\chi }(1+\frac{1}{{\mu }^2})$

and

$E(\chi ,\mu )=\exp ?\left[\frac{\chi C_{GN}^2}{2\min ?\{1,\frac{2}{\chi }\}}\right(\frac{4}{\mu }{6u_{0}6}_{L^1(\mathrm{\Omega })}+\frac{13}{2{\mu }^2}|\mathrm{\Omega }|+{6\mathrm{?}{v}_{0}6}_{L^2(\mathrm{\Omega })}^2\left)\right].$

We notice that these upper bounds are increasing in χ, decreasing in μ, and have only one singularity at $\mu =0$, where the corresponding minimal model (removing the term $u?\mu u^2$ in the first equation) is widely known to possess blow-ups for large initial data.     

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u ₀; R ²) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u ₀; R ²) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u ₀; R ²) < 4π. In this paper, we prove that any solution with m (u ₀; R ²) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u ₀ also in the critical case m (u ₀; R ²) = 8π.     

Multiple positive solutions of the stationary Keller–Segel system

We consider the stationary Keller–Segel equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta v+v=\lambda e^v, \quad v>0 \quad &{} \text {in }\Omega ,\\ \partial _\nu v=0 &{}\text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is a ball. In the regime \(\lambda \rightarrow 0\), we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given \(n\in \mathbb {N}_0\), we build a solution having multiple layers at \(r_1,\ldots ,r_n\) by which we mean that the solutions concentrate on the spheres of radii \(r_i\) as \(\lambda \rightarrow 0\) (for all \(i=1,\ldots ,n\)). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of \(\Omega \) as \(\lambda \rightarrow 0\). Instead they satisfy an optimal partition problem in the limit.