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Displaying similar documents to “Self-similarity in chemotaxis systems”

Asymptotic self-similar blow-up for a model of aggregation

Ignacio Guerra (2004)

Banach Center Publications

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In this article we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see e.g. M. P. Brenner et al. 1999, Nonlinearity 12, 1071-1098); one type is self-similar, and may be viewed as a trade-off between...

Macroscopic models of collective motion and self-organization

Pierre Degond, Amic Frouvelle, Jian-Guo Liu, Sebastien Motsch, Laurent Navoret (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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In this paper, we review recent developments on the derivation and properties of macroscopic models of collective motion and self-organization. The starting point is a model of self-propelled particles interacting with its neighbors through alignment. We successively derive a mean-field model and its hydrodynamic limit. The resulting macroscopic model is the Self-Organized Hydrodynamics (SOH). We review the available existence results and known properties of the SOH model and discuss...

On radially symmetric solutions of some chemotaxis system

Robert Stańczy (2009)

Banach Center Publications

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This paper contains some results concerning self-similar radial solutions for some system of chemotaxis. This kind of solutions describe asymptotic profiles of arbitrary solutions with small mass. Our approach is based on a fixed point analysis for an appropriate integral operator acting on a suitably defined convex subset of some cone in the space of bounded and continuous functions.

On univoque points for self-similar sets

Simon Baker, Karma Dajani, Kan Jiang (2015)

Fundamenta Mathematicae

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Let K ⊆ ℝ be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides...

Nonisothermal systems of self-attracting Fermi-Dirac particles

Piotr Biler, Tadeusz Nadzieja, Robert Stańczy (2004)

Banach Center Publications

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The existence of stationary solutions and blow up of solutions for a system describing the interaction of gravitationally attracting particles that obey the Fermi-Dirac statistics are studied.

A perturbation characterization of compactness of self-adjoint operators

Heydar Radjavi, Ping-Kwan Tam, Kok-Keong Tan (2003)

Studia Mathematica

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A characterization of compactness of a given self-adjoint bounded operator A on a separable infinite-dimensional Hilbert space is established in terms of the spectrum of perturbations. An example is presented to show that without separability, the perturbation condition, which is always necessary, is not sufficient. For non-separable spaces, another condition on the self-adjoint operator A, which is necessary and sufficient for the perturbation, is given.

Global existence versus blow up for some models of interacting particles

Piotr Biler, Lorenzo Brandolese (2006)

Colloquium Mathematicae

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We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and Debye-Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method due to S. Montgomery-Smith.

Existence and asymptotics of solutions of the Debye-Nernst-Planck system in ℝ²

Agnieszka Herczak, Michał Olech (2009)

Banach Center Publications

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We investigate a system describing electrically charged particles in the whole space ℝ². Our main goal is to describe large time behavior of solutions which start their evolution from initial data of small size. This is achieved using radially symmetric self-similar solutions.

The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation

Juan Luis Vázquez (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.