1,577 research outputs found
Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension
This paper is devoted to the analysis of the classical Keller-Segel system
over , . 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.Comment: 44 pages, 2 figure
On nonexistence of Baras--Goldstein type for higher-order parabolic equations with singular potentials
An analogy of nonexistence result by Baras and Goldstein (1984), for the heat
equation with inverse singular potential, is proved for 2mth-order linear
parabolic equations with Hardy-supercritical singular potentials. Extensions to
other linear and nonlinear singular PDEs are discussed.Comment: 22 page
Blow-up dynamics of self-attracting diffusive particles driven by competing convexities
In this paper, we analyze the dynamics of an particles system evolving
according the gradient flow of an energy functional. The particle system is a
consistent approximation of the Lagrangian formulation of a one parameter
family of non-local drift-diffusion equations in one spatial dimension. We
shall prove the global in time existence of the trajectories of the particles
(under a sufficient condition on the initial distribution) and give two blow-up
criteria. All these results are consequences of the competition between the
discrete entropy and the discrete interaction energy. They are also consistent
with the continuous setting, that in turn is a one dimension reformulation of
the parabolic-elliptic Keller-Segel in high dimensions
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