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 Mathematics , 2011, Abstract: A parabolic-parabolic (Patlak-) Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical concentration. For arbitrarily small cross-diffusion coefficients and for suitable exponents of the nonlinear diffusion terms, the global-in-time existence of weak solutions is proved, thus preventing finite-time blow up of the cell density. The global existence result also holds for linear and fast diffusion of the cell density in a certain parameter range in three dimensions. Furthermore, we show $L^\infty$ bounds for the solutions to the parabolic-elliptic system. Sufficient conditions leading to the asymptotic stability of the constant steady state are given for a particular choice of the nonlinear diffusion exponents. Numerical experiments in two and three space dimensions illustrate the theoretical results.
 Mathematics , 2010, Abstract: This paper is devoted to the analysis of the classical Keller-Segel system over $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and in the fully parabolic case. The main results when dealing with the parabolic-elliptic case are: local existence without smallness assumption on the initial density, global existence under an improved smallness condition and comparison of blow-up criteria. A new concentration phenomenon criteria for the fully parabolic case is also given. The analysis is completed by a visualization tool based on the reduction of the parabolic-elliptic system to a finite-dimensional dynamical system of gradient flow type, sharing features similar to the infinite-dimensional system.
 Physics , 2013, Abstract: A degenerate Keller-Segel system with diffusion exponent $2n/(n+2)  Michael Winkler Mathematics , 2011, Abstract: We study the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in\Omega, \ t>0, [1mm] v_t=\Delta v-v+u, \qquad x\in\Omega, \ t>0, where$\Omega$is a ball in${\mathbb{R}}^n$with$n\ge 3$. It is proved that for any prescribed$m>0$there exist radially symmetric positive initial data$(u_0,v_0) \in C^0(\bar\Omega) \times W^{1,\infty}(\Omega)$with$\int_\Omega u_0=m$such that the corresponding solution blows up in finite time. Moreover, by providing an essentially explicit blow-up criterion it is shown that within the space of all radial functions, the set of such blow-up enforcing initial data indeed is large in an appropriate sense; in particular, this set is dense with respect to the topology of$L^p(\Omega) \times W^{1,2}(\Omega)$for any$p \in (1,\frac{2n}{n+2})\$.