Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model [10]. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyse the stationary case showing the existence of a unique, up to translation, global minimizer of the associa...

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic PatlakKeller-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...

This is a continuous study on E. coli chemotaxis under the framework of pathway-based meanfield theory (PBMFT) proposed in [G. Si, M. Tang and X. Yang, Multiscale Model. Simul., 12 (2014), 907–926], following the physical studies in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we derive an augmented Keller-Segel system with macroscopic intercellular ...

We classify the global behavior of the weak solution of the Keller-Segel system of degenerated type. For the stronger degeneracy the weak solution exists globally in time and it shows the time uniform decay under some extra conditions. If the degeneracy is weaker the solution exhibit a finite time blow-up if the data is non-negative. The situation is very similar to the semi-linear case. Some a...

We present a two-dimensional (2D) generalization of the stabilized Kuramoto-Sivashinsky system, based on the Kadomtsev-Petviashvili (KP) equation including dissipation of the generic [Newell-Whitehead-Segel (NWS)] type and gain. The system directly applies to the description of gravity-capillary waves on the surface of a liquid layer flowing down an inclined plane, with a surfactant diffusing a...

Recently, Wang and Hillen [Chaos 17 (3) (2007) 037108–037108] introduced a modified Keller-Segel model with logistic growth that exhibits complex spatiotemporal dynamics of spikes. These dynamics are driven by merging of spikes on one hand, and spike insertion on the other. In this paper we analyse the basic mechanisms that initiate and sustain these events. We identify two distinguished regime...

We study the critical dynamics of the generalized Smoluchowski-Poisson system (for self-gravitating Langevin particles) or generalized Keller-Segel model (for the chemotaxis of bacterial populations). These models [P. H. Chavanis and C. Sire, Phys. Rev. E 69, 016116 (2004)] are based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to poly...

We consider the following Keller-Segel system with gradient dependent chemotactic coefficient: \begin{document}$ \begin{equation*} \begin{cases} u_{t} = \Delta u- \chi \nabla\cdot (uf(|\nabla v|)\nabla v),\\ 0 v -v+g(u), \end{cases} \end{equation*} $\end{document} in smooth bounded domains $ \Omega \subset \mathbb{R}^{n}, \,n\geq 1 f(\xi) \big(\xi^{p-2}\big(1+\xi^{p}\big)^{\frac{q-p}{p}}\big), ...